Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1027 conjugacy classes of subgroups; only the 568 normal subgroups are shown.
This group has 1027 conjugacy classes of subgroups — too many to draw. Only its 568 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2571 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #29231 | C2 | semidirect |
| 2. #29230 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #29231 | 2 |
| #29230 | 2 |
| C2×#2102 | 2 |
| #21184 | 2 |
| #20806 | 2 |
| #20757 | 2 |
| #20445 | 2 |