Finite Group DatabasebetaFinite Group DatabasebetaOrder 31347.
680 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C33×C43 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C33×C43 |
| prime radical | C32×H(3,𝔽3)×C43 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33×C43 ◁ C34×C43 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×3−1+2×C43 | C3 | semidirect |
| 3. C3×C43×C32⋊C9 | C3 | semidirect |
| 4. C43×#6857 | C3 | direct (N × H) |
| 9. C3×3−1+2×C43 | C32 | semidirect |
| 16. C43×C32⋊C9 | C32 | semidirect |
| 35. C3×#6857 | C43 | direct (N × H) |
| 53. C32×3−1+2 | C3×C43 | semidirect |
| 55. C3×C32⋊C9 | C3×C43 | semidirect |
| 56. #6857 | C3×C43 | direct (N × H) |
| 60. C3×C43 | #6857 | direct (N × H) |
| 68. C3×3−1+2 | C32×C43 | semidirect |
| 75. C32⋊C9 | C32×C43 | semidirect |
| 76. C43 | C3×#6857 | direct (N × H) |
| 115. C3 | C43×#6857 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×3−1+2×C43 | 3 |
| C32×H(3,𝔽3)×C43 | 3 |
| C3×C43×C32⋊C9 | 3 |
| C43×#6857 | 3 |
| C3×#6857 | 43 |