← #35224347
Subgroups — #35224347
Order 20864.
416 conjugacy classes of subgroups.
The subgroup lattice has 278 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#19104734 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #41×Dic163 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #23781958 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23781953 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #23781948 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #23781943 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #23781707 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #23781705 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #23781688 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #23781666 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #417×C163 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C22×Dic326 | — | 2 | 1 | no | — |
| 13 | C2×#14579683 | C22 | 1 | 1 | yes | characteristic |
| 14 | #19104734 | — | 2 | 1 | no | — |
| 15 | #19104734 | — | 2 | 1 | no | — |
| 16 | #19104734 | C22 | 1 | 2 | yes | — |
| 17 | #19104734 | C22 | 1 | 2 | yes | — |
| 18 | Q8×Dic163 | C22 | 1 | 1 | yes | characteristic |
| 19 | Q8×Dic163 | C22 | 1 | 1 | yes | characteristic |
| 20 | D4×Dic163 | — | 2 | 1 | no | — |
| 21 | D4×Dic163 | — | 2 | 1 | no | — |
| 22 | D4×Dic163 | C22 | 1 | 1 | yes | characteristic |
| 23 | D4×Dic163 | C22 | 1 | 1 | yes | characteristic |
| 24 | #19104694 | — | 2 | 1 | no | — |
| 25 | #19104694 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×C4⋊Dic163 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×Dic163⋊C4 | — | 2 | 1 | no | — |
| 28 | C2×C4×Dic163 | — | 2 | 1 | no | — |
| 29 | C2×C4×Dic163 | C22 | 1 | 1 | yes | characteristic |
| 30 | #19104669 | C22 | 1 | 2 | yes | — |
| 31 | #19104668 | C22 | 1 | 2 | yes | — |
| 32 | #19104667 | C22 | 1 | 2 | yes | — |
| 33 | #19104666 | C22 | 1 | 2 | yes | — |
| 34 | Dic163⋊C8 | C22 | 1 | 2 | yes | — |
| 35 | Dic163⋊C8 | C22 | 1 | 2 | yes | — |
| 36 | #19104659 | C22 | 1 | 2 | yes | — |
| 37 | #19104656 | C22 | 1 | 2 | yes | — |
| 38 | C2×#41×C163 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×M16×C163 | C22 | 1 | 1 | yes | characteristic |
| 40 | C2×C4⋊C4×C163 | C22 | 1 | 1 | yes | characteristic |
| 41 | #103×C163 | C22 | 1 | 2 | yes | — |
| 42 | #102×C163 | C22 | 1 | 2 | yes | — |
| 43 | C2×Dic326 | — | 2 | 2 | no | — |
| 44 | C2×Dic326 | — | 2 | 2 | no | — |
| 45 | C2×Dic326 | — | 4 | 4 | no | — |
| 46 | C22×Dic163 | — | 2 | 1 | no | — |
| 47 | C22×Dic163 | — | 2 | 1 | no | — |
| 48 | C22×Dic163 | — | 2 | 1 | no | — |
| 49 | C22×Dic163 | D4 | 1 | 1 | yes | characteristic |
| 50 | C22×Dic163 | D4 | 1 | 1 | yes | characteristic |
| 51 | #14579683 | — | 2 | 2 | no | — |
| 52 | C2×C163⋊C8 | C23 | 1 | 2 | yes | — |
| 53 | C4⋊Dic163 | — | 2 | 1 | no | — |
| 54 | C4⋊Dic163 | — | 2 | 1 | no | — |
| 55 | C4⋊Dic163 | — | 2 | 1 | no | — |
| 56 | C4⋊Dic163 | C23 | 1 | 2 | yes | — |
| 57 | C4⋊Dic163 | C23 | 1 | 1 | yes | characteristic |
| 58 | C4⋊Dic163 | C23 | 1 | 1 | yes | characteristic |
| 59 | Dic163⋊C4 | — | 2 | 2 | no | — |
| 60 | Dic163⋊C4 | — | 2 | 2 | no | — |
| 61 | C22⋊Dic163 | — | 2 | 1 | no | — |
| 62 | C22⋊Dic163 | — | 2 | 1 | no | — |
| 63 | C22⋊Dic163 | — | 2 | 1 | no | — |
| 64 | C4×Dic163 | — | 2 | 1 | no | — |
| 65 | C4×Dic163 | — | 2 | 1 | no | — |
| 66 | C4×Dic163 | — | 2 | 1 | no | — |
| 67 | C4×Dic163 | D4 | 1 | 1 | yes | characteristic |
| 68 | C4×Dic163 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×Dic163 | C23 | 1 | 1 | yes | characteristic |
| 70 | C4×Dic163 | C23 | 1 | 1 | yes | characteristic |
| 71 | #41×C163 | — | 2 | 2 | no | — |
| 72 | #41×C163 | Q8 | 1 | 4 | yes | complemented |
| 73 | C2×Q8×C163 | C23 | 1 | 1 | yes | characteristic |
| 74 | C2×D4×C163 | — | 2 | 1 | no | — |
| 75 | C2×D4×C163 | C23 | 1 | 1 | yes | characteristic |
| 76 | C22×C4×C163 | — | 2 | 1 | no | — |
| 77 | C22×C4×C163 | — | 2 | 1 | no | — |
| 78 | C22×C4×C163 | C23 | 1 | 1 | yes | characteristic |
| 79 | M16×C163 | — | 2 | 2 | no | — |
| 80 | C2×C8×C163 | C23 | 1 | 2 | yes | — |
| 81 | C4⋊C4×C163 | — | 2 | 1 | no | — |
| 82 | C4⋊C4×C163 | C23 | 1 | 2 | yes | — |
| 83 | Dic326 | — | 4 | 4 | no | — |
| 84 | Dic326 | — | 4 | 4 | no | — |
| 85 | C2×Dic163 | — | 2 | 2 | no | — |
| 86 | C2×Dic163 | — | 2 | 2 | no | — |
| 87 | C2×Dic163 | — | 2 | 1 | no | — |
| 88 | C2×Dic163 | — | 2 | 1 | no | — |
| 89 | C2×Dic163 | — | 2 | 2 | no | — |
| 90 | C2×Dic163 | — | 4 | 2 | no | — |
| 91 | C2×Dic163 | — | 4 | 2 | no | — |
| 92 | C2×Dic163 | — | 4 | 2 | no | — |
| 93 | C2×Dic163 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×Dic163 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×Dic163 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×Dic163 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C163⋊C8 | — | 2 | 2 | no | — |
| 98 | C23×C163 | — | 2 | 1 | no | — |
| 99 | C23×C163 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | Q8×C163 | — | 2 | 1 | no | — |
| 101 | Q8×C163 | C2×Q8 | 1 | 2 | yes | complemented |
| 102 | D4×C163 | — | 2 | 4 | no | — |
| 103 | D4×C163 | — | 2 | 1 | no | — |
| 104 | D4×C163 | C2×Q8 | 1 | 2 | yes | complemented |
| 105 | C2×C4×C163 | — | 2 | 1 | no | — |
| 106 | C2×C4×C163 | — | 2 | 4 | no | — |
| 107 | C2×C4×C163 | — | 2 | 2 | no | — |
| 108 | C2×C4×C163 | — | 4 | 2 | no | — |
| 109 | C2×C4×C163 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C163 | C2×Q8 | 1 | 2 | yes | — |
| 111 | C2×C4×C163 | #41 | 1 | 2 | yes | — |
| 112 | C2×C4×C163 | C24 | 1 | 1 | yes | characteristic |
| 113 | C8×C163 | — | 2 | 2 | no | — |
| 114 | C22×C163 | — | 2 | 1 | no | — |
| 115 | C22×C163 | — | 2 | 1 | no | — |
| 116 | C22×C163 | — | 2 | 2 | no | — |
| 117 | C22×C163 | — | 4 | 1 | no | — |
| 118 | C22×C163 | — | 4 | 1 | no | — |
| 119 | C22×C163 | C22⋊Q8 | 1 | 2 | yes | — |
| 120 | C22×C163 | C22×D4 | 1 | 1 | yes | characteristic |
| 121 | C4×C163 | — | 2 | 2 | no | — |
| 122 | C4×C163 | — | 4 | 2 | no | — |
| 123 | C4×C163 | C22⋊Q8 | 1 | 2 | yes | — |
| 124 | C4×C163 | C22×Q8 | 1 | 1 | yes | characteristic |
| 125 | C4×C163 | C2×#41 | 1 | 1 | yes | characteristic |
| 126 | Dic163 | — | 2 | 1 | no | — |
| 127 | Dic163 | — | 2 | 1 | no | — |
| 128 | Dic163 | — | 4 | 1 | no | — |
| 129 | Dic163 | — | 4 | 2 | no | — |
| 130 | Dic163 | — | 4 | 1 | no | — |
| 131 | Dic163 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 132 | Dic163 | #137 | 1 | 2 | yes | — |
| 133 | C2×C163 | — | 2 | 2 | no | — |
| 134 | C2×C163 | — | 4 | 2 | no | — |
| 135 | C2×C163 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 136 | C2×C163 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 137 | C2×C163 | C2×#137 | 1 | 1 | yes | characteristic |
| 138 | C163 | #3079 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 139 | #3079 | — | 163 | 1 | no | maximal, Sylow |
| 140 | C2×C4⋊Q8 | — | 163 | 1 | no | — |
| 141 | C4×#41 | — | 163 | 1 | no | — |
| 142 | #477 | — | 163 | 2 | no | — |
| 143 | #476 | — | 163 | 2 | no | — |
| 144 | #475 | — | 163 | 2 | no | — |
| 145 | #474 | — | 163 | 2 | no | — |
| 146 | #428 | — | 163 | 1 | no | — |
| 147 | #428 | — | 163 | 1 | no | — |
| 148 | #423 | — | 163 | 1 | no | — |
| 149 | #417 | — | 163 | 1 | no | — |
| 150 | #417 | D163 | 1 | 1 | yes | characteristic |
| 151 | C2×#41 | C2×D163 | 1 | 1 | yes | characteristic |
| 152 | C22×Q8 | — | 326 | 1 | no | — |
| 153 | C2×M16 | — | 163 | 1 | no | — |
| 154 | C2×M16 | C2×D163 | 1 | 1 | yes | characteristic |
| 155 | C4⋊Q8 | — | 326 | 1 | no | — |
| 156 | C4⋊Q8 | — | 326 | 1 | no | — |
| 157 | C4⋊Q8 | — | 163 | 2 | no | — |
| 158 | C4⋊Q8 | — | 163 | 2 | no | — |
| 159 | C4×Q8 | — | 163 | 1 | no | — |
| 160 | C4×Q8 | — | 163 | 1 | no | — |
| 161 | C4×D4 | — | 326 | 1 | no | — |
| 162 | C4×D4 | — | 326 | 1 | no | — |
| 163 | C4×D4 | — | 163 | 1 | no | — |
| 164 | C4×D4 | — | 163 | 1 | no | — |
| 165 | #117 | — | 326 | 1 | no | — |
| 166 | #117 | — | 163 | 1 | no | — |
| 167 | C2×C4⋊C4 | — | 326 | 1 | no | — |
| 168 | C2×C4⋊C4 | — | 163 | 1 | no | — |
| 169 | C2×C4⋊C4 | C2×D163 | 1 | 1 | yes | characteristic |
| 170 | C2×C42 | — | 326 | 1 | no | — |
| 171 | C2×C42 | — | 163 | 1 | no | — |
| 172 | #107 | — | 163 | 2 | no | — |
| 173 | #107 | — | 163 | 2 | no | — |
| 174 | #106 | — | 163 | 2 | no | — |
| 175 | #106 | — | 163 | 2 | no | — |
| 176 | C4⋊C8 | — | 163 | 2 | no | — |
| 177 | C4⋊C8 | — | 163 | 2 | no | — |
| 178 | #103 | — | 163 | 2 | no | — |
| 179 | #103 | C2×D163 | 1 | 2 | yes | — |
| 180 | #102 | — | 163 | 2 | no | — |
| 181 | #102 | C2×D163 | 1 | 2 | yes | — |
| 182 | #41 | — | 2 | 2 | no | — |
| 183 | #41 | Dic326 | 1 | 4 | yes | complemented |
| 184 | C2×Q8 | — | 326 | 2 | no | — |
| 185 | C2×Q8 | — | 326 | 2 | no | — |
| 186 | C2×Q8 | — | 652 | 4 | no | — |
| 187 | C2×Q8 | C22×D163 | 1 | 1 | yes | characteristic |
| 188 | C2×D4 | — | 2 | 1 | no | — |
| 189 | C2×D4 | C22×D163 | 1 | 1 | yes | characteristic |
| 190 | C22×C4 | — | 2 | 1 | no | — |
| 191 | C22×C4 | — | 2 | 1 | no | — |
| 192 | C22×C4 | — | 326 | 1 | no | — |
| 193 | C22×C4 | — | 326 | 1 | no | — |
| 194 | C22×C4 | — | 326 | 1 | no | — |
| 195 | C22×C4 | — | 163 | 1 | no | — |
| 196 | C22×C4 | — | 163 | 1 | no | — |
| 197 | C22×C4 | C22×D163 | 1 | 1 | yes | characteristic |
| 198 | M16 | — | 2 | 2 | no | — |
| 199 | M16 | — | 326 | 2 | no | — |
| 200 | C2×C8 | — | 163 | 2 | no | — |
| 201 | C2×C8 | C22×D163 | 1 | 2 | yes | — |
| 202 | C4⋊C4 | — | 2 | 1 | no | — |
| 203 | C4⋊C4 | — | 326 | 2 | no | — |
| 204 | C4⋊C4 | — | 326 | 2 | no | — |
| 205 | C4⋊C4 | — | 326 | 1 | no | — |
| 206 | C4⋊C4 | — | 326 | 1 | no | — |
| 207 | C4⋊C4 | — | 326 | 1 | no | — |
| 208 | C4⋊C4 | — | 163 | 2 | no | — |
| 209 | C4⋊C4 | — | 163 | 1 | no | — |
| 210 | C4⋊C4 | — | 163 | 1 | no | — |
| 211 | C4⋊C4 | C22×D163 | 1 | 2 | yes | — |
| 212 | C22⋊C4 | — | 326 | 1 | no | — |
| 213 | C22⋊C4 | — | 326 | 1 | no | — |
| 214 | C22⋊C4 | — | 326 | 1 | no | — |
| 215 | C42 | — | 326 | 1 | no | — |
| 216 | C42 | — | 326 | 1 | no | — |
| 217 | C42 | — | 326 | 1 | no | — |
| 218 | C42 | — | 163 | 1 | no | — |
| 219 | C42 | — | 163 | 1 | no | — |
| 220 | C42 | — | 163 | 1 | no | — |
| 221 | C42 | — | 163 | 1 | no | — |
| 222 | C23 | — | 2 | 1 | no | — |
| 223 | C23 | D4×D163 | 1 | 1 | yes | characteristic |
| 224 | Q8 | — | 2 | 1 | no | — |
| 225 | Q8 | — | 652 | 4 | no | — |
| 226 | Q8 | — | 652 | 4 | no | — |
| 227 | Q8 | C2×Dic326 | 1 | 2 | yes | complemented |
| 228 | D4 | — | 2 | 4 | no | — |
| 229 | D4 | — | 2 | 1 | no | — |
| 230 | D4 | C2×Dic326 | 1 | 2 | yes | complemented |
| 231 | C2×C4 | — | 2 | 1 | no | — |
| 232 | C2×C4 | — | 2 | 4 | no | — |
| 233 | C2×C4 | — | 2 | 2 | no | — |
| 234 | C2×C4 | — | 4 | 2 | no | — |
| 235 | C2×C4 | — | 326 | 2 | no | — |
| 236 | C2×C4 | — | 326 | 2 | no | — |
| 237 | C2×C4 | — | 326 | 1 | no | — |
| 238 | C2×C4 | — | 326 | 1 | no | — |
| 239 | C2×C4 | — | 326 | 2 | no | — |
| 240 | C2×C4 | — | 652 | 2 | no | — |
| 241 | C2×C4 | — | 652 | 2 | no | — |
| 242 | C2×C4 | — | 652 | 2 | no | — |
| 243 | C2×C4 | — | 163 | 1 | no | — |
| 244 | C2×C4 | — | 163 | 1 | no | — |
| 245 | C2×C4 | — | 163 | 1 | no | — |
| 246 | C2×C4 | — | 163 | 1 | no | — |
| 247 | C2×C4 | D4×D163 | 1 | 1 | yes | characteristic |
| 248 | C2×C4 | C2×Dic326 | 1 | 2 | yes | — |
| 249 | C2×C4 | #14579698 | 1 | 2 | yes | — |
| 250 | C2×C4 | C23×D163 | 1 | 1 | yes | characteristic |
| 251 | C8 | — | 2 | 2 | no | — |
| 252 | C8 | — | 326 | 2 | no | — |
| 253 | C22 | — | 2 | 1 | no | — |
| 254 | C22 | — | 2 | 1 | no | — |
| 255 | C22 | — | 2 | 2 | no | — |
| 256 | C22 | — | 4 | 1 | no | — |
| 257 | C22 | — | 4 | 1 | no | — |
| 258 | C22 | #19104712 | 1 | 2 | yes | — |
| 259 | C22 | C2×D4×D163 | 1 | 1 | yes | characteristic |
| 260 | C4 | — | 2 | 2 | no | — |
| 261 | C4 | — | 4 | 2 | no | — |
| 262 | C4 | — | 326 | 1 | no | — |
| 263 | C4 | — | 326 | 1 | no | — |
| 264 | C4 | — | 652 | 2 | no | — |
| 265 | C4 | — | 652 | 1 | no | — |
| 266 | C4 | — | 652 | 1 | no | — |
| 267 | C4 | — | 163 | 2 | no | — |
| 268 | C4 | — | 163 | 2 | no | — |
| 269 | C4 | #19104712 | 1 | 2 | yes | — |
| 270 | C4 | C22×Dic326 | 1 | 1 | yes | characteristic |
| 271 | C4 | C2×#14579698 | 1 | 1 | yes | characteristic |
| 272 | C2 | — | 2 | 2 | no | — |
| 273 | C2 | — | 4 | 2 | no | — |
| 274 | C2 | C2×#19104712 | 1 | 1 | yes | characteristic |
| 275 | C2 | AGL(1,ℤ/8)×D163 | 1 | 1 | yes | characteristic |
| 276 | C2 | #137×D163 | 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