Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
386 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 386 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 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#436 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #436 | D3 |
| hypocenter | C3 | #3354 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ #436 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5563 | C2 | semidirect |
| 2. #5555 | C2 | semidirect |
| 3. C3×#436 | C2 | semidirect |
| 4. #5135 | C2 | semidirect |
| 5. #5121 | C2 | semidirect |
| 6. #4846 | C2 | semidirect |
| 7. #4842 | C2 | semidirect |
| 8. #4737 | C2 | semidirect |
| 9. #4735 | C2 | semidirect |
| 10. C4×D24 | C2 | semidirect |
| 11. #4680 | C2 | semidirect |
| 51. #436 | D3 | semidirect |
| 142. C3×C8 | #41 | semidirect |
| 222. C8 | #237 | semidirect |
| 248. C3 | #3354 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5563 | 2 |
| #5555 | 2 |
| C3×#436 | 2 |
| #5135 | 2 |
| #5121 | 2 |
| #4846 | 2 |
| #4842 | 2 |
| #4737 | 2 |
| #4735 | 2 |
| C4×D24 | 2 |
| #4680 | 2 |
| #3354 | 3 |