Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
556 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 150 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 | C2×#493 | C22 |
| 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. #33261 | C2 | semidirect |
| 3. #13539 | C2 | semidirect |
| 5. #13149 | C2 | semidirect |
| 6. #13147 | C2 | semidirect |
| 7. #12813 | C2 | semidirect |
| 8. #12811 | C2 | semidirect |
| 9. #12654 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33261 | 2 |
| #21697 | 2 |
| #13539 | 2 |
| #13407 | 2 |
| #13149 | 2 |
| #13147 | 2 |
| #12813 | 2 |
| #12811 | 2 |
| #12654 | 2 |