Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
5338 conjugacy classes of subgroups; only the 1602 normal subgroups are shown.
This group has 5338 conjugacy classes of subgroups — too many to draw. Only its 1602 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24×C4 | C23 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C25 | C22×C4 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#1818 | C2 | direct (N × H) |
| 4. C22×#336 | C2 | semidirect |
| 15. #1818 | C22 | direct (N × H) |
| 16. C2×#336 | C22 | semidirect |
| 41. #336 | C23 | semidirect |
| 151. C22 | #1818 | direct (N × H) |
| 163. C2 | C2×#1818 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#117 | 2 |
| C23×C22⋊C8 | 2 |
| C2×#1818 | 2 |
| C22×#336 | 2 |