Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
654 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 654 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 | C32×C42 | C22 |
| hypercenter | C42 | D32 |
| hypocenter | C32 | #532 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#3795 | C2 | semidirect |
| 2. #64078 | C2 | semidirect |
| 3. C3×#874 | C2 | semidirect |
| 4. C3×C42×D3 | C2 | semidirect |
| 5. #63908 | C2 | semidirect |
| 6. Dic3×D12 | C2 | semidirect |
| 7. #63860 | C2 | semidirect |
| 32. #874 | D3 | semidirect |
| 34. C42×D3 | D3 | semidirect |
| 49. C3×C4×D3 | D4 | semidirect |
| 88. C32×C4 | #41 | semidirect |
| 127. C4×D3 | D12 | semidirect |
| 164. C3×C4 | #235 | semidirect |
| 166. C3×C4 | #237 | semidirect |
| 172. C32 | #532 | semidirect |
| 205. C4 | #3795 | semidirect |
| 208. C3 | #5477 | semidirect |
| 209. C3 | #5716 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3795 | 2 |
| #64078 | 2 |
| C3×#874 | 2 |
| C3×C42×D3 | 2 |
| #63908 | 2 |
| Dic3×D12 | 2 |
| #63860 | 2 |
| #5716 | 3 |
| #5477 | 3 |