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Classification of the subgroups of type Pmm2(25) of group Fm-3m(225) with index 48

For the group G = Fm-3m there are 48 different subgroups Hj isomorphic to H = Pmm2 of index 48. These subgroups are distributed in 6 classes of conjugate subgroups with respect to the group G.

In the tables below each table corresponds to one class of conjugate subgroups. Note that the subgroups are specified by their space-group types and transformation matrices. The listed transformation matrices are obtained combining the maximal subgroups transformation matrices from the MAXSUB database for each pair group-maximal subgroup in the chain indicated for each subgroup. Different transformation matrices (e.g. calculated following different chains) might specify the same (identical) subgroup if these transformation matrices are related by an element of the normalizer of the subgroup. The transformations by such transformation matrices of the subgroup operations to the basis of G result in the identical sets of elements of G.

The list of chains and the corresponding transformation matrices that specify the same subgroups can be seen by clicking on the button in the column Identical of the table.

Class 1

Check Chain [indices] Chain with HM symbols Transformation Identical
1 225 221 123 047 025 [4 3 2 2] Fm-3m > Pm-3m > P4/mmm > Pmmm > Pmm2
(
   0   1   0   1/2
   0   0   1   1/2
   1   0   0    0
)
2 225 221 200 047 025 [4 2 3 2] Fm-3m > Pm-3m > Pm-3 > Pmmm > Pmm2
(
   0   0   1    0
   1   0   0   1/2
   0   1   0   1/2
)
3 225 139 069 042 035 025 [3 2 2 2 2] Fm-3m > I4/mmm > Fmmm > Fmm2 > Cmm2 > Pmm2
(
   0  -1   0    0
   1   0   0    0
   0   0   1    0
)




Class 2

Check Chain [indices] Chain with HM symbols Transformation Identical
4 225 216 119 044 025 025 [2 3 2 2 2] Fm-3m > F-43m > I-4m2 > Imm2 > Pmm2 > Pmm2
(
  1/2   1   0    0
 -1/2   1   0    0
   0   0   1    0
)
5 225 216 119 115 025 025 [2 3 2 2 2] Fm-3m > F-43m > I-4m2 > P-4m2 > Pmm2 > Pmm2
(
   0   0   1   1/4
   1  1/2   0   1/2
  -1  1/2   0    0
)
6 225 216 119 044 025 025 [2 3 2 2 2] Fm-3m > F-43m > I-4m2 > Imm2 > Pmm2 > Pmm2
(
   0   0   1    0
   1  1/2   0    0
  -1  1/2   0    0
)
7 225 139 071 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmmm > Pmm2
(
  1/2   1   0   1/4
 -1/2   1   0   1/4
   0   0   1    0
)
8 225 139 107 099 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > I4mm > P4mm > Pmm2 > Pmm2
(
   1  1/2   0   1/4
  -1  1/2   0   -1/4
   0   0   1    0
)
9 225 139 107 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > I4mm > Imm2 > Pmm2 > Pmm2
(
   1  1/2   0    0
  -1  1/2   0    0
   0   0   1    0
)
10 225 139 123 115 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P-4m2 > Pmm2 > Pmm2
(
   0   0   1    0
  1/2   1   0    0
 -1/2   1   0    0
)
11 225 139 123 099 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P4mm > Pmm2 > Pmm2
(
   0   0   1    0
  1/2   1   0   1/4
 -1/2   1   0   1/4
)
12 225 139 123 115 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P-4m2 > Pmm2 > Pmm2
(
  -1  1/2   0   -1/4
   0   0   1    0
   1  1/2   0   1/4
)
13 225 139 071 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Imm2 > Pmm2 > Pmm2
(
 -1/2   1   0   1/4
   0   0   1    0
  1/2   1   0   1/4
)
14 225 139 071 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmm2 > Pmm2
(
  -1  1/2   0    0
   0   0   1    0
   1  1/2   0    0
)
15 225 139 107 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > I4mm > Imm2 > Pmm2 > Pmm2
(
 -1/2   1   0    0
   0   0   1    0
  1/2   1   0    0
)




Class 3

Check Chain [indices] Chain with HM symbols Transformation Identical
16 225 139 123 123 115 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P4/mmm > P-4m2 > Pmm2
(
  1/2  1/2   0    0
 -1/2  1/2   0    0
   0   0   2   1/2
)
17 225 139 071 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmmm > Pmm2
(
   0   0   2    0
  1/2  1/2   0    0
 -1/2  1/2   0    0
)
18 225 139 119 115 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > I-4m2 > P-4m2 > Pmm2 > Pmm2
(
 -1/2  1/2   0    0
   0   0   2    0
  1/2  1/2   0    0
)




Class 4

Check Chain [indices] Chain with HM symbols Transformation Identical
19 225 139 123 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmmm > Pmm2
(
   0  1/2  1/2    0
   0 -1/2  1/2    0
   2   0   0    0
)
20 225 139 123 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmmm > Pmm2
(
   0  1/2  1/2    0
   0 -1/2  1/2    0
   2   0   0   1/2
)
21 225 139 123 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmmm > Pmm2
(
  1/2   0  1/2    0
  1/2   0 -1/2    0
   0   2   0   1/2
)
22 225 139 071 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Imm2 > Pmm2 > Pmm2
(
  1/2   0  1/2    0
  1/2   0 -1/2    0
   0   2   0    0
)
23 225 139 071 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Imm2 > Pmm2 > Pmm2
(
   2   0   0   1/2
   0  1/2  1/2    0
   0 -1/2  1/2    0
)
24 225 139 071 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmm2 > Pmm2
(
   2   0   0    0
   0  1/2  1/2    0
   0 -1/2  1/2    0
)
25 225 139 071 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmmm > Pmm2
(
   0   2   0   1/2
  1/2   0  1/2    0
  1/2   0 -1/2    0
)
26 225 139 123 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmmm > Pmm2
(
   0   2   0    0
  1/2   0  1/2    0
  1/2   0 -1/2    0
)
27 225 139 123 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmmm > Pmm2
(
  1/2   0 -1/2    0
   0   2   0    0
  1/2   0  1/2    0
)
28 225 139 123 123 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > P4/mmm > Pmmm > Pmm2
(
   0 -1/2  1/2    0
   2   0   0    0
   0  1/2  1/2    0
)
29 225 139 071 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Imm2 > Pmm2 > Pmm2
(
  1/2   0 -1/2    0
   0   2   0   1/2
  1/2   0  1/2    0
)
30 225 139 071 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmm2 > Pmm2
(
   0 -1/2  1/2    0
   2   0   0   1/2
   0  1/2  1/2    0
)




Class 5

Check Chain [indices] Chain with HM symbols Transformation Identical
31 225 139 123 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmmm > Pmm2
(
   0  1/2   1   1/4
   0 -1/2   1   1/4
   1   0   0    0
)
32 225 139 071 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Imm2 > Pmm2 > Pmm2
(
  1/2   0   1    0
  1/2   0  -1    0
   0   1   0    0
)
33 225 139 071 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Imm2 > Pmm2 > Pmm2
(
   0   1   0    0
  1/2   0   1    0
  1/2   0  -1    0
)
34 225 139 071 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmm2 > Pmm2
(
   1   0   0    0
   0  1/2   1    0
   0 -1/2   1    0
)
35 225 139 071 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmmm > Pmm2
(
  1/2   0  -1   -1/4
   0   1   0    0
  1/2   0   1   1/4
)
36 225 139 071 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmmm > Pmm2
(
   0 -1/2   1    0
   1   0   0    0
   0  1/2   1    0
)




Class 6

Check Chain [indices] Chain with HM symbols Transformation Identical
37 225 139 123 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmm2 > Pmm2
(
   0   1  1/2    0
   0  -1  1/2    0
   1   0   0    0
)
38 225 139 123 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmm2 > Pmm2
(
   1   0  1/2    0
   1   0 -1/2    0
   0   1   0    0
)
39 225 139 123 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmm2 > Pmm2
(
   1   0  1/2   1/4
   1   0 -1/2   1/4
   0   1   0    0
)
40 225 139 071 044 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Imm2 > Pmm2 > Pmm2
(
   0   1  1/2   1/4
   0  -1  1/2   -1/4
   1   0   0    0
)
41 225 139 071 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmm2 > Pmm2
(
   1   0   0    0
   0   1  1/2    0
   0  -1  1/2    0
)
42 225 139 071 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmmm > Pmm2
(
   1   0   0    0
   0   1  1/2   1/4
   0  -1  1/2   -1/4
)
43 225 139 123 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmm2 > Pmm2
(
   0   1   0    0
   1   0  1/2    0
   1   0 -1/2    0
)
44 225 139 123 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmm2 > Pmm2
(
   0   1   0    0
   1   0  1/2   1/4
   1   0 -1/2   1/4
)
45 225 139 123 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmm2 > Pmm2
(
   0  -1  1/2   -1/4
   1   0   0    0
   0   1  1/2   1/4
)
46 225 139 123 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > P4/mmm > Pmmm > Pmmm > Pmm2
(
   1   0 -1/2   1/4
   0   1   0    0
   1   0  1/2   1/4
)
47 225 139 071 047 025 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmm2 > Pmm2
(
   0  -1  1/2    0
   1   0   0    0
   0   1  1/2    0
)
48 225 139 071 047 047 025 [3 2 2 2 2] Fm-3m > I4/mmm > Immm > Pmmm > Pmmm > Pmm2
(
   1   0 -1/2    0
   0   1   0    0
   1   0  1/2    0
)






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