Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
782 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33312 | C2 | semidirect |
| 2. #29208 | C2 | semidirect |
| 5. #10676 | C2 | semidirect |
| 8. #8820 | C2 | semidirect |
| 9. #8260 | C2 | semidirect |
| 46. #492 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33312 | 2 |
| #29208 | 2 |
| #29139 | 2 |
| C4×#492 | 2 |
| #10676 | 2 |
| #10673 | 2 |
| #9842 | 2 |
| #8820 | 2 |
| #8260 | 2 |