Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6888 conjugacy classes of subgroups.
The subgroup lattice has 390 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. #34336 | C2 | semidirect |
| 3. #34336 | C2 | semidirect |
| 4. #34335 | C2 | semidirect |
| 8. C4×#557 | C2 | semidirect |
| 13. #3502 | C4 | semidirect |
| 71. Q82 | C2×C4 | semidirect |
| 72. #557 | C2×C4 | semidirect |
| 73. #557 | C2×C4 | semidirect |
| 85. #519 | Q8 | semidirect |
| 256. C2×Q8 | C4×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3502 | 2 |
| #34336 | 2 |
| #34335 | 2 |
| #34272 | 2 |
| #34249 | 2 |
| #34215 | 2 |
| C4×#557 | 2 |
| #34145 | 2 |
| C2×#2351 | 2 |
| C4×#519 | 2 |