Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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×D509 |
| commutator, socle, cosocle | C22×C509 | C24 |
| Fitting | C4⋊Q8×C509 | C2 |
| prime radical | C22×D509 | C23 |
| hypercenter | C4⋊Q8 | D509 |
| hypocenter | C509 | #533 |
| LCS | G ▷ C22×C509 ▷ C509 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #30654215 | C2 | semidirect |
| 2. #30654206 | C2 | semidirect |
| 5. Q8×Dic509 | C2 | semidirect |
| 6. #30654180 | C2 | semidirect |
| 9. C4⋊Q8×C509 | C2 | semidirect |
| 35. C4×D509 | Q8 | semidirect |
| 57. C509 | #533 | semidirect |
| 60. C4⋊Q8 | D509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30654215 | 2 |
| #30654206 | 2 |
| #30654195 | 2 |
| #30654191 | 2 |
| Q8×Dic509 | 2 |
| #30654180 | 2 |
| #30654173 | 2 |
| C42×D509 | 2 |
| C4⋊Q8×C509 | 2 |
| #533 | 509 |