Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1225 conjugacy classes of subgroups.
The subgroup lattice has 352 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ 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. #8126 | C2 | semidirect |
| 5. #8125 | C2 | semidirect |
| 6. #8124 | C2 | semidirect |
| 7. #8124 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60284 | 2 |
| #60283 | 2 |
| #60128 | 2 |
| #8126 | 2 |
| #8125 | 2 |
| #8124 | 2 |