Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 456 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 | #384×C251 | C2 |
| prime radical | C22×D1004 | C22 |
| hypercenter | #384 | D251 |
| hypocenter | C251 | #2823 |
| LCS | G ▷ C23×C251 ▷ C251 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22548709 | C2 | semidirect |
| 2. #30631617 | C2 | semidirect |
| 3. #30631609 | C2 | semidirect |
| 5. #384×C251 | C2 | semidirect |
| 29. C42×C251 | D4 | semidirect |
| 58. C251 | #2823 | semidirect |
| 64. #384 | D251 | semidirect |
| 88. C42 | D1004 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548709 | 2 |
| #30631617 | 2 |
| #30631609 | 2 |
| #30631604 | 2 |
| #384×C251 | 2 |
| #2823 | 251 |