Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
398 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 398 conjugacy classes of subgroups — too many to draw. Only its 117 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×#476 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #476 | D3 |
| hypocenter | C3 | #3303 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #476 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5734 | C2 | semidirect |
| 2. #5555 | C2 | semidirect |
| 3. C3×#476 | C2 | semidirect |
| 4. #5093 | C2 | semidirect |
| 5. C4×#213 | C2 | semidirect |
| 6. #4868 | C2 | semidirect |
| 7. #4864 | C2 | semidirect |
| 8. #4841 | C2 | semidirect |
| 9. #4834 | C2 | semidirect |
| 10. #4781 | C2 | semidirect |
| 11. #4772 | C2 | semidirect |
| 46. #476 | D3 | semidirect |
| 254. C3 | #3303 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5734 | 2 |
| #5555 | 2 |
| C3×#476 | 2 |
| #5093 | 2 |
| C4×#213 | 2 |
| #4868 | 2 |
| #4864 | 2 |
| #4841 | 2 |
| #4834 | 2 |
| #4781 | 2 |
| #4772 | 2 |
| #3303 | 3 |