Finite Group DatabasebetaFinite Group DatabasebetaOrder 32576.
238 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D509 |
| commutator, socle, cosocle | C23×C509 | C23 |
| Fitting, prime radical | C22×D4×C509 | C2 |
| hypercenter | C22×D4 | D509 |
| hypocenter | C509 | #394 |
| LCS | G ▷ C23×C509 ▷ C509 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22550840 | C2 | semidirect |
| 2. C2×C22⋊Dic509 | C2 | semidirect |
| 3. C2×C22⋊Dic509 | C2 | semidirect |
| 4. #30654122 | C2 | semidirect |
| 23. C2×Dic509 | D4 | semidirect |
| 57. C509 | #394 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22550840 | 2 |
| C2×C22⋊Dic509 | 2 |
| #30654122 | 2 |
| C22×D4×C509 | 2 |
| #394 | 509 |