Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2008 conjugacy classes of subgroups.
The subgroup lattice has 297 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #22721 | C2 | semidirect |
| 5. Q8×C4⋊C8 | C2 | semidirect |
| 64. #424 | Q8 | direct (N × H) |
| 158. C4⋊C8 | C2×Q8 | semidirect |
| 235. Q8 | #424 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×M16 | 2 |
| C22×C8×Q8 | 2 |
| Q8×#117 | 2 |
| #22721 | 2 |
| Q8×C4⋊C8 | 2 |
| #21145 | 2 |
| #21143 | 2 |
| #20793 | 2 |
| C4×#424 | 2 |