Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4126 conjugacy classes of subgroups; only the 738 normal subgroups are shown.
This group has 4126 conjugacy classes of subgroups — too many to draw. Only its 738 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 |
| prime radical | C22×#466 | C2 |
| 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. C24×Q16 | C2 | semidirect |
| 2. C23×SD32 | C2 | semidirect |
| 4. C2×#2185 | C2 | direct (N × H) |
| 5. C22×#358 | C2 | semidirect |
| 6. C22×C22⋊C16 | C2 | semidirect |
| 12. C23×Q16 | C22 | semidirect |
| 17. C22×SD32 | C22 | semidirect |
| 23. #2185 | C22 | direct (N × H) |
| 24. C2×#358 | C22 | semidirect |
| 25. C2×C22⋊C16 | C22 | semidirect |
| 43. C22×Q16 | D4 | semidirect |
| 45. C22×Q16 | C23 | semidirect |
| 59. C2×SD32 | C23 | semidirect |
| 67. #358 | C23 | semidirect |
| 68. C22⋊C16 | C23 | semidirect |
| 105. C2×Q16 | C2×D4 | semidirect |
| 137. C24 | SD32 | semidirect |
| 177. Q16 | C22×D4 | semidirect |
| 202. C23 | C2×SD32 | semidirect |
| 239. C22 | #2185 | direct (N × H) |
| 242. C22 | C22×SD32 | semidirect |
| 253. C2 | C2×#2185 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q16 | 2 |
| C23×SD32 | 2 |
| C22×#466 | 2 |
| C2×#2185 | 2 |
| C22×#358 | 2 |
| C22×C22⋊C16 | 2 |