Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
5872 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D131 |
| commutator, cosocle | C2×C131 | C26 |
| Frattini | C2 | C25×D131 |
| Fitting | C23×Q8×C131 | C2 |
| socle | C23×C131 | C24 |
| hypercenter | C23×Q8 | D131 |
| hypocenter | C131 | C22×2−1+4 |
| LCS | G ▷ C2×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862294 | C2 | direct (N × H) |
| 2. C22×#14523514 | C2 | semidirect |
| 3. C22×#14523513 | C2 | semidirect |
| 4. C22×Q8×D131 | C2 | semidirect |
| 5. C23×Q8×C131 | C2 | semidirect |
| 6. #16862294 | C22 | direct (N × H) |
| 7. C2×#14523514 | C22 | semidirect |
| 8. C2×#14523513 | C22 | semidirect |
| 10. C2×Q8×D131 | C22 | semidirect |
| 15. C22×Q8×C131 | C22 | semidirect |
| 18. #14523514 | C23 | semidirect |
| 19. #14523513 | C23 | semidirect |
| 21. Q8×D131 | C23 | semidirect |
| 28. C2×Q8×C131 | C23 | semidirect |
| 45. C22×C131 | 2−1+4 | semidirect |
| 53. C2×C131 | C2×2−1+4 | semidirect |
| 56. C131 | C22×2−1+4 | semidirect |
| 62. C23×Q8 | D131 | semidirect |
| 69. C22×Q8 | C2×D131 | semidirect |
| 81. C2×Q8 | C22×D131 | semidirect |
| 104. C22 | #16862294 | direct (N × H) |
| 111. C2 | C2×#16862294 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862294 | 2 |
| C22×#14523514 | 2 |
| C22×#14523513 | 2 |
| C22×Q8×D131 | 2 |
| C23×Q8×C131 | 2 |
| C22×2−1+4 | 131 |