Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2586 conjugacy classes of subgroups.
The subgroup lattice has 240 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22×D4 |
| commutator | C4 | C24×C3×C4 |
| Frattini, cosocle | C2×C4 | C25×C3 |
| socle | C22×C3 | C22×C4⋊C4 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C22×C3×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#2916 | C2 | direct (N × H) |
| 4. C2×C3×#430 | C2 | semidirect |
| 5. C22×C3×#108 | C2 | semidirect |
| 6. C2×#2916 | C3 | direct (N × H) |
| 17. C3×#430 | C22 | semidirect |
| 20. C2×C3×#108 | C22 | semidirect |
| 23. #2916 | C2×C3 | direct (N × H) |
| 24. C2×#430 | C2×C3 | semidirect |
| 25. C22×#108 | C2×C3 | semidirect |
| 62. #430 | C22×C3 | semidirect |
| 65. C2×#108 | C22×C3 | semidirect |
| 218. C2×C3 | #2916 | direct (N × H) |
| 233. C3 | C2×#2916 | direct (N × H) |
| 237. C2 | C3×#2916 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#568 | 2 |
| C22×C3×#131 | 2 |
| C3×#2916 | 2 |
| C2×C3×#430 | 2 |
| C22×C3×#108 | 2 |
| C2×#2916 | 3 |