Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4455 conjugacy classes of subgroups.
The subgroup lattice has 1035 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 |
|---|---|---|
| 201. #557 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #59934 | 2 |
| #59871 | 2 |
| #59870 | 2 |
| #59670 | 2 |
| #59666 | 2 |
| #59445 | 2 |
| #45276 | 2 |
| #44419 | 2 |
| #39978 | 2 |
| #39753 | 2 |
| #39653 | 2 |
| #39619 | 2 |
| #36598 | 2 |
| #36565 | 2 |
| #36563 | 2 |
| Q8×C4⋊Q8 | 2 |
| #36286 | 2 |
| #36285 | 2 |
| #35936 | 2 |
| #34950 | 2 |
| #34335 | 2 |
| #34328 | 2 |
| C4×#557 | 2 |