Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
708 conjugacy classes of subgroups.
The subgroup lattice has 300 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #13516 | C2 | direct (N × H) |
| 291. C2 | #13516 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#367 | 2 |
| C22×C4⋊C16 | 2 |
| C2×#1942 | 2 |
| #13516 | 2 |
| C2×#1380 | 2 |