Finite Group DatabasebetaFinite Group DatabasebetaOrder 16768.
1812 conjugacy classes of subgroups; only the 285 normal subgroups are shown.
This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 285 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 | C23×C131 | C24 |
| Frattini | C23 | C23×D131 |
| Fitting | C2×#95×C131 | C2 |
| socle | C24×C131 | C23 |
| hypercenter | C2×#95 | D131 |
| hypocenter | C131 | C2×#392 |
| LCS | G ▷ C23×C131 ▷ C131 | |
| UCS | C1 ◁ C24 ◁ C2×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D524 | C2 | semidirect |
| 2. C22×(C2×D131)⋊C4 | C2 | semidirect |
| 3. #22553222 | C2 | direct (N × H) |
| 4. C2×#95×C131 | C2 | semidirect |
| 8. C22×D524 | C22 | semidirect |
| 11. C2×(C2×D131)⋊C4 | C22 | semidirect |
| 13. #95×C131 | C22 | semidirect |
| 18. C23×D131 | D4 | semidirect |
| 28. C22×C4×C131 | D4 | semidirect |
| 39. C22×D131 | C2×D4 | semidirect |
| 49. C2×C4×C131 | C2×D4 | semidirect |
| 64. C2×C131 | #392 | semidirect |
| 69. C131 | C2×#392 | semidirect |
| 74. C2×#95 | D131 | semidirect |
| 83. #95 | C2×D131 | semidirect |
| 97. C22×C4 | D524 | semidirect |
| 119. C2×C4 | C2×D524 | semidirect |
| 135. C2 | #22553222 | 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 |
| C22×(C2×D131)⋊C4 | 2 |
| #22553222 | 2 |
| C2×#95×C131 | 2 |
| C2×#392 | 131 |