Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.
This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D127 |
| commutator, socle, cosocle | C22×C127 | C25 |
| Fitting | Q82×C127 | C2 |
| hypercenter | Q82 | D127 |
| hypocenter | C127 | #3483 |
| LCS | G ▷ C22×C127 ▷ C127 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22550592 | C2 | semidirect |
| 2. #22550548 | C2 | semidirect |
| 3. #22550427 | C2 | semidirect |
| 4. C4×#14523249 | C2 | semidirect |
| 5. Q82×C127 | C2 | semidirect |
| 40. Q8×C127 | #41 | semidirect |
| 54. C127 | #3483 | semidirect |
| 55. Q82 | D127 | semidirect |
| 88. Q8 | #14523249 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22550592 | 2 |
| #22550548 | 2 |
| #22550427 | 2 |
| C4×#14523249 | 2 |
| Q82×C127 | 2 |
| #3483 | 127 |