← C26×AGL(1,ℤ/8) #72313896
Order 2048.
20276840 conjugacy classes of subgroups.
The subgroup lattice has 231 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C26×M16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C26×SD16 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C26×D8 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C26×#41 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C25×AGL(1,ℤ/8) | C2 | 1 | 504 | yes | maximal, direct product factor |
| 6 | C27×D4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C29 | — | 2 | 1 | no | — |
| 8 | C25×#41 | — | 2 | 126 | no | — |
| 9 | C25×#41 | C22 | 1 | 252 | yes | complemented |
| 10 | C26×Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C26×D4 | — | 2 | 1 | no | — |
| 12 | C26×D4 | — | 2 | 126 | no | — |
| 13 | C26×D4 | — | 2 | 1 | no | — |
| 14 | C26×D4 | C22 | 1 | 2 | yes | — |
| 15 | C26×D4 | C22 | 1 | 252 | yes | complemented |
| 16 | C26×D4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C27×C4 | — | 2 | 1 | no | — |
| 18 | C27×C4 | C22 | 1 | 1 | yes | characteristic |
| 19 | C25×SD16 | C22 | 1 | 504 | yes | complemented |
| 20 | C24×AGL(1,ℤ/8) | C22 | 1 | 41664 | yes | direct product factor |
| 21 | C25×D8 | C22 | 1 | 504 | yes | complemented |
| 22 | C25×M16 | C22 | 1 | 252 | yes | complemented |
| 23 | C26×C8 | C22 | 1 | 2 | yes | complemented |
| 24 | C28 | — | 2 | 252 | no | — |
| 25 | C28 | — | 2 | 1 | no | — |
| 26 | C28 | — | 2 | 2 | no | — |
| 27 | C28 | — | 4 | 128 | no | — |
| 28 | C28 | D4 | 1 | 1 | yes | characteristic |
| 29 | C24×#41 | — | 2 | 15624 | no | — |
| 30 | C24×#41 | C23 | 1 | 10416 | yes | complemented |
| 31 | C25×Q8 | — | 2 | 63 | no | — |
| 32 | C25×Q8 | C23 | 1 | 126 | yes | complemented |
| 33 | C25×D4 | — | 2 | 252 | no | — |
| 34 | C25×D4 | — | 2 | 126 | no | — |
| 35 | C25×D4 | — | 2 | 252 | no | — |
| 36 | C25×D4 | — | 2 | 15624 | no | — |
| 37 | C25×D4 | — | 2 | 63 | no | — |
| 38 | C25×D4 | C23 | 1 | 252 | yes | — |
| 39 | C25×D4 | C23 | 1 | 10416 | yes | complemented |
| 40 | C25×D4 | C23 | 1 | 126 | yes | complemented |
| 41 | C26×C4 | — | 2 | 1 | no | — |
| 42 | C26×C4 | — | 2 | 252 | no | — |
| 43 | C26×C4 | D4 | 1 | 126 | yes | — |
| 44 | C26×C4 | D4 | 1 | 1 | yes | characteristic |
| 45 | C26×C4 | C23 | 1 | 126 | yes | — |
| 46 | C26×C4 | C23 | 1 | 1 | yes | characteristic |
| 47 | C23×AGL(1,ℤ/8) | C23 | 1 | 714240 | yes | direct product factor |
| 48 | C24×SD16 | C23 | 1 | 20832 | yes | complemented |
| 49 | C24×D8 | C23 | 1 | 20832 | yes | complemented |
| 50 | C24×M16 | C23 | 1 | 10416 | yes | complemented |
| 51 | C25×C8 | C23 | 1 | 252 | yes | complemented |
| 52 | C27 | — | 2 | 64 | no | — |
| 53 | C27 | — | 2 | 126 | no | — |
| 54 | C27 | — | 2 | 252 | no | — |
| 55 | C27 | — | 2 | 10416 | no | — |
| 56 | C27 | — | 4 | 16128 | no | — |
| 57 | C27 | — | 4 | 64 | no | — |
| 58 | C27 | — | 4 | 128 | no | — |
| 59 | C27 | C2×D4 | 1 | 126 | yes | — |
| 60 | C27 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C23×#41 | — | 2 | 312480 | no | — |
| 62 | C23×#41 | C24 | 1 | 89280 | yes | complemented |
| 63 | C24×Q8 | — | 2 | 3906 | no | — |
| 64 | C24×Q8 | C24 | 1 | 2604 | yes | complemented |
| 65 | C24×D4 | — | 2 | 10416 | no | — |
| 66 | C24×D4 | — | 2 | 7812 | no | — |
| 67 | C24×D4 | — | 2 | 10416 | no | — |
| 68 | C24×D4 | — | 2 | 312480 | no | — |
| 69 | C24×D4 | — | 2 | 3906 | no | — |
| 70 | C24×D4 | C24 | 1 | 5208 | yes | — |
| 71 | C24×D4 | C24 | 1 | 89280 | yes | complemented |
| 72 | C24×D4 | C24 | 1 | 2604 | yes | complemented |
| 73 | C25×C4 | — | 2 | 126 | no | — |
| 74 | C25×C4 | — | 2 | 10416 | no | — |
| 75 | C25×C4 | C2×D4 | 1 | 126 | yes | — |
| 76 | C25×C4 | C2×D4 | 1 | 63 | yes | — |
| 77 | C25×C4 | C2×D4 | 1 | 7812 | yes | — |
| 78 | C25×C4 | C24 | 1 | 63 | yes | — |
| 79 | C25×C4 | C24 | 1 | 2604 | yes | — |
| 80 | C22×AGL(1,ℤ/8) | C24 | 1 | 2666496 | yes | direct product factor |
| 81 | C23×SD16 | C24 | 1 | 178560 | yes | complemented |
| 82 | C23×D8 | C24 | 1 | 178560 | yes | complemented |
| 83 | C23×M16 | C24 | 1 | 89280 | yes | complemented |
| 84 | C24×C8 | C24 | 1 | 5208 | yes | complemented |
| 85 | C26 | — | 2 | 4032 | no | — |
| 86 | C26 | — | 2 | 2604 | no | — |
| 87 | C26 | — | 2 | 5208 | no | — |
| 88 | C26 | — | 2 | 89280 | no | — |
| 89 | C26 | — | 4 | 333312 | no | — |
| 90 | C26 | — | 4 | 4032 | no | — |
| 91 | C26 | — | 4 | 8064 | no | — |
| 92 | C26 | AGL(1,ℤ/8) | 1 | 64 | yes | direct product factor |
| 93 | C26 | C22×D4 | 1 | 2604 | yes | — |
| 94 | C26 | C22×D4 | 1 | 63 | yes | — |
| 95 | C22×#41 | — | 2 | 1249920 | no | — |
| 96 | C22×#41 | C25 | 1 | 166656 | yes | complemented |
| 97 | C23×Q8 | — | 2 | 39060 | no | — |
| 98 | C23×Q8 | C25 | 1 | 11160 | yes | complemented |
| 99 | C23×D4 | — | 2 | 89280 | no | — |
| 100 | C23×D4 | — | 2 | 78120 | no | — |
| 101 | C23×D4 | — | 2 | 89280 | no | — |
| 102 | C23×D4 | — | 2 | 1249920 | no | — |
| 103 | C23×D4 | — | 2 | 39060 | no | — |
| 104 | C23×D4 | C25 | 1 | 22320 | yes | — |
| 105 | C23×D4 | C25 | 1 | 166656 | yes | complemented |
| 106 | C23×D4 | C25 | 1 | 11160 | yes | complemented |
| 107 | C24×C4 | — | 2 | 2604 | no | — |
| 108 | C24×C4 | — | 2 | 89280 | no | — |
| 109 | C24×C4 | C22×D4 | 1 | 2604 | yes | — |
| 110 | C24×C4 | C22×D4 | 1 | 1953 | yes | — |
| 111 | C24×C4 | C22×D4 | 1 | 78120 | yes | — |
| 112 | C24×C4 | C25 | 1 | 651 | yes | — |
| 113 | C24×C4 | C25 | 1 | 11160 | yes | — |
| 114 | C2×AGL(1,ℤ/8) | C25 | 1 | 2064384 | yes | direct product factor |
| 115 | C22×SD16 | C25 | 1 | 333312 | yes | complemented |
| 116 | C22×D8 | C25 | 1 | 333312 | yes | complemented |
| 117 | C22×M16 | C25 | 1 | 166656 | yes | complemented |
| 118 | C23×C8 | C25 | 1 | 22320 | yes | complemented |
| 119 | C25 | — | 2 | 41664 | no | — |
| 120 | C25 | — | 2 | 11160 | no | — |
| 121 | C25 | — | 2 | 22320 | no | — |
| 122 | C25 | — | 2 | 166656 | no | — |
| 123 | C25 | — | 4 | 1428480 | no | — |
| 124 | C25 | — | 4 | 41664 | no | — |
| 125 | C25 | — | 4 | 83328 | no | — |
| 126 | C25 | C2×AGL(1,ℤ/8) | 1 | 2016 | yes | direct product factor |
| 127 | C25 | C23×D4 | 1 | 11160 | yes | — |
| 128 | C25 | C23×D4 | 1 | 651 | yes | — |
| 129 | C2×#41 | — | 2 | 999936 | no | — |
| 130 | C2×#41 | C26 | 1 | 64512 | yes | complemented |
| 131 | C22×Q8 | — | 2 | 78120 | no | — |
| 132 | C22×Q8 | C26 | 1 | 10416 | yes | complemented |
| 133 | C22×D4 | — | 2 | 166656 | no | — |
| 134 | C22×D4 | — | 2 | 156240 | no | — |
| 135 | C22×D4 | — | 2 | 166656 | no | — |
| 136 | C22×D4 | — | 2 | 999936 | no | — |
| 137 | C22×D4 | — | 2 | 78120 | no | — |
| 138 | C22×D4 | C26 | 1 | 20832 | yes | — |
| 139 | C22×D4 | C26 | 1 | 64512 | yes | complemented |
| 140 | C22×D4 | C26 | 1 | 10416 | yes | complemented |
| 141 | C23×C4 | — | 2 | 11160 | no | — |
| 142 | C23×C4 | — | 2 | 166656 | no | — |
| 143 | C23×C4 | C23×D4 | 1 | 11160 | yes | — |
| 144 | C23×C4 | C23×D4 | 1 | 9765 | yes | — |
| 145 | C23×C4 | C23×D4 | 1 | 156240 | yes | — |
| 146 | C23×C4 | C26 | 1 | 1395 | yes | — |
| 147 | C23×C4 | C26 | 1 | 10416 | yes | — |
| 148 | AGL(1,ℤ/8) | C26 | 1 | 262144 | yes | direct product factor |
| 149 | C2×SD16 | C26 | 1 | 129024 | yes | complemented |
| 150 | C2×D8 | C26 | 1 | 129024 | yes | complemented |
| 151 | C2×M16 | C26 | 1 | 64512 | yes | complemented |
| 152 | C22×C8 | C26 | 1 | 20832 | yes | complemented |
| 153 | C24 | — | 2 | 89280 | no | — |
| 154 | C24 | — | 2 | 10416 | no | — |
| 155 | C24 | — | 2 | 20832 | no | — |
| 156 | C24 | — | 2 | 64512 | no | — |
| 157 | C24 | — | 4 | 1333248 | no | — |
| 158 | C24 | — | 4 | 89280 | no | — |
| 159 | C24 | — | 4 | 178560 | no | — |
| 160 | C24 | C22×AGL(1,ℤ/8) | 1 | 10416 | yes | direct product factor |
| 161 | C24 | C24×D4 | 1 | 10416 | yes | — |
| 162 | C24 | C24×D4 | 1 | 1395 | yes | — |
| 163 | #41 | — | 2 | 129024 | no | — |
| 164 | #41 | C27 | 1 | 4096 | yes | complemented |
| 165 | C2×Q8 | — | 2 | 31248 | no | — |
| 166 | C2×Q8 | C27 | 1 | 2016 | yes | complemented |
| 167 | C2×D4 | — | 2 | 64512 | no | — |
| 168 | C2×D4 | — | 2 | 62496 | no | — |
| 169 | C2×D4 | — | 2 | 64512 | no | — |
| 170 | C2×D4 | — | 2 | 129024 | no | — |
| 171 | C2×D4 | — | 2 | 31248 | no | — |
| 172 | C2×D4 | C27 | 1 | 4032 | yes | — |
| 173 | C2×D4 | C27 | 1 | 4096 | yes | complemented |
| 174 | C2×D4 | C27 | 1 | 2016 | yes | complemented |
| 175 | C22×C4 | — | 2 | 10416 | no | — |
| 176 | C22×C4 | — | 2 | 64512 | no | — |
| 177 | C22×C4 | C24×D4 | 1 | 10416 | yes | — |
| 178 | C22×C4 | C24×D4 | 1 | 9765 | yes | — |
| 179 | C22×C4 | C24×D4 | 1 | 62496 | yes | — |
| 180 | C22×C4 | C27 | 1 | 651 | yes | — |
| 181 | C22×C4 | C27 | 1 | 2016 | yes | — |
| 182 | SD16 | C27 | 1 | 8192 | yes | complemented |
| 183 | D8 | C27 | 1 | 8192 | yes | complemented |
| 184 | M16 | C27 | 1 | 4096 | yes | complemented |
| 185 | C2×C8 | C27 | 1 | 4032 | yes | complemented |
| 186 | C23 | — | 2 | 41664 | no | — |
| 187 | C23 | — | 2 | 2016 | no | — |
| 188 | C23 | — | 2 | 4032 | no | — |
| 189 | C23 | — | 2 | 4096 | no | — |
| 190 | C23 | — | 4 | 258048 | no | — |
| 191 | C23 | — | 4 | 41664 | no | — |
| 192 | C23 | — | 4 | 83328 | no | — |
| 193 | C23 | C23×AGL(1,ℤ/8) | 1 | 11160 | yes | direct product factor |
| 194 | C23 | C25×D4 | 1 | 2016 | yes | — |
| 195 | C23 | C25×D4 | 1 | 651 | yes | — |
| 196 | Q8 | — | 2 | 2016 | no | — |
| 197 | Q8 | C28 | 1 | 64 | yes | complemented |
| 198 | D4 | — | 2 | 4096 | no | — |
| 199 | D4 | — | 2 | 4032 | no | — |
| 200 | D4 | — | 2 | 4096 | no | — |
| 201 | D4 | — | 2 | 2016 | no | — |
| 202 | D4 | C28 | 1 | 128 | yes | — |
| 203 | D4 | C28 | 1 | 64 | yes | complemented |
| 204 | C2×C4 | — | 2 | 2016 | no | — |
| 205 | C2×C4 | — | 2 | 4096 | no | — |
| 206 | C2×C4 | C25×D4 | 1 | 2016 | yes | — |
| 207 | C2×C4 | C25×D4 | 1 | 1953 | yes | — |
| 208 | C2×C4 | C25×D4 | 1 | 4032 | yes | — |
| 209 | C2×C4 | C28 | 1 | 63 | yes | — |
| 210 | C2×C4 | C28 | 1 | 64 | yes | — |
| 211 | C8 | C28 | 1 | 128 | yes | complemented |
| 212 | C22 | — | 2 | 4032 | no | — |
| 213 | C22 | — | 2 | 64 | no | — |
| 214 | C22 | — | 2 | 128 | no | — |
| 215 | C22 | — | 4 | 8192 | no | — |
| 216 | C22 | — | 4 | 4032 | no | — |
| 217 | C22 | — | 4 | 8064 | no | — |
| 218 | C22 | C24×AGL(1,ℤ/8) | 1 | 2604 | yes | direct product factor |
| 219 | C22 | C26×D4 | 1 | 64 | yes | — |
| 220 | C22 | C26×D4 | 1 | 63 | yes | — |
| 221 | C4 | — | 2 | 64 | no | — |
| 222 | C4 | C26×D4 | 1 | 64 | yes | — |
| 223 | C4 | C26×D4 | 1 | 63 | yes | — |
| 224 | C4 | C29 | 1 | 1 | yes | characteristic |
| 225 | C2 | — | 2 | 64 | no | — |
| 226 | C2 | — | 4 | 64 | no | — |
| 227 | C2 | — | 4 | 128 | no | — |
| 228 | C2 | C27×D4 | 1 | 1 | yes | characteristic |
| 229 | C2 | C25×AGL(1,ℤ/8) | 1 | 126 | yes | direct product factor |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|
| 1. C26×M16 | C2 | semidirect |
| 2. C26×SD16 | C2 | semidirect |
| 3. C26×D8 | C2 | semidirect |
| 4. C26×#41 | C2 | semidirect |
| 5. C25×AGL(1,ℤ/8) | C2 | direct (N × H) |
| 6. C27×D4 | C2 | semidirect |
| 9. C25×#41 | C22 | semidirect |
| 10. C26×Q8 | C22 | semidirect |
| 15. C26×D4 | C22 | semidirect |
| 16. C26×D4 | C22 | semidirect |
| 19. C25×SD16 | C22 | semidirect |
| 20. C24×AGL(1,ℤ/8) | C22 | direct (N × H) |
| 21. C25×D8 | C22 | semidirect |
| 22. C25×M16 | C22 | semidirect |
| 23. C26×C8 | C22 | semidirect |
| 30. C24×#41 | C23 | semidirect |
| 32. C25×Q8 | C23 | semidirect |
| 39. C25×D4 | C23 | semidirect |
| 40. C25×D4 | C23 | semidirect |
| 47. C23×AGL(1,ℤ/8) | C23 | direct (N × H) |
| 48. C24×SD16 | C23 | semidirect |
| 49. C24×D8 | C23 | semidirect |
| 50. C24×M16 | C23 | semidirect |
| 51. C25×C8 | C23 | semidirect |
| 62. C23×#41 | C24 | semidirect |
| 64. C24×Q8 | C24 | semidirect |
| 71. C24×D4 | C24 | semidirect |
| 72. C24×D4 | C24 | semidirect |
| 80. C22×AGL(1,ℤ/8) | C24 | direct (N × H) |
| 81. C23×SD16 | C24 | semidirect |
| 82. C23×D8 | C24 | semidirect |
| 83. C23×M16 | C24 | semidirect |
| 84. C24×C8 | C24 | semidirect |
| 92. C26 | AGL(1,ℤ/8) | direct (N × H) |
| 96. C22×#41 | C25 | semidirect |
| 98. C23×Q8 | C25 | semidirect |
| 105. C23×D4 | C25 | semidirect |
| 106. C23×D4 | C25 | semidirect |
| 114. C2×AGL(1,ℤ/8) | C25 | direct (N × H) |
| 115. C22×SD16 | C25 | semidirect |
| 116. C22×D8 | C25 | semidirect |
| 117. C22×M16 | C25 | semidirect |
| 118. C23×C8 | C25 | semidirect |
| 126. C25 | C2×AGL(1,ℤ/8) | direct (N × H) |
| 130. C2×#41 | C26 | semidirect |
| 132. C22×Q8 | C26 | semidirect |
| 139. C22×D4 | C26 | semidirect |
| 140. C22×D4 | C26 | semidirect |
| 148. AGL(1,ℤ/8) | C26 | direct (N × H) |
| 149. C2×SD16 | C26 | semidirect |
| 150. C2×D8 | C26 | semidirect |
| 151. C2×M16 | C26 | semidirect |
| 152. C22×C8 | C26 | semidirect |
| 160. C24 | C22×AGL(1,ℤ/8) | direct (N × H) |
| 164. #41 | C27 | semidirect |
| 166. C2×Q8 | C27 | semidirect |
| 173. C2×D4 | C27 | semidirect |
| 174. C2×D4 | C27 | semidirect |
| 182. SD16 | C27 | semidirect |
| 183. D8 | C27 | semidirect |
| 184. M16 | C27 | semidirect |
| 185. C2×C8 | C27 | semidirect |
| 193. C23 | C23×AGL(1,ℤ/8) | direct (N × H) |
| 197. Q8 | C28 | semidirect |
| 203. D4 | C28 | semidirect |
| 211. C8 | C28 | semidirect |
| 218. C22 | C24×AGL(1,ℤ/8) | direct (N × H) |
| 229. C2 | C25×AGL(1,ℤ/8) | 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