# List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

## Counts

For $n=1,2,\dots$ the number of nonisomorphic groups of order $n$ is

1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... (sequence A000001 in the OEIS)

For labeled groups, see .

## Glossary

Each group is named by their Small Groups library index as Goi, where o is the order of the group, and i is the index of the group within that order.

Common group names:

The notations Zn and Dihn have the advantage that point groups in three dimensions Cn and Dn do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

The notation G × H denotes the direct product of the two groups; Gn denotes the direct product of a group with itself n times. GH denotes a semidirect product where H acts on G; this may also depend on the choice of action of H on G

Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Zn, for prime n.) The equality sign ("=") denotes isomorphism.

The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.

In the lists of subgroups, the trivial group and the group itself are not listed. Where there are several isomorphic subgroups, the number of such subgroups is indicated in parentheses.

## List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups. The numbers of nonisomorphic abelian groups of orders $n=1,2,\dots$ are

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, ... (sequence A000688 in the OEIS)

For labeled Abelian groups, see .

List of all abelian groups up to order 31
Order ID Goi Group Nontrivial proper Subgroups Cycle
graph
Properties
1 1 G11 Z1 = S1 = A2 Trivial. Cyclic. Alternating. Symmetric. Elementary.
2 2 G21 Z2 = S2 = Dih1 Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.)
3 3 G31 Z3 = A3 Simple. Alternating. Cyclic. Elementary.
4 4 G41 Z4 = Dic1 Z2 Cyclic.
5 G42 Z22 = K4 = Dih2 Z2 (3) Elementary. Product. (Klein four-group. The smallest non-cyclic group.)
5 6 G51 Z5 Simple. Cyclic. Elementary.
6 8 G62 Z6 = Z3 × Z2 Z3, Z2 Cyclic. Product.
7 9 G71 Z7 Simple. Cyclic. Elementary.
8 10 G81 Z8 Z4, Z2 Cyclic.
11 G82 Z4 × Z2 Z22, Z4 (2), Z2 (3) Product.
14 G85 Z23 Z22 (7), Z2 (7) Product. Elementary. (The non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines.)
9 15 G91 Z9 Z3 Cyclic.
16 G92 Z32 Z3 (4) Elementary. Product.
10 18 G102 Z10 = Z5 × Z2 Z5, Z2 Cyclic. Product.
11 19 G111 Z11 Simple. Cyclic. Elementary.
12 21 G122 Z12 = Z4 × Z3 Z6, Z4, Z3, Z2 Cyclic. Product.
24 G125 Z6 × Z2 = Z3 × Z22 Z6 (3), Z3, Z2 (3), Z22 Product.
13 25 G131 Z13 Simple. Cyclic. Elementary.
14 27 G142 Z14 = Z7 × Z2 Z7, Z2 Cyclic. Product.
15 28 G151 Z15 = Z5 × Z3 Z5, Z3 Cyclic. Product.
16 29 G161 Z16 Z8, Z4, Z2 Cyclic.
30 G162 Z42 Z2 (3), Z4 (6), Z22, Z4 × Z2 (3) Product.
33 G165 Z8 × Z2 Z2 (3), Z4 (2), Z22, Z8 (2), Z4 × Z2 Product.
38 G1610 Z4 × Z22 Z2 (7), Z4 (4), Z22 (7), Z23, Z4 × Z2 (6) Product.
42 G1614 Z24 = K42 Z2 (15), Z22 (35), Z23 (15) Product. Elementary.
17 43 G171 Z17 Simple. Cyclic. Elementary.
18 45 G182 Z18 = Z9 × Z2 Z9, Z6, Z3, Z2 Cyclic. Product.
48 G185 Z6 × Z3 = Z32 × Z2Z6, Z3, Z2Product.
19 49 G191 Z19 Simple. Cyclic. Elementary.
20 51 G202 Z20 = Z5 × Z4 Z10, Z5, Z4, Z2 Cyclic. Product.
54 G205 Z10 × Z2 = Z5 × Z22Z5, Z2 Product.
21 56 G212 Z21 = Z7 × Z3 Z7, Z3 Cyclic. Product.
22 58 G222 Z22 = Z11 × Z2 Z11, Z2 Cyclic. Product.
23 59 G231 Z23 Simple. Cyclic. Elementary.
24 61 G242 Z24 = Z8 × Z3 Z12, Z8, Z6, Z4, Z3, Z2 Cyclic. Product.
68 G249 Z12 × Z2 = Z6 × Z4
= Z4 × Z3 × Z2
Z12, Z6, Z4, Z3, Z2 Product.
74 G2415 Z6 × Z22 = Z3 × Z23 Z6, Z3, Z2 Product.
25 75 G251 Z25 Z5 Cyclic.
76 G252 Z52 Z5 Product. Elementary.
26 78 G262 Z26 = Z13 × Z2 Z13, Z2 Cyclic. Product.
27 79 G271 Z27Z9, Z3 Cyclic.
80 G272 Z9 × Z3 Z9, Z3 Product.
83 G275 Z33Z3Product. Elementary.
28 85 G282 Z28 = Z7 × Z4Z14, Z7, Z4, Z2Cyclic. Product.
87 G284 Z14 × Z2 = Z7 × Z22Z14, Z7, Z4, Z2 Product.
29 88 G291 Z29 Simple. Cyclic. Elementary.
30 92 G304 Z30 = Z15 × Z2 = Z10 × Z3
= Z6 × Z5 = Z5 × Z3 × Z2
Z15, Z10, Z6, Z5, Z3, Z2 Cyclic. Product.
31 93 G311 Z31 Simple. Cyclic. Elementary.

## List of small non-abelian groups

The numbers of non-abelian groups, by order, are counted by (sequence A060689 in the OEIS). However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are

6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, ... (sequence A060652 in the OEIS)
List of all nonabelian groups up to order 31
Order ID Goi Group Nontrivial proper Subgroups Cycle
graph
Properties
6 7 G61 Dih3 = S3 = D6 Z3, Z2 (3) Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group
8 12 G83 Dih4 = D8 Z4, Z22 (2), Z2 (5) Dihedral group. Extraspecial group. Nilpotent.
13 G84 Q8 = Dic2 = <2,2,2> Z4 (3), Z2 Quaternion group, Hamiltonian group. all subgroups are normal without the group being abelian. The smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G. Extraspecial group Binary dihedral group. Nilpotent.
10 17 G101 Dih5 = D10 Z5, Z2 (5) Dihedral group, Frobenius group
12 20 G121 Q12 = Dic3 = <3,2,2>
= Z3 ⋊ Z4
Z2, Z3, Z4 (3), Z6 Binary dihedral group
22 G123 A4 Z22, Z3 (4), Z2 (3) Alternating group. No subgroups of order 6, although 6 divides its order. Frobenius group
23 G124 Dih6 = D12 = Dih3 × Z2 Z6, Dih3 (2), Z22 (3), Z3, Z2 (7) Dihedral group, product
14 26 G141 Dih7 = D14 Z7, Z2 (7) Dihedral group, Frobenius group
16 31 G163 G4,4 = K4 ⋊ Z4
(Z4×Z2) ⋊ Z2
E8, Z4 × Z2 (2), Z4 (4), K4 (6), Z2 (6) Has the same number of elements of every order as the Pauli group. Nilpotent.
32 G164 Z4 ⋊ Z4 The squares of elements do not form a subgroup. Has the same number of elements of every order as Q8 × Z2. Nilpotent.
34 G166 Z8 ⋊ Z2 Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q8 × Z2 are also modular. Nilpotent.
35 G167 Dih8 = D16 Z8, Dih4 (2), Z22 (4), Z4, Z2 (9) Dihedral group. Nilpotent.
36 G168 QD16 The order 16 quasidihedral group. Nilpotent.
37 G169 Q16 = Dic4 = <4,2,2> generalized quaternion group, binary dihedral group. Nilpotent.
39 G1611 Dih4 × Z2 Dih4 (4), Z4 × Z2, Z23 (2), Z22 (13), Z4 (2), Z2 (11) Product. Nilpotent.
40 G1612 Q8 × Z2 Hamiltonian, product. Nilpotent.
41 G1613 (Z4 × Z2) ⋊ Z2 The Pauli group generated by the Pauli matrices. Nilpotent.
18 44 G181 Dih9 = D18Dihedral group, Frobenius group
46 G183 S3 × Z3Product
47 G184 (Z3 × Z3) ⋊ Z2Frobenius group
20 50 G201 Q20 = Dic5 = <5,2,2>Binary dihedral group
52 G203 Z5 ⋊ Z4Frobenius group
53 G204 Dih10 = Dih5 × Z2 = D20Dihedral group, product
21 55 G211 Z7 ⋊ Z3Z7, Z3 (7)Smallest non-abelian group of odd order. Frobenius group
22 57 G221 Dih11 = D22 Z11, Z2 (11) Dihedral group, Frobenius group
24 60 G241 Z3 ⋊ Z8Central extension of S3
62 G243 SL(2,3) = 2T = Q8 ⋊ Z3Binary tetrahedral group
63 G244 Q24 = Dic6 = <6,2,2> = Z3 ⋊ Q8Binary dihedral
64 G245 Z4 × S3Product
65 G246 Dih12Dihedral group
66 G247 Dic3 × Z2 = Z2 × (Z3 ⋊ Z4)Product
67 G248 (Z6 × Z2) ⋊ Z2 = Z3 ⋊ Dih4Double cover of dihedral group
69 G2410 Dih4 × Z3Product. Nilpotent.
70 G2411 Q8 × Z3Product. Nilpotent.
71 G2412 S4Symmetric group. Has no normal Sylow subgroups.
72 G2413 A4 × Z2Product
73 G2414 D12× Z2Product
26 77 G261 Dih13Dihedral group, Frobenius group
27 81 G273 Z32 ⋊ Z3All non-trivial elements have order 3. Extraspecial group. Nilpotent.
82 G274 Z9 ⋊ Z3Extraspecial group. Nilpotent.
28 84 G281 Z7 ⋊ Z4Binary dihedral group
86 G283 Dih14Dihedral group, product
30 89 G301 Z5 × S3Product
90 G302 Z3 × Dih5Product
91 G303 Dih15Dihedral group, Frobenius group

## Classifying groups of small order

Small groups of prime power order pn are given as follows:

• Order p: The only group is cyclic.
• Order p2: There are just two groups, both abelian.
• Order p3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
• Order p4: The classification is complicated, and gets much harder as the exponent of p increases.

Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p complement include:

• Order 24: The symmetric group S4
• Order 48: The binary octahedral group and the product S4 × Z2
• Order 60: The alternating group A5.

## Small groups library

The group theoretical computer algebra system GAP contains the "Small Groups library" which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:

• those of order at most 2000;
• those of cubefree order at most 50000 (395 703 groups);
• those of squarefree order;
• those of order pn for n at most 6 and p prime;
• those of order p7 for p = 3, 5, 7, 11 (907 489 groups);
• those of order pqn where qn divides 28, 36, 55 or 74 and p is an arbitrary prime which differs from q;
• those whose orders factorise into at most 3 primes (not necessarily distinct).

It contains explicit descriptions of the available groups in computer readable format.

The smallest order for which the SmallGroups library does not have information is 2048.

## See also

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