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.

SymbolsEdit

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. This is somtimes supplemented with a symbol of the form   which specifies the bravais lattice. Here   is the lattice system, and   is the centering type.[1]

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 TriclinicEdit

Triclinic Bravais lattice
 
Triclinic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
1 1   P1 P 1   1s    
2 1   P1 P 1   2s    

List of MonoclinicEdit

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)
3 2   P2 P 1 2 1 P 1 1 2   3s      
4 P21 P 1 21 1 P 1 1 21   1a      
5 C2 C 1 2 1 B 1 1 2   4s      ,  
6 m   Pm P 1 m 1 P 1 1 m   5s      
7 Pc P 1 c 1 P 1 1 b   1h      ,  
8 Cm C 1 m 1 B 1 1 m   6s      ,  
9 Cc C 1 c 1 B 1 1 b   2h      ,  
10 2/m   P2/m P 1 2/m 1 P 1 1 2/m   7s      
11 P21/m P 1 21/m 1 P 1 1 21/m   2a      
12 C2/m C 1 2/m 1 B 1 1 2/m   8s      ,  
13 P2/c P 1 2/c 1 P 1 1 2/b   3h      ,  
14 P21/c P 1 21/c 1 P 1 1 21/b   3a      ,  
15 C2/c C 1 2/c 1 B 1 1 2/b   4h      ,  

List of OrthorhombicEdit

Orthorhombic Bravais lattice
Simple
(P)
Body
(I)
Face
(F)
Base
(A or C)
       
Orthorhombic crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)
16 222   P222 P 2 2 2   9s    
17 P2221 P 2 2 21   4a      
18 P21212 P 21 21 2   7a          
19 P212121 P 21 21 21   8a        
20 C2221 C 2 2 21   5a      
21 C222 C 2 2 2   10s      
22 F222 F 2 2 2   12s    
23 I222 I 2 2 2   11s    
24 I212121 I 21 21 21   6a    
25 mm2   Pmm2 P m m 2   13s      
26 Pmc21 P m c 21   9a      ,  
27 Pcc2 P c c 2   5h      
28 Pma2 P m a 2   6h      ,  
29 Pca21 P c a 21   11a      
30 Pnc2 P n c 2   7h      ,  
31 Pmn21 P m n 21   10a      ,  
32 Pba2 P b a 2   9h      
33 Pna21 P n a 21   12a      ,  
34 Pnn2 P n n 2   8h      
35 Cmm2 C m m 2   14s      
36 Cmc21 C m c 21   13a      ,  
37 Ccc2 C c c 2   10h      
38 Amm2 A m m 2   15s      ,  
39 Aem2 A b m 2   11h      ,  
40 Ama2 A m a 2   12h      ,  
41 Aea2 A b a 2   13h      ,  
42 Fmm2 F m m 2   17s      
43 Fdd2 F dd2   16h      
44 Imm2 I m m 2   16s      
45 Iba2 I b a 2   15h      
46 Ima2 I m a 2   14h      ,  
47     Pmmm P 2/m 2/m 2/m   18s    
48 Pnnn P 2/n 2/n 2/n   19h    
49 Pccm P 2/c 2/c 2/m   17h      
50 Pban P 2/b 2/a 2/n   18h      
51 Pmma P 21/m 2/m 2/a   14a      ,  
52 Pnna P 2/n 21/n 2/a   17a      ,  
53 Pmna P 2/m 2/n 21/a   15a      ,  
54 Pcca P 21/c 2/c 2/a   16a      ,  
55 Pbam P 21/b 21/a 2/m   22a      
56 Pccn P 21/c 21/c 2/n   27a      
57 Pbcm P 2/b 21/c 21/m   23a      ,  
58 Pnnm P 21/n 21/n 2/m   25a      
59 Pmmn P 21/m 21/m 2/n   24a      
60 Pbcn P 21/b 2/c 21/n   26a      ,  
61 Pbca P 21/b 21/c 21/a   29a    
62 Pnma P 21/n 21/m 21/a   28a      ,  
63 Cmcm C 2/m 2/c 21/m   18a      ,  
64 Cmca C 2/m 2/c 21/a   19a      ,  
65 Cmmm C 2/m 2/m 2/m   19s      
66 Cccm C 2/c 2/c 2/m   20h      
67 Cmme C 2/m 2/m 2/e   21h      
68 Ccce C 2/c 2/c 2/e   22h      
69 Fmmm F 2/m 2/m 2/m   21s    
70 Fddd F 2/d 2/d 2/d   24h    
71 Immm I 2/m 2/m 2/m   20s    
72 Ibam I 2/b 2/a 2/m   23h      
73 Ibca I 2/b 2/c 2/a   21a    
74 Imma I 2/m 2/m 2/a   20a      

List of TetragonalEdit

Tetragonal Bravais lattice
Simple
(P)
Body
(I)
   
Tetragonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
75 4   P4 P 4   22s    
76 P41 P 41   30a    
77 P42 P 42   33a    
78 P43 P 43   31a    
79 I4 I 4   23s    
80 I41 I 41   32a    
81 4   P4 P 4   26s    
82 I4 I 4   27s    
83 4/m   P4/m P 4/m   28s    
84 P42/m P 42/m   41a    
85 P4/n P 4/n   29h    
86 P42/n P 42/n   42a    
87 I4/m I 4/m   29s    
88 I41/a I 41/a   40a    
89 422   P422 P 4 2 2   30s    
90 P4212 P4212   43a        
91 P4122 P 41 2 2   44a    
92 P41212 P 41 21 2   48a        
93 P4222 P 42 2 2   47a    
94 P42212 P 42 21 2   50a        
95 P4322 P 43 2 2   45a    
96 P43212 P 43 21 2   49a        
97 I422 I 4 2 2   31s    
98 I4122 I 41 2 2   46a    
99 4mm   P4mm P 4 m m   24s    
100 P4bm P 4 b m   26h    
101 P42cm P 42 c m   37a    
102 P42nm P 42 n m   38a    
103 P4cc P 4 c c   25h    
104 P4nc P 4 n c   27h    
105 P42mc P 42 m c   36a    
106 P42bc P 42 b c   39a    
107 I4mm I 4 m m   25s    
108 I4cm I 4 c m   28h    
109 I41md I 41 m d   34a    
110 I41cd I 41 c d   35a    
111 42m   P42m P 4 2 m   32s    
112 P42c P 4 2 c   30h        
113 P421m P 4 21 m   52a    
114 P421c P 4 21 c   53a    
115 P4m2 P 4 m 2   33s    
116 P4c2 P 4 c 2   31h    
117 P4b2 P 4 b 2   32h    
118 P4n2 P 4 n 2   33h    
119 I4m2 I 4 m 2   35s    
120 I4c2 I 4 c 2   34h    
121 I42m I 4 2 m   34s    
122 I42d I 4 2 d   51a    
123 4/m 2/m 2/m   P4/mmm P 4/m 2/m 2/m   36s    
124 P4/mcc P 4/m 2/c 2/c   35h    
125 P4/nbm P 4/n 2/b 2/m   36h    
126 P4/nnc P 4/n 2/n 2/c   37h    
127 P4/mbm P 4/m 21/b 2/m   54a    
128 P4/mnc P 4/m 21/n 2/c   56a    
129 P4/nmm P 4/n 21/m 2/m   55a    
130 P4/ncc P 4/n 21/c 2/c   57a    
131 P42/mmc P 42/m 2/m 2/c   60a    
132 P42/mcm P 42/m 2/c 2/m   61a    
133 P42/nbc P 42/n 2/b 2/c   63a    
134 P42/nnm P 42/n 2/n 2/m   62a    
135 P42/mbc P 42/m 21/b 2/c   66a    
136 P42/mnm P 42/m 21/n 2/m   65a    
137 P42/nmc P 42/n 21/m 2/c   67a    
138 P42/ncm P 42/n 21/c 2/m   65a    
139 I4/mmm I 4/m 2/m 2/m   37s    
140 I4/mcm I 4/m 2/c 2/m   38h    
141 I41/amd I 41/a 2/m 2/d   59a    
142 I41/acd I 41/a 2/c 2/d   58a    

List of TrigonalEdit

Trigonal Bravais lattice
Rhombohedral
(R)
Hexagonal
(P)
   
Trigonal crystal system
Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold
143 3   P3 P 3   38s    
144 P31 P 31   68a    
145 P32 P 32   69a    
146 R3 R 3   39s    
147 3   P3 P 3   51s    
148 R3 R 3   52s    
149 32   P312 P 3 1 2   45s    
150 P321 P 3 2 1   44s    
151 P3112 P 31 1 2   72a    
152 P3121 P 31 2 1   70a    
153 P3212 P 32 1 2   73a    
154 P3221 P 32 2 1   71a    
155 R32 R 3 2   46s    
156 3m   P3m1 P 3 m 1   40s    
157 P31m P 3 1 m   41s    
158 P3c1 P 3 c 1   39h    
159 P31c P 3 1 c   40h    
160 R3m R 3 m   42s    
161 R3c R 3 c   41h    
162 3 2/m   P31m P 3 1 2/m   56s    
163 P31c P 3 1 2/c   46h    
164 P3m1 P 3 2/m 1   55s