Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2694 conjugacy classes of subgroups; only the 1562 normal subgroups are shown.
This group has 2694 conjugacy classes of subgroups — too many to draw. Only its 1562 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C22 | C23×C42 |
| prime radical | C25 | C24 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13829 | C2 | semidirect |
| 3. #13806 | C2 | semidirect |
| 6. #2283 | C4 | semidirect |
| 53. C2×C42 | C42 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2283 | 2 |
| #13829 | 2 |
| #13806 | 2 |
| #13780 | 2 |