Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1787 conjugacy classes of subgroups.
The subgroup lattice has 285 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C4⋊C4 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #22733 | C2 | semidirect |
| 4. Q8×#108 | C2 | semidirect |
| 59. #430 | Q8 | direct (N × H) |
| 139. #108 | C2×Q8 | semidirect |
| 225. Q8 | #430 | 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 |
| #22733 | 2 |
| Q8×#108 | 2 |
| #21177 | 2 |
| #20806 | 2 |
| C4×#430 | 2 |