Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
946 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 946 conjugacy classes of subgroups — too many to draw. Only its 322 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C42 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. #1744 | C4 | semidirect |
| 12. C4×#97 | C4 | semidirect |
| 14. #1274 | C4 | semidirect |
| 73. C4×C8 | C42 | semidirect |
| 113. C42 | #97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20463 | 2 |
| C4×#376 | 2 |
| C2×#1274 | 2 |