Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
839 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 839 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2247 | C2 | semidirect |
| 3. C2×#2245 | C2 | semidirect |
| 6. #13281 | C2 | semidirect |
| 7. #13207 | C2 | semidirect |
| 8. #13015 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3177 | 2 |
| C2×#2247 | 2 |
| C2×#2245 | 2 |
| #33180 | 2 |
| #13306 | 2 |
| #13281 | 2 |
| #13207 | 2 |
| #13015 | 2 |
| #12600 | 2 |