Finite Group DatabasebetaFinite Group DatabasebetaOrder 1512.
240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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. C2×C3×D3×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C32×C4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C3×C7×D12 | C3 | semidirect |
| 5. ΓL(1,𝔽8)×D12 | C3 | direct (N × H) |
| 8. C2×C3×D3×C7 | C2×C3 | semidirect |
| 10. C2×D3×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 11. C32×C4×C7 | C2×C3 | semidirect |
| 14. C3×C4×ΓL(1,𝔽8) | D3 | semidirect |
| 15. C3×C4×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 17. C32×ΓL(1,𝔽8) | D4 | semidirect |
| 19. C7×D12 | C32 | semidirect |
| 32. C2×D3×C7 | C2×C32 | semidirect |
| 35. C3×C4×C7 | C3×D3 | semidirect |
| 36. C3×C4×C7 | C2×C32 | semidirect |
| 38. C4×ΓL(1,𝔽8) | C3×D3 | semidirect |
| 41. C3×D12 | ΓL(1,𝔽8) | direct (N × H) |
| 42. C32×C7 | C3×D4 | semidirect |
| 45. C3×ΓL(1,𝔽8) | D12 | direct (N × H) |
| 46. C3×ΓL(1,𝔽8) | C3×D4 | semidirect |
| 60. C2×C3×D3 | C2×ΓL(1,𝔽8) | semidirect |
| 65. C32×C4 | C2×ΓL(1,𝔽8) | semidirect |
| 67. C4×C7 | C32×D3 | semidirect |
| 71. D12 | C3×ΓL(1,𝔽8) | direct (N × H) |
| 73. C3×C7 | C3×D12 | semidirect |
| 74. C3×C7 | C32×D4 | semidirect |
| 76. ΓL(1,𝔽8) | C3×D12 | direct (N × H) |
| 89. C2×D3 | C2×C3×ΓL(1,𝔽8) | semidirect |
| 93. C3×C4 | D3×ΓL(1,𝔽8) | semidirect |
| 94. C3×C4 | C2×C3×ΓL(1,𝔽8) | semidirect |
| 99. C32 | D4×ΓL(1,𝔽8) | semidirect |
| 101. C7 | C32×D12 | semidirect |
| 111. C4 | C3×D3×ΓL(1,𝔽8) | semidirect |
| 115. C3 | ΓL(1,𝔽8)×D12 | direct (N × H) |
| 116. C3 | C3×D4×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D3×ΓL(1,𝔽8) | 2 |
| C32×C4×ΓL(1,𝔽8) | 2 |
| C3×C7×D12 | 3 |
| C3×D4×ΓL(1,𝔽8) | 3 |
| ΓL(1,𝔽8)×D12 | 3 |
| C32×D12 | 7 |