Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
188 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11858 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. #2253 | C4 | semidirect |
| 8. #2247 | C4 | semidirect |
| 12. #2176 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33741 | 2 |
| #13643 | 2 |
| #12893 | 2 |
| #7425 | 2 |
| #7423 | 2 |