Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
496 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 131 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 | #3035 |
| 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. #5725 | C2 | semidirect |
| 3. C4×#239 | C2 | semidirect |
| 4. C3×#462 | C2 | semidirect |
| 5. #5185 | C2 | semidirect |
| 6. #5083 | C2 | semidirect |
| 7. #5027 | C2 | semidirect |
| 8. #4843 | C2 | semidirect |
| 9. D3×C4⋊C8 | C2 | semidirect |
| 10. #4819 | C2 | semidirect |
| 11. #4804 | C2 | semidirect |
| 50. #462 | D3 | semidirect |
| 61. #239 | D4 | semidirect |
| 225. C3×C4 | #135 | semidirect |
| 291. C4 | #919 | semidirect |
| 294. C3 | #3035 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#919 | 2 |
| #5725 | 2 |
| C4×#239 | 2 |
| C3×#462 | 2 |
| #5185 | 2 |
| #5083 | 2 |
| #5027 | 2 |
| #4843 | 2 |
| D3×C4⋊C8 | 2 |
| #4819 | 2 |
| #4804 | 2 |
| #3035 | 3 |