Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
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×D131 |
| commutator, socle, cosocle | C23×C131 | C24 |
| Fitting | #381×C131 | C2 |
| hypercenter | #381 | D131 |
| hypocenter | C131 | #2404 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #381 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#14523513 | C2 | semidirect |
| 2. C2×#16862216 | C2 | semidirect |
| 3. C22×(C2×D131)⋊C4 | C2 | semidirect |
| 4. #22553225 | C2 | semidirect |
| 5. #22553222 | C2 | semidirect |
| 6. #22553190 | C2 | semidirect |
| 7. #22553189 | C2 | semidirect |
| 8. #381×C131 | C2 | semidirect |
| 12. C22×Dic262 | C22 | semidirect |
| 14. C22×D524 | C22 | semidirect |
| 17. C22×C4×D131 | C22 | semidirect |
| 23. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 24. C131⋊#95 | C22 | semidirect |
| 27. #95×C131 | C22 | semidirect |
| 28. #95×C131 | C22 | semidirect |
| 45. C2×C22⋊D131 | D4 | semidirect |
| 59. C22×C4×C131 | D4 | semidirect |
| 77. C22×D131 | #41 | semidirect |
| 133. C131 | #2404 | semidirect |
| 142. #381 | D131 | semidirect |
| 161. #95 | C2×D131 | semidirect |
| 162. #95 | C2×D131 | semidirect |
| 189. C22×C4 | D524 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#14523513 | 2 |
| C2×#16862216 | 2 |
| C22×(C2×D131)⋊C4 | 2 |
| #22553225 | 2 |
| #22553222 | 2 |
| #22553190 | 2 |
| #22553189 | 2 |
| #381×C131 | 2 |
| #2404 | 131 |