Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
418 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #376 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1434 |
| prime radical | C24 | #124 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. C2×#322 | C4 | semidirect |
| 13. C2×#322 | C4 | semidirect |
| 14. C2×#322 | C4 | semidirect |
| 15. C2×C82 | C4 | semidirect |
| 19. #1274 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1454 | 2 |
| #8006 | 2 |
| #7980 | 2 |
| #7956 | 2 |
| #7857 | 2 |