Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2646 conjugacy classes of subgroups; only the 650 normal subgroups are shown.
This group has 2646 conjugacy classes of subgroups — too many to draw. Only its 650 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator | C23 | C23×C8 |
| socle | C24 | C23×C4 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1954 | C2 | semidirect |
| 3. C2×#1842 | C2 | semidirect |
| 4. C2×C42×C8 | C2 | semidirect |
| 24. C2×#127 | C8 | semidirect |
| 39. C2×C4×C8 | D4 | semidirect |
| 117. C2×C8 | #127 | semidirect |
| 121. C42 | C22⋊C8 | semidirect |
| 172. C4 | #1954 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#127 | 2 |
| C2×#1954 | 2 |
| C2×#1842 | 2 |
| C2×C42×C8 | 2 |