Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D251 |
| commutator, socle, cosocle | C23×C251 | C24 |
| Fitting | C2×#120×C251 | C2 |
| hypercenter | C2×#120 | D251 |
| hypocenter | C251 | #2812 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ C2×#120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22548684 | C2 | semidirect |
| 2. #30631615 | C2 | semidirect |
| 3. #30631602 | C2 | semidirect |
| 4. #30631601 | C2 | semidirect |
| 5. #30631595 | C2 | semidirect |
| 6. #30631543 | C2 | semidirect |
| 7. C2×#120×C251 | C2 | semidirect |
| 9. C22×C22⋊D251 | C22 | semidirect |
| 11. C2×Dic251⋊C4 | C22 | semidirect |
| 15. C2×C22⋊Dic251 | C22 | semidirect |
| 16. C2×(C2×D251)⋊C4 | C22 | semidirect |
| 17. C251⋊#95 | C22 | semidirect |
| 18. C251⋊#95 | C22 | semidirect |
| 39. C24×C251 | D4 | semidirect |
| 104. C251 | #2812 | semidirect |
| 107. C2×#120 | D251 | semidirect |
| 131. C24 | C22⋊D251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548684 | 2 |
| #30631615 | 2 |
| #30631602 | 2 |
| #30631601 | 2 |
| #30631595 | 2 |
| #30631543 | 2 |
| C2×#120×C251 | 2 |
| #2812 | 251 |