Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
356 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C8×C22⋊C4 |
| prime radical | #525 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #12611 | C2 | semidirect |
| 5. #7202 | C2 | semidirect |
| 6. C2×#504 | C4 | semidirect |
| 11. #2151 | C4 | semidirect |
| 12. #2113 | C4 | semidirect |
| 15. #2108 | C4 | semidirect |
| 20. #1323 | C4 | semidirect |
| 21. #1315 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33432 | 2 |
| #12611 | 2 |
| #11468 | 2 |
| #8238 | 2 |
| #7202 | 2 |