Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4673 conjugacy classes of subgroups.
The subgroup lattice has 815 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 161. #557 | Q8 | semidirect |
| 707. Q8 | #557 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #59872 | 2 |
| #59670 | 2 |
| #59667 | 2 |
| #59587 | 2 |
| #59467 | 2 |
| #59456 | 2 |
| #36600 | 2 |
| #36574 | 2 |
| #36567 | 2 |
| Q8×#125 | 2 |
| #36535 | 2 |
| #36497 | 2 |
| #36475 | 2 |
| #36466 | 2 |
| #36447 | 2 |
| C4×#557 | 2 |