Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
552 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33249 | C2 | semidirect |
| 3. #13450 | C2 | semidirect |
| 5. #13366 | C2 | semidirect |
| 7. #13364 | C2 | semidirect |
| 8. #13330 | C2 | semidirect |
| 11. #12880 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33249 | 2 |
| #33247 | 2 |
| #13450 | 2 |
| #13449 | 2 |
| #13366 | 2 |
| #13365 | 2 |
| #13364 | 2 |
| #13330 | 2 |
| #13329 | 2 |
| #13327 | 2 |
| #12880 | 2 |