Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1726 conjugacy classes of subgroups.
The subgroup lattice has 974 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #387 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1441 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. #11248 | C2 | direct (N × H) |
| 25. C2×#430 | C4 | semidirect |
| 29. C22×#108 | C4 | semidirect |
| 164. C2×#108 | C2×C4 | semidirect |
| 971. C2 | #11248 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#430 | 2 |
| C22×#344 | 2 |
| C2×#1742 | 2 |
| C22×#338 | 2 |
| #11248 | 2 |