Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
458 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13542 | C2 | semidirect |
| 4. #13521 | C2 | semidirect |
| 5. #13457 | C2 | semidirect |
| 46. #427 | Q8 | semidirect |
| 49. C2×#107 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33213 | 2 |
| #21163 | 2 |
| #13542 | 2 |
| #13521 | 2 |
| #13457 | 2 |
| #12928 | 2 |
| #12870 | 2 |
| #12789 | 2 |