Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
328 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | #33600 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×M64 | C2 | semidirect |
| 3. C2×#1418 | C2 | semidirect |
| 4. #12835 | C2 | semidirect |
| 5. #7477 | C2 | semidirect |
| 15. #2216 | C4 | semidirect |
| 16. #2216 | C4 | semidirect |
| 17. #2215 | C4 | semidirect |
| 18. #2215 | C4 | semidirect |
| 53. M64 | D4 | semidirect |
| 224. C22 | #1418 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M64 | 2 |
| #33600 | 2 |
| C2×#1418 | 2 |
| #12835 | 2 |
| #7477 | 2 |