Finite Group DatabasebetaFinite Group DatabasebetaOrder 1458.
1244 conjugacy classes of subgroups; only the 518 normal subgroups are shown.
This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 518 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 | C34 |
| commutator, Frattini, cosocle | C32 | C2×C34 |
| prime radical | C2×C32×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C2×C32 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽3)×3−1+2 | C2 | direct (N × H) |
| 2. C2×C32×3−1+2 | C3 | semidirect |
| 4. C2×C9⋊H(3,𝔽3) | C3 | semidirect |
| 5. C2×#6891 | C3 | semidirect |
| 6. C2×C9×H(3,𝔽3) | C3 | semidirect |
| 7. C32×3−1+2 | C2×C3 | semidirect |
| 9. C9⋊H(3,𝔽3) | C2×C3 | semidirect |
| 10. #6891 | C2×C3 | semidirect |
| 11. C9×H(3,𝔽3) | C2×C3 | semidirect |
| 16. C2×C3×3−1+2 | C32 | semidirect |
| 20. C2×C32×C9 | C32 | semidirect |
| 21. C2×C32⋊C9 | C32 | semidirect |
| 22. C2×C32⋊C9 | C32 | semidirect |
| 27. C3×3−1+2 | C2×C32 | semidirect |
| 31. C32×C9 | C2×C32 | semidirect |
| 32. C32⋊C9 | C2×C32 | semidirect |
| 33. C32⋊C9 | C2×C32 | semidirect |
| 42. C2×3−1+2 | H(3,𝔽3) | direct (N × H) |
| 44. C2×H(3,𝔽3) | 3−1+2 | direct (N × H) |
| 56. 3−1+2 | C2×H(3,𝔽3) | direct (N × H) |
| 58. H(3,𝔽3) | C2×3−1+2 | direct (N × H) |
| 71. C2×C32 | C3×3−1+2 | semidirect |
| 74. C2×C9 | C3×H(3,𝔽3) | semidirect |
| 84. C32 | C2×C3×3−1+2 | semidirect |
| 87. C9 | C2×C3×H(3,𝔽3) | semidirect |
| 100. C2 | H(3,𝔽3)×3−1+2 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×3−1+2 | 2 |
| C2×C32×3−1+2 | 3 |
| C2×C32×H(3,𝔽3) | 3 |
| C2×C9⋊H(3,𝔽3) | 3 |
| C2×#6891 | 3 |
| C2×C9×H(3,𝔽3) | 3 |