Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
254 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #382 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 55. #322 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #8807 | 2 |
| #8785 | 2 |
| #7843 | 2 |
| #7018 | 2 |