Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1607 conjugacy classes of subgroups.
The subgroup lattice has 354 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #18068 | C2 | semidirect |
| 4. M16×Q16 | C2 | semidirect |
| 5. #17998 | C2 | semidirect |
| 6. #17958 | C2 | semidirect |
| 7. #17903 | C2 | semidirect |
| 8. C2×C8×Q16 | C2 | semidirect |
| 150. #131 | Q16 | direct (N × H) |
| 250. Q16 | #131 | direct (N × H) |
| 253. M16 | C2×Q16 | semidirect |
| 263. C2×C8 | C2×Q16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q16×#41 | 2 |
| Q8×#131 | 2 |
| #18068 | 2 |
| M16×Q16 | 2 |
| #17998 | 2 |
| #17958 | 2 |
| #17903 | 2 |
| C2×C8×Q16 | 2 |
| #17828 | 2 |
| #17509 | 2 |
| C8×#131 | 2 |