Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
317 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #340 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| socle | C22 | #1279 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. C2×C82 | C4 | semidirect |
| 13. #1376 | C4 | semidirect |
| 32. #341 | C8 | semidirect |
| 34. #340 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #12788 | 2 |
| #10818 | 2 |
| C8×C22⋊C8 | 2 |