Finite Group DatabasebetaFinite Group DatabasebetaOrder 336.
80 conjugacy classes of subgroups.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C3×C7 |
| socle | C2×C3×C7 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C7×Q8 | C2 | semidirect |
| 2. C3×C7×D4 | C2 | semidirect |
| 3. C2×C3×C4×C7 | C2 | semidirect |
| 4. C7×#41 | C3 | direct (N × H) |
| 8. C7×Q8 | C2×C3 | semidirect |
| 9. C7×D4 | C2×C3 | semidirect |
| 10. C2×C4×C7 | C2×C3 | semidirect |
| 11. C3×#41 | C7 | direct (N × H) |
| 17. C3×Q8 | C2×C7 | semidirect |
| 18. C3×D4 | C2×C7 | semidirect |
| 19. C2×C3×C4 | C2×C7 | semidirect |
| 20. C3×C7 | #41 | direct (N × H) |
| 21. #41 | C3×C7 | direct (N × H) |
| 27. Q8 | C2×C3×C7 | semidirect |
| 28. D4 | C2×C3×C7 | semidirect |
| 29. C2×C4 | C2×C3×C7 | semidirect |
| 30. C7 | C3×#41 | direct (N × H) |
| 36. C3 | C7×#41 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7×Q8 | 2 |
| C3×C7×D4 | 2 |
| C2×C3×C4×C7 | 2 |
| C7×#41 | 3 |
| C3×#41 | 7 |