Finite Group DatabasebetaFinite Group DatabasebetaOrder 672.
120 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 120 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. Q8×AGL(1,𝔽7) | C2 | semidirect |
| 3. SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. #1144⋊C3 | C2 | semidirect |
| 5. #1142⋊C3 | C2 | semidirect |
| 6. Dic28⋊C3 | C2 | semidirect |
| 7. #1131⋊C3 | C2 | semidirect |
| 8. #6478 | C3 | semidirect |
| 20. Q8×D7 | C2×C3 | semidirect |
| 22. C7×SD16 | C2×C3 | semidirect |
| 23. #1144 | C2×C3 | semidirect |
| 24. #1142 | C2×C3 | semidirect |
| 25. Dic28 | C2×C3 | semidirect |
| 26. #1131 | C2×C3 | semidirect |
| 75. ΓL(1,𝔽8) | #137 | semidirect |
| 81. SD16 | AGL(1,𝔽7) | semidirect |
| 104. C7 | C3×#137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×AGL(1,𝔽7) | 2 |
| #1159⋊C3 | 2 |
| SD16×ΓL(1,𝔽8) | 2 |
| #1144⋊C3 | 2 |
| #1142⋊C3 | 2 |
| Dic28⋊C3 | 2 |
| #1131⋊C3 | 2 |
| #6478 | 3 |
| C3×#137 | 7 |