Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
292 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 144 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 | C2×D4 |
| commutator | C2×C4 | C22×C4×C131 |
| Frattini, cosocle | C22×C4 | C23×C131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C131 ◁ C2×C42×C131 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C4⋊Q8×C131 | C4 | semidirect |
| 29. C4⋊C4×C131 | Q8 | semidirect |
| 67. C131 | #1987 | direct (N × H) |
| 68. #1987 | C131 | direct (N × H) |
| 76. C4⋊Q8 | C4×C131 | semidirect |
| 97. 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 |
| C2×C4⋊C8×C131 | 2 |
| C4×C4⋊C4×C131 | 2 |
| #340×C131 | 2 |
| #1987 | 131 |