Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
490 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 490 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#432 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #432 | D3 |
| hypocenter | C3 | #3217 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #432 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5479 | C2 | semidirect |
| 2. C2×#874 | C2 | semidirect |
| 3. C3×#432 | C2 | semidirect |
| 4. #4848 | C2 | semidirect |
| 5. #4843 | C2 | semidirect |
| 6. #4732 | C2 | semidirect |
| 7. #4731 | C2 | semidirect |
| 8. #4713 | C2 | semidirect |
| 9. #4712 | C2 | semidirect |
| 19. C2×D24 | C22 | semidirect |
| 20. C2×#204 | C22 | semidirect |
| 22. #875 | C22 | semidirect |
| 27. C4×D12 | C22 | semidirect |
| 29. #870 | C22 | semidirect |
| 30. C4×Dic6 | C22 | semidirect |
| 31. C3×C4⋊C8 | C22 | semidirect |
| 33. C3×#97 | C22 | semidirect |
| 34. #821 | C22 | semidirect |
| 35. #821 | C22 | semidirect |
| 44. #432 | D3 | semidirect |
| 101. C4⋊C8 | C2×D3 | semidirect |
| 105. #97 | C2×D3 | semidirect |
| 248. C3 | #3217 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5479 | 2 |
| C2×#874 | 2 |
| C3×#432 | 2 |
| #4848 | 2 |
| #4843 | 2 |
| #4732 | 2 |
| #4731 | 2 |
| #4713 | 2 |
| #4712 | 2 |
| #3217 | 3 |