Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
696 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 222 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 | C22×D8 | C23 |
| 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 |
|---|---|---|
| 2. #33312 | C2 | semidirect |
| 4. #13276 | C2 | semidirect |
| 5. #13177 | C2 | semidirect |
| 6. #12898 | C2 | semidirect |
| 40. C4×D8 | Q8 | semidirect |
| 143. D8 | C4⋊Q8 | semidirect |
| 230. C4 | #2226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2226 | 2 |
| #33312 | 2 |
| C42×D8 | 2 |
| #13276 | 2 |
| #13177 | 2 |
| #12898 | 2 |
| #12811 | 2 |