Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1335 conjugacy classes of subgroups.
The subgroup lattice has 391 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ 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. #7962 | C2 | semidirect |
| 5. #7961 | C2 | semidirect |
| 6. #7958 | C2 | semidirect |
| 7. #7957 | C2 | semidirect |
| 10. #2991 | C4 | semidirect |
| 12. #2990 | C4 | semidirect |
| 17. #2965 | C4 | semidirect |
| 18. C8×#41 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60278 | 2 |
| #60263 | 2 |
| #34193 | 2 |
| #7962 | 2 |
| #7961 | 2 |
| #7958 | 2 |
| #7957 | 2 |