Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
527 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 527 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#3802 |
| commutator, cosocle | C32×C4 | C24 |
| Frattini | C4 | C22×D32 |
| Fitting | C2×C32×C8 | C22 |
| socle | C2×C32 | C22×D4 |
| hypercenter | C2×C8 | D32 |
| hypocenter | C32 | #578 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #64297 | C2 | semidirect |
| 2. #64110 | C2 | semidirect |
| 3. C2×C3×Dic12 | C2 | semidirect |
| 4. C3×#904 | C2 | semidirect |
| 5. #63827 | C2 | semidirect |
| 6. #63811 | C2 | semidirect |
| 7. #63810 | C2 | semidirect |
| 8. #63802 | C2 | semidirect |
| 42. C2×Dic12 | D3 | semidirect |
| 43. #904 | D3 | semidirect |
| 216. C32 | #578 | semidirect |
| 256. C3 | #5766 | semidirect |
| 257. C3 | #5771 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64297 | 2 |
| #64110 | 2 |
| C2×C3×Dic12 | 2 |
| C3×#904 | 2 |
| #63827 | 2 |
| #63811 | 2 |
| #63810 | 2 |
| #63802 | 2 |
| #5771 | 3 |
| #5766 | 3 |