Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
486 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C16 | C23 |
| commutator, socle | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13105 | C2 | semidirect |
| 4. #12895 | C2 | semidirect |
| 5. C16×C4⋊C4 | C2 | semidirect |
| 7. #12539 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21668 | 2 |
| #13121 | 2 |
| #13105 | 2 |
| #12895 | 2 |
| C16×C4⋊C4 | 2 |
| C16×C22⋊C4 | 2 |
| #12539 | 2 |