Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2630 conjugacy classes of subgroups.
The subgroup lattice has 391 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #1446 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ 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 |
|---|---|---|
| 4. #11395 | C2 | direct (N × H) |
| 37. C22×M16 | Q8 | semidirect |
| 38. C22×M16 | Q8 | semidirect |
| 39. C22×M16 | Q8 | semidirect |
| 43. C23×C8 | Q8 | semidirect |
| 129. C2×M16 | C2×Q8 | semidirect |
| 139. C22×C8 | C2×Q8 | semidirect |
| 388. C2 | #11395 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3458 | 2 |
| C22×#344 | 2 |
| #11395 | 2 |