List of space groups

There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point group of the unit cell.

Symbols

In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions:

  • P primitive
  • I body centered (from the German "Innenzentriert")
  • F face centered (from the German "Flächenzentriert")
  • A centered on A faces only
  • B centered on B faces only
  • C centered on C faces only
  • R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

  • , , or glide translation along half the lattice vector of this face
  • glide translation along with half a face diagonal
  • glide planes with translation along a quarter of a face diagonal.
  • two glides with the same glide plane and translation along two (different) half-lattice vectors.

A gyration point can be replaced by a screw axis denoted by a number, n, where the angle of rotation is . The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (twofold) rotation followed by a translation of ½ of the lattice vector. 31 is a 120° (threefold) rotation followed by a translation of ⅓ of the lattice vector.

The possible screw axes are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.

In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups.

In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups. Symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. All the other space groups are asymmorphic. Example for point group 4/mmm (): the symmorphic space groups are P4/mmm (, 36s) and I4/mmm (, 37s); hemisymmorphic space groups should contain axial combination 422, these are P4/mcc (, 35h), P4/nbm (, 36h), P4/nnc (, 37h), and I4/mcm (, 38h).

List of Triclinic

Triclinic Bravais lattice
Triclinic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
11P1P 11s
21P1P 12s

List of Monoclinic

Monoclinic Bravais lattice
Simple
(P)
Base
(C)
Monoclinic crystal system
Number Point group Orbifold Short name Full name(s) Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)
32P2P 1 2 1P 1 1 23s
4P21P 1 21 1P 1 1 211a
5C2C 1 2 1B 1 1 24s,
6mPmP 1 m 1P 1 1 m5s
7PcP 1 c 1P 1 1 b1h,
8CmC 1 m 1B 1 1 m6s,
9CcC 1 c 1B 1 1 b2h,
102/mP2/mP 1 2/m 1P 1 1 2/m7s
11P21/mP 1 21/m 1P 1 1 21/m2a
12C2/mC 1 2/m 1B 1 1 2/m8s,
13P2/cP 1 2/c 1P 1 1 2/b3h,
14P21/cP 1 21/c 1P 1 1 21/b3a,
15C2/cC 1 2/c 1B 1 1 2/b4h,

List of Orthorhombic

Orthorhombic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)
16222P222P 2 2 29s
17P2221P 2 2 214a
18P21212P 21 21 27a
19P212121P 21 21 218a
20C2221C 2 2 215a
21C222C 2 2 210s
22F222F 2 2 212s
23I222I 2 2 211s
24I212121I 21 21 216a
25mm2Pmm2P m m 213s
26Pmc21P m c 219a,
27Pcc2P c c 25h
28Pma2P m a 26h,
29Pca21P c a 2111a
30Pnc2P n c 27h,
31Pmn21P m n 2110a,
32Pba2P b a 29h
33Pna21P n a 2112a,
34Pnn2P n n 28h
35Cmm2C m m 214s
36Cmc21C m c 2113a,
37Ccc2C c c 210h
38Amm2A m m 215s,
39Aem2A b m 211h,
40Ama2A m a 212h,
41Aea2A b a 213h,
42Fmm2F m m 217s
43Fdd2F dd216h
44Imm2I m m 216s
45Iba2I b a 215h
46Ima2I m a 214h,
47PmmmP 2/m 2/m 2/m18s
48PnnnP 2/n 2/n 2/n19h
49PccmP 2/c 2/c 2/m17h
50PbanP 2/b 2/a 2/n18h
51PmmaP 21/m 2/m 2/a14a,
52PnnaP 2/n 21/n 2/a17a,
53PmnaP 2/m 2/n 21/a15a,
54PccaP 21/c 2/c 2/a16a,
55PbamP 21/b 21/a 2/m22a
56PccnP 21/c 21/c 2/n27a
57PbcmP 2/b 21/c 21/m23a,
58PnnmP 21/n 21/n 2/m25a
59PmmnP 21/m 21/m 2/n24a
60PbcnP 21/b 2/c 21/n26a,
61PbcaP 21/b 21/c 21/a29a
62PnmaP 21/n 21/m 21/a28a,
63CmcmC 2/m 2/c 21/m18a,
64CmcaC 2/m 2/c 21/a19a,
65CmmmC 2/m 2/m 2/m19s
66CccmC 2/c 2/c 2/m20h
67CmmeC 2/m 2/m 2/e21h
68CcceC 2/c 2/c 2/e22h
69FmmmF 2/m 2/m 2/m21s
70FdddF 2/d 2/d 2/d24h
71ImmmI 2/m 2/m 2/m20s
72IbamI 2/b 2/a 2/m23h
73IbcaI 2/b 2/c 2/a21a
74ImmaI 2/m 2/m 2/a20a

List of Tetragonal

Tetragonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
754P4P 422s
76P41P 4130a
77P42P 4233a
78P43P 4331a
79I4I 423s
80I41I 4132a
814P4P 426s
82I4I 427s
834/mP4/mP 4/m28s
84P42/mP 42/m41a
85P4/nP 4/n29h
86P42/nP 42/n42a
87I4/mI 4/m29s
88I41/aI 41/a40a
89422P422P 4 2 230s
90P4212P421243a
91P4122P 41 2 244a
92P41212P 41 21 248a
93P4222P 42 2 247a
94P42212P 42 21 250a
95P4322P 43 2 245a
96P43212P 43 21 249a
97I422I 4 2 231s
98I4122I 41 2 246a
994mmP4mmP 4 m m24s
100P4bmP 4 b m26h
101P42cmP 42 c m37a
102P42nmP 42 n m38a
103P4ccP 4 c c25h
104P4ncP 4 n c27h
105P42mcP 42 m c36a
106P42bcP 42 b c39a
107I4mmI 4 m m25s
108I4cmI 4 c m28h
109I41mdI 41 m d34a
110I41cdI 41 c d35a
11142mP42mP 4 2 m32s
112P42cP 4 2 c30h
113P421mP 4 21 m52a
114P421cP 4 21 c53a
115P4m2P 4 m 233s
116P4c2P 4 c 231h
117P4b2P 4 b 232h
118P4n2P 4 n 233h
119I4m2I 4 m 235s
120I4c2I 4 c 234h
121I42mI 4 2 m34s
122I42dI 4 2 d51a
1234/m 2/m 2/mP4/mmmP 4/m 2/m 2/m36s
124P4/mccP 4/m 2/c 2/c35h
125P4/nbmP 4/n 2/b 2/m36h
126P4/nncP 4/n 2/n 2/c37h
127P4/mbmP 4/m 21/b 2/m54a
128P4/mncP 4/m 21/n 2/c56a
129P4/nmmP 4/n 21/m 2/m55a
130P4/nccP 4/n 21/c 2/c57a
131P42/mmcP 42/m 2/m 2/c60a
132P42/mcmP 42/m 2/c 2/m61a
133P42/nbcP 42/n 2/b 2/c63a
134P42/nnmP 42/n 2/n 2/m62a
135P42/mbcP 42/m 21/b 2/c66a
136P42/mnmP 42/m 21/n 2/m65a
137P42/nmcP 42/n 21/m 2/c67a
138P42/ncmP 42/n 21/c 2/m65a
139I4/mmmI 4/m 2/m 2/m37s
140I4/mcmI 4/m 2/c 2/m38h
141I41/amdI 41/a 2/m 2/d59a
142I41/acdI 41/a 2/c 2/d58a

List of Trigonal

Unit cells for trigonal crystal system
Rhombohedral
(R)
Hexagonal
(P)
Trigonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1433P3P 338s
144P31P 3168a
145P32P 3269a
146R3R 339s
1473P3P 351s
148R3R 352s
14932P312P 3 1 245s
150P321P 3 2 144s
151P3112P 31 1 272a
152P3121P 31 2 170a
153P3212P 32 1 273a
154P3221P 32 2 171a
155R32R 3 246s
1563mP3m1P 3 m 140s
157P31mP 3 1 m41s
158P3c1P 3 c 139h
159P31cP 3 1 c40h
160R3mR 3 m42s
161R3cR 3 c41h
1623 2/mP31mP 3 1 2/m56s
163P31cP 3 1 2/c46h
164P3m1P 3 2/m 155s
165P3c1P 3 2/c 145h
166R3mR 3 2/m57s
167R3cR 3 2/c47h

List of Hexagonal

Hexagonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1686P6P 649s
169P61P 6174a
170P65P 6575a
171P62P 6276a
172P64P 6477a
173P63P 6378a
1746P6P 643s
1756/mP6/mP 6/m53s
176P63/mP 63/m81a
177622P622P 6 2 254s
178P6122P 61 2 282a
179P6522P 65 2 283a
180P6222P 62 2 284a
181P6422P 64 2 285a
182P6322P 63 2 286a
1836mmP6mmP 6 m m50s
184P6ccP 6 c c44h
185P63cmP 63 c m80a
186P63mcP 63 m c79a
1876m2P6m2P 6 m 248s
188P6c2P 6 c 243h
189P62mP 6 2 m47s
190P62cP 6 2 c42h
1916/m 2/m 2/mP6/mmmP 6/m 2/m 2/m58s
192P6/mccP 6/m 2/c 2/c48h
193P63/mcmP 63/m 2/c 2/m87a
194P63/mmcP 63/m 2/m 2/c88a

List of Cubic

Cubic Bravais lattice
Simple
(P)
Body centered
(I)
Face centered
(F)
Cubic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Conway Fibrifold (preserving ) Fibrifold (preserving , , )
19523P23P 2 359s
196F23F 2 361s
197I23I 2 360s
198P213P 21 389a
199I213I 21 390a
2002/m 3Pm3P 2/m 362s
201Pn3P 2/n 349h
202Fm3F 2/m 364s
203Fd3F 2/d 350h
204Im3I 2/m 363s
205Pa3P 21/a 391a
206Ia3I 21/a 392a
207432P432P 4 3 268s
208P4232P 42 3 298a
209F432F 4 3 270s
210F4132F 41 3 297a
211I432I 4 3 269s
212P4332P 43 3 294a
213P4132P 41 3 295a
214I4132I 41 3 296a
21543mP43mP 4 3 m65s
216F43mF 4 3 m67s
217I43mI 4 3 m66s
218P43nP 4 3 n51h
219F43cF 4 3 c52h
220I43dI 4 3 d93a
2214/m 3 2/mPm3mP 4/m 3 2/m71s
222Pn3nP 4/n 3 2/n53h
223Pm3nP 42/m 3 2/n102a
224Pn3mP 42/n 3 2/m103a
225Fm3mF 4/m 3 2/m73s
226Fm3cF 4/m 3 2/c54h
227Fd3mF 41/d 3 2/m100a
228Fd3cF 41/d 3 2/c101a
229Im3mI 4/m 3 2/m72s