Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
206 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 66 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 | #13528 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13638 | C2 | semidirect |
| 2. #13636 | C2 | semidirect |
| 6. #12911 | C2 | semidirect |
| 7. #7469 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13638 | 2 |
| #13636 | 2 |
| C2×#1419 | 2 |
| #13617 | 2 |
| #13245 | 2 |
| #12911 | 2 |
| #7469 | 2 |