Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
564 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 137 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×#496 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #496 | D3 |
| hypocenter | C3 | #3150 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #496 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#916 | C2 | semidirect |
| 2. C2×#912 | C2 | semidirect |
| 3. #5716 | C2 | semidirect |
| 4. #5668 | C2 | semidirect |
| 5. C3×#496 | C2 | semidirect |
| 6. #5167 | C2 | semidirect |
| 7. #5154 | C2 | semidirect |
| 8. #5074 | C2 | semidirect |
| 9. #5060 | C2 | semidirect |
| 10. #4715 | C2 | semidirect |
| 11. D3×#97 | C2 | semidirect |
| 51. #496 | D3 | semidirect |
| 91. D3×C8 | D4 | semidirect |
| 260. C3 | #3150 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#916 | 2 |
| C2×#912 | 2 |
| #5716 | 2 |
| #5668 | 2 |
| C3×#496 | 2 |
| #5167 | 2 |
| #5154 | 2 |
| #5074 | 2 |
| #5060 | 2 |
| #4715 | 2 |
| D3×#97 | 2 |
| #3150 | 3 |