Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1370 conjugacy classes of subgroups.
The subgroup lattice has 368 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. #8073 | C2 | semidirect |
| 5. #8071 | C2 | semidirect |
| 6. #8070 | C2 | semidirect |
| 7. #8068 | C2 | semidirect |
| 9. #2989 | C4 | semidirect |
| 10. #2988 | C4 | semidirect |
| 14. #2971 | C4 | semidirect |
| 16. #2970 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60283 | 2 |
| M16×#41 | 2 |
| #34125 | 2 |
| #8073 | 2 |
| #8071 | 2 |
| #8070 | 2 |
| #8068 | 2 |