Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
292 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D7 |
| commutator, socle, cosocle | C2×C3×C7 | C24 |
| Fitting | C3×C7×Q8 | C22 |
| hypercenter | Q8 | D3×D7 |
| hypocenter | C3×C7 | 2−1+4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×D21 | C2 | semidirect |
| 2. D3×C7×Q8 | C2 | semidirect |
| 3. C3×#1161 | C2 | semidirect |
| 4. #66675 | C2 | semidirect |
| 5. #66670 | C2 | semidirect |
| 6. #66669 | C2 | semidirect |
| 7. D7×Dic6 | C2 | semidirect |
| 23. #1161 | D3 | semidirect |
| 50. D3×Q8 | D7 | semidirect |
| 79. C3×C7 | 2−1+4 | semidirect |
| 110. C7 | #1010 | semidirect |
| 122. C3 | #6551 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D21 | 2 |
| D3×C7×Q8 | 2 |
| C3×#1161 | 2 |
| #66675 | 2 |
| #66670 | 2 |
| #66669 | 2 |
| D7×Dic6 | 2 |
| #6551 | 3 |
| #1010 | 7 |