Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
222 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1347 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C22×M32 | C22 |
| socle | C2 | #7258 |
| prime radical | C24 | SD32 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ C22×M32 ◁ G | |
| LECS | G ▷ C22×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C2×#368 | C4 | semidirect |
| 8. C2×#368 | C4 | semidirect |
| 11. #1400 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1400 | 2 |
| #12838 | 2 |
| C2×#1384 | 2 |