Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6494 conjugacy classes of subgroups.
The subgroup lattice has 320 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C26 |
| commutator, socle | C22 | C25×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60195 | C2 | semidirect |
| 3. #60193 | C2 | semidirect |
| 4. #60192 | C2 | semidirect |
| 5. #60191 | C2 | semidirect |
| 6. C2×#2961 | C2 | semidirect |
| 23. Q8×M16 | C22 | semidirect |
| 24. #2962 | C22 | semidirect |
| 25. #2962 | C22 | semidirect |
| 30. #2961 | C22 | semidirect |
| 31. #2961 | C22 | semidirect |
| 33. #2960 | C22 | semidirect |
| 34. C2×#446 | C22 | semidirect |
| 53. C2×#131 | Q8 | semidirect |
| 139. #131 | C2×Q8 | semidirect |
| 142. C2×M16 | C2×Q8 | semidirect |
| 148. C22×C8 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62813 | 2 |
| #60195 | 2 |
| #60193 | 2 |
| #60192 | 2 |
| #60191 | 2 |
| C2×#2961 | 2 |
| #60100 | 2 |
| #60099 | 2 |
| #60058 | 2 |
| C2×#2875 | 2 |