Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
136 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #66659 | C2 | semidirect |
| 2. D4×AGL(1,𝔽7) | C2 | semidirect |
| 3. SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. #1143⋊C3 | C2 | semidirect |
| 5. #1141⋊C3 | C2 | semidirect |
| 6. D56⋊C3 | C2 | semidirect |
| 7. #1131⋊C3 | C2 | semidirect |
| 8. #6477 | C3 | semidirect |
| 10. Q8×ΓL(1,𝔽8) | C22 | semidirect |
| 15. C4⋊AGL(1,𝔽7) | C22 | semidirect |
| 18. C8×ΓL(1,𝔽8) | C22 | semidirect |
| 19. ΓL(1,𝔽8)⋊C8 | C22 | semidirect |
| 20. #1161 | C2×C3 | semidirect |
| 21. D4×D7 | C2×C3 | semidirect |
| 22. C7×SD16 | C2×C3 | semidirect |
| 23. #1143 | C2×C3 | semidirect |
| 24. #1141 | C2×C3 | semidirect |
| 25. D56 | C2×C3 | semidirect |
| 26. #1131 | C2×C3 | semidirect |
| 37. C7×Q8 | C22×C3 | semidirect |
| 42. D28 | C22×C3 | semidirect |
| 45. C7×C8 | C22×C3 | semidirect |
| 46. C7⋊C8 | C22×C3 | semidirect |
| 79. ΓL(1,𝔽8) | AGL(1,ℤ/8) | semidirect |
| 83. SD16 | AGL(1,𝔽7) | semidirect |
| 101. Q8 | C2×AGL(1,𝔽7) | semidirect |
| 110. C8 | C2×AGL(1,𝔽7) | semidirect |
| 111. C7 | C3×AGL(1,ℤ/8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #66659 | 2 |
| D4×AGL(1,𝔽7) | 2 |
| SD16×ΓL(1,𝔽8) | 2 |
| #1143⋊C3 | 2 |
| #1141⋊C3 | 2 |
| D56⋊C3 | 2 |
| #1131⋊C3 | 2 |
| #6477 | 3 |
| C3×AGL(1,ℤ/8) | 7 |