Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
774 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 282 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 | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C24 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #13618 | C2 | direct (N × H) |
| 5. C2×#1416 | C2 | semidirect |
| 10. C22×Q32 | C4 | semidirect |
| 22. #1416 | C22 | semidirect |
| 39. C2×Q32 | C2×C4 | semidirect |
| 182. C2 | #13618 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q32 | 2 |
| C22×M64 | 2 |
| C2×#2158 | 2 |
| #13618 | 2 |
| C2×#1416 | 2 |