Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
3006 conjugacy classes of subgroups.
The subgroup lattice has 222 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25×C3 | D4 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C3 ◁ C24×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C3×#340 | C2 | direct (N × H) |
| 4. C22×#340 | C3 | direct (N × H) |
| 8. C23×C3×C8 | C4 | semidirect |
| 12. C22×C3×C4⋊C4 | C4 | semidirect |
| 14. C3×#340 | C22 | direct (N × H) |
| 17. C2×#340 | C2×C3 | direct (N × H) |
| 31. C22×C3×C8 | C2×C4 | semidirect |
| 36. C2×C3×C4⋊C4 | C2×C4 | semidirect |
| 41. C23×C8 | C3×C4 | semidirect |
| 45. C22×C4⋊C4 | C3×C4 | semidirect |
| 47. #340 | C22×C3 | direct (N × H) |
| 72. C2×C3×C8 | C22×C4 | semidirect |
| 75. C3×C4⋊C4 | C22×C4 | semidirect |
| 89. C22×C8 | C2×C3×C4 | semidirect |
| 94. C2×C4⋊C4 | C2×C3×C4 | semidirect |
| 146. C2×C8 | C22×C3×C4 | semidirect |
| 149. C4⋊C4 | C22×C3×C4 | semidirect |
| 150. C22×C3 | #340 | direct (N × H) |
| 192. C2×C3 | C2×#340 | direct (N × H) |
| 198. C22 | C3×#340 | direct (N × H) |
| 214. C3 | C22×#340 | direct (N × H) |
| 215. C2 | C2×C3×#340 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C8 | 2 |
| C23×C3×C4⋊C4 | 2 |
| C2×C3×#340 | 2 |
| C22×#340 | 3 |