Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4965 conjugacy classes of subgroups.
The subgroup lattice has 372 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×#568 | C2 | semidirect |
| 4. #20399 | C2 | semidirect |
| 5. #20397 | C2 | semidirect |
| 6. C4×#413 | C2 | semidirect |
| 7. C2×C4×#101 | C2 | semidirect |
| 8. #20358 | C2 | semidirect |
| 23. #2889 | C4 | direct (N × H) |
| 46. #568 | C2×C4 | semidirect |
| 93. #413 | C2×C4 | semidirect |
| 94. C2×#101 | C2×C4 | semidirect |
| 362. C4 | #2889 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×2−1+4 | 2 |
| C2×#2889 | 2 |
| C4×#568 | 2 |
| #20399 | 2 |
| #20397 | 2 |
| C4×#413 | 2 |
| C2×C4×#101 | 2 |
| #20358 | 2 |