Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
782 conjugacy classes of subgroups.
The subgroup lattice has 132 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | D8 |
| commutator | C8 | C23×C3×C4 |
| Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C3×#366 | C2 | direct (N × H) |
| 4. C22×#366 | C3 | direct (N × H) |
| 7. C22×C3×C16 | C4 | semidirect |
| 11. C3×#366 | C22 | direct (N × H) |
| 14. C2×#366 | C2×C3 | direct (N × H) |
| 23. C2×C3×C16 | C2×C4 | semidirect |
| 29. C22×C16 | C3×C4 | semidirect |
| 33. #366 | C22×C3 | direct (N × H) |
| 48. C3×C16 | C22×C4 | semidirect |
| 57. C2×C16 | C2×C3×C4 | semidirect |
| 89. C16 | C22×C3×C4 | semidirect |
| 90. C22×C3 | #366 | direct (N × H) |
| 113. C2×C3 | C2×#366 | direct (N × H) |
| 117. C22 | C3×#366 | direct (N × H) |
| 126. C3 | C22×#366 | direct (N × H) |
| 127. C2 | C2×C3×#366 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C16 | 2 |
| C22×C3×#107 | 2 |
| C2×C3×#366 | 2 |
| C22×#366 | 3 |