← C3×#63091 #10681698
Subgroups — C3×#63091 #10681698
Order 768.
10894 conjugacy classes of subgroups.
The subgroup lattice has 196 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×H(7,𝔽2) | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#3585 | C2 | 1 | 30 | yes | maximal, complemented |
| 3 | C3×#3584 | C2 | 1 | 15 | yes | maximal, complemented |
| 4 | C2×C3×#576 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | C3×#3573 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #63091 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 7 | C3×#585 | — | 2 | 12 | no | — |
| 8 | C3×#585 | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×#585 | C22 | 1 | 30 | yes | complemented |
| 10 | C2×C3×H(5,𝔽2) | — | 2 | 20 | no | — |
| 11 | C2×C3×H(5,𝔽2) | C22 | 1 | 30 | yes | complemented |
| 12 | C3×#578 | C22 | 1 | 20 | yes | complemented |
| 13 | C3×#577 | C22 | 1 | 120 | yes | complemented |
| 14 | C3×#576 | C22 | 1 | 60 | yes | complemented |
| 15 | C3×#576 | C22 | 1 | 120 | yes | complemented |
| 16 | C3×#575 | C22 | 1 | 90 | yes | complemented |
| 17 | C2×C3×AGL(1,ℤ/8) | C22 | 1 | 90 | yes | complemented |
| 18 | C2×C3×#135 | C22 | 1 | 45 | yes | complemented |
| 19 | C22×C3×D8 | C22 | 1 | 15 | yes | complemented |
| 20 | C3×#568 | C22 | 1 | 15 | yes | complemented |
| 21 | C2×C3×#131 | C22 | 1 | 15 | yes | complemented |
| 22 | H(7,𝔽2) | C2×C3 | 1 | 2 | yes | complemented |
| 23 | #3585 | C2×C3 | 1 | 30 | yes | complemented |
| 24 | #3584 | C2×C3 | 1 | 15 | yes | complemented |
| 25 | C2×#576 | C2×C3 | 1 | 15 | yes | complemented |
| 26 | #3573 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C3×2−1+4 | — | 2 | 30 | no | — |
| 28 | C3×2−1+4 | D4 | 1 | 6 | yes | — |
| 29 | C3×2−1+4 | C23 | 1 | 40 | yes | complemented |
| 30 | C3×H(5,𝔽2) | — | 2 | 120 | no | — |
| 31 | C3×H(5,𝔽2) | — | 2 | 90 | no | — |
| 32 | C3×H(5,𝔽2) | D4 | 1 | 10 | yes | — |
| 33 | C3×H(5,𝔽2) | C23 | 1 | 120 | yes | complemented |
| 34 | C2×C3×#41 | — | 2 | 120 | no | — |
| 35 | C2×C3×#41 | — | 2 | 30 | no | — |
| 36 | C2×C3×#41 | C23 | 1 | 15 | yes | — |
| 37 | C2×C3×#41 | C23 | 1 | 90 | yes | complemented |
| 38 | C22×C3×D4 | — | 2 | 90 | no | — |
| 39 | C22×C3×D4 | C23 | 1 | 30 | yes | complemented |
| 40 | C3×#137 | C23 | 1 | 120 | yes | complemented |
| 41 | C3×AGL(1,ℤ/8) | C23 | 1 | 360 | yes | complemented |
| 42 | C3×#135 | C23 | 1 | 60 | yes | complemented |
| 43 | C3×#135 | C23 | 1 | 180 | yes | complemented |
| 44 | C2×C3×Q16 | C23 | 1 | 15 | yes | complemented |
| 45 | C2×C3×SD16 | C23 | 1 | 90 | yes | complemented |
| 46 | C2×C3×D8 | C23 | 1 | 45 | yes | complemented |
| 47 | C2×C3×D8 | C23 | 1 | 90 | yes | complemented |
| 48 | C3×#131 | C23 | 1 | 20 | yes | complemented |
| 49 | C3×#131 | C23 | 1 | 60 | yes | complemented |
| 50 | C2×C3×M16 | C23 | 1 | 45 | yes | complemented |
| 51 | C22×C3×C8 | C23 | 1 | 15 | yes | complemented |
| 52 | #585 | — | 2 | 12 | no | — |
| 53 | #585 | C22×C3 | 1 | 1 | yes | characteristic |
| 54 | #585 | C22×C3 | 1 | 30 | yes | complemented |
| 55 | C2×H(5,𝔽2) | — | 2 | 20 | no | — |
| 56 | C2×H(5,𝔽2) | C22×C3 | 1 | 30 | yes | complemented |
| 57 | #578 | C22×C3 | 1 | 20 | yes | complemented |
| 58 | #577 | C22×C3 | 1 | 120 | yes | complemented |
| 59 | #576 | C22×C3 | 1 | 60 | yes | complemented |
| 60 | #576 | C22×C3 | 1 | 120 | yes | complemented |
| 61 | #575 | C22×C3 | 1 | 90 | yes | complemented |
| 62 | C2×AGL(1,ℤ/8) | C22×C3 | 1 | 90 | yes | complemented |
| 63 | C2×#135 | C22×C3 | 1 | 45 | yes | complemented |
| 64 | C22×D8 | C22×C3 | 1 | 15 | yes | complemented |
| 65 | #568 | C22×C3 | 1 | 15 | yes | complemented |
| 66 | C2×#131 | C22×C3 | 1 | 15 | yes | complemented |
| 67 | C24×C3 | — | 2 | 30 | no | — |
| 68 | C3×#41 | — | 2 | 120 | no | — |
| 69 | C3×#41 | — | 2 | 120 | no | — |
| 70 | C3×#41 | — | 2 | 180 | no | — |
| 71 | C3×#41 | C2×D4 | 1 | 60 | yes | — |
| 72 | C3×#41 | C24 | 1 | 120 | yes | — |
| 73 | C3×#41 | C24 | 1 | 20 | yes | — |
| 74 | C2×C3×Q8 | — | 2 | 40 | no | — |
| 75 | C2×C3×Q8 | C2×D4 | 1 | 15 | yes | — |
| 76 | C2×C3×Q8 | C24 | 1 | 30 | yes | — |
| 77 | C2×C3×D4 | — | 2 | 90 | no | — |
| 78 | C2×C3×D4 | — | 2 | 360 | no | — |
| 79 | C2×C3×D4 | — | 2 | 90 | no | — |
| 80 | C2×C3×D4 | C2×D4 | 1 | 45 | yes | — |
| 81 | C2×C3×D4 | C24 | 1 | 90 | yes | — |
| 82 | C22×C3×C4 | — | 2 | 90 | no | — |
| 83 | C22×C3×C4 | C24 | 1 | 15 | yes | — |
| 84 | C3×Q16 | C24 | 1 | 6 | yes | — |
| 85 | C3×Q16 | C24 | 1 | 30 | yes | — |
| 86 | C3×SD16 | C24 | 1 | 120 | yes | — |
| 87 | C3×D8 | C24 | 1 | 10 | yes | — |
| 88 | C3×D8 | C24 | 1 | 90 | yes | — |
| 89 | C3×M16 | C24 | 1 | 60 | yes | — |
| 90 | C2×C3×C8 | C24 | 1 | 15 | yes | — |
| 91 | C2×C3×C8 | C24 | 1 | 45 | yes | — |
| 92 | 2−1+4 | — | 2 | 30 | no | — |
| 93 | 2−1+4 | C3×D4 | 1 | 6 | yes | — |
| 94 | 2−1+4 | C23×C3 | 1 | 40 | yes | complemented |
| 95 | H(5,𝔽2) | — | 2 | 120 | no | — |
| 96 | H(5,𝔽2) | — | 2 | 90 | no | — |
| 97 | H(5,𝔽2) | C3×D4 | 1 | 10 | yes | — |
| 98 | H(5,𝔽2) | C23×C3 | 1 | 120 | yes | complemented |
| 99 | C2×#41 | — | 2 | 120 | no | — |
| 100 | C2×#41 | — | 2 | 30 | no | — |
| 101 | C2×#41 | C23×C3 | 1 | 15 | yes | — |
| 102 | C2×#41 | C23×C3 | 1 | 90 | yes | complemented |
| 103 | C22×D4 | — | 2 | 90 | no | — |
| 104 | C22×D4 | C23×C3 | 1 | 30 | yes | complemented |
| 105 | #137 | C23×C3 | 1 | 120 | yes | complemented |
| 106 | AGL(1,ℤ/8) | C23×C3 | 1 | 360 | yes | complemented |
| 107 | #135 | C23×C3 | 1 | 60 | yes | complemented |
| 108 | #135 | C23×C3 | 1 | 180 | yes | complemented |
| 109 | C2×Q16 | C23×C3 | 1 | 15 | yes | complemented |
| 110 | C2×SD16 | C23×C3 | 1 | 90 | yes | complemented |
| 111 | C2×D8 | C23×C3 | 1 | 45 | yes | complemented |
| 112 | C2×D8 | C23×C3 | 1 | 90 | yes | complemented |
| 113 | #131 | C23×C3 | 1 | 20 | yes | complemented |
| 114 | #131 | C23×C3 | 1 | 60 | yes | complemented |
| 115 | C2×M16 | C23×C3 | 1 | 45 | yes | complemented |
| 116 | C22×C8 | C23×C3 | 1 | 15 | yes | complemented |
| 117 | C23×C3 | — | 2 | 90 | no | — |
| 118 | C23×C3 | — | 16 | 30 | no | — |
| 119 | C23×C3 | C22×D4 | 1 | 15 | yes | — |
| 120 | C3×Q8 | — | 2 | 30 | no | — |
| 121 | C3×Q8 | C22×D4 | 1 | 20 | yes | — |
| 122 | C3×Q8 | C25 | 1 | 12 | yes | — |
| 123 | C3×D4 | — | 2 | 120 | no | — |
| 124 | C3×D4 | — | 2 | 90 | no | — |
| 125 | C3×D4 | C22×D4 | 1 | 60 | yes | — |
| 126 | C3×D4 | C25 | 1 | 20 | yes | — |
| 127 | C2×C3×C4 | — | 2 | 30 | no | — |
| 128 | C2×C3×C4 | — | 2 | 120 | no | — |
| 129 | C2×C3×C4 | C22×D4 | 1 | 45 | yes | — |
| 130 | C2×C3×C4 | C25 | 1 | 15 | yes | — |
| 131 | C3×C8 | C25 | 1 | 1 | yes | characteristic |
| 132 | C3×C8 | C25 | 1 | 15 | yes | — |
| 133 | C24 | — | 2 | 30 | no | — |
| 134 | #41 | — | 2 | 120 | no | — |
| 135 | #41 | — | 2 | 120 | no | — |
| 136 | #41 | — | 2 | 180 | no | — |
| 137 | #41 | C2×C3×D4 | 1 | 60 | yes | — |
| 138 | #41 | C24×C3 | 1 | 120 | yes | — |
| 139 | #41 | C24×C3 | 1 | 20 | yes | — |
| 140 | C2×Q8 | — | 2 | 40 | no | — |
| 141 | C2×Q8 | C2×C3×D4 | 1 | 15 | yes | — |
| 142 | C2×Q8 | C24×C3 | 1 | 30 | yes | — |
| 143 | C2×D4 | — | 2 | 90 | no | — |
| 144 | C2×D4 | — | 2 | 360 | no | — |
| 145 | C2×D4 | — | 2 | 90 | no | — |
| 146 | C2×D4 | C2×C3×D4 | 1 | 45 | yes | — |
| 147 | C2×D4 | C24×C3 | 1 | 90 | yes | — |
| 148 | C22×C4 | — | 2 | 90 | no | — |
| 149 | C22×C4 | C24×C3 | 1 | 15 | yes | — |
| 150 | Q16 | C24×C3 | 1 | 6 | yes | — |
| 151 | Q16 | C24×C3 | 1 | 30 | yes | — |
| 152 | SD16 | C24×C3 | 1 | 120 | yes | — |
| 153 | D8 | C24×C3 | 1 | 10 | yes | — |
| 154 | D8 | C24×C3 | 1 | 90 | yes | — |
| 155 | M16 | C24×C3 | 1 | 60 | yes | — |
| 156 | C2×C8 | C24×C3 | 1 | 15 | yes | — |
| 157 | C2×C8 | C24×C3 | 1 | 45 | yes | — |
| 158 | C22×C3 | — | 2 | 20 | no | — |
| 159 | C22×C3 | — | 4 | 15 | no | — |
| 160 | C22×C3 | — | 8 | 90 | no | — |
| 161 | C22×C3 | C23×D4 | 1 | 15 | yes | — |
| 162 | C3×C4 | — | 2 | 12 | no | — |
| 163 | C3×C4 | C23×D4 | 1 | 15 | yes | — |
| 164 | C3×C4 | C26 | 1 | 1 | yes | characteristic |
| 165 | C23 | — | 2 | 90 | no | — |
| 166 | C23 | — | 16 | 30 | no | — |
| 167 | C23 | C22×C3×D4 | 1 | 15 | yes | — |
| 168 | Q8 | — | 2 | 30 | no | — |
| 169 | Q8 | C22×C3×D4 | 1 | 20 | yes | — |
| 170 | Q8 | C25×C3 | 1 | 12 | yes | — |
| 171 | D4 | — | 2 | 120 | no | — |
| 172 | D4 | — | 2 | 90 | no | — |
| 173 | D4 | C22×C3×D4 | 1 | 60 | yes | — |
| 174 | D4 | C25×C3 | 1 | 20 | yes | — |
| 175 | C2×C4 | — | 2 | 30 | no | — |
| 176 | C2×C4 | — | 2 | 120 | no | — |
| 177 | C2×C4 | C22×C3×D4 | 1 | 45 | yes | — |
| 178 | C2×C4 | C25×C3 | 1 | 15 | yes | — |
| 179 | C8 | C25×C3 | 1 | 1 | yes | characteristic |
| 180 | C8 | C25×C3 | 1 | 15 | yes | — |
| 181 | C2×C3 | — | 2 | 15 | no | — |
| 182 | C2×C3 | — | 4 | 20 | no | — |
| 183 | C2×C3 | C24×D4 | 1 | 1 | yes | characteristic |
| 184 | C22 | — | 2 | 20 | no | — |
| 185 | C22 | — | 4 | 15 | no | — |
| 186 | C22 | — | 8 | 90 | no | — |
| 187 | C22 | C23×C3×D4 | 1 | 15 | yes | — |
| 188 | C4 | — | 2 | 12 | no | — |
| 189 | C4 | C23×C3×D4 | 1 | 15 | yes | — |
| 190 | C4 | C26×C3 | 1 | 1 | yes | characteristic |
| 191 | C3 | #63091 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 192 | C2 | — | 2 | 15 | no | — |
| 193 | C2 | — | 4 | 20 | no | — |
| 194 | C2 | C24×C3×D4 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C3×H(7,𝔽2) | C2 | semidirect |
| 2. C3×#3585 | C2 | semidirect |
| 3. C3×#3584 | C2 | semidirect |
| 4. C2×C3×#576 | C2 | semidirect |
| 5. C3×#3573 | C2 | semidirect |
| 6. #63091 | C3 | direct (N × H) |
| 9. C3×#585 | C22 | semidirect |
| 11. C2×C3×H(5,𝔽2) | C22 | semidirect |
| 12. C3×#578 | C22 | semidirect |
| 13. C3×#577 | C22 | semidirect |
| 14. C3×#576 | C22 | semidirect |
| 15. C3×#576 | C22 | semidirect |
| 16. C3×#575 | C22 | semidirect |
| 17. C2×C3×AGL(1,ℤ/8) | C22 | semidirect |
| 18. C2×C3×#135 | C22 | semidirect |
| 19. C22×C3×D8 | C22 | semidirect |
| 20. C3×#568 | C22 | semidirect |
| 21. C2×C3×#131 | C22 | semidirect |
| 22. H(7,𝔽2) | C2×C3 | semidirect |
| 23. #3585 | C2×C3 | semidirect |
| 24. #3584 | C2×C3 | semidirect |
| 25. C2×#576 | C2×C3 | semidirect |
| 26. #3573 | C2×C3 | semidirect |
| 29. C3×2−1+4 | C23 | semidirect |
| 33. C3×H(5,𝔽2) | C23 | semidirect |
| 37. C2×C3×#41 | C23 | semidirect |
| 39. C22×C3×D4 | C23 | semidirect |
| 40. C3×#137 | C23 | semidirect |
| 41. C3×AGL(1,ℤ/8) | C23 | semidirect |
| 42. C3×#135 | C23 | semidirect |
| 43. C3×#135 | C23 | semidirect |
| 44. C2×C3×Q16 | C23 | semidirect |
| 45. C2×C3×SD16 | C23 | semidirect |
| 46. C2×C3×D8 | C23 | semidirect |
| 47. C2×C3×D8 | C23 | semidirect |
| 48. C3×#131 | C23 | semidirect |
| 49. C3×#131 | C23 | semidirect |
| 50. C2×C3×M16 | C23 | semidirect |
| 51. C22×C3×C8 | C23 | semidirect |
| 54. #585 | C22×C3 | semidirect |
| 56. C2×H(5,𝔽2) | C22×C3 | semidirect |
| 57. #578 | C22×C3 | semidirect |
| 58. #577 | C22×C3 | semidirect |
| 59. #576 | C22×C3 | semidirect |
| 60. #576 | C22×C3 | semidirect |
| 61. #575 | C22×C3 | semidirect |
| 62. C2×AGL(1,ℤ/8) | C22×C3 | semidirect |
| 63. C2×#135 | C22×C3 | semidirect |
| 64. C22×D8 | C22×C3 | semidirect |
| 65. #568 | C22×C3 | semidirect |
| 66. C2×#131 | C22×C3 | semidirect |
| 94. 2−1+4 | C23×C3 | semidirect |
| 98. H(5,𝔽2) | C23×C3 | semidirect |
| 102. C2×#41 | C23×C3 | semidirect |
| 104. C22×D4 | C23×C3 | semidirect |
| 105. #137 | C23×C3 | semidirect |
| 106. AGL(1,ℤ/8) | C23×C3 | semidirect |
| 107. #135 | C23×C3 | semidirect |
| 108. #135 | C23×C3 | semidirect |
| 109. C2×Q16 | C23×C3 | semidirect |
| 110. C2×SD16 | C23×C3 | semidirect |
| 111. C2×D8 | C23×C3 | semidirect |
| 112. C2×D8 | C23×C3 | semidirect |
| 113. #131 | C23×C3 | semidirect |
| 114. #131 | C23×C3 | semidirect |
| 115. C2×M16 | C23×C3 | semidirect |
| 116. C22×C8 | C23×C3 | semidirect |
| 191. C3 | #63091 | direct (N × H) |
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta