Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
396 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 82 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 | C2×D16 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2215 | C2 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C23×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. C22×SD64 | C2 | semidirect |
| 2. C2×#2216 | C2 | semidirect |
| 4. C2×#1416 | C2 | semidirect |
| 5. C2×#1415 | C2 | semidirect |
| 6. C2×C22⋊C32 | C2 | semidirect |
| 7. #7471 | C2 | semidirect |
| 39. C2×Q32 | D4 | semidirect |
| 68. C2×C32 | D4 | semidirect |
| 72. #366 | D4 | semidirect |
| 211. C23 | SD64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD64 | 2 |
| C2×#2216 | 2 |
| C2×#2215 | 2 |
| C2×#1416 | 2 |
| C2×#1415 | 2 |
| C2×C22⋊C32 | 2 |
| #7471 | 2 |