Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6036 conjugacy classes of subgroups.
The subgroup lattice has 736 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C24 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #34530 | C2 | semidirect |
| 3. #34490 | C2 | semidirect |
| 4. #34490 | C2 | semidirect |
| 5. #34488 | C2 | semidirect |
| 12. #34336 | C2 | semidirect |
| 13. #34328 | C2 | semidirect |
| 18. #3556 | C4 | semidirect |
| 140. #564 | C2×C4 | semidirect |
| 141. #557 | C2×C4 | semidirect |
| 142. #557 | C2×C4 | semidirect |
| 143. #557 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3556 | 2 |
| #34530 | 2 |
| #34490 | 2 |
| #34488 | 2 |
| #34455 | 2 |
| #34445 | 2 |
| #34421 | 2 |
| #34420 | 2 |
| #34394 | 2 |
| #34375 | 2 |
| #34336 | 2 |
| #34328 | 2 |
| #34281 | 2 |
| #34221 | 2 |
| C4×#541 | 2 |
| #34074 | 2 |