Finite Group DatabasebetaFinite Group DatabasebetaOrder 7056.
1160 conjugacy classes of subgroups; only the 297 normal subgroups are shown.
This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 297 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D12×#3861 | C2 | direct (N × H) |
| 2. C2×C3×C4×#3861 | C2 | semidirect |
| 3. C2×C72×D12 | C3 | semidirect |
| 4. C22×D3×#3861 | C2 | semidirect |
| 8. C2×D3×#3861 | C22 | semidirect |
| 9. C3×C4×#3861 | C22 | semidirect |
| 10. C22×D3×C72 | C2×C3 | semidirect |
| 12. C2×C3×C4×C72 | C2×C3 | semidirect |
| 13. C72×D12 | C2×C3 | semidirect |
| 16. C2×C4×#3861 | D3 | semidirect |
| 18. C2×C3×#3861 | D4 | semidirect |
| 24. C2×D3×C72 | C22×C3 | semidirect |
| 29. C3×C4×C72 | C22×C3 | semidirect |
| 31. C4×#3861 | C2×D3 | semidirect |
| 35. C3×#3861 | C2×D4 | semidirect |
| 38. C2×C4×C72 | C3×D3 | semidirect |
| 39. C2×C7×D12 | ΓL(1,𝔽8) | semidirect |
| 41. C2×C3×C72 | C3×D4 | semidirect |
| 47. C2×#3861 | D12 | direct (N × H) |
| 56. C4×C72 | C2×C3×D3 | semidirect |
| 57. C22×D3×C7 | C2×ΓL(1,𝔽8) | semidirect |
| 59. C2×C3×C4×C7 | C2×ΓL(1,𝔽8) | semidirect |
| 60. C7×D12 | C2×ΓL(1,𝔽8) | semidirect |
| 64. C3×C72 | C2×C3×D4 | semidirect |
| 66. #3861 | C2×D12 | direct (N × H) |
| 73. C2×C72 | C3×D12 | semidirect |
| 77. C2×D3×C7 | C22×ΓL(1,𝔽8) | semidirect |
| 82. C3×C4×C7 | C22×ΓL(1,𝔽8) | semidirect |
| 91. C2×C4×C7 | D3×ΓL(1,𝔽8) | semidirect |
| 92. C72 | C2×C3×D12 | semidirect |
| 94. C2×D12 | #3861 | direct (N × H) |
| 95. C2×C3×C7 | D4×ΓL(1,𝔽8) | semidirect |
| 110. C4×C7 | C2×D3×ΓL(1,𝔽8) | semidirect |
| 112. C22×D3 | C2×#3861 | semidirect |
| 116. C2×C3×C4 | C2×#3861 | semidirect |
| 117. D12 | C2×#3861 | direct (N × H) |
| 118. C3×C7 | C2×D4×ΓL(1,𝔽8) | semidirect |
| 126. C2×C7 | ΓL(1,𝔽8)×D12 | semidirect |
| 134. C2×D3 | C22×#3861 | semidirect |
| 137. C3×C4 | C22×#3861 | semidirect |
| 141. C2×C4 | D3×#3861 | semidirect |
| 142. C7 | C2×ΓL(1,𝔽8)×D12 | semidirect |
| 148. C2×C3 | D4×#3861 | semidirect |
| 154. C4 | C2×D3×#3861 | semidirect |
| 157. C3 | C2×D4×#3861 | semidirect |
| 160. C2 | D12×#3861 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D12×#3861 | 2 |
| C2×C3×C4×#3861 | 2 |
| C2×C72×D12 | 3 |
| C22×D3×#3861 | 2 |
| C2×D4×#3861 | 3 |
| C2×ΓL(1,𝔽8)×D12 | 7 |