Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
322 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#97 |
| prime radical | C23 | C4×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. #1313 | C4 | semidirect |
| 15. #1312 | C4 | semidirect |
| 16. #1311 | C4 | semidirect |
| 17. C8×C16 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12594 | 2 |
| #11404 | 2 |
| #11402 | 2 |
| C8×M32 | 2 |
| #7846 | 2 |