Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
348 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #99 |
| commutator | C2×C4 | C4×C16 |
| Frattini, cosocle | C22×C4⋊C8 | C22 |
| socle | C23 | #323 |
| prime radical | C24 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. #1322 | C4 | semidirect |
| 51. C2×C4⋊C4 | C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1322 | 2 |
| #10834 | 2 |