Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
672 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 186 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 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ 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×#2252 | C2 | semidirect |
| 3. C2×#2243 | C2 | semidirect |
| 4. #33247 | C2 | semidirect |
| 5. #13238 | C2 | semidirect |
| 6. #13236 | C2 | semidirect |
| 8. #12539 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2252 | 2 |
| C2×#2249 | 2 |
| C2×#2243 | 2 |
| #33247 | 2 |
| #13238 | 2 |
| #13236 | 2 |
| #13235 | 2 |
| #12539 | 2 |