Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
205 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 205 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1391 |
| commutator | C22×C8 | C42 |
| Frattini, cosocle | #2931 | C22 |
| socle | C2 | #7237 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ #2931 ◁ G | |
| LECS | G ▷ #2931 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #2150 | C4 | semidirect |
| 7. C2×#346 | C4 | semidirect |
| 9. #1391 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13192 | 2 |
| #12807 | 2 |