Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
8034 conjugacy classes of subgroups; only the 3490 normal subgroups are shown.
This group has 8034 conjugacy classes of subgroups — too many to draw. Only its 3490 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C26 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C24×Q16 | C2 | semidirect |
| 3. C22×#510 | C2 | direct (N × H) |
| 4. C23×Q32 | C2 | semidirect |
| 5. C23×SD32 | C2 | semidirect |
| 6. C23×M32 | C2 | semidirect |
| 15. C23×Q16 | C22 | semidirect |
| 19. C2×#510 | C22 | direct (N × H) |
| 20. C22×Q32 | C22 | semidirect |
| 21. C22×SD32 | C22 | semidirect |
| 22. C22×M32 | C22 | semidirect |
| 44. C22×Q16 | C23 | semidirect |
| 52. #510 | C23 | direct (N × H) |
| 53. C2×Q32 | C23 | semidirect |
| 54. C2×SD32 | C23 | semidirect |
| 55. C2×M32 | C23 | semidirect |
| 87. C2×Q16 | C24 | semidirect |
| 96. Q32 | C24 | semidirect |
| 97. SD32 | C24 | semidirect |
| 98. M32 | C24 | semidirect |
| 143. C23 | #510 | direct (N × H) |
| 169. C22 | C2×#510 | direct (N × H) |
| 179. C2 | C22×#510 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#135 | 2 |
| C24×Q16 | 2 |
| C22×#510 | 2 |
| C23×Q32 | 2 |
| C23×SD32 | 2 |
| C23×M32 | 2 |