← #15757271
Subgroups — #15757271
Order 3328.
376 conjugacy classes of subgroups.
The subgroup lattice has 338 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | M16×Dic26 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #13115611 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13113592 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #13113266 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#11718048 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #13112732 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C13×#1278 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×C4×Dic26 | C22 | 1 | 1 | yes | characteristic |
| 9 | C13×#431 | C22 | 1 | 1 | yes | characteristic |
| 10 | C4×C13×M16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×C8×C13 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#88062 | — | 2 | 1 | no | — |
| 13 | #11718779 | — | 2 | 1 | no | — |
| 14 | M16×Dic13 | — | 2 | 1 | no | — |
| 15 | #11718585 | C22 | 1 | 1 | yes | characteristic |
| 16 | #11718585 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×C4⋊#1055 | C22 | 1 | 1 | yes | characteristic |
| 18 | #11718362 | — | 2 | 2 | no | — |
| 19 | #11718340 | — | 2 | 1 | no | — |
| 20 | C8×Dic26 | — | 2 | 1 | no | — |
| 21 | #11718150 | — | 2 | 1 | no | — |
| 22 | #11718049 | — | 2 | 2 | no | — |
| 23 | #11718048 | — | 2 | 1 | no | — |
| 24 | #11718048 | — | 2 | 1 | no | — |
| 25 | #11718048 | C4 | 1 | 1 | yes | characteristic |
| 26 | #11718048 | C4 | 1 | 1 | yes | characteristic |
| 27 | #11718048 | C4 | 1 | 1 | yes | characteristic |
| 28 | #11718048 | C4 | 1 | 1 | yes | characteristic |
| 29 | C22×Dic26 | Q8 | 1 | 1 | yes | characteristic |
| 30 | C2×C13×M16 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C13×M16 | D4 | 1 | 1 | yes | characteristic |
| 32 | C22×C8×C13 | — | 2 | 1 | no | — |
| 33 | C2×C42×C13 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C4⋊Dic13 | Q8 | 1 | 1 | yes | characteristic |
| 35 | C2×#6101 | — | 2 | 1 | no | — |
| 36 | C2×C4×Dic13 | — | 2 | 1 | no | — |
| 37 | C2×#6099 | — | 2 | 1 | no | — |
| 38 | C2×#6099 | — | 2 | 1 | no | — |
| 39 | C22×#1055 | — | 2 | 1 | no | — |
| 40 | C4×Dic26 | — | 2 | 1 | no | — |
| 41 | C4×Dic26 | — | 2 | 1 | no | — |
| 42 | C4×Dic26 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C4×Dic26 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C4×Dic26 | D4 | 1 | 1 | yes | characteristic |
| 45 | C4×Dic26 | D4 | 1 | 1 | yes | characteristic |
| 46 | C13×C4⋊C8 | — | 2 | 1 | no | — |
| 47 | C13×C22⋊C8 | — | 2 | 1 | no | — |
| 48 | C13×#97 | — | 2 | 1 | no | — |
| 49 | C4×C8×C13 | — | 2 | 1 | no | — |
| 50 | C4×C8×C13 | — | 2 | 1 | no | — |
| 51 | C4×C8×C13 | — | 2 | 1 | no | — |
| 52 | C4×C8×C13 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | C4×C8×C13 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | #88062 | — | 4 | 2 | no | — |
| 55 | #88055 | — | 2 | 2 | no | — |
| 56 | #88054 | — | 2 | 2 | no | — |
| 57 | #88054 | — | 2 | 2 | no | — |
| 58 | C8×Dic13 | — | 2 | 2 | no | — |
| 59 | C4⋊#1055 | — | 2 | 1 | no | — |
| 60 | C4⋊#1055 | — | 2 | 1 | no | — |
| 61 | C4⋊#1055 | — | 2 | 1 | no | — |
| 62 | C4⋊#1055 | — | 2 | 1 | no | — |
| 63 | C4⋊#1055 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | C4⋊#1055 | C2×C4 | 1 | 1 | yes | characteristic |
| 65 | C4⋊#1055 | D4 | 1 | 2 | yes | — |
| 66 | #8909 | — | 13 | 1 | no | maximal, Sylow |
| 67 | C22×C4×C13 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C22×C4×C13 | C2×Q8 | 1 | 1 | yes | characteristic |
| 69 | C22×C4×C13 | #41 | 1 | 1 | yes | characteristic |
| 70 | C22×Dic13 | — | 2 | 1 | no | — |
| 71 | C22×Dic13 | — | 2 | 1 | no | — |
| 72 | C2×Dic26 | — | 2 | 2 | no | — |
| 73 | C2×Dic26 | — | 2 | 1 | no | — |
| 74 | C2×Dic26 | — | 4 | 1 | no | — |
| 75 | C2×Dic26 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 76 | C2×Dic26 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C13×M16 | — | 2 | 1 | no | — |
| 78 | C13×M16 | — | 2 | 1 | no | — |
| 79 | C13×M16 | — | 2 | 1 | no | — |
| 80 | C13×M16 | SD16 | 1 | 2 | yes | — |
| 81 | C2×C8×C13 | — | 2 | 1 | no | — |
| 82 | C2×C8×C13 | — | 2 | 1 | no | — |
| 83 | C2×C8×C13 | — | 2 | 1 | no | — |
| 84 | C2×C8×C13 | — | 2 | 1 | no | — |
| 85 | C2×C8×C13 | — | 4 | 2 | no | — |
| 86 | C2×C8×C13 | — | 4 | 1 | no | — |
| 87 | C42×C13 | C22×C4 | 1 | 1 | yes | characteristic |
| 88 | C42×C13 | C2×D4 | 1 | 1 | yes | characteristic |
| 89 | C42×C13 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C42×C13 | #41 | 1 | 1 | yes | characteristic |
| 91 | C4⋊Dic13 | — | 2 | 1 | no | — |
| 92 | C4⋊Dic13 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 93 | C4⋊Dic13 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 94 | #6101 | — | 2 | 1 | no | — |
| 95 | #6101 | — | 2 | 1 | no | — |
| 96 | #6101 | — | 2 | 1 | no | — |
| 97 | #6101 | — | 2 | 1 | no | — |
| 98 | C4×Dic13 | — | 2 | 1 | no | — |
| 99 | C4×Dic13 | — | 2 | 1 | no | — |
| 100 | C4×Dic13 | — | 2 | 1 | no | — |
| 101 | C4×Dic13 | — | 2 | 1 | no | — |
| 102 | #6099 | — | 4 | 1 | no | — |
| 103 | #6099 | — | 4 | 1 | no | — |
| 104 | #6099 | — | 4 | 1 | no | — |
| 105 | #6099 | — | 4 | 1 | no | — |
| 106 | C2×#1055 | — | 2 | 1 | no | — |
| 107 | C2×#1055 | — | 2 | 1 | no | — |
| 108 | C2×#1055 | — | 2 | 2 | no | — |
| 109 | C2×#1055 | — | 4 | 1 | no | — |
| 110 | C2×#1055 | — | 4 | 1 | no | — |
| 111 | C2×#1055 | — | 4 | 1 | no | — |
| 112 | Q8×M16 | — | 13 | 1 | no | — |
| 113 | #1950 | — | 13 | 1 | no | — |
| 114 | #1566 | — | 13 | 1 | no | — |
| 115 | #1487 | — | 13 | 1 | no | — |
| 116 | C2×Q8⋊C8 | — | 13 | 1 | no | — |
| 117 | #1301 | — | 13 | 1 | no | — |
| 118 | #1278 | D13 | 1 | 1 | yes | characteristic |
| 119 | C23×C13 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C13 | — | 2 | 1 | no | — |
| 121 | C2×C4×C13 | — | 2 | 1 | no | — |
| 122 | C2×C4×C13 | — | 4 | 1 | no | — |
| 123 | C2×C4×C13 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C13 | C4×D4 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C13 | C4×D4 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C13 | #120 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C13 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C13 | #123 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C13 | C2×SD16 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C13 | #135 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C13 | #137 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C13 | #137 | 1 | 1 | yes | characteristic |
| 133 | C2×Dic13 | — | 2 | 1 | no | — |
| 134 | C2×Dic13 | — | 2 | 1 | no | — |
| 135 | C2×Dic13 | — | 2 | 1 | no | — |
| 136 | C2×Dic13 | — | 2 | 1 | no | — |
| 137 | C2×Dic13 | — | 4 | 2 | no | — |
| 138 | C2×Dic13 | — | 8 | 1 | no | — |
| 139 | Dic26 | — | 2 | 1 | no | — |
| 140 | Dic26 | — | 2 | 1 | no | — |
| 141 | Dic26 | — | 2 | 1 | no | — |
| 142 | Dic26 | — | 4 | 2 | no | — |
| 143 | Dic26 | #108 | 1 | 2 | yes | complemented |
| 144 | C8×C13 | — | 2 | 2 | no | — |
| 145 | C8×C13 | — | 2 | 2 | no | — |
| 146 | C8×C13 | — | 4 | 1 | no | — |
| 147 | C8×C13 | — | 4 | 1 | no | — |
| 148 | C8×C13 | — | 8 | 1 | no | — |
| 149 | #1055 | — | 4 | 1 | no | — |
| 150 | #1055 | — | 4 | 1 | no | — |
| 151 | #1055 | — | 4 | 2 | no | — |
| 152 | #1055 | — | 8 | 1 | no | — |
| 153 | C2×C4×Q8 | — | 13 | 1 | no | — |
| 154 | #446 | — | 26 | 1 | no | — |
| 155 | #446 | — | 26 | 2 | no | — |
| 156 | C8×Q8 | — | 26 | 1 | no | — |
| 157 | #431 | C2×D13 | 1 | 1 | yes | characteristic |
| 158 | C2×#108 | — | 26 | 1 | no | — |
| 159 | #423 | — | 13 | 1 | no | — |
| 160 | #423 | — | 13 | 1 | no | — |
| 161 | #423 | — | 26 | 1 | no | — |
| 162 | C2×C4⋊C8 | — | 13 | 1 | no | — |
| 163 | C4×M16 | — | 26 | 1 | no | — |
| 164 | C4×M16 | C2×D13 | 1 | 1 | yes | characteristic |
| 165 | C2×C4×C8 | C2×D13 | 1 | 1 | yes | characteristic |
| 166 | #341 | — | 26 | 1 | no | — |
| 167 | #334 | — | 26 | 2 | no | — |
| 168 | Q8⋊C8 | — | 13 | 1 | no | — |
| 169 | Q8⋊C8 | — | 13 | 1 | no | — |
| 170 | Q8⋊C8 | — | 13 | 1 | no | — |
| 171 | Q8⋊C8 | — | 13 | 1 | no | — |
| 172 | Q8⋊C8 | — | 26 | 1 | no | — |
| 173 | Q8⋊C8 | — | 26 | 1 | no | — |
| 174 | C22×C13 | — | 2 | 1 | no | — |
| 175 | C22×C13 | — | 2 | 1 | no | — |
| 176 | C22×C13 | #385 | 1 | 1 | yes | characteristic |
| 177 | C22×C13 | #475 | 1 | 1 | yes | characteristic |
| 178 | C22×C13 | #479 | 1 | 1 | yes | characteristic |
| 179 | C4×C13 | — | 2 | 1 | no | — |
| 180 | C4×C13 | — | 2 | 1 | no | — |
| 181 | C4×C13 | — | 2 | 1 | no | — |
| 182 | C4×C13 | C2×#108 | 1 | 1 | yes | characteristic |
| 183 | C4×C13 | #430 | 1 | 1 | yes | characteristic |
| 184 | C4×C13 | C4×SD16 | 1 | 1 | yes | characteristic |
| 185 | C4×C13 | #441 | 1 | 1 | yes | characteristic |
| 186 | C4×C13 | #448 | 1 | 1 | yes | characteristic |
| 187 | C4×C13 | #452 | 1 | 1 | yes | characteristic |
| 188 | Dic13 | — | 2 | 2 | no | — |
| 189 | Dic13 | — | 4 | 1 | no | — |
| 190 | Dic13 | — | 4 | 1 | no | — |
| 191 | Dic13 | — | 4 | 1 | no | — |
| 192 | C22×Q8 | — | 13 | 1 | no | — |
| 193 | C2×M16 | — | 26 | 1 | no | — |
| 194 | C2×M16 | — | 26 | 1 | no | — |
| 195 | C2×M16 | D52 | 1 | 1 | yes | characteristic |
| 196 | C2×M16 | D52 | 1 | 1 | yes | characteristic |
| 197 | C22×C8 | — | 2 | 1 | no | — |
| 198 | C22×C8 | — | 26 | 1 | no | — |
| 199 | C4×Q8 | — | 13 | 1 | no | — |
| 200 | C4×Q8 | — | 13 | 1 | no | — |
| 201 | C4×Q8 | — | 13 | 1 | no | — |
| 202 | C4×Q8 | — | 13 | 1 | no | — |
| 203 | C4×Q8 | — | 26 | 1 | no | — |
| 204 | C4×Q8 | — | 26 | 1 | no | — |
| 205 | C2×C4⋊C4 | — | 13 | 1 | no | — |
| 206 | C2×C4⋊C4 | — | 26 | 1 | no | — |
| 207 | C2×C42 | — | 26 | 1 | no | — |
| 208 | C2×C42 | C22×D13 | 1 | 1 | yes | characteristic |
| 209 | #108 | — | 52 | 2 | no | — |
| 210 | C4⋊C8 | — | 2 | 1 | no | — |
| 211 | C4⋊C8 | — | 13 | 1 | no | — |
| 212 | C4⋊C8 | — | 13 | 1 | no | — |
| 213 | C4⋊C8 | — | 13 | 2 | no | — |
| 214 | C4⋊C8 | — | 26 | 2 | no | — |
| 215 | C4⋊C8 | — | 26 | 2 | no | — |
| 216 | C4⋊C8 | — | 26 | 1 | no | — |
| 217 | C4⋊C8 | — | 26 | 1 | no | — |
| 218 | C4⋊C8 | — | 26 | 1 | no | — |
| 219 | C4⋊C8 | — | 26 | 1 | no | — |
| 220 | C22⋊C8 | — | 2 | 1 | no | — |
| 221 | #97 | — | 2 | 1 | no | — |
| 222 | #97 | — | 26 | 2 | no | — |
| 223 | C4×C8 | — | 2 | 1 | no | — |
| 224 | C4×C8 | — | 2 | 1 | no | — |
| 225 | C4×C8 | — | 2 | 1 | no | — |
| 226 | C4×C8 | — | 26 | 2 | no | — |
| 227 | C4×C8 | C4×D13 | 1 | 1 | yes | characteristic |
| 228 | C4×C8 | C4×D13 | 1 | 1 | yes | characteristic |
| 229 | C2×C13 | — | 2 | 1 | no | — |
| 230 | C2×C13 | — | 2 | 1 | no | — |
| 231 | C2×C13 | #1855 | 1 | 1 | yes | characteristic |
| 232 | C2×C13 | #1861 | 1 | 1 | yes | characteristic |
| 233 | C2×C13 | #1865 | 1 | 1 | yes | characteristic |
| 234 | C2×Q8 | — | 13 | 1 | no | — |
| 235 | C2×Q8 | — | 13 | 1 | no | — |
| 236 | C2×Q8 | — | 26 | 2 | no | — |
| 237 | C2×Q8 | — | 26 | 1 | no | — |
| 238 | C2×Q8 | — | 52 | 1 | no | — |
| 239 | C22×C4 | — | 26 | 1 | no | — |
| 240 | C22×C4 | — | 26 | 1 | no | — |
| 241 | C22×C4 | C2×D52 | 1 | 1 | yes | characteristic |
| 242 | C22×C4 | #6129 | 1 | 1 | yes | characteristic |
| 243 | C22×C4 | Q8×D13 | 1 | 1 | yes | characteristic |
| 244 | M16 | — | 2 | 1 | no | — |
| 245 | M16 | — | 2 | 1 | no | — |
| 246 | M16 | — | 2 | 1 | no | — |
| 247 | M16 | — | 52 | 1 | no | — |
| 248 | M16 | — | 52 | 1 | no | — |
| 249 | M16 | — | 52 | 1 | no | — |
| 250 | M16 | — | 52 | 1 | no | — |
| 251 | M16 | #6095 | 1 | 2 | yes | — |
| 252 | C2×C8 | — | 2 | 1 | no | — |
| 253 | C2×C8 | — | 2 | 1 | no | — |
| 254 | C2×C8 | — | 2 | 1 | no | — |
| 255 | C2×C8 | — | 2 | 1 | no | — |
| 256 | C2×C8 | — | 4 | 2 | no | — |
| 257 | C2×C8 | — | 4 | 1 | no | — |
| 258 | C2×C8 | — | 26 | 1 | no | — |
| 259 | C2×C8 | — | 26 | 1 | no | — |
| 260 | C2×C8 | — | 26 | 2 | no | — |
| 261 | C2×C8 | — | 52 | 1 | no | — |
| 262 | C2×C8 | — | 52 | 1 | no | — |
| 263 | C2×C8 | — | 52 | 1 | no | — |
| 264 | C4⋊C4 | — | 13 | 1 | no | — |
| 265 | C4⋊C4 | — | 13 | 1 | no | — |
| 266 | C4⋊C4 | — | 26 | 1 | no | — |
| 267 | C4⋊C4 | — | 26 | 1 | no | — |
| 268 | C4⋊C4 | — | 26 | 1 | no | — |
| 269 | C4⋊C4 | — | 26 | 1 | no | — |
| 270 | C4⋊C4 | — | 26 | 1 | no | — |
| 271 | C42 | — | 26 | 1 | no | — |
| 272 | C42 | — | 26 | 1 | no | — |
| 273 | C42 | — | 26 | 1 | no | — |
| 274 | C42 | — | 26 | 1 | no | — |
| 275 | C42 | C2×C4×D13 | 1 | 1 | yes | characteristic |
| 276 | C42 | #6127 | 1 | 1 | yes | characteristic |
| 277 | C42 | D4×D13 | 1 | 1 | yes | characteristic |
| 278 | C42 | D4×D13 | 1 | 1 | yes | characteristic |
| 279 | C13 | #8909 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 280 | C23 | #88151 | 1 | 1 | yes | characteristic |
| 281 | Q8 | — | 13 | 2 | no | — |
| 282 | Q8 | — | 26 | 1 | no | — |
| 283 | Q8 | — | 26 | 1 | no | — |
| 284 | Q8 | — | 26 | 1 | no | — |
| 285 | Q8 | — | 52 | 2 | no | — |
| 286 | C2×C4 | — | 2 | 1 | no | — |
| 287 | C2×C4 | — | 2 | 1 | no | — |
| 288 | C2×C4 | — | 4 | 1 | no | — |
| 289 | C2×C4 | — | 26 | 1 | no | — |
| 290 | C2×C4 | — | 26 | 1 | no | — |
| 291 | C2×C4 | — | 26 | 1 | no | — |
| 292 | C2×C4 | — | 26 | 1 | no | — |
| 293 | C2×C4 | — | 52 | 2 | no | — |
| 294 | C2×C4 | — | 104 | 1 | no | — |
| 295 | C2×C4 | C4×D52 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | #88135 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #88136 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #88137 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | C4⋊C4×D13 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #88150 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | C2×#6095 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | #88158 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | #88163 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | #88163 | 1 | 1 | yes | characteristic |
| 305 | C8 | — | 2 | 2 | no | — |
| 306 | C8 | — | 2 | 2 | no | — |
| 307 | C8 | — | 4 | 1 | no | — |
| 308 | C8 | — | 4 | 1 | no | — |
| 309 | C8 | — | 8 | 1 | no | — |
| 310 | C8 | — | 52 | 1 | no | — |
| 311 | C8 | — | 52 | 1 | no | — |
| 312 | C8 | — | 52 | 2 | no | — |
| 313 | C8 | — | 104 | 1 | no | — |
| 314 | C22 | — | 2 | 1 | no | — |
| 315 | C22 | — | 2 | 1 | no | — |
| 316 | C22 | #11718329 | 1 | 1 | yes | characteristic |
| 317 | C22 | #11718490 | 1 | 1 | yes | characteristic |
| 318 | C22 | #11718512 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 2 | 1 | no | — |
| 320 | C4 | — | 2 | 1 | no | — |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 26 | 2 | no | — |
| 323 | C4 | — | 52 | 1 | no | — |
| 324 | C4 | — | 52 | 1 | no | — |
| 325 | C4 | — | 52 | 1 | no | — |
| 326 | C4 | C4×#6095 | 1 | 1 | yes | characteristic |
| 327 | C4 | #11718377 | 1 | 1 | yes | characteristic |
| 328 | C4 | #11718394 | 1 | 1 | yes | characteristic |
| 329 | C4 | #11718399 | 1 | 1 | yes | characteristic |
| 330 | C4 | D13×#108 | 1 | 1 | yes | characteristic |
| 331 | C4 | #11718554 | 1 | 1 | yes | characteristic |
| 332 | C2 | — | 2 | 1 | no | — |
| 333 | C2 | — | 2 | 1 | no | — |
| 334 | C2 | #13115042 | 1 | 1 | yes | characteristic |
| 335 | C2 | #13115077 | 1 | 1 | yes | characteristic |
| 336 | C2 | #13115107 | 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