Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
2528 conjugacy classes of subgroups; only the 725 normal subgroups are shown.
This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 725 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C22×D131 |
| commutator, cosocle | C22×C131 | C25 |
| Frattini | C22 | C24×D131 |
| Fitting | C24×C4×C131 | C2 |
| socle | C24×C131 | C23 |
| hypercenter | C24×C4 | D131 |
| hypocenter | C131 | C22×#121 |
| LCS | G ▷ C22×C131 ▷ C131 | |
| UCS | C1 ◁ C24 ◁ C24×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D524 | C2 | semidirect |
| 2. C23×C22⋊D131 | C2 | semidirect |
| 3. C2×#16862220 | C2 | direct (N × H) |
| 4. C22×C4⋊Dic131 | C2 | semidirect |
| 5. C22×(C2×D131)⋊C4 | C2 | semidirect |
| 6. C24×C4×C131 | C2 | semidirect |
| 9. C22×D524 | C22 | semidirect |
| 11. C22×C22⋊D131 | C22 | semidirect |
| 13. #16862220 | C22 | direct (N × H) |
| 14. C2×C4⋊Dic131 | C22 | semidirect |
| 15. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 20. C23×C4×C131 | C22 | semidirect |
| 26. C2×D524 | C23 | semidirect |
| 28. C2×C22⋊D131 | C23 | semidirect |
| 31. C4⋊Dic131 | C23 | semidirect |
| 32. (C2×D131)⋊C4 | C23 | semidirect |
| 35. C24×C131 | D4 | semidirect |
| 40. C22×C4×C131 | D4 | semidirect |
| 43. C22×C4×C131 | C23 | semidirect |
| 58. C23×C131 | C2×D4 | semidirect |
| 64. C2×C4×C131 | C2×D4 | semidirect |
| 70. C22×C131 | #121 | semidirect |
| 73. C22×C131 | C22×D4 | semidirect |
| 81. C4×C131 | C22×D4 | semidirect |
| 84. C2×C131 | C2×#121 | semidirect |
| 89. C131 | C22×#121 | semidirect |
| 93. C24×C4 | D131 | semidirect |
| 107. C23×C4 | C2×D131 | semidirect |
| 117. C24 | D524 | semidirect |
| 128. C22×C4 | C22⋊D131 | semidirect |
| 131. C22×C4 | C22×D131 | semidirect |
| 144. C23 | C2×D524 | semidirect |
| 154. C2×C4 | C2×C22⋊D131 | semidirect |
| 164. C22 | #16862220 | direct (N × H) |
| 167. C22 | C22×D524 | semidirect |
| 172. C4 | C22×C22⋊D131 | semidirect |
| 175. C2 | C2×#16862220 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D524 | 2 |
| C23×C22⋊D131 | 2 |
| C2×#16862220 | 2 |
| C22×C4⋊Dic131 | 2 |
| C22×(C2×D131)⋊C4 | 2 |
| C24×C4×C131 | 2 |
| C22×#121 | 131 |