Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
637 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 637 conjugacy classes of subgroups — too many to draw. Only its 108 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 | C2×D16 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×#357 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1417 | C2 | semidirect |
| 3. C2×C22⋊C32 | C2 | semidirect |
| 4. #12837 | C2 | semidirect |
| 15. C22×C32 | C22 | semidirect |
| 16. C2×#368 | C22 | semidirect |
| 233. C22 | #1417 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#508 | 2 |
| C2×#1417 | 2 |
| C2×C22⋊C32 | 2 |
| #12837 | 2 |