Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
348 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 348 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×D32 |
| commutator, socle, cosocle | C2×C32 | C24 |
| Fitting | C32×Q8 | C22 |
| hypercenter | Q8 | D32 |
| hypocenter | C32 | C2×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3832 | C2 | semidirect |
| 2. C3×#239 | C2 | semidirect |
| 3. C3×D3×Q8 | C2 | semidirect |
| 4. D3×D12 | C2 | semidirect |
| 5. C4×D32 | C2 | semidirect |
| 6. #3799 | C2 | semidirect |
| 7. #3796 | C2 | semidirect |
| 11. C32×Q8 | C22 | semidirect |
| 12. C4⋊(C3⋊D3) | C22 | semidirect |
| 13. C4×C3⋊D3 | C22 | semidirect |
| 14. C3×D12 | C22 | semidirect |
| 16. C3×C4×D3 | C22 | semidirect |
| 17. C3×Dic6 | C22 | semidirect |
| 18. #626 | C22 | semidirect |
| 19. #624 | C22 | semidirect |
| 21. D3×Dic3 | C22 | semidirect |
| 29. #239 | D3 | direct (N × H) |
| 30. D3×Q8 | D3 | semidirect |
| 50. C3×Q8 | C2×D3 | semidirect |
| 51. C3×Q8 | C2×D3 | semidirect |
| 62. D12 | C2×D3 | semidirect |
| 71. C4×D3 | C2×D3 | semidirect |
| 72. Dic6 | C2×D3 | semidirect |
| 76. C3×D3 | #41 | semidirect |
| 103. C32 | C2×#41 | semidirect |
| 106. Q8 | D32 | semidirect |
| 125. D3 | #239 | direct (N × H) |
| 135. C3 | C2×#239 | semidirect |
| 136. C3 | D3×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3832 | 2 |
| C3×#239 | 2 |
| C3×D3×Q8 | 2 |
| D3×D12 | 2 |
| C4×D32 | 2 |
| #3799 | 2 |
| #3796 | 2 |
| D3×#41 | 3 |
| C2×#239 | 3 |