Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
419 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2206 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2263 | C2 | semidirect |
| 2. C2×#2246 | C2 | semidirect |
| 3. C2×#1419 | C2 | semidirect |
| 4. #13241 | C2 | semidirect |
| 5. #7467 | C2 | semidirect |
| 27. C2×Q32 | D4 | semidirect |
| 38. C2×D16 | D4 | semidirect |
| 50. M64 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2263 | 2 |
| C2×#2246 | 2 |
| C2×#1419 | 2 |
| #13241 | 2 |
| #7467 | 2 |