Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
453 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3253 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #27626 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ 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. #20828 | C2 | semidirect |
| 3. #13498 | C2 | semidirect |
| 5. #13188 | C2 | semidirect |
| 6. #13182 | C2 | semidirect |
| 7. #13137 | C2 | semidirect |
| 8. #13117 | C2 | semidirect |
| 9. #12825 | C2 | semidirect |
| 10. #12822 | C2 | semidirect |
| 11. #12789 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21545 | 2 |
| #20828 | 2 |
| #13498 | 2 |
| #13395 | 2 |
| #13188 | 2 |
| #13182 | 2 |
| #13137 | 2 |
| #13117 | 2 |
| #12825 | 2 |
| #12822 | 2 |
| #12789 | 2 |