Finite Group DatabasebetaFinite Group DatabasebetaOrder 1056.
328 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D11 |
| commutator, socle, cosocle | C2×C3×C11 | C24 |
| Fitting | C3×Q8×C11 | C22 |
| hypercenter | Q8 | D3×D11 |
| hypocenter | C3×C11 | C2×#41 |
| LCS | G ▷ C2×C3×C11 ▷ C3×C11 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10587283 | C2 | semidirect |
| 2. C11×#239 | C2 | semidirect |
| 3. C3×Q8×D11 | C2 | semidirect |
| 4. D11×D12 | C2 | semidirect |
| 5. C4×D3×D11 | C2 | semidirect |
| 6. #10587231 | C2 | semidirect |
| 7. #10587230 | C2 | semidirect |
| 10. C3×Q8×C11 | C22 | semidirect |
| 11. D132 | C22 | semidirect |
| 12. C4×D33 | C22 | semidirect |
| 13. C11×D12 | C22 | semidirect |
| 14. C4×D3×C11 | C22 | semidirect |
| 16. C3×Dic22 | C22 | semidirect |
| 17. #63142 | C22 | semidirect |
| 18. D33⋊C4 | C22 | semidirect |
| 19. D3×Dic11 | C22 | semidirect |
| 23. Q8×D11 | D3 | semidirect |
| 38. Q8×C11 | C2×D3 | semidirect |
| 48. Dic22 | C2×D3 | semidirect |
| 51. C3×D11 | #41 | semidirect |
| 57. #239 | D11 | direct (N × H) |
| 68. C3×C11 | C2×#41 | semidirect |
| 72. C3×Q8 | C2×D11 | semidirect |
| 77. D12 | C2×D11 | semidirect |
| 81. C4×D3 | C2×D11 | semidirect |
| 85. D11 | #239 | direct (N × H) |
| 101. C11 | C2×#239 | semidirect |
| 104. Q8 | D3×D11 | semidirect |
| 126. C3 | #41×D11 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10587283 | 2 |
| C11×#239 | 2 |
| C3×Q8×D11 | 2 |
| D11×D12 | 2 |
| C4×D3×D11 | 2 |
| #10587231 | 2 |
| #10587230 | 2 |
| #41×D11 | 3 |
| C2×#239 | 11 |