Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 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, prime radical | C23×C3×C7 | C22 |
| hypercenter | C23 | D3×D7 |
| hypocenter | C3×C7 | C22⋊Q8 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#4248 | C2 | semidirect |
| 5. Dic3⋊Dic7 | C2 | semidirect |
| 6. #66577 | C2 | semidirect |
| 7. Dic7⋊Dic3 | C2 | semidirect |
| 28. C22×C3×C7 | Q8 | semidirect |
| 59. C22×C7 | Dic6 | semidirect |
| 76. C3×C7 | C22⋊Q8 | semidirect |
| 90. C22×C3 | Dic14 | semidirect |
| 107. C7 | #924 | semidirect |
| 114. C22 | #4248 | semidirect |
| 122. C3 | #6441 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Dic7 | 2 |
| C2×#4248 | 2 |
| #66636 | 2 |
| C7×#217 | 2 |
| Dic3⋊Dic7 | 2 |
| #66577 | 2 |
| Dic7⋊Dic3 | 2 |
| #6441 | 3 |
| #924 | 7 |