← #31914039
Subgroups — #31914039
Order 18176.
394 conjugacy classes of subgroups.
The subgroup lattice has 306 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22596935 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22590113 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#17985822 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22588394 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22586446 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #22586442 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C71×#2117 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #17986482 | — | 2 | 1 | no | — |
| 9 | C2×#14556064 | C22 | 1 | 1 | yes | characteristic |
| 10 | #17986041 | — | 2 | 2 | no | — |
| 11 | #17986041 | — | 2 | 1 | no | — |
| 12 | #17986037 | — | 2 | 1 | no | — |
| 13 | #17986037 | C22 | 1 | 1 | yes | characteristic |
| 14 | #17986037 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#14556060 | — | 2 | 1 | no | — |
| 16 | C2×#14556060 | C22 | 1 | 1 | yes | characteristic |
| 17 | #17986016 | — | 2 | 1 | no | — |
| 18 | #17985822 | — | 2 | 2 | no | — |
| 19 | #17985822 | — | 2 | 1 | no | — |
| 20 | #17985822 | — | 2 | 1 | no | — |
| 21 | #17985771 | — | 2 | 1 | no | — |
| 22 | C2×#131×C71 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×M32×C71 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×M32×C71 | C22 | 1 | 1 | yes | characteristic |
| 25 | #350×C71 | C4 | 1 | 4 | yes | complemented |
| 26 | C2×#11765849 | — | 2 | 1 | no | — |
| 27 | Q8×Dic71 | — | 4 | 1 | no | — |
| 28 | D4×Dic71 | — | 4 | 2 | no | — |
| 29 | D4×Dic71 | — | 4 | 1 | no | — |
| 30 | #14556087 | — | 2 | 2 | no | — |
| 31 | #14556087 | — | 2 | 1 | no | — |
| 32 | #14556087 | — | 2 | 1 | no | — |
| 33 | C2×C4⋊Dic71 | — | 2 | 2 | no | — |
| 34 | C2×C4⋊Dic71 | — | 2 | 1 | no | — |
| 35 | #14556064 | — | 2 | 1 | no | — |
| 36 | #14556064 | — | 2 | 1 | no | — |
| 37 | #14556062 | — | 2 | 2 | no | — |
| 38 | #14556062 | — | 2 | 1 | no | — |
| 39 | #14556062 | — | 2 | 1 | no | — |
| 40 | #14556060 | — | 2 | 2 | no | — |
| 41 | #14556060 | — | 2 | 1 | no | — |
| 42 | #14556060 | — | 2 | 1 | no | — |
| 43 | #14556060 | D4 | 1 | 1 | yes | characteristic |
| 44 | #14556060 | D4 | 1 | 2 | yes | — |
| 45 | #14556060 | D4 | 1 | 1 | yes | characteristic |
| 46 | #14556055 | — | 4 | 2 | no | — |
| 47 | C2×#41×C71 | D4 | 1 | 1 | yes | characteristic |
| 48 | #131×C71 | — | 2 | 1 | no | — |
| 49 | #131×C71 | — | 2 | 1 | no | — |
| 50 | #131×C71 | C2×C4 | 1 | 2 | yes | — |
| 51 | #131×C71 | Q8 | 1 | 2 | yes | — |
| 52 | C2×M16×C71 | — | 2 | 1 | no | — |
| 53 | C2×M16×C71 | D4 | 1 | 1 | yes | characteristic |
| 54 | C22×C8×C71 | — | 2 | 1 | no | — |
| 55 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 56 | M32×C71 | — | 2 | 1 | no | — |
| 57 | M32×C71 | — | 2 | 1 | no | — |
| 58 | M32×C71 | — | 2 | 1 | no | — |
| 59 | M32×C71 | C2×C4 | 1 | 2 | yes | complemented |
| 60 | C2×C16×C71 | — | 2 | 1 | no | — |
| 61 | C2×C16×C71 | C2×C4 | 1 | 2 | yes | complemented |
| 62 | C22×Dic71 | — | 4 | 2 | no | — |
| 63 | C22×Dic71 | — | 4 | 1 | no | — |
| 64 | #11765849 | — | 4 | 1 | no | — |
| 65 | #11765849 | — | 4 | 1 | no | — |
| 66 | C2×C71⋊C8 | — | 4 | 1 | no | — |
| 67 | C4⋊Dic71 | — | 2 | 1 | no | — |
| 68 | C4⋊Dic71 | — | 2 | 2 | no | — |
| 69 | C4⋊Dic71 | — | 2 | 1 | no | — |
| 70 | C4⋊Dic71 | — | 4 | 2 | no | — |
| 71 | C4⋊Dic71 | — | 4 | 1 | no | — |
| 72 | C4⋊Dic71 | — | 4 | 1 | no | — |
| 73 | C22⋊Dic71 | — | 4 | 2 | no | — |
| 74 | C22⋊Dic71 | — | 4 | 1 | no | — |
| 75 | C22⋊Dic71 | — | 4 | 1 | no | — |
| 76 | C4×Dic71 | — | 4 | 2 | no | — |
| 77 | C4×Dic71 | — | 4 | 1 | no | — |
| 78 | C4×Dic71 | — | 4 | 1 | no | — |
| 79 | #41×C71 | — | 2 | 1 | no | — |
| 80 | #41×C71 | — | 2 | 1 | no | — |
| 81 | #41×C71 | C4⋊C4 | 1 | 2 | yes | — |
| 82 | #41×C71 | SD16 | 1 | 2 | yes | — |
| 83 | C2×Q8×C71 | SD16 | 1 | 1 | yes | characteristic |
| 84 | C2×D4×C71 | — | 2 | 1 | no | — |
| 85 | C2×D4×C71 | SD16 | 1 | 1 | yes | characteristic |
| 86 | C22×C4×C71 | — | 2 | 1 | no | — |
| 87 | C22×C4×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 88 | M16×C71 | — | 2 | 1 | no | — |
| 89 | M16×C71 | — | 2 | 2 | no | — |
| 90 | M16×C71 | — | 2 | 2 | no | — |
| 91 | M16×C71 | C4⋊C4 | 1 | 2 | yes | — |
| 92 | C2×C8×C71 | — | 2 | 1 | no | — |
| 93 | C2×C8×C71 | — | 2 | 2 | no | — |
| 94 | C2×C8×C71 | — | 2 | 2 | no | — |
| 95 | C2×C8×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C8×C71 | C2×Q8 | 1 | 1 | yes | characteristic |
| 99 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 100 | C2×C8×C71 | #41 | 1 | 1 | yes | characteristic |
| 101 | C16×C71 | — | 2 | 2 | no | — |
| 102 | C16×C71 | — | 2 | 2 | no | — |
| 103 | C23×C71 | — | 2 | 1 | no | — |
| 104 | C23×C71 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 105 | Q8×C71 | — | 2 | 1 | no | — |
| 106 | Q8×C71 | #106 | 1 | 2 | yes | — |
| 107 | D4×C71 | — | 2 | 2 | no | — |
| 108 | D4×C71 | — | 2 | 2 | no | — |
| 109 | D4×C71 | — | 2 | 1 | no | — |
| 110 | D4×C71 | #106 | 1 | 2 | yes | — |
| 111 | C2×C4×C71 | — | 2 | 1 | no | — |
| 112 | C2×C4×C71 | — | 2 | 2 | no | — |
| 113 | C2×C4×C71 | — | 2 | 2 | no | — |
| 114 | C2×C4×C71 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C71 | #120 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C71 | #123 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C71 | C2×SD16 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C71 | C2×SD16 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C71 | #137 | 1 | 1 | yes | characteristic |
| 120 | C2×Dic71 | — | 4 | 1 | no | — |
| 121 | C2×Dic71 | — | 4 | 2 | no | — |
| 122 | C2×Dic71 | — | 4 | 1 | no | — |
| 123 | C2×Dic71 | — | 8 | 1 | no | — |
| 124 | C2×Dic71 | — | 16 | 1 | no | — |
| 125 | C2×Dic71 | — | 16 | 2 | no | — |
| 126 | C8×C71 | — | 2 | 2 | no | — |
| 127 | C8×C71 | C4×D4 | 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 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 131 | C71⋊C8 | — | 8 | 1 | no | — |
| 132 | C22×C71 | — | 2 | 2 | no | — |
| 133 | C22×C71 | — | 4 | 1 | no | — |
| 134 | C22×C71 | — | 4 | 2 | no | — |
| 135 | C22×C71 | #428 | 1 | 1 | yes | characteristic |
| 136 | C22×C71 | #450 | 1 | 1 | yes | characteristic |
| 137 | C22×C71 | #481 | 1 | 1 | yes | characteristic |
| 138 | C4×C71 | — | 2 | 2 | no | — |
| 139 | C4×C71 | #385 | 1 | 1 | yes | characteristic |
| 140 | C4×C71 | C2×#106 | 1 | 1 | yes | characteristic |
| 141 | C4×C71 | #448 | 1 | 1 | yes | characteristic |
| 142 | C4×C71 | #483 | 1 | 1 | yes | characteristic |
| 143 | Dic71 | — | 8 | 1 | no | — |
| 144 | Dic71 | — | 8 | 2 | no | — |
| 145 | Dic71 | — | 8 | 1 | no | — |
| 146 | Dic71 | — | 16 | 1 | no | — |
| 147 | #12934 | — | 71 | 1 | no | maximal, Sylow |
| 148 | C2×C71 | — | 2 | 1 | no | — |
| 149 | C2×C71 | — | 2 | 1 | no | — |
| 150 | C2×C71 | — | 2 | 1 | no | — |
| 151 | C2×C71 | — | 4 | 2 | no | — |
| 152 | C2×C71 | #1853 | 1 | 1 | yes | characteristic |
| 153 | #2912 | — | 71 | 1 | no | — |
| 154 | #2155 | — | 71 | 1 | no | — |
| 155 | C2×#365 | — | 71 | 1 | no | — |
| 156 | #2117 | D71 | 1 | 1 | yes | characteristic |
| 157 | #1814 | — | 71 | 1 | no | — |
| 158 | #1386 | — | 71 | 1 | no | — |
| 159 | #1384 | — | 71 | 1 | no | — |
| 160 | C71 | #12934 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 161 | C2×#131 | C2×D71 | 1 | 1 | yes | characteristic |
| 162 | #520 | — | 142 | 1 | no | — |
| 163 | C2×M32 | C2×D71 | 1 | 1 | yes | characteristic |
| 164 | C2×M32 | C2×D71 | 1 | 1 | yes | characteristic |
| 165 | C2×#108 | — | 71 | 1 | no | — |
| 166 | #428 | — | 142 | 2 | no | — |
| 167 | #428 | — | 142 | 1 | no | — |
| 168 | #427 | — | 71 | 1 | no | — |
| 169 | #427 | — | 71 | 1 | no | — |
| 170 | #427 | — | 142 | 1 | no | — |
| 171 | C2×#106 | — | 71 | 1 | no | — |
| 172 | C2×#106 | — | 142 | 1 | no | — |
| 173 | #421 | — | 142 | 1 | no | — |
| 174 | #365 | — | 142 | 2 | no | — |
| 175 | #365 | — | 142 | 1 | no | — |
| 176 | #365 | — | 142 | 1 | no | — |
| 177 | #350 | Dic71 | 1 | 4 | yes | complemented |
| 178 | #344 | — | 142 | 1 | no | — |
| 179 | C2×#41 | D284 | 1 | 1 | yes | characteristic |
| 180 | #131 | — | 2 | 1 | no | — |
| 181 | #131 | — | 2 | 1 | no | — |
| 182 | #131 | Dic142 | 1 | 2 | yes | — |
| 183 | #131 | C2×Dic71 | 1 | 2 | yes | — |
| 184 | C2×M16 | — | 2 | 1 | no | — |
| 185 | C2×M16 | — | 142 | 1 | no | — |
| 186 | C2×M16 | D284 | 1 | 1 | yes | characteristic |
| 187 | C22×C8 | — | 2 | 1 | no | — |
| 188 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 189 | C4×Q8 | — | 284 | 1 | no | — |
| 190 | C4×D4 | — | 284 | 2 | no | — |
| 191 | C4×D4 | — | 284 | 1 | no | — |
| 192 | #117 | — | 142 | 1 | no | — |
| 193 | #117 | — | 142 | 2 | no | — |
| 194 | #117 | — | 142 | 1 | no | — |
| 195 | C2×C4⋊C4 | — | 142 | 2 | no | — |
| 196 | C2×C4⋊C4 | — | 142 | 1 | no | — |
| 197 | M32 | — | 2 | 1 | no | — |
| 198 | M32 | — | 2 | 1 | no | — |
| 199 | M32 | — | 2 | 1 | no | — |
| 200 | M32 | C2×Dic71 | 1 | 2 | yes | complemented |
| 201 | C2×C16 | — | 2 | 1 | no | — |
| 202 | C2×C16 | C2×Dic71 | 1 | 2 | yes | complemented |
| 203 | #108 | — | 142 | 1 | no | — |
| 204 | #108 | — | 142 | 1 | no | — |
| 205 | #107 | — | 142 | 1 | no | — |
| 206 | #107 | — | 142 | 2 | no | — |
| 207 | #107 | — | 142 | 1 | no | — |
| 208 | #106 | — | 71 | 2 | no | — |
| 209 | #106 | — | 71 | 1 | no | — |
| 210 | #106 | — | 71 | 1 | no | — |
| 211 | #106 | — | 142 | 1 | no | — |
| 212 | #106 | — | 142 | 2 | no | — |
| 213 | #106 | — | 142 | 1 | no | — |
| 214 | #104 | — | 284 | 2 | no | — |
| 215 | #41 | — | 2 | 1 | no | — |
| 216 | #41 | — | 2 | 1 | no | — |
| 217 | #41 | C4⋊Dic71 | 1 | 2 | yes | — |
| 218 | #41 | #11765854 | 1 | 2 | yes | — |
| 219 | C2×Q8 | #11765854 | 1 | 1 | yes | characteristic |
| 220 | C2×D4 | — | 2 | 1 | no | — |
| 221 | C2×D4 | #11765854 | 1 | 1 | yes | characteristic |
| 222 | C22×C4 | — | 2 | 1 | no | — |
| 223 | C22×C4 | — | 284 | 2 | no | — |
| 224 | C22×C4 | — | 284 | 1 | no | — |
| 225 | C22×C4 | C2×D284 | 1 | 1 | yes | characteristic |
| 226 | M16 | — | 2 | 1 | no | — |
| 227 | M16 | — | 2 | 2 | no | — |
| 228 | M16 | — | 2 | 2 | no | — |
| 229 | M16 | — | 284 | 1 | no | — |
| 230 | M16 | — | 284 | 1 | no | — |
| 231 | M16 | C4⋊Dic71 | 1 | 2 | yes | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 2 | no | — |
| 234 | C2×C8 | — | 2 | 2 | no | — |
| 235 | C2×C8 | — | 284 | 1 | no | — |
| 236 | C2×C8 | C22×Dic71 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | D4×D71 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | D4×D71 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | C2×Dic142 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | #11765864 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | #11765864 | 1 | 1 | yes | characteristic |
| 242 | C4⋊C4 | — | 284 | 2 | no | — |
| 243 | C4⋊C4 | — | 284 | 1 | no | — |
| 244 | C4⋊C4 | — | 284 | 1 | no | — |
| 245 | C4⋊C4 | — | 142 | 2 | no | — |
| 246 | C4⋊C4 | — | 142 | 1 | no | — |
| 247 | C4⋊C4 | — | 142 | 1 | no | — |
| 248 | C22⋊C4 | — | 284 | 2 | no | — |
| 249 | C22⋊C4 | — | 284 | 1 | no | — |
| 250 | C22⋊C4 | — | 284 | 1 | no | — |
| 251 | C42 | — | 284 | 2 | no | — |
| 252 | C42 | — | 284 | 1 | no | — |
| 253 | C42 | — | 284 | 1 | no | — |
| 254 | C16 | — | 2 | 2 | no | — |
| 255 | C16 | — | 2 | 2 | no | — |
| 256 | C23 | — | 2 | 1 | no | — |
| 257 | C23 | #14556156 | 1 | 1 | yes | characteristic |
| 258 | Q8 | — | 2 | 1 | no | — |
| 259 | Q8 | #14556060 | 1 | 2 | yes | — |
| 260 | D4 | — | 2 | 2 | no | — |
| 261 | D4 | — | 2 | 2 | no | — |
| 262 | D4 | — | 2 | 1 | no | — |
| 263 | D4 | #14556060 | 1 | 2 | yes | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 2 | 2 | no | — |
| 267 | C2×C4 | — | 284 | 1 | no | — |
| 268 | C2×C4 | — | 284 | 2 | no | — |
| 269 | C2×C4 | — | 284 | 1 | no | — |
| 270 | C2×C4 | — | 568 | 1 | no | — |
| 271 | C2×C4 | — | 1136 | 1 | no | — |
| 272 | C2×C4 | — | 1136 | 2 | no | — |
| 273 | C2×C4 | C2×C4⋊Dic71 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | #14556098 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | #14556112 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | C2×#11765854 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C2×#11765854 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | #14556161 | 1 | 1 | yes | characteristic |
| 279 | C8 | — | 2 | 2 | no | — |
| 280 | C8 | — | 568 | 1 | no | — |
| 281 | C8 | D4×Dic71 | 1 | 1 | yes | characteristic |
| 282 | C8 | D4×Dic71 | 1 | 1 | yes | characteristic |
| 283 | C8 | #14556105 | 1 | 1 | yes | characteristic |
| 284 | C8 | #14556105 | 1 | 1 | yes | characteristic |
| 285 | C22 | — | 2 | 2 | no | — |
| 286 | C22 | — | 4 | 1 | no | — |
| 287 | C22 | — | 4 | 2 | no | — |
| 288 | C22 | #17986041 | 1 | 1 | yes | characteristic |
| 289 | C22 | #17986152 | 1 | 1 | yes | characteristic |
| 290 | C22 | #17986318 | 1 | 1 | yes | characteristic |
| 291 | C4 | — | 2 | 2 | no | — |
| 292 | C4 | — | 568 | 2 | no | — |
| 293 | C4 | — | 568 | 1 | no | — |
| 294 | C4 | — | 568 | 1 | no | — |
| 295 | C4 | — | 1136 | 1 | no | — |
| 296 | C4 | #17985874 | 1 | 1 | yes | characteristic |
| 297 | C4 | C2×#14556060 | 1 | 1 | yes | characteristic |
| 298 | C4 | #17986140 | 1 | 1 | yes | characteristic |
| 299 | C4 | #17986330 | 1 | 1 | yes | characteristic |
| 300 | C2 | — | 2 | 1 | no | — |
| 301 | C2 | — | 2 | 1 | no | — |
| 302 | C2 | — | 2 | 1 | no | — |
| 303 | C2 | — | 4 | 2 | no | — |
| 304 | C2 | #22588535 | 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