Finite Group DatabasebetaFinite Group DatabasebetaOrder 3402.
336 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 80 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. ΓL(1,𝔽8)×#732 | C2 | semidirect |
| 2. H(3,𝔽3)×AGL(1,𝔽7) | C3 | semidirect |
| 3. C3×C9×AGL(1,𝔽7) | C3 | semidirect |
| 4. D7×#732 | C3 | semidirect |
| 5. #11765591 | C3 | semidirect |
| 6. #11765587 | C3 | semidirect |
| 7. ΓL(1,𝔽8)×H(3,𝔽3) | C2×C3 | semidirect |
| 8. C3×C9×ΓL(1,𝔽8) | C2×C3 | semidirect |
| 9. C7×#732 | C2×C3 | semidirect |
| 10. #10588042 | C2×C3 | semidirect |
| 11. #10588038 | C2×C3 | semidirect |
| 15. D7×H(3,𝔽3) | C32 | semidirect |
| 16. C3×C9×D7 | C32 | semidirect |
| 18. D7⋊H(3,𝔽3) | C32 | semidirect |
| 19. C3×D7⋊C9 | C32 | semidirect |
| 25. C7×H(3,𝔽3) | C2×C32 | semidirect |
| 26. C3×C7×C9 | C2×C32 | semidirect |
| 28. C7⋊H(3,𝔽3) | C2×C32 | semidirect |
| 29. C3×C7⋊C9 | C2×C32 | semidirect |
| 50. #732 | AGL(1,𝔽7) | direct (N × H) |
| 74. AGL(1,𝔽7) | #732 | direct (N × H) |
| 79. H(3,𝔽3) | C3×AGL(1,𝔽7) | semidirect |
| 83. C3×C9 | C3×AGL(1,𝔽7) | semidirect |
| 89. ΓL(1,𝔽8) | C2×#732 | semidirect |
| 99. D7 | C3×#732 | semidirect |
| 109. C7 | C2×C3×#732 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#732 | 2 |
| H(3,𝔽3)×AGL(1,𝔽7) | 3 |
| C3×C9×AGL(1,𝔽7) | 3 |
| D7×#732 | 3 |
| #11765591 | 3 |
| #11765587 | 3 |
| C2×C3×#732 | 7 |