Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
756 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 218 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2242 | C2 | semidirect |
| 2. #33311 | C2 | semidirect |
| 4. #13207 | C2 | semidirect |
| 6. #13205 | C2 | semidirect |
| 7. C4×#346 | C2 | semidirect |
| 70. C4×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2242 | 2 |
| #33311 | 2 |
| #33303 | 2 |
| #13207 | 2 |
| #13206 | 2 |
| #13205 | 2 |
| C4×#346 | 2 |