Finite Group DatabasebetaFinite Group DatabasebetaOrder 12032.
4086 conjugacy classes of subgroups.
The subgroup lattice has 160 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C43×C47 | C22 |
| commutator | C2 | C23×C42×C47 |
| Frattini, cosocle | C23 | C25×C47 |
| socle | C23×C47 | C25 |
| prime radical | C22×#41×C47 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C43×C47 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C42×Q8×C47 | C2 | semidirect |
| 3. C42×D4×C47 | C2 | semidirect |
| 4. C4×#117×C47 | C2 | semidirect |
| 5. C2×C43×C47 | C2 | semidirect |
| 7. C4×#41×C47 | C4 | direct (N × H) |
| 24. C4×Q8×C47 | C2×C4 | semidirect |
| 25. C4×D4×C47 | C2×C4 | semidirect |
| 26. #117×C47 | C2×C4 | semidirect |
| 30. C2×C42×C47 | C2×C4 | semidirect |
| 38. #41×C47 | C42 | direct (N × H) |
| 54. C42×C47 | #41 | direct (N × H) |
| 58. Q8×C47 | C2×C42 | semidirect |
| 59. D4×C47 | C2×C42 | semidirect |
| 62. C2×C4×C47 | C2×C42 | semidirect |
| 68. C42×#41 | C47 | direct (N × H) |
| 76. C4×C47 | C4×#41 | direct (N × H) |
| 78. C42×Q8 | C2×C47 | semidirect |
| 79. C42×D4 | C2×C47 | semidirect |
| 80. C4×#117 | C2×C47 | semidirect |
| 81. C2×C43 | C2×C47 | semidirect |
| 86. C4×#41 | C4×C47 | direct (N × H) |
| 98. C47 | C42×#41 | direct (N × H) |
| 104. C4×Q8 | C2×C4×C47 | semidirect |
| 105. C4×D4 | C2×C4×C47 | semidirect |
| 106. #117 | C2×C4×C47 | semidirect |
| 110. C2×C42 | C2×C4×C47 | semidirect |
| 118. #41 | C42×C47 | direct (N × H) |
| 134. C42 | #41×C47 | direct (N × H) |
| 138. Q8 | C2×C42×C47 | semidirect |
| 139. D4 | C2×C42×C47 | semidirect |
| 142. C2×C4 | C2×C42×C47 | semidirect |
| 155. C4 | C4×#41×C47 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#41×C47 | 2 |
| C42×Q8×C47 | 2 |
| C42×D4×C47 | 2 |
| C4×#117×C47 | 2 |
| C2×C43×C47 | 2 |
| C42×#41 | 47 |