Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
541 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3227 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #27645 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #20806 | C2 | semidirect |
| 4. #13511 | C2 | semidirect |
| 5. #13486 | C2 | semidirect |
| 6. #13188 | C2 | semidirect |
| 7. #12876 | C2 | semidirect |
| 8. #12825 | C2 | semidirect |
| 9. #12791 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33656 | 2 |
| #21549 | 2 |
| #20806 | 2 |
| #13511 | 2 |
| #13486 | 2 |
| #13188 | 2 |
| #12876 | 2 |
| #12825 | 2 |
| #12791 | 2 |