Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1056 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 169 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×D127 |
| commutator, socle, cosocle | C23×C127 | C24 |
| Fitting | #381×C127 | C2 |
| hypercenter | #381 | D127 |
| hypocenter | C127 | #2404 |
| LCS | G ▷ C23×C127 ▷ C127 | |
| UCS | C1 ◁ C23 ◁ #381 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14523248 | C2 | semidirect |
| 2. C2×#16861179 | C2 | semidirect |
| 3. C22×(C2×D127)⋊C4 | C2 | semidirect |
| 4. #22549789 | C2 | semidirect |
| 5. #22549786 | C2 | semidirect |
| 6. #22549754 | C2 | semidirect |
| 7. #22549753 | C2 | semidirect |
| 8. #381×C127 | C2 | semidirect |
| 12. C22×Dic254 | C22 | semidirect |
| 14. C22×D508 | C22 | semidirect |
| 17. C22×C4×D127 | C22 | semidirect |
| 23. C2×(C2×D127)⋊C4 | C22 | semidirect |
| 24. C127⋊#95 | C22 | semidirect |
| 27. #95×C127 | C22 | semidirect |
| 28. #95×C127 | C22 | semidirect |
| 45. C2×C22⋊D127 | D4 | semidirect |
| 59. C22×C4×C127 | D4 | semidirect |
| 77. C22×D127 | #41 | semidirect |
| 134. C127 | #2404 | semidirect |
| 142. #381 | D127 | semidirect |
| 161. #95 | C2×D127 | semidirect |
| 162. #95 | C2×D127 | semidirect |
| 189. C22×C4 | D508 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14523248 | 2 |
| C2×#16861179 | 2 |
| C22×(C2×D127)⋊C4 | 2 |
| #22549789 | 2 |
| #22549786 | 2 |
| #22549754 | 2 |
| #22549753 | 2 |
| #381×C127 | 2 |
| #2404 | 127 |