Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2220 conjugacy classes of subgroups.
The subgroup lattice has 392 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C24 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| socle, prime radical | C23×C3 | C25 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. C4×#557 | C3 | direct (N × H) |
| 13. C3×#557 | C4 | direct (N × H) |
| 70. C3×C4×Q8 | Q8 | semidirect |
| 103. #557 | C3×C4 | direct (N × H) |
| 192. C4×Q8 | C3×Q8 | semidirect |
| 227. C3×Q8 | C4×Q8 | semidirect |
| 321. C3×C4 | #557 | direct (N × H) |
| 324. Q8 | C3×C4×Q8 | semidirect |
| 382. C4 | C3×#557 | direct (N × H) |
| 383. C3 | C4×#557 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#557 | 2 |
| C3×#2370 | 2 |
| C3×#2365 | 2 |
| C3×#2355 | 2 |
| C3×#2351 | 2 |
| C3×Q8×C4⋊C4 | 2 |
| C3×C4×C4⋊Q8 | 2 |
| C3×C4×#125 | 2 |
| C3×C42×Q8 | 2 |
| C4×#557 | 3 |