Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1135 conjugacy classes of subgroups; only the 302 normal subgroups are shown.
This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 302 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator, socle | C23 | C23×C8 |
| prime radical | C2×#120 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C16 | C2 | semidirect |
| 2. C2×#2168 | C2 | semidirect |
| 4. #12976 | C2 | semidirect |
| 5. #12912 | C2 | semidirect |
| 6. #12679 | C2 | semidirect |
| 7. #12611 | C2 | semidirect |
| 9. C22×M32 | C22 | semidirect |
| 16. C2×C4⋊C16 | C22 | semidirect |
| 19. C2×C22⋊C16 | C22 | semidirect |
| 20. C2×#346 | C22 | semidirect |
| 25. #1374 | C22 | semidirect |
| 39. C22×C16 | D4 | semidirect |
| 109. C24 | M32 | semidirect |
| 153. C16 | #120 | semidirect |
| 196. C22 | #2168 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C16 | 2 |
| C2×#2168 | 2 |
| C8×#120 | 2 |
| #12976 | 2 |
| #12912 | 2 |
| #12679 | 2 |
| #12611 | 2 |