Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
784 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 784 conjugacy classes of subgroups — too many to draw. Only its 162 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 ◁ C2×#117 ◁ C2×#405 ◁ 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×#2218 | C2 | semidirect |
| 2. C2×#2215 | C2 | semidirect |
| 3. #33179 | C2 | semidirect |
| 4. #13528 | C2 | semidirect |
| 5. #13325 | C2 | semidirect |
| 6. #12545 | C2 | semidirect |
| 42. C2×M32 | D4 | semidirect |
| 45. C22×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2218 | 2 |
| C2×#2215 | 2 |
| #33179 | 2 |
| #13528 | 2 |
| #13325 | 2 |
| #12545 | 2 |