Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
356 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 80 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 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1862 |
| prime radical | #3050 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ #1729 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12922 | C2 | semidirect |
| 5. #11272 | C2 | semidirect |
| 6. #7261 | C2 | semidirect |
| 7. #7186 | C2 | semidirect |
| 27. #1350 | C4 | semidirect |
| 28. #1349 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22206 | 2 |
| #12990 | 2 |
| #12922 | 2 |
| #11709 | 2 |
| #11272 | 2 |
| #7261 | 2 |
| #7186 | 2 |