← #35322196
Subgroups — #35322196
Order 21248.
306 conjugacy classes of subgroups.
The subgroup lattice has 286 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #24909219 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #24909091 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #24909084 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #24909068 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #24905549 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #24905545 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C83×#1764 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#14581732 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#14581732 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#14581732 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#14581732 | C22 | 1 | 1 | yes | characteristic |
| 12 | C4⋊(C83⋊C16) | — | 2 | 2 | no | — |
| 13 | #19127485 | — | 2 | 2 | no | — |
| 14 | #19127483 | — | 2 | 2 | no | — |
| 15 | #19127479 | — | 2 | 2 | no | — |
| 16 | C2×#131×C83 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#104×C83 | C4 | 1 | 1 | yes | characteristic |
| 18 | #416×C83 | C4 | 1 | 1 | yes | characteristic |
| 19 | #405×C83 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×C8×C83 | C22 | 1 | 1 | yes | characteristic |
| 21 | #341×C83 | C4 | 1 | 1 | yes | characteristic |
| 22 | #339×C83 | C4 | 1 | 1 | yes | characteristic |
| 23 | #14581732 | — | 2 | 1 | no | — |
| 24 | #14581732 | — | 2 | 1 | no | — |
| 25 | #14581732 | — | 2 | 1 | no | — |
| 26 | #14581732 | — | 2 | 1 | no | — |
| 27 | #14581732 | — | 2 | 1 | no | — |
| 28 | #14581732 | — | 2 | 1 | no | — |
| 29 | #14581732 | — | 2 | 1 | no | — |
| 30 | #14581732 | D4 | 1 | 2 | yes | — |
| 31 | C2×C83⋊C16 | — | 2 | 1 | no | — |
| 32 | C2×C83⋊C16 | — | 2 | 1 | no | — |
| 33 | C2×C83⋊C16 | — | 2 | 1 | no | — |
| 34 | C2×C83⋊C16 | D4 | 1 | 2 | yes | — |
| 35 | C2×#41×C83 | C2×C4 | 1 | 1 | yes | characteristic |
| 36 | #131×C83 | — | 2 | 1 | no | — |
| 37 | #131×C83 | — | 2 | 1 | no | — |
| 38 | #131×C83 | — | 2 | 1 | no | — |
| 39 | #131×C83 | — | 2 | 1 | no | — |
| 40 | C2×M16×C83 | — | 2 | 1 | no | — |
| 41 | C2×M16×C83 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×M16×C83 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | C22×C8×C83 | — | 2 | 1 | no | — |
| 44 | C22×C8×C83 | C2×C4 | 1 | 1 | yes | characteristic |
| 45 | C22×C8×C83 | C23 | 1 | 1 | yes | characteristic |
| 46 | #117×C83 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C2×C42×C83 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | #104×C83 | — | 2 | 1 | no | — |
| 49 | #104×C83 | — | 2 | 1 | no | — |
| 50 | #103×C83 | — | 2 | 1 | no | — |
| 51 | #102×C83 | — | 2 | 1 | no | — |
| 52 | #97×C83 | — | 2 | 1 | no | — |
| 53 | C4×C8×C83 | — | 2 | 1 | no | — |
| 54 | C4×C8×C83 | — | 2 | 1 | no | — |
| 55 | C4×C8×C83 | — | 2 | 1 | no | — |
| 56 | C83⋊C16 | — | 2 | 2 | no | — |
| 57 | C83⋊C16 | — | 2 | 2 | no | — |
| 58 | C83⋊C16 | — | 4 | 1 | no | — |
| 59 | C83⋊C16 | — | 4 | 1 | no | — |
| 60 | #41×C83 | — | 2 | 1 | no | — |
| 61 | #41×C83 | — | 2 | 1 | no | — |
| 62 | #41×C83 | — | 2 | 1 | no | — |
| 63 | #41×C83 | — | 2 | 1 | no | — |
| 64 | C2×Q8×C83 | M16 | 1 | 1 | yes | characteristic |
| 65 | C2×D4×C83 | — | 2 | 1 | no | — |
| 66 | C2×D4×C83 | M16 | 1 | 1 | yes | characteristic |
| 67 | C22×C4×C83 | — | 2 | 1 | no | — |
| 68 | C22×C4×C83 | — | 2 | 1 | no | — |
| 69 | C22×C4×C83 | C22×C4 | 1 | 1 | yes | characteristic |
| 70 | M16×C83 | — | 2 | 1 | no | — |
| 71 | M16×C83 | — | 2 | 1 | no | — |
| 72 | M16×C83 | — | 2 | 1 | no | — |
| 73 | M16×C83 | — | 2 | 1 | no | — |
| 74 | M16×C83 | — | 4 | 1 | no | — |
| 75 | M16×C83 | — | 4 | 1 | no | — |
| 76 | C2×C8×C83 | — | 2 | 1 | no | — |
| 77 | C2×C8×C83 | — | 2 | 1 | no | — |
| 78 | C2×C8×C83 | — | 2 | 1 | no | — |
| 79 | C2×C8×C83 | — | 2 | 1 | no | — |
| 80 | C2×C8×C83 | — | 2 | 1 | no | — |
| 81 | C2×C8×C83 | — | 4 | 1 | no | — |
| 82 | C2×C8×C83 | — | 4 | 1 | no | — |
| 83 | C2×C8×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C8×C83 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C8×C83 | #41 | 1 | 1 | yes | characteristic |
| 86 | C2×C8×C83 | #41 | 1 | 1 | yes | characteristic |
| 87 | C2×C8×C83 | #41 | 1 | 1 | yes | characteristic |
| 88 | C2×C8×C83 | #41 | 1 | 1 | yes | characteristic |
| 89 | C4⋊C4×C83 | — | 2 | 1 | no | — |
| 90 | C22⋊C4×C83 | — | 2 | 1 | no | — |
| 91 | C42×C83 | — | 2 | 1 | no | — |
| 92 | C42×C83 | — | 2 | 1 | no | — |
| 93 | C42×C83 | M16 | 1 | 1 | yes | characteristic |
| 94 | C42×C83 | M16 | 1 | 1 | yes | characteristic |
| 95 | C23×C83 | — | 2 | 1 | no | — |
| 96 | C23×C83 | #131 | 1 | 1 | yes | characteristic |
| 97 | Q8×C83 | — | 2 | 1 | no | — |
| 98 | Q8×C83 | — | 2 | 1 | no | — |
| 99 | D4×C83 | — | 2 | 1 | no | — |
| 100 | D4×C83 | — | 2 | 1 | no | — |
| 101 | D4×C83 | — | 4 | 1 | no | — |
| 102 | D4×C83 | — | 4 | 1 | no | — |
| 103 | C2×C4×C83 | — | 2 | 1 | no | — |
| 104 | C2×C4×C83 | — | 2 | 1 | no | — |
| 105 | C2×C4×C83 | — | 2 | 1 | no | — |
| 106 | C2×C4×C83 | — | 4 | 2 | no | — |
| 107 | C2×C4×C83 | — | 4 | 1 | no | — |
| 108 | C2×C4×C83 | — | 4 | 1 | no | — |
| 109 | C2×C4×C83 | — | 4 | 1 | no | — |
| 110 | C2×C4×C83 | #117 | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C83 | C4×D4 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C83 | C4×D4 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C83 | C2×M16 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C83 | C2×M16 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C83 | #131 | 1 | 1 | yes | characteristic |
| 116 | C8×C83 | — | 4 | 1 | no | — |
| 117 | C8×C83 | — | 4 | 1 | no | — |
| 118 | C8×C83 | — | 4 | 1 | no | — |
| 119 | C8×C83 | #121 | 1 | 1 | yes | characteristic |
| 120 | C8×C83 | #123 | 1 | 1 | yes | characteristic |
| 121 | C8×C83 | #124 | 1 | 1 | yes | characteristic |
| 122 | C8×C83 | #126 | 1 | 1 | yes | characteristic |
| 123 | C22×C83 | — | 4 | 1 | no | — |
| 124 | C22×C83 | — | 4 | 1 | no | — |
| 125 | C22×C83 | — | 8 | 1 | no | — |
| 126 | C22×C83 | #433 | 1 | 1 | yes | characteristic |
| 127 | C22×C83 | #435 | 1 | 1 | yes | characteristic |
| 128 | C22×C83 | #435 | 1 | 1 | yes | characteristic |
| 129 | C4×C83 | — | 2 | 2 | no | — |
| 130 | C4×C83 | — | 4 | 1 | no | — |
| 131 | C4×C83 | — | 4 | 1 | no | — |
| 132 | C4×C83 | — | 8 | 1 | no | — |
| 133 | C4×C83 | #388 | 1 | 1 | yes | characteristic |
| 134 | C4×C83 | #432 | 1 | 1 | yes | characteristic |
| 135 | C4×C83 | #435 | 1 | 1 | yes | characteristic |
| 136 | C4×C83 | #436 | 1 | 1 | yes | characteristic |
| 137 | #13129 | — | 83 | 1 | no | maximal, Sylow |
| 138 | C2×C83 | — | 2 | 1 | no | — |
| 139 | C2×C83 | — | 2 | 1 | no | — |
| 140 | C2×C83 | — | 2 | 1 | no | — |
| 141 | C2×C83 | — | 8 | 1 | no | — |
| 142 | C2×C83 | #1924 | 1 | 1 | yes | characteristic |
| 143 | #2150 | — | 83 | 1 | no | — |
| 144 | #2117 | — | 83 | 1 | no | — |
| 145 | #2115 | — | 83 | 1 | no | — |
| 146 | #2110 | — | 83 | 1 | no | — |
| 147 | #1764 | D83 | 1 | 1 | yes | characteristic |
| 148 | #1378 | — | 83 | 1 | no | — |
| 149 | #1376 | — | 83 | 1 | no | — |
| 150 | C83 | #13129 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 151 | C2×#131 | C2×D83 | 1 | 1 | yes | characteristic |
| 152 | C2×M32 | — | 83 | 1 | no | — |
| 153 | C2×M32 | — | 83 | 1 | no | — |
| 154 | C2×M32 | — | 83 | 1 | no | — |
| 155 | C2×M32 | — | 83 | 1 | no | — |
| 156 | C2×#104 | Dic83 | 1 | 1 | yes | characteristic |
| 157 | #416 | Dic83 | 1 | 1 | yes | characteristic |
| 158 | #405 | C2×D83 | 1 | 1 | yes | characteristic |
| 159 | C2×C4×C8 | C2×D83 | 1 | 1 | yes | characteristic |
| 160 | C4⋊C16 | — | 166 | 2 | no | — |
| 161 | #350 | — | 166 | 2 | no | — |
| 162 | #349 | — | 166 | 2 | no | — |
| 163 | #347 | — | 166 | 2 | no | — |
| 164 | #341 | Dic83 | 1 | 1 | yes | characteristic |
| 165 | #339 | Dic83 | 1 | 1 | yes | characteristic |
| 166 | C2×#41 | C2×Dic83 | 1 | 1 | yes | characteristic |
| 167 | #131 | — | 2 | 1 | no | — |
| 168 | #131 | — | 2 | 1 | no | — |
| 169 | #131 | — | 2 | 1 | no | — |
| 170 | #131 | — | 2 | 1 | no | — |
| 171 | C2×M16 | — | 2 | 1 | no | — |
| 172 | C2×M16 | C2×Dic83 | 1 | 1 | yes | characteristic |
| 173 | C2×M16 | C2×Dic83 | 1 | 1 | yes | characteristic |
| 174 | C22×C8 | — | 2 | 1 | no | — |
| 175 | C22×C8 | C2×Dic83 | 1 | 1 | yes | characteristic |
| 176 | C22×C8 | C22×D83 | 1 | 1 | yes | characteristic |
| 177 | #117 | C2×Dic83 | 1 | 1 | yes | characteristic |
| 178 | C2×C42 | C2×Dic83 | 1 | 1 | yes | characteristic |
| 179 | M32 | — | 83 | 2 | no | — |
| 180 | M32 | — | 166 | 1 | no | — |
| 181 | M32 | — | 166 | 1 | no | — |
| 182 | M32 | — | 166 | 1 | no | — |
| 183 | M32 | — | 166 | 1 | no | — |
| 184 | M32 | — | 166 | 1 | no | — |
| 185 | M32 | — | 166 | 1 | no | — |
| 186 | M32 | — | 166 | 1 | no | — |
| 187 | C2×C16 | — | 83 | 2 | no | — |
| 188 | C2×C16 | — | 166 | 1 | no | — |
| 189 | C2×C16 | — | 166 | 1 | no | — |
| 190 | C2×C16 | — | 166 | 1 | no | — |
| 191 | #104 | — | 2 | 1 | no | — |
| 192 | #104 | — | 2 | 1 | no | — |
| 193 | #103 | — | 2 | 1 | no | — |
| 194 | #102 | — | 2 | 1 | no | — |
| 195 | #97 | — | 2 | 1 | no | — |
| 196 | C4×C8 | — | 2 | 1 | no | — |
| 197 | C4×C8 | — | 2 | 1 | no | — |
| 198 | C4×C8 | — | 2 | 1 | no | — |
| 199 | #41 | — | 2 | 1 | no | — |
| 200 | #41 | — | 2 | 1 | no | — |
| 201 | #41 | — | 2 | 1 | no | — |
| 202 | #41 | — | 2 | 1 | no | — |
| 203 | C2×Q8 | #13050574 | 1 | 1 | yes | characteristic |
| 204 | C2×D4 | — | 2 | 1 | no | — |
| 205 | C2×D4 | #13050574 | 1 | 1 | yes | characteristic |
| 206 | C22×C4 | — | 2 | 1 | no | — |
| 207 | C22×C4 | — | 2 | 1 | no | — |
| 208 | C22×C4 | C22×Dic83 | 1 | 1 | yes | characteristic |
| 209 | M16 | — | 2 | 1 | no | — |
| 210 | M16 | — | 2 | 1 | no | — |
| 211 | M16 | — | 2 | 1 | no | — |
| 212 | M16 | — | 2 | 1 | no | — |
| 213 | M16 | — | 4 | 1 | no | — |
| 214 | M16 | — | 4 | 1 | no | — |
| 215 | C2×C8 | — | 2 | 1 | no | — |
| 216 | C2×C8 | — | 2 | 1 | no | — |
| 217 | C2×C8 | — | 2 | 1 | no | — |
| 218 | C2×C8 | — | 2 | 1 | no | — |
| 219 | C2×C8 | — | 2 | 1 | no | — |
| 220 | C2×C8 | — | 4 | 1 | no | — |
| 221 | C2×C8 | — | 4 | 1 | no | — |
| 222 | C2×C8 | D4×D83 | 1 | 1 | yes | characteristic |
| 223 | C2×C8 | D4×D83 | 1 | 1 | yes | characteristic |
| 224 | C2×C8 | #13050589 | 1 | 1 | yes | characteristic |
| 225 | C2×C8 | #13050589 | 1 | 1 | yes | characteristic |
| 226 | C2×C8 | #13050590 | 1 | 1 | yes | characteristic |
| 227 | C2×C8 | #13050590 | 1 | 1 | yes | characteristic |
| 228 | C4⋊C4 | — | 2 | 1 | no | — |
| 229 | C22⋊C4 | — | 2 | 1 | no | — |
| 230 | C42 | — | 2 | 1 | no | — |
| 231 | C42 | — | 2 | 1 | no | — |
| 232 | C42 | #13050574 | 1 | 1 | yes | characteristic |
| 233 | C42 | #13050574 | 1 | 1 | yes | characteristic |
| 234 | C16 | — | 332 | 1 | no | — |
| 235 | C16 | — | 332 | 1 | no | — |
| 236 | C16 | — | 166 | 2 | no | — |
| 237 | C16 | — | 166 | 2 | no | — |
| 238 | C23 | — | 2 | 1 | no | — |
| 239 | C23 | #14581799 | 1 | 1 | yes | characteristic |
| 240 | Q8 | — | 2 | 1 | no | — |
| 241 | Q8 | — | 2 | 1 | no | — |
| 242 | D4 | — | 2 | 1 | no | — |
| 243 | D4 | — | 2 | 1 | no | — |
| 244 | D4 | — | 4 | 1 | no | — |
| 245 | D4 | — | 4 | 1 | no | — |
| 246 | C2×C4 | — | 2 | 1 | no | — |
| 247 | C2×C4 | — | 2 | 1 | no | — |
| 248 | C2×C4 | — | 2 | 1 | no | — |
| 249 | C2×C4 | — | 4 | 2 | no | — |
| 250 | C2×C4 | — | 4 | 1 | no | — |
| 251 | C2×C4 | — | 4 | 1 | no | — |
| 252 | C2×C4 | — | 4 | 1 | no | — |
| 253 | C2×C4 | #14581751 | 1 | 1 | yes | characteristic |
| 254 | C2×C4 | D4×Dic83 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | D4×Dic83 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | C2×#13050574 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | C2×#13050574 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | #14581799 | 1 | 1 | yes | characteristic |
| 259 | C8 | — | 4 | 1 | no | — |
| 260 | C8 | — | 4 | 1 | no | — |
| 261 | C8 | — | 4 | 1 | no | — |
| 262 | C8 | #14581765 | 1 | 1 | yes | characteristic |
| 263 | C8 | #14581774 | 1 | 1 | yes | characteristic |
| 264 | C8 | #14581778 | 1 | 1 | yes | characteristic |
| 265 | C8 | #14581785 | 1 | 1 | yes | characteristic |
| 266 | C22 | — | 4 | 1 | no | — |
| 267 | C22 | — | 4 | 1 | no | — |
| 268 | C22 | — | 8 | 1 | no | — |
| 269 | C22 | #19127762 | 1 | 1 | yes | characteristic |
| 270 | C22 | #19127773 | 1 | 1 | yes | characteristic |
| 271 | C22 | #19127773 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 2 | no | — |
| 273 | C4 | — | 4 | 1 | no | — |
| 274 | C4 | — | 4 | 1 | no | — |
| 275 | C4 | — | 8 | 1 | no | — |
| 276 | C4 | #19127594 | 1 | 1 | yes | characteristic |
| 277 | C4 | #19127758 | 1 | 1 | yes | characteristic |
| 278 | C4 | #19127773 | 1 | 1 | yes | characteristic |
| 279 | C4 | #19127779 | 1 | 1 | yes | characteristic |
| 280 | C2 | — | 2 | 1 | no | — |
| 281 | C2 | — | 2 | 1 | no | — |
| 282 | C2 | — | 2 | 1 | no | — |
| 283 | C2 | — | 8 | 1 | no | — |
| 284 | C2 | #24908078 | 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