Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
662 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 662 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C2×C32×Q8 | C22 |
| hypercenter | C2×Q8 | D32 |
| hypocenter | C32 | #548 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3832 | C2 | semidirect |
| 2. C2×C3×D3×Q8 | C2 | semidirect |
| 3. C3×Q8×Dic3 | C2 | semidirect |
| 4. #63909 | C2 | semidirect |
| 5. C4×#626 | C2 | semidirect |
| 6. #63871 | C2 | semidirect |
| 7. #63864 | C2 | semidirect |
| 28. C2×D3×Q8 | D3 | semidirect |
| 32. Q8×Dic3 | D3 | semidirect |
| 41. C32×Q8 | D4 | semidirect |
| 91. C2×C3×D3 | #41 | semidirect |
| 111. C3×Q8 | D12 | semidirect |
| 112. C3×Q8 | #67 | semidirect |
| 172. C2×D3 | #239 | semidirect |
| 185. C32 | #548 | semidirect |
| 189. Q8 | #626 | semidirect |
| 226. C3 | #5576 | semidirect |
| 227. C3 | #5817 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3832 | 2 |
| C2×C3×D3×Q8 | 2 |
| C3×Q8×Dic3 | 2 |
| #63909 | 2 |
| C4×#626 | 2 |
| #63871 | 2 |
| #63864 | 2 |
| #5817 | 3 |
| #5576 | 3 |