Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
355 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 355 conjugacy classes of subgroups — too many to draw. Only its 110 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×D4 | C22 |
| hypercenter | D4 | D32 |
| hypocenter | C32 | C2×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D4×C3⋊D3 | C2 | semidirect |
| 2. C3×#237 | C2 | semidirect |
| 3. #3811 | C2 | semidirect |
| 4. C2×#624 | C2 | semidirect |
| 5. #3804 | C2 | semidirect |
| 6. C4×D32 | C2 | semidirect |
| 7. #3799 | C2 | semidirect |
| 8. #3797 | C2 | semidirect |
| 12. C32×D4 | C22 | semidirect |
| 13. #638 | C22 | semidirect |
| 14. C4⋊(C3⋊D3) | C22 | semidirect |
| 16. C3×#67 | C22 | semidirect |
| 17. C2×C3×Dic3 | C22 | semidirect |
| 18. C3×C4×D3 | C22 | semidirect |
| 19. C3×Dic6 | C22 | semidirect |
| 20. #627 | C22 | semidirect |
| 21. #626 | C22 | semidirect |
| 24. #624 | C22 | semidirect |
| 25. D3×Dic3 | C22 | semidirect |
| 30. #237 | D3 | semidirect |
| 55. C3×D4 | C2×D3 | semidirect |
| 61. #67 | C2×D3 | semidirect |
| 63. C2×Dic3 | C2×D3 | semidirect |
| 74. C4×D3 | C2×D3 | semidirect |
| 76. Dic6 | C2×D3 | semidirect |
| 80. C3⋊D3 | #41 | semidirect |
| 109. C32 | C2×#41 | semidirect |
| 118. D4 | D32 | semidirect |
| 147. C3 | D3×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D4×C3⋊D3 | 2 |
| C3×#237 | 2 |
| #3811 | 2 |
| C2×#624 | 2 |
| #3804 | 2 |
| C4×D32 | 2 |
| #3799 | 2 |
| #3797 | 2 |
| D3×#41 | 3 |