Finite Group DatabasebetaFinite Group DatabasebetaOrder 1458.
368 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C32 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C2×C33 |
| prime radical | C2×C3×AΣL(1,𝔽27) | C3 |
| LCS | G ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ C2×C32×C9 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624798 | C2 | direct (N × H) |
| 2. C2×C32×3−1+2 | C3 | semidirect |
| 4. C2×#6894 | C3 | semidirect |
| 5. C2×C9⋊H(3,𝔽3) | C3 | semidirect |
| 6. C2×C9×3−1+2 | C3 | semidirect |
| 7. C2×3−1+2⋊C9 | C3 | semidirect |
| 8. C2×3−1+2⋊C9 | C3 | semidirect |
| 9. C2×H(3,𝔽3)⋊C9 | C3 | semidirect |
| 10. C2×#6866 | C3 | semidirect |
| 11. C32×3−1+2 | C2×C3 | semidirect |
| 13. #6894 | C2×C3 | semidirect |
| 14. C9⋊H(3,𝔽3) | C2×C3 | semidirect |
| 15. C9×3−1+2 | C2×C3 | semidirect |
| 16. 3−1+2⋊C9 | C2×C3 | semidirect |
| 17. 3−1+2⋊C9 | C2×C3 | semidirect |
| 18. H(3,𝔽3)⋊C9 | C2×C3 | semidirect |
| 19. #6866 | C2×C3 | semidirect |
| 71. C2×3−1+2 | 3−1+2 | semidirect |
| 97. 3−1+2 | C2×3−1+2 | semidirect |
| 130. C2×C9 | AΣL(1,𝔽27) | semidirect |
| 151. C9 | C2×AΣL(1,𝔽27) | semidirect |
| 166. C2 | #10624798 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624798 | 2 |
| C2×C32×3−1+2 | 3 |
| C2×C3×AΣL(1,𝔽27) | 3 |
| C2×#6894 | 3 |
| C2×C9⋊H(3,𝔽3) | 3 |
| C2×C9×3−1+2 | 3 |
| C2×3−1+2⋊C9 | 3 |
| C2×H(3,𝔽3)⋊C9 | 3 |
| C2×#6866 | 3 |