Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
588 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 171 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 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#123 | C2 |
| prime radical | C2×C32×D4 | C22 |
| hypercenter | #123 | C3⋊D3 |
| hypocenter | C32 | #563 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #64133 | C2 | semidirect |
| 5. #64098 | C2 | semidirect |
| 6. #64097 | C2 | semidirect |
| 7. #64088 | C2 | semidirect |
| 8. #64088 | C2 | semidirect |
| 9. #64087 | C2 | semidirect |
| 149. C32 | #563 | semidirect |
| 180. C3 | #5651 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#123 | 2 |
| #64145 | 2 |
| #64144 | 2 |
| #64133 | 2 |
| #64098 | 2 |
| #64097 | 2 |
| #64088 | 2 |
| #64087 | 2 |
| #5651 | 3 |