Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
328 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 110 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 | C2×#41 |
| 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. C7×#239 | C2 | semidirect |
| 3. C3×#1161 | C2 | semidirect |
| 4. #66681 | C2 | semidirect |
| 5. C4×D3×D7 | C2 | semidirect |
| 6. #66676 | C2 | semidirect |
| 7. #66669 | C2 | semidirect |
| 10. C3×C7×Q8 | C22 | semidirect |
| 12. Dic42 | C22 | semidirect |
| 13. C7×D12 | C22 | semidirect |
| 14. C4×D3×C7 | C22 | semidirect |
| 15. C3×D28 | C22 | semidirect |
| 16. C3×C4×D7 | C22 | semidirect |
| 17. #4245 | C22 | semidirect |
| 19. D3×Dic7 | C22 | semidirect |
| 20. Dic3×D7 | C22 | semidirect |
| 22. #1161 | D3 | semidirect |
| 38. C7×Q8 | C2×D3 | semidirect |
| 43. D28 | C2×D3 | semidirect |
| 47. C4×D7 | C2×D3 | semidirect |
| 50. #239 | D7 | semidirect |
| 57. D21 | #41 | semidirect |
| 70. C3×Q8 | C2×D7 | semidirect |
| 75. D12 | C2×D7 | semidirect |
| 79. C4×D3 | C2×D7 | semidirect |
| 81. C3×C7 | C2×#41 | semidirect |
| 103. Q8 | D3×D7 | semidirect |
| 113. C7 | D3×#41 | semidirect |
| 126. C3 | D7×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D21 | 2 |
| C7×#239 | 2 |
| C3×#1161 | 2 |
| #66681 | 2 |
| C4×D3×D7 | 2 |
| #66676 | 2 |
| #66669 | 2 |
| D7×#41 | 3 |
| D3×#41 | 7 |