Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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 | C2×C3×C7×D4 | C2 |
| hypercenter | C2×D4 | D21 |
| hypocenter | C3×C7 | #121 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C7×D4 | C2 | semidirect |
| 2. C2×#4268 | C2 | semidirect |
| 3. C2×#4268 | C2 | semidirect |
| 4. C22×Dic21 | C2 | semidirect |
| 5. #66636 | C2 | semidirect |
| 6. #66631 | C2 | semidirect |
| 7. #66629 | C2 | semidirect |
| 20. C2×C7×D4 | D3 | semidirect |
| 33. C22×C3×C7 | D4 | semidirect |
| 41. Dic21 | D4 | semidirect |
| 53. C2×C3×D4 | D7 | semidirect |
| 72. C22×C7 | #67 | semidirect |
| 92. C3×C7 | #121 | semidirect |
| 95. C2×D4 | D21 | semidirect |
| 111. C22×C3 | C22⋊D7 | semidirect |
| 131. C7 | #939 | semidirect |
| 146. C22 | #4268 | semidirect |
| 152. C3 | #6502 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C7×D4 | 2 |
| C2×#4268 | 2 |
| C22×Dic21 | 2 |
| #66636 | 2 |
| #66631 | 2 |
| #66629 | 2 |
| #6502 | 3 |
| #939 | 7 |