Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6525 conjugacy classes of subgroups.
The subgroup lattice has 440 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. #60195 | C2 | semidirect |
| 4. Q8×#131 | C2 | semidirect |
| 5. #60193 | C2 | semidirect |
| 6. #60192 | C2 | semidirect |
| 7. C2×#2962 | C2 | semidirect |
| 28. Q8×M16 | C22 | semidirect |
| 29. Q8×M16 | C22 | semidirect |
| 34. #2962 | C22 | semidirect |
| 35. #2962 | C22 | semidirect |
| 36. #2962 | C22 | semidirect |
| 37. #2962 | C22 | semidirect |
| 38. #2962 | C22 | semidirect |
| 39. #2961 | C22 | semidirect |
| 40. #2961 | C22 | semidirect |
| 41. #2960 | C22 | semidirect |
| 42. #2960 | C22 | semidirect |
| 43. C2×#446 | C22 | semidirect |
| 44. C2×#446 | C22 | semidirect |
| 45. C2×C8×Q8 | C22 | semidirect |
| 66. #568 | Q8 | semidirect |
| 187. #131 | C2×Q8 | semidirect |
| 190. C2×M16 | 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 |
| Q8×#131 | 2 |
| #60193 | 2 |
| #60192 | 2 |
| C2×#2962 | 2 |
| #60100 | 2 |
| C2×#2906 | 2 |
| #60058 | 2 |
| C4×#568 | 2 |