Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
188 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 54 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 | C2×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C23 |
| Fitting | C2×C3×C4×C7 | C22 |
| hypercenter | C2×C4 | D3×D7 |
| hypocenter | C3×C7 | #121 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C4×D21 | C2 | semidirect |
| 2. C2×C3×D28 | C2 | semidirect |
| 3. C2×#4246 | C2 | semidirect |
| 4. C7×C4⋊Dic3 | C2 | semidirect |
| 5. (C2×D7)⋊Dic3 | C2 | semidirect |
| 16. C2×D28 | D3 | semidirect |
| 24. C2×D21 | D4 | semidirect |
| 28. C3×C4×C7 | D4 | semidirect |
| 43. C4⋊Dic3 | D7 | semidirect |
| 58. C4×C7 | #67 | semidirect |
| 68. C3×C7 | #121 | semidirect |
| 86. C3×C4 | D28 | semidirect |
| 97. C7 | #938 | semidirect |
| 112. C4 | #4246 | semidirect |
| 113. C3 | #6458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D21 | 2 |
| C2×C3×D28 | 2 |
| C2×#4246 | 2 |
| C7×C4⋊Dic3 | 2 |
| (C2×D7)⋊Dic3 | 2 |
| #6458 | 3 |
| #938 | 7 |