Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
240 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C4 | C2×C4×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×C4⋊C16 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ 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 |
|---|---|---|
| 13. #1422 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1422 | 2 |
| C4×M64 | 2 |
| #12539 | 2 |
| #7467 | 2 |