Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
382 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23350 |
| prime radical | #575 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #13270 | C2 | semidirect |
| 3. #13267 | C2 | semidirect |
| 4. #13126 | C2 | semidirect |
| 5. #13125 | C2 | semidirect |
| 7. #12918 | C2 | semidirect |
| 8. #12916 | C2 | semidirect |
| 9. #12885 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23833 | 2 |
| #13270 | 2 |
| #13267 | 2 |
| #13126 | 2 |
| #13125 | 2 |
| #12979 | 2 |
| #12918 | 2 |
| #12916 | 2 |
| #12885 | 2 |