Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D16 |
| commutator, cosocle | C3×C16 | C23 |
| Frattini | C16 | C22×D3 |
| Fitting | C3×D32 | C2 |
| socle | C2×C3 | C2×D16 |
| hypercenter | D32 | D3 |
| hypocenter | C3 | #2263 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #4912 | C2 | semidirect |
| 2. D3×D16 | C2 | semidirect |
| 3. C3×D32 | C2 | semidirect |
| 4. #4520 | C2 | semidirect |
| 5. #4519 | C2 | semidirect |
| 6. #4449 | C2 | semidirect |
| 7. #4447 | C2 | semidirect |
| 12. C3×D16 | C22 | semidirect |
| 15. Dic24 | C22 | semidirect |
| 18. C3×C32 | C22 | semidirect |
| 19. C3⋊C32 | C22 | semidirect |
| 25. D32 | D3 | semidirect |
| 45. D16 | C2×D3 | semidirect |
| 48. C32 | C2×D3 | semidirect |
| 103. C3 | #2263 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #4912 | 2 |
| D3×D16 | 2 |
| C3×D32 | 2 |
| #4520 | 2 |
| #4519 | 2 |
| #4449 | 2 |
| #4447 | 2 |
| #2263 | 3 |