Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
556 conjugacy classes of subgroups; only the 280 normal subgroups are shown.
This group has 556 conjugacy classes of subgroups — too many to draw. Only its 280 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C131 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C131 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 28. C4⋊C4×C131 | Q8 | semidirect |
| 30. C42×C131 | Q8 | semidirect |
| 56. C4×C131 | C4⋊Q8 | semidirect |
| 62. C131 | #2551 | direct (N × H) |
| 63. #2551 | C131 | direct (N × H) |
| 91. C4⋊C4 | Q8×C131 | semidirect |
| 93. C42 | Q8×C131 | semidirect |
| 119. C4 | C4⋊Q8×C131 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊Q8×C131 | 2 |
| #395×C131 | 2 |
| #389×C131 | 2 |
| #384×C131 | 2 |
| C4×C4⋊C4×C131 | 2 |
| #2551 | 131 |