← #13272016
Subgroups — #13272016
Order 1792.
514 conjugacy classes of subgroups.
The subgroup lattice has 340 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#89927 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C7×#357 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#89357 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#89298 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×C16×Dic7 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #11728361 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#89287 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×C7×D8 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#6499 | — | 2 | 1 | no | — |
| 10 | C22×#1142 | — | 2 | 1 | no | — |
| 11 | C22×Dic28 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×C7×C16 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C7×#107 | C22 | 1 | 1 | yes | characteristic |
| 14 | #89927 | — | 2 | 2 | no | — |
| 15 | #89927 | C4 | 1 | 4 | yes | complemented |
| 16 | C2×#6392 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×C8×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22×C7⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 19 | #89749 | — | 2 | 1 | no | — |
| 20 | C2×#6382 | — | 2 | 1 | no | — |
| 21 | C7×#357 | — | 2 | 2 | no | — |
| 22 | #89357 | — | 2 | 2 | no | — |
| 23 | #89298 | — | 2 | 2 | no | — |
| 24 | C16×Dic7 | — | 2 | 2 | no | — |
| 25 | #89287 | — | 2 | 2 | no | — |
| 26 | #13204 | — | 7 | 1 | no | maximal, Sylow |
| 27 | C22×C7×D4 | — | 2 | 1 | no | — |
| 28 | C22×Dic14 | — | 2 | 1 | no | — |
| 29 | C2×C7×D8 | — | 2 | 1 | no | — |
| 30 | C2×C7×D8 | — | 2 | 1 | no | — |
| 31 | C2×C7×D8 | — | 2 | 1 | no | — |
| 32 | C2×C7×D8 | — | 2 | 1 | no | — |
| 33 | C2×C7×D8 | — | 2 | 1 | no | — |
| 34 | C2×C7×D8 | C2×C4 | 1 | 2 | yes | — |
| 35 | C22×C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 36 | C2×C7×C4⋊C4 | — | 2 | 1 | no | — |
| 37 | C2×C22⋊Dic7 | — | 4 | 2 | no | — |
| 38 | #6499 | — | 2 | 4 | no | — |
| 39 | #6499 | — | 4 | 2 | no | — |
| 40 | C2×#1142 | — | 2 | 4 | no | — |
| 41 | C2×#1142 | — | 4 | 2 | no | — |
| 42 | C2×#1142 | — | 4 | 2 | no | — |
| 43 | C2×C4⋊Dic7 | — | 2 | 1 | no | — |
| 44 | C2×C4×Dic7 | D4 | 1 | 1 | yes | characteristic |
| 45 | C22×C7⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×Dic28 | — | 2 | 1 | no | — |
| 47 | C2×Dic28 | — | 2 | 1 | no | — |
| 48 | C2×Dic28 | — | 2 | 1 | no | — |
| 49 | C2×Dic28 | — | 2 | 1 | no | — |
| 50 | C2×Dic28 | — | 2 | 1 | no | — |
| 51 | C2×Dic28 | C2×C4 | 1 | 2 | yes | — |
| 52 | C2×C7×C16 | — | 2 | 1 | no | — |
| 53 | C2×C7×C16 | — | 2 | 1 | no | — |
| 54 | C2×C7×C16 | D4 | 1 | 2 | yes | — |
| 55 | C7×#107 | — | 2 | 1 | no | — |
| 56 | C7×#107 | — | 2 | 1 | no | — |
| 57 | C7⋊#95 | — | 4 | 2 | no | — |
| 58 | #6392 | — | 2 | 1 | no | — |
| 59 | #6392 | — | 2 | 1 | no | — |
| 60 | C8×Dic7 | C2×C4 | 1 | 2 | yes | — |
| 61 | C8×Dic7 | D4 | 1 | 2 | yes | — |
| 62 | C2×C7⋊C16 | — | 2 | 1 | no | — |
| 63 | C2×C7⋊C16 | — | 2 | 1 | no | — |
| 64 | C2×C7⋊C16 | D4 | 1 | 2 | yes | — |
| 65 | #6382 | — | 4 | 2 | no | — |
| 66 | C2×#495 | — | 7 | 1 | no | — |
| 67 | C2×#358 | — | 7 | 1 | no | — |
| 68 | C2×#358 | — | 7 | 1 | no | — |
| 69 | C2×#357 | — | 7 | 1 | no | — |
| 70 | C2×#357 | D7 | 1 | 1 | yes | characteristic |
| 71 | C2×C4×C16 | — | 7 | 1 | no | — |
| 72 | #1846 | — | 7 | 1 | no | — |
| 73 | C24×C7 | — | 4 | 1 | no | — |
| 74 | C2×C7×D4 | — | 2 | 2 | no | — |
| 75 | C2×C7×D4 | — | 4 | 1 | no | — |
| 76 | C2×C7×D4 | — | 4 | 1 | no | — |
| 77 | C2×C7×D4 | — | 4 | 1 | no | — |
| 78 | C2×C7×D4 | — | 4 | 1 | no | — |
| 79 | C2×C7×D4 | — | 4 | 1 | no | — |
| 80 | C22×C4×C7 | — | 4 | 1 | no | — |
| 81 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 82 | C22×Dic7 | — | 4 | 1 | no | — |
| 83 | C22×Dic7 | — | 4 | 1 | no | — |
| 84 | C22×Dic7 | D8 | 1 | 2 | yes | — |
| 85 | C2×Dic14 | — | 2 | 2 | no | — |
| 86 | C2×Dic14 | — | 4 | 1 | no | — |
| 87 | C2×Dic14 | — | 4 | 1 | no | — |
| 88 | C2×Dic14 | — | 4 | 1 | no | — |
| 89 | C2×Dic14 | — | 4 | 1 | no | — |
| 90 | C2×Dic14 | — | 4 | 1 | no | — |
| 91 | C7×D8 | — | 2 | 2 | no | — |
| 92 | C7×D8 | — | 2 | 2 | no | — |
| 93 | C7×D8 | — | 4 | 1 | no | — |
| 94 | C7×D8 | — | 4 | 1 | no | — |
| 95 | C2×C7×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C7×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 100 | C2×C7×C8 | #41 | 1 | 1 | yes | characteristic |
| 101 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 102 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 103 | C22⋊Dic7 | — | 4 | 8 | no | — |
| 104 | #1142 | — | 4 | 8 | no | — |
| 105 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 106 | C4⋊Dic7 | — | 4 | 1 | no | — |
| 107 | C4×Dic7 | C22⋊C4 | 1 | 2 | yes | — |
| 108 | C4×Dic7 | SD16 | 1 | 2 | yes | — |
| 109 | C2×C7⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 110 | C2×C7⋊C8 | D8 | 1 | 2 | yes | — |
| 111 | C2×C7⋊C8 | SD16 | 1 | 2 | yes | — |
| 112 | Dic28 | — | 2 | 2 | no | — |
| 113 | Dic28 | — | 2 | 2 | no | — |
| 114 | Dic28 | — | 4 | 1 | no | — |
| 115 | Dic28 | — | 4 | 1 | no | — |
| 116 | C7×C16 | — | 2 | 2 | no | — |
| 117 | C7⋊C16 | — | 2 | 2 | no | — |
| 118 | C22×Q16 | — | 7 | 1 | no | — |
| 119 | C22×SD16 | — | 14 | 1 | no | — |
| 120 | C22×D8 | C2×D7 | 1 | 1 | yes | characteristic |
| 121 | C2×#124 | — | 14 | 1 | no | — |
| 122 | C22×C16 | — | 7 | 1 | no | — |
| 123 | C22×C16 | C2×D7 | 1 | 1 | yes | characteristic |
| 124 | #495 | — | 7 | 4 | no | — |
| 125 | #495 | — | 14 | 2 | no | — |
| 126 | C2×#107 | — | 7 | 1 | no | — |
| 127 | C2×#107 | C2×D7 | 1 | 1 | yes | characteristic |
| 128 | C2×#106 | — | 14 | 1 | no | — |
| 129 | C2×C4×C8 | — | 7 | 1 | no | — |
| 130 | #382 | — | 14 | 1 | no | — |
| 131 | #358 | — | 14 | 2 | no | — |
| 132 | #358 | — | 14 | 2 | no | — |
| 133 | #357 | — | 2 | 2 | no | — |
| 134 | #357 | — | 14 | 2 | no | — |
| 135 | C4×C16 | — | 14 | 2 | no | — |
| 136 | C23×C7 | — | 4 | 2 | no | — |
| 137 | C23×C7 | — | 8 | 1 | no | — |
| 138 | C23×C7 | — | 8 | 1 | no | — |
| 139 | C23×C7 | — | 8 | 2 | no | — |
| 140 | C23×C7 | — | 8 | 2 | no | — |
| 141 | C23×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 142 | C7×D4 | — | 4 | 2 | no | — |
| 143 | C7×D4 | — | 4 | 2 | no | — |
| 144 | C7×D4 | — | 8 | 1 | no | — |
| 145 | C7×D4 | — | 8 | 1 | no | — |
| 146 | C2×C4×C7 | — | 8 | 1 | no | — |
| 147 | C2×C4×C7 | — | 8 | 2 | no | — |
| 148 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 149 | C2×C4×C7 | #124 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×C7 | #127 | 1 | 1 | yes | characteristic |
| 151 | C2×C4×C7 | C2×D8 | 1 | 1 | yes | characteristic |
| 152 | C2×C4×C7 | C2×SD16 | 1 | 1 | yes | characteristic |
| 153 | C2×C4×C7 | C2×SD16 | 1 | 1 | yes | characteristic |
| 154 | C2×Dic7 | — | 2 | 2 | no | — |
| 155 | C2×Dic7 | — | 2 | 2 | no | — |
| 156 | C2×Dic7 | — | 4 | 2 | no | — |
| 157 | C2×Dic7 | — | 8 | 1 | no | — |
| 158 | C2×Dic7 | — | 8 | 1 | no | — |
| 159 | C2×Dic7 | — | 8 | 1 | no | — |
| 160 | C2×Dic7 | — | 8 | 2 | no | — |
| 161 | C2×Dic7 | #102 | 1 | 4 | yes | — |
| 162 | Dic14 | — | 4 | 2 | no | — |
| 163 | Dic14 | — | 4 | 2 | no | — |
| 164 | Dic14 | — | 8 | 1 | no | — |
| 165 | Dic14 | — | 8 | 1 | no | — |
| 166 | C7×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 167 | C7×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 168 | C7×C8 | #121 | 1 | 1 | yes | characteristic |
| 169 | C7×C8 | #121 | 1 | 1 | yes | characteristic |
| 170 | C7⋊C8 | #102 | 1 | 4 | yes | — |
| 171 | C22×Q8 | — | 14 | 1 | no | — |
| 172 | C22×D4 | — | 2 | 1 | no | — |
| 173 | C2×Q16 | — | 7 | 2 | no | — |
| 174 | C2×Q16 | — | 14 | 1 | no | — |
| 175 | C2×Q16 | — | 14 | 1 | no | — |
| 176 | C2×Q16 | — | 14 | 1 | no | — |
| 177 | C2×Q16 | — | 14 | 1 | no | — |
| 178 | C2×Q16 | — | 14 | 1 | no | — |
| 179 | C2×SD16 | — | 14 | 4 | no | — |
| 180 | C2×SD16 | — | 28 | 2 | no | — |
| 181 | C2×SD16 | — | 28 | 2 | no | — |
| 182 | C2×D8 | — | 2 | 1 | no | — |
| 183 | C2×D8 | — | 2 | 1 | no | — |
| 184 | C2×D8 | — | 2 | 1 | no | — |
| 185 | C2×D8 | — | 2 | 1 | no | — |
| 186 | C2×D8 | — | 2 | 1 | no | — |
| 187 | C2×D8 | C4×D7 | 1 | 2 | yes | — |
| 188 | C22×C8 | — | 7 | 1 | no | — |
| 189 | C22×C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 190 | #124 | — | 14 | 4 | no | — |
| 191 | #124 | — | 28 | 2 | no | — |
| 192 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 193 | C2×C4⋊C4 | — | 14 | 1 | no | — |
| 194 | C2×C22⋊C4 | — | 28 | 2 | no | — |
| 195 | C2×C42 | — | 7 | 1 | no | — |
| 196 | C2×C16 | — | 2 | 1 | no | — |
| 197 | C2×C16 | — | 2 | 1 | no | — |
| 198 | C2×C16 | — | 7 | 2 | no | — |
| 199 | C2×C16 | — | 14 | 1 | no | — |
| 200 | C2×C16 | — | 14 | 1 | no | — |
| 201 | C2×C16 | C22⋊D7 | 1 | 2 | yes | — |
| 202 | #107 | — | 2 | 1 | no | — |
| 203 | #107 | — | 2 | 1 | no | — |
| 204 | #107 | — | 14 | 1 | no | — |
| 205 | #107 | — | 14 | 1 | no | — |
| 206 | #106 | — | 28 | 2 | no | — |
| 207 | C4×C8 | — | 7 | 2 | no | — |
| 208 | C4×C8 | — | 7 | 2 | no | — |
| 209 | #95 | — | 28 | 2 | no | — |
| 210 | C22×C7 | — | 8 | 4 | no | — |
| 211 | C22×C7 | — | 8 | 2 | no | — |
| 212 | C22×C7 | — | 16 | 2 | no | — |
| 213 | C22×C7 | — | 16 | 1 | no | — |
| 214 | C22×C7 | — | 16 | 1 | no | — |
| 215 | C22×C7 | C2×#102 | 1 | 1 | yes | characteristic |
| 216 | C22×C7 | #486 | 1 | 1 | yes | characteristic |
| 217 | C22×C7 | #493 | 1 | 1 | yes | characteristic |
| 218 | C22×C7 | #508 | 1 | 2 | yes | — |
| 219 | C22×C7 | #508 | 1 | 2 | yes | — |
| 220 | C4×C7 | — | 16 | 1 | no | — |
| 221 | C4×C7 | #390 | 1 | 1 | yes | characteristic |
| 222 | C4×C7 | C2×#102 | 1 | 1 | yes | characteristic |
| 223 | C4×C7 | #486 | 1 | 1 | yes | characteristic |
| 224 | C4×C7 | #492 | 1 | 1 | yes | characteristic |
| 225 | Dic7 | — | 2 | 4 | no | — |
| 226 | Dic7 | — | 8 | 2 | no | — |
| 227 | Dic7 | — | 16 | 1 | no | — |
| 228 | C24 | — | 4 | 1 | no | — |
| 229 | C2×Q8 | — | 14 | 2 | no | — |
| 230 | C2×Q8 | — | 28 | 1 | no | — |
| 231 | C2×Q8 | — | 28 | 1 | no | — |
| 232 | C2×Q8 | — | 28 | 1 | no | — |
| 233 | C2×Q8 | — | 28 | 1 | no | — |
| 234 | C2×Q8 | — | 28 | 1 | no | — |
| 235 | C2×D4 | — | 2 | 2 | no | — |
| 236 | C2×D4 | — | 4 | 1 | no | — |
| 237 | C2×D4 | — | 4 | 1 | no | — |
| 238 | C2×D4 | — | 4 | 1 | no | — |
| 239 | C2×D4 | — | 4 | 1 | no | — |
| 240 | C2×D4 | — | 4 | 1 | no | — |
| 241 | C22×C4 | — | 4 | 1 | no | — |
| 242 | C22×C4 | — | 7 | 2 | no | — |
| 243 | C22×C4 | — | 28 | 1 | no | — |
| 244 | C22×C4 | — | 28 | 1 | no | — |
| 245 | C22×C4 | D4×D7 | 1 | 1 | yes | characteristic |
| 246 | Q16 | — | 14 | 2 | no | — |
| 247 | Q16 | — | 14 | 2 | no | — |
| 248 | Q16 | — | 28 | 1 | no | — |
| 249 | Q16 | — | 28 | 1 | no | — |
| 250 | SD16 | — | 28 | 8 | no | — |
| 251 | D8 | — | 2 | 2 | no | — |
| 252 | D8 | — | 2 | 2 | no | — |
| 253 | D8 | — | 4 | 1 | no | — |
| 254 | D8 | — | 4 | 1 | no | — |
| 255 | C2×C8 | — | 7 | 2 | no | — |
| 256 | C2×C8 | — | 7 | 2 | no | — |
| 257 | C2×C8 | — | 7 | 2 | no | — |
| 258 | C2×C8 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | #1157 | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 261 | C2×C8 | D4×D7 | 1 | 1 | yes | characteristic |
| 262 | C2×C8 | #1159 | 1 | 1 | yes | characteristic |
| 263 | C2×C8 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 264 | C4⋊C4 | — | 4 | 1 | no | — |
| 265 | C4⋊C4 | — | 4 | 1 | no | — |
| 266 | C4⋊C4 | — | 28 | 1 | no | — |
| 267 | C4⋊C4 | — | 28 | 1 | no | — |
| 268 | C22⋊C4 | — | 28 | 8 | no | — |
| 269 | C42 | — | 7 | 2 | no | — |
| 270 | C42 | — | 7 | 2 | no | — |
| 271 | C16 | — | 2 | 2 | no | — |
| 272 | C16 | — | 14 | 2 | no | — |
| 273 | C2×C7 | — | 16 | 2 | no | — |
| 274 | C2×C7 | #1957 | 1 | 1 | yes | characteristic |
| 275 | C2×C7 | #2138 | 1 | 2 | yes | — |
| 276 | C2×C7 | #2242 | 1 | 2 | yes | — |
| 277 | C2×C7 | #2249 | 1 | 1 | yes | characteristic |
| 278 | C2×C7 | #2249 | 1 | 1 | yes | characteristic |
| 279 | C23 | — | 4 | 2 | no | — |
| 280 | C23 | — | 8 | 1 | no | — |
| 281 | C23 | — | 8 | 2 | no | — |
| 282 | C23 | — | 8 | 2 | no | — |
| 283 | C23 | — | 8 | 1 | no | — |
| 284 | C23 | D7×D8 | 1 | 1 | yes | characteristic |
| 285 | Q8 | — | 28 | 2 | no | — |
| 286 | Q8 | — | 28 | 2 | no | — |
| 287 | Q8 | — | 56 | 1 | no | — |
| 288 | Q8 | — | 56 | 1 | no | — |
| 289 | D4 | — | 4 | 2 | no | — |
| 290 | D4 | — | 4 | 2 | no | — |
| 291 | D4 | — | 8 | 1 | no | — |
| 292 | D4 | — | 8 | 1 | no | — |
| 293 | C2×C4 | — | 7 | 4 | no | — |
| 294 | C2×C4 | — | 8 | 1 | no | — |
| 295 | C2×C4 | — | 8 | 2 | no | — |
| 296 | C2×C4 | — | 14 | 2 | no | — |
| 297 | C2×C4 | — | 14 | 2 | no | — |
| 298 | C2×C4 | — | 28 | 2 | no | — |
| 299 | C2×C4 | — | 56 | 1 | no | — |
| 300 | C2×C4 | — | 56 | 1 | no | — |
| 301 | C2×C4 | — | 56 | 1 | no | — |
| 302 | C2×C4 | — | 56 | 2 | no | — |
| 303 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | #6448 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | D7×D8 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | D7×SD16 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | D7×SD16 | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | #6503 | 1 | 1 | yes | characteristic |
| 309 | C8 | — | 7 | 4 | no | — |
| 310 | C8 | (C22⋊D7)⋊C4 | 1 | 1 | yes | characteristic |
| 311 | C8 | #6447 | 1 | 1 | yes | characteristic |
| 312 | C8 | C4×C22⋊D7 | 1 | 1 | yes | characteristic |
| 313 | C8 | #6502 | 1 | 1 | yes | characteristic |
| 314 | C7 | #13204 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 315 | C22 | — | 8 | 2 | no | — |
| 316 | C22 | — | 8 | 4 | no | — |
| 317 | C22 | — | 16 | 1 | no | — |
| 318 | C22 | — | 16 | 1 | no | — |
| 319 | C22 | — | 16 | 2 | no | — |
| 320 | C22 | #89532 | 1 | 1 | yes | characteristic |
| 321 | C22 | D7×#102 | 1 | 1 | yes | characteristic |
| 322 | C22 | #89684 | 1 | 2 | yes | — |
| 323 | C22 | #89688 | 1 | 2 | yes | — |
| 324 | C22 | #89923 | 1 | 1 | yes | characteristic |
| 325 | C4 | — | 14 | 4 | no | — |
| 326 | C4 | — | 16 | 1 | no | — |
| 327 | C4 | — | 56 | 2 | no | — |
| 328 | C4 | — | 112 | 1 | no | — |
| 329 | C4 | #89532 | 1 | 1 | yes | characteristic |
| 330 | C4 | D7×#102 | 1 | 1 | yes | characteristic |
| 331 | C4 | #89729 | 1 | 1 | yes | characteristic |
| 332 | C4 | #89947 | 1 | 1 | yes | characteristic |
| 333 | C2 | — | 16 | 2 | no | — |
| 334 | C2 | #11726938 | 1 | 2 | yes | — |
| 335 | C2 | #11726942 | 1 | 2 | yes | — |
| 336 | C2 | #11727911 | 1 | 1 | yes | characteristic |
| 337 | C2 | #11729219 | 1 | 1 | yes | characteristic |
| 338 | C2 | #11729234 | 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