Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
468 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 65. #346 | Q8 | semidirect |
| 66. C4×C16 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21725 | 2 |
| #13507 | 2 |
| #13452 | 2 |
| #13451 | 2 |
| #12674 | 2 |
| C4×#367 | 2 |
| C4×#366 | 2 |
| #12592 | 2 |