Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
948 conjugacy classes of subgroups; only the 205 normal subgroups are shown.
This group has 948 conjugacy classes of subgroups — too many to draw. Only its 205 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 | C4×C4⋊C4×C131 | C2 |
| hypercenter | C4×C4⋊C4 | D131 |
| hypocenter | C131 | #2541 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16862243 | C2 | semidirect |
| 2. C2×#16862243 | C2 | semidirect |
| 3. C2×#16862221 | C2 | semidirect |
| 4. C2×C4×D524 | C2 | semidirect |
| 5. #22553222 | C2 | semidirect |
| 6. #22553214 | C2 | semidirect |
| 7. C4×C4⋊C4×C131 | C2 | semidirect |
| 39. C2×D524 | D4 | semidirect |
| 50. C4⋊C4×C131 | D4 | semidirect |
| 52. C42×C131 | D4 | semidirect |
| 99. C4×C131 | #121 | semidirect |
| 100. C4×C131 | #127 | semidirect |
| 109. C131 | #2541 | semidirect |
| 117. C4×C4⋊C4 | D131 | semidirect |
| 158. C4⋊C4 | D524 | semidirect |
| 162. C42 | D524 | semidirect |
| 210. C4 | #16862221 | semidirect |
| 211. C4 | #16862243 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16862243 | 2 |
| C2×#16862221 | 2 |
| C2×C4×D524 | 2 |
| #22553222 | 2 |
| #22553214 | 2 |
| C4×C4⋊C4×C131 | 2 |
| #2541 | 131 |