Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
607 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 607 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 |
| prime radical | C2×C4⋊(C3⋊D3) | C22 |
| hypercenter | C42 | D32 |
| hypocenter | C32 | D4×Q8 |
| 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×#3797 | C2 | semidirect |
| 3. C3×#869 | C2 | semidirect |
| 4. C3×C4×Dic6 | C2 | semidirect |
| 5. #63918 | C2 | semidirect |
| 6. #63895 | C2 | semidirect |
| 8. #63871 | C2 | semidirect |
| 43. #869 | D3 | semidirect |
| 44. C4×Dic6 | D3 | semidirect |
| 50. C4⋊(C3⋊D3) | Q8 | semidirect |
| 56. C3×Dic6 | D4 | semidirect |
| 102. C2×C3⋊D3 | C2×Q8 | semidirect |
| 105. C32×C4 | C2×Q8 | semidirect |
| 152. Dic6 | D12 | semidirect |
| 196. C3×C4 | D3×Q8 | semidirect |
| 197. C3×C4 | D3×Q8 | semidirect |
| 204. C32 | D4×Q8 | semidirect |
| 240. C4 | #3797 | semidirect |
| 243. C3 | Q8×D12 | semidirect |
| 244. C3 | #5727 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3797 | 2 |
| C4×C4⋊(C3⋊D3) | 2 |
| C3×#869 | 2 |
| C3×C4×Dic6 | 2 |
| #63918 | 2 |
| #63895 | 2 |
| #63894 | 2 |
| #63871 | 2 |
| #5727 | 3 |
| Q8×D12 | 3 |