Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
478 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 120 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 | #562 |
| 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. #64084 | C2 | semidirect |
| 2. C3×#875 | C2 | semidirect |
| 3. C3×#870 | C2 | semidirect |
| 4. #63910 | C2 | semidirect |
| 5. #63896 | C2 | semidirect |
| 6. #63890 | C2 | semidirect |
| 7. #63888 | C2 | semidirect |
| 8. #63886 | C2 | semidirect |
| 9. #63854 | C2 | semidirect |
| 50. #875 | D3 | semidirect |
| 51. #870 | D3 | semidirect |
| 243. C32 | #562 | semidirect |
| 290. C3 | #5684 | semidirect |
| 291. C3 | #5700 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #64084 | 2 |
| C3×#875 | 2 |
| C3×#870 | 2 |
| #63910 | 2 |
| #63896 | 2 |
| #63890 | 2 |
| #63888 | 2 |
| #63886 | 2 |
| #63854 | 2 |
| #5700 | 3 |
| #5684 | 3 |