Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
132 conjugacy classes of subgroups.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×D7 |
| commutator, hypocenter | C22×C7 | C2×C3×C4 |
| Frattini | C2 | C2×A4×D7 |
| Fitting, socle | C24×C7 | C2×C3 |
| prime radical | C22×C7×A4 | C2 |
| cosocle | C23×C7 | C22×C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. A4×Dic7 | C2 | direct (N × H) |
| 3. C23×Dic7 | C3 | semidirect |
| 4. C2×C7×A4 | C4 | semidirect |
| 11. C22×Dic7 | C2×C3 | semidirect |
| 14. C7×A4 | C2×C4 | semidirect |
| 19. C23×C7 | C3×C4 | semidirect |
| 26. C2×Dic7 | A4 | direct (N × H) |
| 39. C22×C7 | C2×C3×C4 | semidirect |
| 42. Dic7 | C2×A4 | direct (N × H) |
| 43. C2×A4 | Dic7 | direct (N × H) |
| 55. C2×C7 | C4×A4 | semidirect |
| 58. A4 | C2×Dic7 | direct (N × H) |
| 63. C23 | C3×Dic7 | semidirect |
| 71. C7 | C2×C4×A4 | semidirect |
| 81. C22 | C2×C3×Dic7 | semidirect |
| 89. C2 | A4×Dic7 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C7×A4 | 2 |
| A4×Dic7 | 2 |
| C23×Dic7 | 3 |
| C2×C3×Dic7 | 4 |
| C2×C4×A4 | 7 |