Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
272 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C4×D16 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1425 | C2 | semidirect |
| 3. C2×#1417 | C2 | semidirect |
| 4. C2×#1397 | C2 | semidirect |
| 5. #13241 | C2 | semidirect |
| 9. C2×Q64 | C4 | semidirect |
| 10. C2×SD64 | C4 | semidirect |
| 11. C2×D32 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2262 | 2 |
| C2×#1425 | 2 |
| C2×#1417 | 2 |
| C2×#1397 | 2 |
| #13241 | 2 |