Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
492 conjugacy classes of subgroups; only the 129 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#447 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #447 | D3 |
| hypocenter | C3 | #3100 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #447 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×Dic3 | C2 | semidirect |
| 2. #5596 | C2 | semidirect |
| 3. C3×#447 | C2 | semidirect |
| 4. #5217 | C2 | semidirect |
| 5. #5150 | C2 | semidirect |
| 6. Dic3×D8 | C2 | semidirect |
| 7. #5033 | C2 | semidirect |
| 8. #4759 | C2 | semidirect |
| 9. #4756 | C2 | semidirect |
| 10. #4736 | C2 | semidirect |
| 11. #4718 | C2 | semidirect |
| 19. #940 | C22 | semidirect |
| 25. D4×Dic3 | C22 | semidirect |
| 30. #893 | C22 | semidirect |
| 31. #888 | C22 | semidirect |
| 37. #832 | C22 | semidirect |
| 39. #821 | C22 | semidirect |
| 40. #818 | C22 | semidirect |
| 41. Dic3⋊C8 | C22 | semidirect |
| 42. C8×Dic3 | C22 | semidirect |
| 43. #807 | C22 | semidirect |
| 51. #447 | D3 | semidirect |
| 137. C3×D4 | #41 | semidirect |
| 152. C2×Dic3 | D8 | semidirect |
| 241. D4 | #237 | semidirect |
| 291. C3 | #3100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×Dic3 | 2 |
| #5596 | 2 |
| C3×#447 | 2 |
| #5217 | 2 |
| #5150 | 2 |
| Dic3×D8 | 2 |
| #5033 | 2 |
| #4759 | 2 |
| #4756 | 2 |
| #4736 | 2 |
| #4718 | 2 |
| #3100 | 3 |