Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6852 conjugacy classes of subgroups.
The subgroup lattice has 279 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. Q8×#131 | C2 | direct (N × H) |
| 3. C2×Q8×M16 | C2 | semidirect |
| 4. C2×#2962 | C2 | semidirect |
| 5. C2×#2960 | C2 | semidirect |
| 6. C22×C8×Q8 | C2 | semidirect |
| 20. Q8×M16 | C22 | semidirect |
| 21. #2962 | C22 | semidirect |
| 22. #2960 | C22 | semidirect |
| 30. C2×C8×Q8 | C22 | semidirect |
| 43. C2×#131 | Q8 | direct (N × H) |
| 115. #131 | C2×Q8 | direct (N × H) |
| 117. C2×M16 | C2×Q8 | semidirect |
| 121. C22×C8 | C2×Q8 | semidirect |
| 178. C2×Q8 | #131 | direct (N × H) |
| 200. M16 | C22×Q8 | semidirect |
| 206. C2×C8 | C22×Q8 | semidirect |
| 232. Q8 | C2×#131 | direct (N × H) |
| 274. C2 | Q8×#131 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×#41 | 2 |
| Q8×#131 | 2 |
| C2×Q8×M16 | 2 |
| C2×#2962 | 2 |
| C2×#2960 | 2 |
| C22×C8×Q8 | 2 |
| C2×#2906 | 2 |
| C2×C4×#131 | 2 |