Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
504 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 504 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×#460 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #460 | D3 |
| hypocenter | C3 | #3287 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #460 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#916 | C2 | semidirect |
| 2. #5718 | C2 | semidirect |
| 3. #5577 | C2 | semidirect |
| 4. C3×#460 | C2 | semidirect |
| 5. #5172 | C2 | semidirect |
| 6. #5073 | C2 | semidirect |
| 7. C4×#216 | C2 | semidirect |
| 8. #4848 | C2 | semidirect |
| 9. #4834 | C2 | semidirect |
| 10. #4781 | C2 | semidirect |
| 11. #4775 | C2 | semidirect |
| 49. #460 | D3 | semidirect |
| 81. #216 | D4 | semidirect |
| 212. C3×C4 | #135 | semidirect |
| 275. C4 | #916 | semidirect |
| 278. C3 | #3287 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#916 | 2 |
| #5718 | 2 |
| #5577 | 2 |
| C3×#460 | 2 |
| #5172 | 2 |
| #5073 | 2 |
| C4×#216 | 2 |
| #4848 | 2 |
| #4834 | 2 |
| #4781 | 2 |
| #4775 | 2 |
| #3287 | 3 |