Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
405 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini, socle | C22×C3 | C23×D3 |
| Fitting | C22×Q8×C9 | C2 |
| prime radical | C2×#3697 | C2 |
| hypercenter | C22×Q8 | D9 |
| hypocenter | C9 | #536 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×C9 | C2 | semidirect |
| 3. #63509 | C2 | semidirect |
| 4. #63508 | C2 | semidirect |
| 5. #63506 | C2 | semidirect |
| 6. #63491 | C2 | semidirect |
| 7. #63488 | C2 | semidirect |
| 73. C22×Q8 | D9 | semidirect |
| 123. C9 | #536 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×C9 | 2 |
| C2×#3697 | 2 |
| #63509 | 2 |
| #63508 | 2 |
| #63506 | 2 |
| #63491 | 2 |
| #63488 | 2 |
| #5816 | 3 |