← #31914186
Subgroups — #31914186
Order 18176.
408 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#17986121 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#17985822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#17985809 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #22589079 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #22586448 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22586419 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C71×#2139 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#14556135 | — | 2 | 1 | no | — |
| 9 | C2×#14556068 | C22 | 1 | 1 | yes | characteristic |
| 10 | #17986121 | — | 2 | 2 | no | — |
| 11 | #17986121 | — | 2 | 2 | no | — |
| 12 | C2×#14556064 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#14556060 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#14556055 | — | 2 | 1 | no | — |
| 15 | C2×#14556038 | C22 | 1 | 1 | yes | characteristic |
| 16 | #17985898 | — | 2 | 1 | no | — |
| 17 | #17985822 | — | 2 | 2 | no | — |
| 18 | #17985809 | — | 2 | 2 | no | — |
| 19 | #17985761 | — | 2 | 1 | no | — |
| 20 | C2×#135×C71 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×M32×C71 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#107×C71 | C22 | 1 | 1 | yes | characteristic |
| 23 | #358×C71 | C4 | 1 | 2 | yes | complemented |
| 24 | #357×C71 | C4 | 1 | 2 | yes | complemented |
| 25 | #14556135 | — | 2 | 4 | no | — |
| 26 | #14556135 | — | 4 | 2 | no | — |
| 27 | C2×#11765849 | — | 2 | 1 | no | — |
| 28 | C2×#11765849 | — | 4 | 1 | no | — |
| 29 | C22×C71⋊C8 | — | 4 | 1 | no | — |
| 30 | C22×C71⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C4⋊Dic71 | — | 2 | 1 | no | — |
| 32 | C2×C4⋊Dic71 | — | 4 | 1 | no | — |
| 33 | C2×C4×Dic71 | D4 | 1 | 1 | yes | characteristic |
| 34 | #14556068 | — | 2 | 1 | no | — |
| 35 | #14556068 | — | 2 | 1 | no | — |
| 36 | C2×C71⋊C16 | — | 2 | 1 | no | — |
| 37 | #14556064 | — | 2 | 1 | no | — |
| 38 | #14556064 | D4 | 1 | 2 | yes | — |
| 39 | #14556060 | — | 2 | 1 | no | — |
| 40 | #14556060 | — | 2 | 1 | no | — |
| 41 | #14556055 | — | 2 | 4 | no | — |
| 42 | #14556038 | — | 2 | 1 | no | — |
| 43 | #14556038 | — | 2 | 1 | no | — |
| 44 | C71⋊#95 | — | 4 | 1 | no | — |
| 45 | C2×#41×C71 | — | 2 | 1 | no | — |
| 46 | #135×C71 | — | 2 | 1 | no | — |
| 47 | #135×C71 | — | 2 | 1 | no | — |
| 48 | #135×C71 | D4 | 1 | 2 | yes | — |
| 49 | #135×C71 | Q8 | 1 | 2 | yes | — |
| 50 | C2×Q16×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C2×SD16×C71 | — | 2 | 1 | no | — |
| 52 | C2×D8×C71 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 54 | C2×C4⋊C4×C71 | — | 2 | 1 | no | — |
| 55 | M32×C71 | — | 2 | 1 | no | — |
| 56 | M32×C71 | — | 2 | 1 | no | — |
| 57 | C2×C16×C71 | C2×C4 | 1 | 2 | yes | — |
| 58 | #107×C71 | — | 2 | 1 | no | — |
| 59 | #107×C71 | C2×C4 | 1 | 2 | yes | — |
| 60 | C22×Dic71 | — | 2 | 1 | no | — |
| 61 | C22×Dic71 | — | 4 | 1 | no | — |
| 62 | #11765849 | — | 2 | 2 | no | — |
| 63 | #11765849 | — | 4 | 4 | no | — |
| 64 | #11765849 | — | 4 | 1 | no | — |
| 65 | C2×C71⋊C8 | — | 2 | 1 | no | — |
| 66 | C2×C71⋊C8 | — | 2 | 1 | no | — |
| 67 | C2×C71⋊C8 | — | 4 | 4 | no | — |
| 68 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 69 | C2×C71⋊C8 | SD16 | 1 | 2 | yes | — |
| 70 | C4⋊Dic71 | — | 4 | 1 | no | — |
| 71 | C4⋊Dic71 | — | 4 | 1 | no | — |
| 72 | C4⋊Dic71 | — | 8 | 2 | no | — |
| 73 | C4×Dic71 | — | 2 | 1 | no | — |
| 74 | C4×Dic71 | — | 2 | 1 | no | — |
| 75 | C71⋊C16 | — | 4 | 1 | no | — |
| 76 | #41×C71 | — | 2 | 2 | no | — |
| 77 | #41×C71 | — | 2 | 2 | no | — |
| 78 | #41×C71 | — | 4 | 1 | no | — |
| 79 | #41×C71 | — | 4 | 1 | no | — |
| 80 | C2×Q8×C71 | — | 2 | 1 | no | — |
| 81 | C2×D4×C71 | — | 2 | 1 | no | — |
| 82 | C2×D4×C71 | — | 4 | 1 | no | — |
| 83 | C22×C4×C71 | — | 4 | 1 | no | — |
| 84 | C22×C4×C71 | — | 4 | 1 | no | — |
| 85 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | Q16×C71 | — | 2 | 1 | no | — |
| 87 | Q16×C71 | C4⋊C4 | 1 | 2 | yes | complemented |
| 88 | SD16×C71 | — | 2 | 2 | no | — |
| 89 | SD16×C71 | — | 2 | 2 | no | — |
| 90 | D8×C71 | — | 2 | 1 | no | — |
| 91 | D8×C71 | C4⋊C4 | 1 | 2 | yes | complemented |
| 92 | C2×C8×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C8×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 96 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 97 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 98 | C4⋊C4×C71 | — | 2 | 2 | no | — |
| 99 | C4⋊C4×C71 | — | 4 | 1 | no | — |
| 100 | C16×C71 | — | 2 | 2 | no | — |
| 101 | C23×C71 | — | 4 | 1 | no | — |
| 102 | C23×C71 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 103 | Q8×C71 | — | 2 | 2 | no | — |
| 104 | Q8×C71 | — | 4 | 1 | no | — |
| 105 | D4×C71 | — | 2 | 2 | no | — |
| 106 | D4×C71 | — | 4 | 2 | no | — |
| 107 | D4×C71 | — | 4 | 2 | no | — |
| 108 | D4×C71 | — | 4 | 1 | no | — |
| 109 | C2×C4×C71 | — | 4 | 1 | no | — |
| 110 | C2×C4×C71 | — | 4 | 2 | no | — |
| 111 | C2×C4×C71 | — | 4 | 2 | no | — |
| 112 | C2×C4×C71 | — | 4 | 2 | no | — |
| 113 | C2×C4×C71 | — | 8 | 1 | no | — |
| 114 | C2×C4×C71 | — | 8 | 1 | no | — |
| 115 | C2×C4×C71 | C4×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C71 | #120 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C71 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C71 | C2×SD16 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C71 | #135 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C71 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 121 | C2×Dic71 | — | 4 | 1 | no | — |
| 122 | C2×Dic71 | — | 4 | 1 | no | — |
| 123 | C2×Dic71 | — | 4 | 1 | no | — |
| 124 | C2×Dic71 | — | 8 | 1 | no | — |
| 125 | C2×Dic71 | — | 8 | 1 | no | — |
| 126 | C2×Dic71 | — | 8 | 1 | no | — |
| 127 | C8×C71 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 128 | C8×C71 | C4×D4 | 1 | 1 | yes | characteristic |
| 129 | C8×C71 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 130 | C8×C71 | #123 | 1 | 1 | yes | characteristic |
| 131 | C71⋊C8 | — | 2 | 2 | no | — |
| 132 | C71⋊C8 | — | 4 | 2 | no | — |
| 133 | C22×C71 | — | 2 | 1 | no | — |
| 134 | C22×C71 | — | 2 | 1 | no | — |
| 135 | C22×C71 | — | 4 | 2 | no | — |
| 136 | C22×C71 | — | 8 | 1 | no | — |
| 137 | C22×C71 | — | 8 | 1 | no | — |
| 138 | C22×C71 | #442 | 1 | 1 | yes | characteristic |
| 139 | C22×C71 | #449 | 1 | 1 | yes | characteristic |
| 140 | C22×C71 | #476 | 1 | 1 | yes | characteristic |
| 141 | C4×C71 | — | 4 | 2 | no | — |
| 142 | C4×C71 | — | 8 | 2 | no | — |
| 143 | C4×C71 | #385 | 1 | 1 | yes | characteristic |
| 144 | C4×C71 | C4×SD16 | 1 | 1 | yes | characteristic |
| 145 | C4×C71 | #450 | 1 | 1 | yes | characteristic |
| 146 | C4×C71 | #478 | 1 | 1 | yes | characteristic |
| 147 | Dic71 | — | 8 | 1 | no | — |
| 148 | Dic71 | — | 16 | 1 | no | — |
| 149 | #12955 | — | 71 | 1 | no | maximal, Sylow |
| 150 | C2×C71 | — | 2 | 1 | no | — |
| 151 | C2×C71 | — | 2 | 1 | no | — |
| 152 | C2×C71 | — | 8 | 2 | no | — |
| 153 | C2×C71 | #1864 | 1 | 1 | yes | characteristic |
| 154 | C2×C71 | #2180 | 1 | 1 | yes | characteristic |
| 155 | C2×C71 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 156 | C2×#444 | — | 71 | 1 | no | — |
| 157 | C2×#365 | — | 71 | 1 | no | — |
| 158 | C2×#360 | — | 71 | 1 | no | — |
| 159 | #2139 | D71 | 1 | 1 | yes | characteristic |
| 160 | #1943 | — | 71 | 1 | no | — |
| 161 | #1387 | — | 71 | 1 | no | — |
| 162 | #1375 | — | 71 | 1 | no | — |
| 163 | C71 | #12955 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 164 | C2×#135 | C2×D71 | 1 | 1 | yes | characteristic |
| 165 | C2×#131 | — | 142 | 1 | no | — |
| 166 | C2×M32 | — | 71 | 1 | no | — |
| 167 | C2×M32 | C2×D71 | 1 | 1 | yes | characteristic |
| 168 | #444 | — | 142 | 2 | no | — |
| 169 | #444 | — | 142 | 2 | no | — |
| 170 | C2×#108 | — | 71 | 1 | no | — |
| 171 | C2×#107 | C2×D71 | 1 | 1 | yes | characteristic |
| 172 | C2×#106 | — | 71 | 1 | no | — |
| 173 | C2×#104 | — | 142 | 1 | no | — |
| 174 | C2×#97 | — | 71 | 1 | no | — |
| 175 | #389 | — | 142 | 1 | no | — |
| 176 | #365 | — | 142 | 2 | no | — |
| 177 | #360 | — | 142 | 2 | no | — |
| 178 | #358 | Dic71 | 1 | 2 | yes | complemented |
| 179 | #357 | Dic71 | 1 | 2 | yes | complemented |
| 180 | #340 | — | 142 | 1 | no | — |
| 181 | C2×#41 | — | 2 | 1 | no | — |
| 182 | #135 | — | 2 | 1 | no | — |
| 183 | #135 | — | 2 | 1 | no | — |
| 184 | #135 | Dic142 | 1 | 2 | yes | — |
| 185 | #135 | D284 | 1 | 2 | yes | — |
| 186 | C2×Q16 | C2×Dic71 | 1 | 1 | yes | characteristic |
| 187 | C2×SD16 | — | 2 | 1 | no | — |
| 188 | C2×D8 | C2×Dic71 | 1 | 1 | yes | characteristic |
| 189 | #131 | — | 284 | 2 | no | — |
| 190 | #131 | — | 142 | 4 | no | — |
| 191 | C2×M16 | — | 284 | 1 | no | — |
| 192 | C2×M16 | — | 142 | 1 | no | — |
| 193 | C22×C8 | — | 284 | 1 | no | — |
| 194 | C22×C8 | — | 71 | 1 | no | — |
| 195 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 196 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 197 | C2×C4⋊C4 | — | 284 | 1 | no | — |
| 198 | C2×C4⋊C4 | — | 142 | 1 | no | — |
| 199 | C2×C42 | — | 71 | 1 | no | — |
| 200 | M32 | — | 2 | 1 | no | — |
| 201 | M32 | — | 2 | 1 | no | — |
| 202 | M32 | — | 142 | 1 | no | — |
| 203 | M32 | — | 142 | 1 | no | — |
| 204 | C2×C16 | — | 142 | 1 | no | — |
| 205 | C2×C16 | C2×Dic71 | 1 | 2 | yes | — |
| 206 | #108 | — | 71 | 2 | no | — |
| 207 | #108 | — | 142 | 1 | no | — |
| 208 | #107 | — | 2 | 1 | no | — |
| 209 | #107 | C2×Dic71 | 1 | 2 | yes | — |
| 210 | #106 | — | 142 | 1 | no | — |
| 211 | #106 | — | 142 | 1 | no | — |
| 212 | #104 | — | 142 | 4 | no | — |
| 213 | #97 | — | 142 | 1 | no | — |
| 214 | #97 | — | 142 | 1 | no | — |
| 215 | #95 | — | 284 | 1 | no | — |
| 216 | #41 | — | 2 | 2 | no | — |
| 217 | #41 | — | 2 | 2 | no | — |
| 218 | #41 | — | 4 | 1 | no | — |
| 219 | #41 | — | 4 | 1 | no | — |
| 220 | C2×Q8 | — | 2 | 1 | no | — |
| 221 | C2×D4 | — | 2 | 1 | no | — |
| 222 | C2×D4 | — | 4 | 1 | no | — |
| 223 | C22×C4 | — | 4 | 1 | no | — |
| 224 | C22×C4 | — | 4 | 1 | no | — |
| 225 | C22×C4 | — | 284 | 1 | no | — |
| 226 | C22×C4 | — | 142 | 1 | no | — |
| 227 | C22×C4 | D4×D71 | 1 | 1 | yes | characteristic |
| 228 | Q16 | — | 2 | 1 | no | — |
| 229 | Q16 | C4⋊Dic71 | 1 | 2 | yes | complemented |
| 230 | SD16 | — | 2 | 2 | no | — |
| 231 | SD16 | — | 2 | 2 | no | — |
| 232 | D8 | — | 2 | 1 | no | — |
| 233 | D8 | C4⋊Dic71 | 1 | 2 | yes | complemented |
| 234 | M16 | — | 284 | 1 | no | — |
| 235 | M16 | — | 284 | 4 | no | — |
| 236 | M16 | — | 142 | 2 | no | — |
| 237 | C2×C8 | — | 284 | 1 | no | — |
| 238 | C2×C8 | — | 284 | 4 | no | — |
| 239 | C2×C8 | — | 71 | 2 | no | — |
| 240 | C2×C8 | — | 142 | 1 | no | — |
| 241 | C2×C8 | — | 142 | 1 | no | — |
| 242 | C2×C8 | C22×Dic71 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | D4×D71 | 1 | 1 | yes | characteristic |
| 244 | C2×C8 | C2×D284 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | C2×Dic142 | 1 | 1 | yes | characteristic |
| 246 | C2×C8 | #11765864 | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | #11765864 | 1 | 1 | yes | characteristic |
| 248 | C4⋊C4 | — | 2 | 2 | no | — |
| 249 | C4⋊C4 | — | 4 | 1 | no | — |
| 250 | C4⋊C4 | — | 284 | 1 | no | — |
| 251 | C4⋊C4 | — | 284 | 1 | no | — |
| 252 | C4⋊C4 | — | 568 | 2 | no | — |
| 253 | C42 | — | 142 | 1 | no | — |
| 254 | C42 | — | 142 | 1 | no | — |
| 255 | C16 | — | 2 | 2 | no | — |
| 256 | C16 | — | 284 | 1 | no | — |
| 257 | C23 | — | 4 | 1 | no | — |
| 258 | C23 | #14556152 | 1 | 1 | yes | characteristic |
| 259 | Q8 | — | 2 | 2 | no | — |
| 260 | Q8 | — | 4 | 1 | no | — |
| 261 | D4 | — | 2 | 2 | no | — |
| 262 | D4 | — | 4 | 2 | no | — |
| 263 | D4 | — | 4 | 2 | no | — |
| 264 | D4 | — | 4 | 1 | no | — |
| 265 | C2×C4 | — | 4 | 1 | no | — |
| 266 | C2×C4 | — | 4 | 2 | no | — |
| 267 | C2×C4 | — | 4 | 2 | no | — |
| 268 | C2×C4 | — | 4 | 2 | no | — |
| 269 | C2×C4 | — | 8 | 1 | no | — |
| 270 | C2×C4 | — | 8 | 1 | no | — |
| 271 | C2×C4 | — | 284 | 1 | no | — |
| 272 | C2×C4 | — | 284 | 1 | no | — |
| 273 | C2×C4 | — | 284 | 1 | no | — |
| 274 | C2×C4 | — | 568 | 1 | no | — |
| 275 | C2×C4 | — | 568 | 1 | no | — |
| 276 | C2×C4 | — | 568 | 1 | no | — |
| 277 | C2×C4 | D4×Dic71 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | #14556098 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | #14556105 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | SD16×D71 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #14556149 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | #14556152 | 1 | 1 | yes | characteristic |
| 283 | C8 | — | 284 | 2 | no | — |
| 284 | C8 | — | 142 | 2 | no | — |
| 285 | C8 | C2×C4⋊Dic71 | 1 | 1 | yes | characteristic |
| 286 | C8 | D4×Dic71 | 1 | 1 | yes | characteristic |
| 287 | C8 | #14556105 | 1 | 1 | yes | characteristic |
| 288 | C8 | #14556112 | 1 | 1 | yes | characteristic |
| 289 | C22 | — | 2 | 1 | no | — |
| 290 | C22 | — | 2 | 1 | no | — |
| 291 | C22 | — | 4 | 2 | no | — |
| 292 | C22 | — | 8 | 1 | no | — |
| 293 | C22 | — | 8 | 1 | no | — |
| 294 | C22 | #17986111 | 1 | 1 | yes | characteristic |
| 295 | C22 | #17986145 | 1 | 1 | yes | characteristic |
| 296 | C22 | #17986287 | 1 | 1 | yes | characteristic |
| 297 | C4 | — | 4 | 2 | no | — |
| 298 | C4 | — | 8 | 2 | no | — |
| 299 | C4 | — | 568 | 1 | no | — |
| 300 | C4 | — | 1136 | 1 | no | — |
| 301 | C4 | #17985874 | 1 | 1 | yes | characteristic |
| 302 | C4 | SD16×Dic71 | 1 | 1 | yes | characteristic |
| 303 | C4 | #17986151 | 1 | 1 | yes | characteristic |
| 304 | C4 | #17986301 | 1 | 1 | yes | characteristic |
| 305 | C2 | — | 2 | 1 | no | — |
| 306 | C2 | — | 2 | 1 | no | — |
| 307 | C2 | — | 8 | 2 | no | — |
| 308 | C2 | #22588603 | 1 | 1 | yes | characteristic |
| 309 | C2 | #22590232 | 1 | 1 | yes | characteristic |
| 310 | C2 | #22590239 | 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