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×#462 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #462 | D3 |
| hypocenter | C3 | #3284 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #462 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#919 | C2 | semidirect |
| 2. #5726 | C2 | semidirect |
| 3. #5577 | C2 | semidirect |
| 4. C3×#462 | C2 | semidirect |
| 5. #5187 | C2 | semidirect |
| 6. #5085 | C2 | semidirect |
| 7. C4×#215 | C2 | semidirect |
| 8. #4843 | C2 | semidirect |
| 9. #4834 | C2 | semidirect |
| 10. #4810 | C2 | semidirect |
| 11. #4807 | C2 | semidirect |
| 48. #462 | D3 | semidirect |
| 82. #215 | D4 | semidirect |
| 211. C3×C4 | #135 | semidirect |
| 271. C4 | #919 | 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×#919 | 2 |
| #5726 | 2 |
| #5577 | 2 |
| C3×#462 | 2 |
| #5187 | 2 |
| #5085 | 2 |
| C4×#215 | 2 |
| #4843 | 2 |
| #4834 | 2 |
| #4810 | 2 |
| #4807 | 2 |
| #3284 | 3 |