Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
419 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #29912 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2873 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13545 | C2 | semidirect |
| 4. #13524 | C2 | semidirect |
| 5. #13512 | C2 | semidirect |
| 6. #13491 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33126 | 2 |
| #20757 | 2 |
| #13545 | 2 |
| #13524 | 2 |
| #13512 | 2 |
| #13491 | 2 |
| #13009 | 2 |
| #13000 | 2 |
| #12932 | 2 |
| #12924 | 2 |
| #12851 | 2 |