Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
606 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 606 conjugacy classes of subgroups — too many to draw. Only its 274 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#382 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #8795 | C2 | semidirect |
| 19. #1451 | C4 | semidirect |
| 56. #322 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20801 | 2 |
| #20769 | 2 |
| #17319 | 2 |
| #8795 | 2 |
| #7957 | 2 |
| C2×#1286 | 2 |