Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 494 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 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | #3284 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#912 | C2 | semidirect |
| 2. #5718 | C2 | semidirect |
| 3. #5555 | C2 | semidirect |
| 4. C3×#459 | C2 | semidirect |
| 5. #5158 | C2 | semidirect |
| 6. #5076 | C2 | semidirect |
| 7. C4×#214 | C2 | semidirect |
| 8. #4849 | C2 | semidirect |
| 9. #4834 | C2 | semidirect |
| 10. #4783 | C2 | semidirect |
| 11. #4778 | C2 | semidirect |
| 49. #459 | D3 | semidirect |
| 80. #214 | D4 | semidirect |
| 209. C3×C4 | #135 | semidirect |
| 271. C4 | #912 | semidirect |
| 274. C3 | #3284 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#912 | 2 |
| #5718 | 2 |
| #5555 | 2 |
| C3×#459 | 2 |
| #5158 | 2 |
| #5076 | 2 |
| C4×#214 | 2 |
| #4849 | 2 |
| #4834 | 2 |
| #4783 | 2 |
| #4778 | 2 |
| #3284 | 3 |