Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
918 conjugacy classes of subgroups.
The subgroup lattice has 358 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | #121 |
| commutator | C2×C4 | C23×C3×C4 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| prime radical | C24×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C3×#1952 | C2 | direct (N × H) |
| 7. C2×#1952 | C3 | direct (N × H) |
| 13. C2×C3×#108 | C4 | semidirect |
| 32. #1952 | C2×C3 | direct (N × H) |
| 66. C3×#108 | C2×C4 | semidirect |
| 78. C2×#108 | C3×C4 | semidirect |
| 171. #108 | C2×C3×C4 | semidirect |
| 323. C2×C3 | #1952 | direct (N × H) |
| 349. C3 | C2×#1952 | direct (N × H) |
| 353. C2 | C3×#1952 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×#108 | 2 |
| C2×C3×#428 | 2 |
| C22×C3×#107 | 2 |
| C2×C3×#405 | 2 |
| C3×#1952 | 2 |
| C2×C3×#343 | 2 |
| C2×#1952 | 3 |