Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
704 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 218 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 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C4×#127 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #21734 | C2 | semidirect |
| 4. #17960 | C2 | semidirect |
| 6. #9855 | C2 | semidirect |
| 7. #9854 | C2 | semidirect |
| 8. #8826 | C2 | semidirect |
| 9. #7844 | C2 | semidirect |
| 25. #1716 | C4 | semidirect |
| 26. #1715 | C4 | semidirect |
| 27. #1714 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21734 | 2 |
| #21706 | 2 |
| #19608 | 2 |
| #17960 | 2 |
| #9856 | 2 |
| #9855 | 2 |
| #9854 | 2 |
| #8826 | 2 |
| #7844 | 2 |