Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
670 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 160 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 | C22×C32×C4 | C22 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #537 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3828 | C2 | semidirect |
| 2. C2×C3×#235 | C2 | semidirect |
| 3. C2×C3×C4⋊Dic3 | C2 | semidirect |
| 4. #63978 | C2 | semidirect |
| 5. #63916 | C2 | semidirect |
| 6. #63909 | C2 | semidirect |
| 7. #63871 | C2 | semidirect |
| 8. #63857 | C2 | semidirect |
| 33. C2×#235 | D3 | semidirect |
| 40. C2×C4⋊Dic3 | D3 | semidirect |
| 49. C2×C32×C4 | D4 | semidirect |
| 149. C2×C3×C4 | D12 | semidirect |
| 150. C2×C3×C4 | #67 | semidirect |
| 243. C32 | #537 | semidirect |
| 260. C2×C4 | #626 | semidirect |
| 296. C3 | #5510 | semidirect |
| 297. C3 | #5833 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3828 | 2 |
| C2×C3×#235 | 2 |
| C2×C3×C4⋊Dic3 | 2 |
| #63978 | 2 |
| #63916 | 2 |
| #63909 | 2 |
| #63871 | 2 |
| #63857 | 2 |
| #5833 | 3 |
| #5510 | 3 |