Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
620 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 620 conjugacy classes of subgroups — too many to draw. Only its 181 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 | #384×C131 | C2 |
| prime radical | C22×D524 | C22 |
| hypercenter | #384 | D131 |
| hypocenter | C131 | #2538 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ #384 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862226 | C2 | semidirect |
| 2. C2×C4×Dic262 | C2 | semidirect |
| 4. #22553234 | C2 | semidirect |
| 6. #22553211 | C2 | semidirect |
| 7. #22553169 | C2 | semidirect |
| 8. #384×C131 | C2 | semidirect |
| 31. C2×Dic262 | D4 | semidirect |
| 34. C2×D524 | Q8 | semidirect |
| 106. C4×C131 | C22⋊Q8 | semidirect |
| 118. C131 | #2538 | semidirect |
| 126. #384 | D131 | semidirect |
| 228. C4 | #16862226 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862226 | 2 |
| C2×C4×Dic262 | 2 |
| C2×C4×D524 | 2 |
| #22553234 | 2 |
| #22553213 | 2 |
| #22553211 | 2 |
| #22553169 | 2 |
| #384×C131 | 2 |
| #2538 | 131 |