Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
934 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 934 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×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. C2×#2246 | C2 | semidirect |
| 2. C2×#2218 | C2 | semidirect |
| 3. C2×C4×M32 | C2 | semidirect |
| 4. #33171 | C2 | semidirect |
| 5. #13239 | C2 | semidirect |
| 6. #13202 | C2 | semidirect |
| 7. #12906 | C2 | semidirect |
| 40. C2×M32 | D4 | semidirect |
| 64. C4×C16 | D4 | semidirect |
| 176. C16 | #121 | semidirect |
| 234. C4 | #2218 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2246 | 2 |
| C2×#2218 | 2 |
| C2×C4×M32 | 2 |
| #33171 | 2 |
| #13239 | 2 |
| #13202 | 2 |
| #12906 | 2 |