Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
806 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 806 conjugacy classes of subgroups — too many to draw. Only its 282 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 | #357 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×M64 | C2 | semidirect |
| 3. C22×#367 | C2 | semidirect |
| 4. C2×#1418 | C2 | direct (N × H) |
| 8. C2×#508 | C4 | semidirect |
| 10. C22×Q32 | C4 | semidirect |
| 12. C22×D16 | C4 | semidirect |
| 16. C2×M64 | C22 | semidirect |
| 19. C2×#367 | C22 | semidirect |
| 20. #1418 | C22 | direct (N × H) |
| 34. #508 | C2×C4 | semidirect |
| 37. C2×Q32 | C2×C4 | semidirect |
| 41. C2×D16 | C2×C4 | semidirect |
| 51. M64 | C23 | semidirect |
| 54. #367 | C23 | semidirect |
| 85. Q32 | C22×C4 | semidirect |
| 89. D16 | C22×C4 | semidirect |
| 181. C22 | #1418 | direct (N × H) |
| 194. C2 | C2×#1418 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#508 | 2 |
| C22×M64 | 2 |
| C22×#367 | 2 |
| C2×#1418 | 2 |