Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2430 conjugacy classes of subgroups; only the 698 normal subgroups are shown.
This group has 2430 conjugacy classes of subgroups — too many to draw. Only its 698 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 | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D16 | C2 | semidirect |
| 2. C24×C16 | C2 | semidirect |
| 3. C22×#466 | C2 | semidirect |
| 4. C2×#2215 | C2 | direct (N × H) |
| 5. C22×#366 | C2 | semidirect |
| 6. C22×#357 | C2 | semidirect |
| 11. C22×D16 | C22 | semidirect |
| 15. C23×C16 | C22 | semidirect |
| 17. C2×#466 | C22 | semidirect |
| 20. #2215 | C22 | direct (N × H) |
| 21. C2×#366 | C22 | semidirect |
| 22. C2×#357 | C22 | semidirect |
| 41. C2×D16 | C23 | semidirect |
| 44. C22×C16 | D4 | semidirect |
| 47. C22×C16 | C23 | semidirect |
| 49. #466 | C23 | semidirect |
| 53. #366 | C23 | semidirect |
| 54. #357 | C23 | semidirect |
| 94. C2×C16 | C2×D4 | semidirect |
| 106. C24 | D16 | semidirect |
| 146. C16 | C22×D4 | semidirect |
| 157. C23 | C2×D16 | semidirect |
| 188. C22 | #2215 | direct (N × H) |
| 191. C22 | C22×D16 | semidirect |
| 201. C2 | C2×#2215 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D16 | 2 |
| C24×C16 | 2 |
| C22×#466 | 2 |
| C2×#2215 | 2 |
| C22×#366 | 2 |
| C22×#357 | 2 |