Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1284 conjugacy classes of subgroups.
The subgroup lattice has 355 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #8089 | C2 | semidirect |
| 5. #8087 | C2 | semidirect |
| 6. #8086 | C2 | semidirect |
| 7. #8084 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60283 | 2 |
| C2×#2971 | 2 |
| #60127 | 2 |
| #8089 | 2 |
| #8087 | 2 |
| #8086 | 2 |
| #8084 | 2 |