Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
188 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 63 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 | C22×C3×C4×C7 | 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. C22×C3×C4×C7 | C2 | semidirect |
| 2. C2×#4268 | C2 | semidirect |
| 3. C2×D84 | C2 | semidirect |
| 4. #66631 | C2 | semidirect |
| 5. C4⋊Dic21 | C2 | semidirect |
| 15. C22×C4×C7 | D3 | semidirect |
| 23. C22×C3×C7 | D4 | semidirect |
| 29. C3×C4×C7 | D4 | semidirect |
| 39. C22×C3×C4 | D7 | semidirect |
| 52. C22×C7 | D12 | semidirect |
| 58. C4×C7 | #67 | semidirect |
| 68. C3×C7 | #121 | semidirect |
| 71. C22×C4 | D21 | semidirect |
| 81. C22×C3 | D28 | semidirect |
| 87. C3×C4 | C22⋊D7 | semidirect |
| 97. C7 | #930 | semidirect |
| 108. C22 | D84 | semidirect |
| 112. C4 | #4268 | semidirect |
| 113. C3 | #6493 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C4×C7 | 2 |
| C2×#4268 | 2 |
| C2×D84 | 2 |
| #66631 | 2 |
| C4⋊Dic21 | 2 |
| #6493 | 3 |
| #930 | 7 |