Finite Group DatabasebetaFinite Group DatabasebetaOrder 31347.
992 conjugacy classes of subgroups; only the 476 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C43 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C34×C43 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C43 ◁ C32×C9×C43 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C43×#737 | C3 | semidirect |
| 2. C32×3−1+2×C43 | C3 | semidirect |
| 3. C43×#6913 | C3 | direct (N × H) |
| 4. C3×C43×#733 | C3 | semidirect |
| 5. C3×C43×#732 | C3 | semidirect |
| 6. C3×C43×#731 | C3 | semidirect |
| 7. C3×C43×AΣL(1,𝔽27) | C3 | semidirect |
| 10. C43×#737 | C32 | semidirect |
| 15. C3×3−1+2×C43 | C32 | semidirect |
| 21. C43×#733 | C32 | semidirect |
| 22. C43×#732 | C32 | semidirect |
| 23. C43×#731 | C32 | semidirect |
| 24. C43×AΣL(1,𝔽27) | C32 | semidirect |
| 48. C3×#6913 | C43 | direct (N × H) |
| 63. C3×#737 | C3×C43 | semidirect |
| 64. C32×3−1+2 | C3×C43 | semidirect |
| 65. #6913 | C3×C43 | direct (N × H) |
| 66. C3×#733 | C3×C43 | semidirect |
| 67. C3×#732 | C3×C43 | semidirect |
| 68. C3×#731 | C3×C43 | semidirect |
| 69. C3×AΣL(1,𝔽27) | C3×C43 | semidirect |
| 74. C3×C43 | #6913 | direct (N × H) |
| 78. #737 | C32×C43 | semidirect |
| 83. C3×3−1+2 | C32×C43 | semidirect |
| 89. #733 | C32×C43 | semidirect |
| 90. #732 | C32×C43 | semidirect |
| 91. #731 | C32×C43 | semidirect |
| 92. AΣL(1,𝔽27) | C32×C43 | semidirect |
| 93. C43 | C3×#6913 | direct (N × H) |
| 135. C3 | C43×#6913 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C43×#737 | 3 |
| C32×3−1+2×C43 | 3 |
| C43×#6913 | 3 |
| C3×C43×#733 | 3 |
| C3×C43×#732 | 3 |
| C3×C43×#731 | 3 |
| C3×C43×AΣL(1,𝔽27) | 3 |
| C3×#6913 | 43 |