← #31965726
Subgroups — #31965726
Order 18176.
872 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22602166 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22597360 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22586977 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #22586943 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #22586941 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #22586934 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C71×#2971 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #17986725 | — | 2 | 1 | no | — |
| 9 | #17986725 | — | 2 | 1 | no | — |
| 10 | #17986708 | — | 2 | 3 | no | — |
| 11 | #17986708 | — | 2 | 3 | no | — |
| 12 | #17986627 | C4 | 1 | 3 | yes | — |
| 13 | #17986627 | C4 | 1 | 3 | yes | — |
| 14 | #17986556 | C22 | 1 | 3 | yes | — |
| 15 | C2×#14556120 | C22 | 1 | 3 | yes | — |
| 16 | #17986482 | C4 | 1 | 1 | yes | characteristic |
| 17 | #17986482 | C4 | 1 | 1 | yes | characteristic |
| 18 | Q8×C71⋊C8 | — | 2 | 1 | no | — |
| 19 | #17986077 | — | 2 | 3 | no | — |
| 20 | #17986056 | C22 | 1 | 3 | yes | — |
| 21 | C4×#11765849 | C22 | 1 | 3 | yes | — |
| 22 | #17985740 | C22 | 1 | 1 | yes | characteristic |
| 23 | #17985740 | C22 | 1 | 3 | yes | — |
| 24 | #17985740 | C22 | 1 | 3 | yes | — |
| 25 | #17985740 | C22 | 1 | 3 | yes | — |
| 26 | #17985740 | C22 | 1 | 6 | yes | — |
| 27 | C4×#41×C71 | C22 | 1 | 1 | yes | characteristic |
| 28 | C8×Q8×C71 | — | 2 | 1 | no | — |
| 29 | #436×C71 | — | 2 | 3 | no | — |
| 30 | #432×C71 | C22 | 1 | 3 | yes | — |
| 31 | C4×M16×C71 | C22 | 1 | 3 | yes | — |
| 32 | C2×#11765849 | — | 2 | 3 | no | — |
| 33 | #14556128 | C2×C4 | 1 | 3 | yes | — |
| 34 | #14556128 | C2×C4 | 1 | 3 | yes | — |
| 35 | #14556127 | — | 2 | 3 | no | — |
| 36 | #14556127 | — | 2 | 3 | no | — |
| 37 | #14556120 | — | 2 | 3 | no | — |
| 38 | #14556120 | — | 2 | 3 | no | — |
| 39 | #14556120 | C2×C4 | 1 | 6 | yes | — |
| 40 | #14556120 | C23 | 1 | 3 | yes | — |
| 41 | #14556120 | C23 | 1 | 1 | yes | characteristic |
| 42 | #14556119 | — | 2 | 3 | no | — |
| 43 | #14556119 | — | 2 | 3 | no | — |
| 44 | #14556107 | — | 2 | 3 | no | — |
| 45 | #14556107 | — | 2 | 3 | no | — |
| 46 | #14556105 | — | 2 | 6 | no | — |
| 47 | #14556105 | — | 2 | 6 | no | — |
| 48 | Q8×Dic71 | — | 2 | 1 | no | — |
| 49 | Q8×Dic71 | — | 2 | 1 | no | — |
| 50 | C4×Dic142 | — | 2 | 3 | no | — |
| 51 | C4×Dic142 | — | 2 | 3 | no | — |
| 52 | D4×Dic71 | — | 2 | 3 | no | — |
| 53 | D4×Dic71 | — | 2 | 3 | no | — |
| 54 | #14556087 | C2×C4 | 1 | 3 | yes | — |
| 55 | #14556087 | C2×C4 | 1 | 3 | yes | — |
| 56 | C2×C4⋊Dic71 | C2×C4 | 1 | 3 | yes | — |
| 57 | C2×C4⋊Dic71 | C2×C4 | 1 | 3 | yes | — |
| 58 | C2×Dic71⋊C4 | — | 2 | 3 | no | — |
| 59 | C2×Dic71⋊C4 | — | 2 | 3 | no | — |
| 60 | C4⋊(C71⋊C8) | — | 2 | 3 | no | — |
| 61 | C22⋊(C71⋊C8) | — | 2 | 3 | no | — |
| 62 | #14556039 | C23 | 1 | 3 | yes | — |
| 63 | #14556039 | C23 | 1 | 1 | yes | characteristic |
| 64 | C4×C71⋊C8 | — | 2 | 3 | no | — |
| 65 | C2×#41×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | C2×M16×C71 | — | 2 | 3 | no | — |
| 67 | C4×Q8×C71 | D4 | 1 | 1 | yes | characteristic |
| 68 | C4×Q8×C71 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×D4×C71 | D4 | 1 | 3 | yes | — |
| 70 | C4×D4×C71 | D4 | 1 | 3 | yes | — |
| 71 | #117×C71 | C2×C4 | 1 | 3 | yes | — |
| 72 | C2×C42×C71 | C23 | 1 | 3 | yes | — |
| 73 | C4⋊C8×C71 | — | 2 | 3 | no | — |
| 74 | C22⋊C8×C71 | — | 2 | 3 | no | — |
| 75 | #97×C71 | C23 | 1 | 3 | yes | — |
| 76 | #97×C71 | C23 | 1 | 1 | yes | characteristic |
| 77 | C4×C8×C71 | — | 2 | 3 | no | — |
| 78 | C2×Dic142 | — | 2 | 3 | no | — |
| 79 | C2×Dic142 | — | 2 | 3 | no | — |
| 80 | C22×Dic71 | — | 2 | 3 | no | — |
| 81 | C22×Dic71 | — | 2 | 3 | no | — |
| 82 | #11765849 | — | 4 | 6 | no | — |
| 83 | C2×C71⋊C8 | — | 2 | 1 | no | — |
| 84 | C2×C71⋊C8 | — | 2 | 3 | no | — |
| 85 | C4⋊Dic71 | — | 2 | 3 | no | — |
| 86 | C4⋊Dic71 | — | 2 | 3 | no | — |
| 87 | C4⋊Dic71 | — | 2 | 3 | no | — |
| 88 | C4⋊Dic71 | — | 2 | 3 | no | — |
| 89 | C4⋊Dic71 | C22×C4 | 1 | 3 | yes | — |
| 90 | C4⋊Dic71 | C22×C4 | 1 | 3 | yes | — |
| 91 | C4⋊Dic71 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C4⋊Dic71 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | Dic71⋊C4 | — | 2 | 6 | no | — |
| 94 | Dic71⋊C4 | — | 2 | 6 | no | — |
| 95 | Dic71⋊C4 | — | 2 | 3 | no | — |
| 96 | Dic71⋊C4 | — | 2 | 3 | no | — |
| 97 | Dic71⋊C4 | — | 2 | 1 | no | — |
| 98 | Dic71⋊C4 | — | 2 | 1 | no | — |
| 99 | C22⋊Dic71 | — | 2 | 3 | no | — |
| 100 | C22⋊Dic71 | — | 2 | 3 | no | — |
| 101 | C4×Dic71 | — | 2 | 3 | no | — |
| 102 | C4×Dic71 | — | 2 | 3 | no | — |
| 103 | #41×C71 | — | 2 | 4 | no | — |
| 104 | C2×Q8×C71 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 105 | C2×D4×C71 | C22⋊C4 | 1 | 3 | yes | — |
| 106 | C22×C4×C71 | C22×C4 | 1 | 3 | yes | — |
| 107 | C22×C4×C71 | C2×D4 | 1 | 3 | yes | — |
| 108 | C22×C4×C71 | C2×D4 | 1 | 3 | yes | — |
| 109 | M16×C71 | — | 4 | 6 | no | — |
| 110 | C2×C8×C71 | — | 2 | 1 | no | — |
| 111 | C2×C8×C71 | — | 2 | 3 | no | — |
| 112 | C4⋊C4×C71 | C22⋊C4 | 1 | 3 | yes | — |
| 113 | C4⋊C4×C71 | C22⋊C4 | 1 | 3 | yes | — |
| 114 | C22⋊C4×C71 | C22⋊C4 | 1 | 6 | yes | — |
| 115 | C42×C71 | C22×C4 | 1 | 3 | yes | — |
| 116 | C42×C71 | C2×D4 | 1 | 3 | yes | — |
| 117 | C42×C71 | C2×D4 | 1 | 3 | yes | — |
| 118 | C42×C71 | C24 | 1 | 1 | yes | characteristic |
| 119 | C23×C71 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 120 | Q8×C71 | #101 | 1 | 4 | yes | — |
| 121 | D4×C71 | — | 2 | 6 | no | — |
| 122 | C2×C4×C71 | — | 2 | 6 | no | — |
| 123 | C2×C4×C71 | — | 4 | 3 | no | — |
| 124 | C2×C4×C71 | — | 4 | 3 | no | — |
| 125 | C2×C4×C71 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 126 | C2×C4×C71 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 127 | C2×C4×C71 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 128 | C2×C4×C71 | C23×C4 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 131 | C2×Dic71 | — | 2 | 1 | no | — |
| 132 | C2×Dic71 | — | 2 | 1 | no | — |
| 133 | C2×Dic71 | — | 2 | 3 | no | — |
| 134 | C2×Dic71 | — | 2 | 3 | no | — |
| 135 | C2×Dic71 | — | 8 | 3 | no | — |
| 136 | C2×Dic71 | — | 8 | 3 | no | — |
| 137 | Dic142 | — | 8 | 3 | no | — |
| 138 | Dic142 | — | 8 | 3 | no | — |
| 139 | C8×C71 | — | 2 | 2 | no | — |
| 140 | C8×C71 | — | 4 | 3 | no | — |
| 141 | C71⋊C8 | — | 2 | 2 | no | — |
| 142 | C71⋊C8 | — | 4 | 3 | no | — |
| 143 | C22×C71 | — | 2 | 3 | no | — |
| 144 | C22×C71 | — | 2 | 3 | no | — |
| 145 | C22×C71 | — | 2 | 3 | no | — |
| 146 | C22×C71 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 147 | C4×C71 | — | 2 | 1 | no | — |
| 148 | C4×C71 | — | 2 | 1 | no | — |
| 149 | C4×C71 | — | 2 | 1 | no | — |
| 150 | C4×C71 | — | 2 | 3 | no | — |
| 151 | C4×C71 | — | 2 | 3 | no | — |
| 152 | C4×C71 | C2×#101 | 1 | 6 | yes | — |
| 153 | Dic71 | — | 4 | 1 | no | — |
| 154 | Dic71 | — | 4 | 1 | no | — |
| 155 | Dic71 | — | 4 | 3 | no | — |
| 156 | Dic71 | — | 4 | 3 | no | — |
| 157 | #17692 | — | 71 | 1 | no | maximal, Sylow |
| 158 | C2×C71 | — | 4 | 3 | no | — |
| 159 | C2×C71 | C22×#101 | 1 | 1 | yes | characteristic |
| 160 | C2×C71 | #2901 | 1 | 1 | yes | characteristic |
| 161 | C2×C71 | #2901 | 1 | 1 | yes | characteristic |
| 162 | #3504 | — | 71 | 1 | no | — |
| 163 | #2971 | — | 71 | 1 | no | — |
| 164 | #2971 | D71 | 1 | 1 | yes | characteristic |
| 165 | #1537 | — | 71 | 3 | no | — |
| 166 | #1528 | — | 71 | 3 | no | — |
| 167 | #1528 | — | 71 | 3 | no | — |
| 168 | #1526 | — | 71 | 3 | no | — |
| 169 | C71 | #17692 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 170 | #557 | — | 142 | 1 | no | — |
| 171 | #557 | — | 142 | 1 | no | — |
| 172 | #554 | — | 142 | 3 | no | — |
| 173 | #554 | — | 142 | 3 | no | — |
| 174 | #544 | — | 71 | 3 | no | — |
| 175 | #544 | — | 71 | 3 | no | — |
| 176 | #533 | — | 71 | 3 | no | — |
| 177 | C2×#125 | — | 71 | 3 | no | — |
| 178 | #520 | — | 71 | 1 | no | — |
| 179 | #520 | — | 71 | 1 | no | — |
| 180 | C4×#41 | C2×D71 | 1 | 1 | yes | characteristic |
| 181 | C8×Q8 | — | 2 | 1 | no | — |
| 182 | C8×Q8 | — | 142 | 1 | no | — |
| 183 | #436 | — | 2 | 3 | no | — |
| 184 | #436 | — | 142 | 3 | no | — |
| 185 | #432 | — | 71 | 3 | no | — |
| 186 | #432 | C2×D71 | 1 | 3 | yes | — |
| 187 | C4×M16 | — | 71 | 3 | no | — |
| 188 | C4×M16 | C2×D71 | 1 | 3 | yes | — |
| 189 | #330 | — | 71 | 3 | no | — |
| 190 | #330 | — | 71 | 6 | no | — |
| 191 | #330 | — | 71 | 1 | no | — |
| 192 | #330 | — | 71 | 3 | no | — |
| 193 | #330 | — | 71 | 3 | no | — |
| 194 | C2×#41 | C4×D71 | 1 | 1 | yes | characteristic |
| 195 | C2×M16 | — | 2 | 3 | no | — |
| 196 | C2×M16 | — | 142 | 3 | no | — |
| 197 | C4⋊Q8 | — | 71 | 3 | no | — |
| 198 | C4⋊Q8 | — | 71 | 3 | no | — |
| 199 | C4⋊Q8 | — | 142 | 3 | no | — |
| 200 | C4⋊Q8 | — | 142 | 3 | no | — |
| 201 | #125 | — | 71 | 6 | no | — |
| 202 | #125 | — | 71 | 3 | no | — |
| 203 | #125 | — | 71 | 1 | no | — |
| 204 | #125 | — | 142 | 3 | no | — |
| 205 | #125 | — | 142 | 3 | no | — |
| 206 | #125 | — | 142 | 3 | no | — |
| 207 | #125 | — | 142 | 3 | no | — |
| 208 | C22⋊Q8 | — | 142 | 3 | no | — |
| 209 | C22⋊Q8 | — | 142 | 3 | no | — |
| 210 | C22⋊Q8 | — | 142 | 6 | no | — |
| 211 | C22⋊Q8 | — | 142 | 6 | no | — |
| 212 | C4×Q8 | — | 142 | 1 | no | — |
| 213 | C4×Q8 | — | 142 | 1 | no | — |
| 214 | C4×Q8 | — | 142 | 3 | no | — |
| 215 | C4×Q8 | — | 142 | 3 | no | — |
| 216 | C4×Q8 | D284 | 1 | 1 | yes | characteristic |
| 217 | C4×Q8 | C22⋊D71 | 1 | 1 | yes | characteristic |
| 218 | C4×D4 | — | 142 | 3 | no | — |
| 219 | C4×D4 | — | 142 | 3 | no | — |
| 220 | C4×D4 | D284 | 1 | 3 | yes | — |
| 221 | C4×D4 | C22⋊D71 | 1 | 3 | yes | — |
| 222 | #117 | — | 71 | 3 | no | — |
| 223 | #117 | — | 71 | 3 | no | — |
| 224 | #117 | C4×D71 | 1 | 3 | yes | — |
| 225 | C2×C4⋊C4 | — | 71 | 3 | no | — |
| 226 | C2×C4⋊C4 | — | 71 | 3 | no | — |
| 227 | C2×C4⋊C4 | — | 142 | 3 | no | — |
| 228 | C2×C4⋊C4 | — | 142 | 3 | no | — |
| 229 | C2×C42 | C22×D71 | 1 | 3 | yes | — |
| 230 | C4⋊C8 | — | 2 | 3 | no | — |
| 231 | C4⋊C8 | — | 142 | 3 | no | — |
| 232 | C22⋊C8 | — | 2 | 3 | no | — |
| 233 | C22⋊C8 | — | 142 | 3 | no | — |
| 234 | #97 | — | 71 | 1 | no | — |
| 235 | #97 | — | 71 | 3 | no | — |
| 236 | #97 | C22×D71 | 1 | 1 | yes | characteristic |
| 237 | #97 | C22×D71 | 1 | 3 | yes | — |
| 238 | C4×C8 | — | 2 | 3 | no | — |
| 239 | C4×C8 | — | 142 | 3 | no | — |
| 240 | #41 | — | 2 | 4 | no | — |
| 241 | C2×Q8 | — | 142 | 3 | no | — |
| 242 | C2×Q8 | — | 142 | 3 | no | — |
| 243 | C2×Q8 | (C2×D71)⋊C4 | 1 | 1 | yes | characteristic |
| 244 | C2×D4 | (C2×D71)⋊C4 | 1 | 3 | yes | — |
| 245 | C22×C4 | — | 142 | 3 | no | — |
| 246 | C22×C4 | — | 142 | 3 | no | — |
| 247 | C22×C4 | C2×C4×D71 | 1 | 3 | yes | — |
| 248 | C22×C4 | C2×C22⋊D71 | 1 | 3 | yes | — |
| 249 | C22×C4 | C2×D284 | 1 | 3 | yes | — |
| 250 | M16 | — | 4 | 6 | no | — |
| 251 | M16 | — | 284 | 6 | no | — |
| 252 | C2×C8 | — | 2 | 1 | no | — |
| 253 | C2×C8 | — | 2 | 3 | no | — |
| 254 | C2×C8 | — | 142 | 3 | no | — |
| 255 | C2×C8 | — | 142 | 1 | no | — |
| 256 | C4⋊C4 | — | 71 | 3 | no | — |
| 257 | C4⋊C4 | — | 71 | 3 | no | — |
| 258 | C4⋊C4 | — | 71 | 1 | no | — |
| 259 | C4⋊C4 | — | 71 | 1 | no | — |
| 260 | C4⋊C4 | — | 142 | 3 | no | — |
| 261 | C4⋊C4 | — | 142 | 1 | no | — |
| 262 | C4⋊C4 | — | 142 | 3 | no | — |
| 263 | C4⋊C4 | — | 142 | 1 | no | — |
| 264 | C4⋊C4 | — | 142 | 6 | no | — |
| 265 | C4⋊C4 | — | 142 | 6 | no | — |
| 266 | C4⋊C4 | — | 142 | 3 | no | — |
| 267 | C4⋊C4 | — | 142 | 3 | no | — |
| 268 | C4⋊C4 | — | 142 | 3 | no | — |
| 269 | C4⋊C4 | — | 142 | 3 | no | — |
| 270 | C4⋊C4 | (C2×D71)⋊C4 | 1 | 3 | yes | — |
| 271 | C4⋊C4 | (C2×D71)⋊C4 | 1 | 3 | yes | — |
| 272 | C22⋊C4 | — | 142 | 3 | no | — |
| 273 | C22⋊C4 | — | 142 | 3 | no | — |
| 274 | C22⋊C4 | (C2×D71)⋊C4 | 1 | 6 | yes | — |
| 275 | C42 | — | 142 | 3 | no | — |
| 276 | C42 | — | 142 | 3 | no | — |
| 277 | C42 | C2×C4×D71 | 1 | 3 | yes | — |
| 278 | C42 | C2×C22⋊D71 | 1 | 3 | yes | — |
| 279 | C42 | C2×D284 | 1 | 3 | yes | — |
| 280 | C42 | C23×D71 | 1 | 1 | yes | characteristic |
| 281 | C23 | C2×(C2×D71)⋊C4 | 1 | 3 | yes | — |
| 282 | Q8 | — | 568 | 3 | no | — |
| 283 | Q8 | — | 568 | 3 | no | — |
| 284 | Q8 | #14556046 | 1 | 4 | yes | — |
| 285 | D4 | — | 2 | 6 | no | — |
| 286 | C2×C4 | — | 2 | 6 | no | — |
| 287 | C2×C4 | — | 4 | 3 | no | — |
| 288 | C2×C4 | — | 4 | 3 | no | — |
| 289 | C2×C4 | — | 568 | 3 | no | — |
| 290 | C2×C4 | — | 568 | 3 | no | — |
| 291 | C2×C4 | — | 142 | 1 | no | — |
| 292 | C2×C4 | — | 142 | 1 | no | — |
| 293 | C2×C4 | — | 142 | 3 | no | — |
| 294 | C2×C4 | — | 142 | 3 | no | — |
| 295 | C2×C4 | C2×(C2×D71)⋊C4 | 1 | 3 | yes | — |
| 296 | C2×C4 | C2×(C2×D71)⋊C4 | 1 | 3 | yes | — |
| 297 | C2×C4 | C2×(C2×D71)⋊C4 | 1 | 3 | yes | — |
| 298 | C2×C4 | C22×C4×D71 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | C22×C22⋊D71 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | C22×D284 | 1 | 1 | yes | characteristic |
| 301 | C8 | — | 2 | 2 | no | — |
| 302 | C8 | — | 4 | 3 | no | — |
| 303 | C8 | — | 284 | 3 | no | — |
| 304 | C8 | — | 142 | 2 | no | — |
| 305 | C22 | — | 2 | 3 | no | — |
| 306 | C22 | — | 2 | 3 | no | — |
| 307 | C22 | — | 2 | 3 | no | — |
| 308 | C22 | C22×(C2×D71)⋊C4 | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 2 | 1 | no | — |
| 310 | C4 | — | 2 | 3 | no | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 2 | 3 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 284 | 3 | no | — |
| 315 | C4 | — | 284 | 1 | no | — |
| 316 | C4 | — | 284 | 3 | no | — |
| 317 | C4 | — | 284 | 1 | no | — |
| 318 | C4 | C2×#14556046 | 1 | 6 | yes | — |
| 319 | C2 | — | 4 | 3 | no | — |
| 320 | C2 | C22×#14556046 | 1 | 1 | yes | characteristic |
| 321 | C2 | #22596885 | 1 | 1 | yes | characteristic |
| 322 | C2 | #22596888 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta