Finite Group DatabasebetaFinite Group DatabasebetaOrder 32128.
924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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 |
| hypercenter | #384 | D251 |
| hypocenter | C251 | #2856 |
| 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×#22548712 | C2 | semidirect |
| 2. C2×#22548690 | C2 | semidirect |
| 3. C2×C42×D251 | C2 | semidirect |
| 4. #30631588 | C2 | semidirect |
| 5. #384×C251 | C2 | semidirect |
| 25. C2×C4×D251 | D4 | semidirect |
| 31. C42×C251 | D4 | semidirect |
| 57. C2×D251 | #127 | semidirect |
| 59. C4×C251 | #121 | semidirect |
| 66. C251 | #2856 | semidirect |
| 72. #384 | D251 | semidirect |
| 98. C42 | D1004 | semidirect |
| 127. C4 | #22548690 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22548712 | 2 |
| C2×#22548690 | 2 |
| C2×C42×D251 | 2 |
| #30631588 | 2 |
| #384×C251 | 2 |
| #2856 | 251 |