Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
422 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 422 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #390 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1954 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12788 | C2 | semidirect |
| 3. C4×C22⋊C16 | C2 | semidirect |
| 4. #11444 | C2 | semidirect |
| 5. #11441 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1937 | 2 |
| #12788 | 2 |
| C4×C22⋊C16 | 2 |
| #11444 | 2 |
| #11441 | 2 |
| #11422 | 2 |
| #11419 | 2 |