Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
268 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2215 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13235 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13633 | C2 | semidirect |
| 3. #13622 | C2 | semidirect |
| 4. #13549 | C2 | semidirect |
| 5. #13243 | C2 | semidirect |
| 11. C2×Q64 | C4 | semidirect |
| 12. C2×SD64 | C4 | semidirect |
| 13. C2×D32 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2262 | 2 |
| #13633 | 2 |
| #13622 | 2 |
| #13549 | 2 |
| #13243 | 2 |