Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
720 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 202 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 | #19616 | C2 |
| 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. #22280 | C2 | semidirect |
| 3. #17981 | C2 | semidirect |
| 4. #17975 | C2 | semidirect |
| 5. #9860 | C2 | semidirect |
| 6. #9849 | C2 | semidirect |
| 7. #9844 | C2 | semidirect |
| 8. #8827 | C2 | semidirect |
| 9. #7842 | C2 | semidirect |
| 20. #1722 | C4 | semidirect |
| 21. #1721 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22280 | 2 |
| #19616 | 2 |
| #17981 | 2 |
| #17975 | 2 |
| #9860 | 2 |
| #9849 | 2 |
| #9844 | 2 |
| #8827 | 2 |
| #7842 | 2 |