Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1710 conjugacy classes of subgroups.
The subgroup lattice has 387 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. #22737 | C2 | semidirect |
| 5. #22736 | C2 | semidirect |
| 6. #22723 | C2 | semidirect |
| 7. #22722 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2987 | 2 |
| C2×#2961 | 2 |
| #34154 | 2 |
| #22737 | 2 |
| #22736 | 2 |
| #22723 | 2 |
| #22722 | 2 |
| #21980 | 2 |
| #21978 | 2 |
| #21357 | 2 |
| #21355 | 2 |