Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
336 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #99 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | C8×#99 |
| prime radical | #525 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12613 | C2 | semidirect |
| 5. #7207 | C2 | semidirect |
| 7. C23×C16 | C4 | semidirect |
| 11. #2151 | C4 | semidirect |
| 15. #2112 | C4 | semidirect |
| 17. #2108 | C4 | semidirect |
| 22. #1328 | C4 | semidirect |
| 23. #1320 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33432 | 2 |
| #12623 | 2 |
| #12613 | 2 |
| #11483 | 2 |
| #7207 | 2 |