Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
33492 conjugacy classes of subgroups.
The subgroup lattice has 182 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×2−1+4 | C2 | direct (N × H) |
| 2. C2×#3503 | C2 | semidirect |
| 3. C2×#3502 | C2 | semidirect |
| 4. C2×Q8×#41 | C2 | semidirect |
| 5. C22×Q82 | C2 | semidirect |
| 8. #3503 | C22 | semidirect |
| 9. #3502 | C22 | semidirect |
| 10. Q8×#41 | C22 | semidirect |
| 13. C2×Q82 | C22 | semidirect |
| 28. C2×2−1+4 | Q8 | direct (N × H) |
| 67. 2−1+4 | C2×Q8 | direct (N × H) |
| 68. C2×#41 | C2×Q8 | semidirect |
| 72. C22×Q8 | C2×Q8 | semidirect |
| 111. #41 | C22×Q8 | semidirect |
| 117. C2×Q8 | C22×Q8 | semidirect |
| 118. C2×Q8 | 2−1+4 | direct (N × H) |
| 151. Q8 | C2×2−1+4 | direct (N × H) |
| 177. C2 | Q8×2−1+4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×2−1+4 | 2 |
| C2×#3503 | 2 |
| C2×#3502 | 2 |
| C2×Q8×#41 | 2 |
| C22×Q82 | 2 |
| C2×C4×2−1+4 | 2 |