Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
188 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C23 |
| Fitting | C3×C7×C22⋊C4 | C2 |
| hypercenter | C22⋊C4 | D21 |
| hypocenter | C3×C7 | #121 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4268 | C2 | semidirect |
| 2. C2×#4268 | C2 | semidirect |
| 3. C2×D84 | C2 | semidirect |
| 4. C2×C4×D21 | C2 | semidirect |
| 5. C3×C7×C22⋊C4 | C2 | semidirect |
| 6. #66631 | C2 | semidirect |
| 7. #66629 | C2 | semidirect |
| 24. C7×C22⋊C4 | D3 | semidirect |
| 37. C2×D21 | D4 | semidirect |
| 41. Dic21 | D4 | semidirect |
| 57. C3×C22⋊C4 | D7 | semidirect |
| 92. C3×C7 | #121 | semidirect |
| 99. C22⋊C4 | D21 | semidirect |
| 131. C7 | #884 | semidirect |
| 152. C3 | #6447 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4268 | 2 |
| C2×D84 | 2 |
| C2×C4×D21 | 2 |
| C3×C7×C22⋊C4 | 2 |
| #66631 | 2 |
| #66629 | 2 |
| #6447 | 3 |
| #884 | 7 |