← #24921874
Subgroups — #24921874
Order 10624.
1002 conjugacy classes of subgroups.
The subgroup lattice has 316 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #19128455 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #19128436 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | Q8×Dic166 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #19128425 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #19128424 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | Q8×D332 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #19128366 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #19128324 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #19128322 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #19128274 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #19128258 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #19128251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 13 | #533×C83 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | C2×#13050590 | C22 | 1 | 2 | yes | — |
| 15 | C2×Q8×D83 | C22 | 1 | 4 | yes | — |
| 16 | #14581792 | C22 | 1 | 4 | yes | — |
| 17 | #14581792 | C22 | 1 | 1 | yes | characteristic |
| 18 | #14581792 | C22 | 1 | 1 | yes | characteristic |
| 19 | #14581792 | C22 | 1 | 1 | yes | characteristic |
| 20 | #14581791 | C22 | 1 | 4 | yes | — |
| 21 | #14581791 | C22 | 1 | 8 | yes | — |
| 22 | #14581789 | C22 | 1 | 1 | yes | characteristic |
| 23 | #14581787 | C22 | 1 | 8 | yes | — |
| 24 | #14581787 | C22 | 1 | 8 | yes | — |
| 25 | #14581784 | C22 | 1 | 2 | yes | — |
| 26 | #14581781 | C22 | 1 | 4 | yes | — |
| 27 | #14581780 | C22 | 1 | 4 | yes | — |
| 28 | #14581780 | C22 | 1 | 2 | yes | — |
| 29 | #14581778 | C22 | 1 | 8 | yes | — |
| 30 | #14581777 | C22 | 1 | 8 | yes | — |
| 31 | #14581774 | C22 | 1 | 8 | yes | — |
| 32 | #14581772 | C22 | 1 | 8 | yes | — |
| 33 | #14581772 | C22 | 1 | 4 | yes | — |
| 34 | #14581771 | C22 | 1 | 2 | yes | — |
| 35 | #14581771 | C22 | 1 | 2 | yes | — |
| 36 | #14581770 | C22 | 1 | 8 | yes | — |
| 37 | #14581768 | C22 | 1 | 4 | yes | — |
| 38 | #14581766 | C22 | 1 | 2 | yes | — |
| 39 | #14581766 | C22 | 1 | 2 | yes | — |
| 40 | Q8×Dic83 | C22 | 1 | 4 | yes | — |
| 41 | C4×Dic166 | C22 | 1 | 4 | yes | — |
| 42 | C4×Dic166 | C22 | 1 | 2 | yes | — |
| 43 | C4×Dic166 | C22 | 1 | 2 | yes | — |
| 44 | C4×D332 | C22 | 1 | 4 | yes | — |
| 45 | C4×D332 | C22 | 1 | 2 | yes | — |
| 46 | C4×D332 | C22 | 1 | 2 | yes | — |
| 47 | #14581754 | C22 | 1 | 4 | yes | — |
| 48 | #14581751 | C22 | 1 | 2 | yes | — |
| 49 | #14581750 | C22 | 1 | 4 | yes | — |
| 50 | C4⋊Q8×C83 | C22 | 1 | 1 | yes | characteristic |
| 51 | C4⋊Q8×C83 | C22 | 1 | 1 | yes | characteristic |
| 52 | #125×C83 | C22 | 1 | 2 | yes | — |
| 53 | C22⋊Q8×C83 | C22 | 1 | 2 | yes | — |
| 54 | C22⋊Q8×C83 | C22 | 1 | 2 | yes | — |
| 55 | C4×Q8×C83 | C22 | 1 | 2 | yes | — |
| 56 | C4×Q8×C83 | C22 | 1 | 2 | yes | — |
| 57 | #117×C83 | C22 | 1 | 2 | yes | — |
| 58 | C2×C42×C83 | C22 | 1 | 1 | yes | characteristic |
| 59 | #13050590 | — | 4 | 4 | no | — |
| 60 | C2×Dic166 | C23 | 1 | 2 | yes | — |
| 61 | C2×Dic166 | C23 | 1 | 2 | yes | — |
| 62 | C2×Dic166 | C23 | 1 | 8 | yes | — |
| 63 | C2×Dic166 | C23 | 1 | 2 | yes | — |
| 64 | Q8×D83 | — | 4 | 8 | no | — |
| 65 | C2×D332 | C23 | 1 | 2 | yes | — |
| 66 | C2×D332 | C23 | 1 | 2 | yes | — |
| 67 | C2×D332 | C23 | 1 | 2 | yes | — |
| 68 | C2×C22⋊D83 | C23 | 1 | 4 | yes | — |
| 69 | C2×C4×D83 | C23 | 1 | 4 | yes | — |
| 70 | C2×C4×D83 | C23 | 1 | 8 | yes | — |
| 71 | C4⋊Dic83 | C23 | 1 | 2 | yes | — |
| 72 | C4⋊Dic83 | C23 | 1 | 2 | yes | — |
| 73 | C4⋊Dic83 | C23 | 1 | 2 | yes | — |
| 74 | C4⋊Dic83 | C23 | 1 | 2 | yes | — |
| 75 | C4⋊Dic83 | C23 | 1 | 2 | yes | — |
| 76 | C4⋊Dic83 | C23 | 1 | 8 | yes | — |
| 77 | C4⋊Dic83 | C23 | 1 | 2 | yes | — |
| 78 | Dic83⋊C4 | C23 | 1 | 8 | yes | — |
| 79 | Dic83⋊C4 | C23 | 1 | 4 | yes | — |
| 80 | Dic83⋊C4 | C23 | 1 | 4 | yes | — |
| 81 | C22⋊Dic83 | C23 | 1 | 4 | yes | — |
| 82 | (C2×D83)⋊C4 | C23 | 1 | 4 | yes | — |
| 83 | (C2×D83)⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | (C2×D83)⋊C4 | C23 | 1 | 8 | yes | — |
| 85 | (C2×D83)⋊C4 | C23 | 1 | 8 | yes | — |
| 86 | (C2×D83)⋊C4 | C23 | 1 | 8 | yes | — |
| 87 | C4×Dic83 | C23 | 1 | 8 | yes | — |
| 88 | C4×Dic83 | C23 | 1 | 4 | yes | — |
| 89 | C2×Q8×C83 | C23 | 1 | 2 | yes | — |
| 90 | C2×Q8×C83 | C23 | 1 | 2 | yes | — |
| 91 | C22×C4×C83 | C23 | 1 | 1 | yes | characteristic |
| 92 | C22×C4×C83 | C23 | 1 | 1 | yes | characteristic |
| 93 | C22×C4×C83 | C23 | 1 | 1 | yes | characteristic |
| 94 | C4⋊C4×C83 | C23 | 1 | 2 | yes | — |
| 95 | C4⋊C4×C83 | C23 | 1 | 4 | yes | — |
| 96 | C4⋊C4×C83 | C23 | 1 | 2 | yes | — |
| 97 | C4⋊C4×C83 | C23 | 1 | 4 | yes | — |
| 98 | C4⋊C4×C83 | C23 | 1 | 2 | yes | — |
| 99 | C4⋊C4×C83 | C23 | 1 | 2 | yes | — |
| 100 | C22⋊C4×C83 | C23 | 1 | 4 | yes | — |
| 101 | C42×C83 | C23 | 1 | 1 | yes | characteristic |
| 102 | C42×C83 | C23 | 1 | 4 | yes | — |
| 103 | C42×C83 | C23 | 1 | 1 | yes | characteristic |
| 104 | C42×C83 | C23 | 1 | 1 | yes | characteristic |
| 105 | C42×C83 | C23 | 1 | 1 | yes | characteristic |
| 106 | C23×C83 | C24 | 1 | 1 | yes | characteristic |
| 107 | C22×D83 | C24 | 1 | 4 | yes | — |
| 108 | Q8×C83 | — | 2 | 4 | no | — |
| 109 | Q8×C83 | — | 2 | 4 | no | — |
| 110 | C2×C4×C83 | — | 2 | 2 | no | — |
| 111 | C2×C4×C83 | — | 2 | 2 | no | — |
| 112 | C2×C4×C83 | — | 2 | 2 | no | — |
| 113 | C2×C4×C83 | C24 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C83 | C24 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C83 | C24 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C83 | C24 | 1 | 4 | yes | — |
| 117 | C2×C4×C83 | C24 | 1 | 4 | yes | — |
| 118 | C2×C4×C83 | C24 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C83 | C24 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C83 | C24 | 1 | 1 | yes | characteristic |
| 121 | C22⋊D83 | — | 4 | 4 | no | — |
| 122 | C2×Dic83 | C24 | 1 | 4 | yes | — |
| 123 | C2×Dic83 | C24 | 1 | 4 | yes | — |
| 124 | C2×Dic83 | C24 | 1 | 4 | yes | — |
| 125 | D332 | — | 2 | 4 | no | — |
| 126 | D332 | — | 2 | 4 | no | — |
| 127 | D332 | — | 4 | 2 | no | — |
| 128 | C4×D83 | — | 2 | 8 | no | — |
| 129 | C4×D83 | — | 4 | 8 | no | — |
| 130 | Dic166 | — | 2 | 4 | no | — |
| 131 | Dic166 | — | 2 | 4 | no | — |
| 132 | Dic166 | — | 4 | 8 | no | — |
| 133 | Dic166 | — | 4 | 2 | no | — |
| 134 | C22×C83 | — | 2 | 1 | no | — |
| 135 | C22×C83 | — | 2 | 1 | no | — |
| 136 | C22×C83 | — | 2 | 1 | no | — |
| 137 | C22×C83 | C25 | 1 | 1 | yes | characteristic |
| 138 | C2×D83 | — | 2 | 4 | no | — |
| 139 | C2×D83 | — | 2 | 4 | no | — |
| 140 | C2×D83 | — | 2 | 4 | no | — |
| 141 | C4×C83 | — | 2 | 4 | no | — |
| 142 | C4×C83 | — | 2 | 1 | no | — |
| 143 | C4×C83 | — | 2 | 1 | no | — |
| 144 | C4×C83 | — | 2 | 1 | no | — |
| 145 | C4×C83 | — | 2 | 4 | no | — |
| 146 | C4×C83 | H(5,𝔽2) | 1 | 2 | yes | — |
| 147 | C4×C83 | 2−1+4 | 1 | 2 | yes | — |
| 148 | C4×C83 | 2−1+4 | 1 | 2 | yes | — |
| 149 | Dic83 | — | 2 | 4 | no | — |
| 150 | Dic83 | — | 2 | 4 | no | — |
| 151 | Dic83 | — | 2 | 4 | no | — |
| 152 | C2×C83 | — | 4 | 1 | no | — |
| 153 | C2×C83 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 154 | C2×C83 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 155 | C2×C83 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 156 | D83 | — | 4 | 4 | no | — |
| 157 | #3550 | — | 83 | 1 | no | maximal, Sylow |
| 158 | C83 | #3550 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 159 | #563 | — | 83 | 4 | no | — |
| 160 | #559 | — | 83 | 2 | no | — |
| 161 | Q82 | — | 83 | 2 | no | — |
| 162 | #556 | — | 83 | 2 | no | — |
| 163 | #556 | — | 83 | 2 | no | — |
| 164 | D4×Q8 | — | 83 | 2 | no | — |
| 165 | D4×Q8 | — | 83 | 2 | no | — |
| 166 | #542 | — | 83 | 4 | no | — |
| 167 | #542 | — | 83 | 4 | no | — |
| 168 | #536 | — | 83 | 4 | no | — |
| 169 | #533 | — | 83 | 1 | no | — |
| 170 | #533 | D83 | 1 | 1 | yes | characteristic, complemented |
| 171 | #532 | — | 83 | 1 | no | — |
| 172 | C2×#41 | — | 83 | 2 | no | — |
| 173 | C22×Q8 | — | 83 | 4 | no | — |
| 174 | C4⋊Q8 | — | 83 | 8 | no | — |
| 175 | C4⋊Q8 | — | 83 | 4 | no | — |
| 176 | C4⋊Q8 | — | 83 | 1 | no | — |
| 177 | C4⋊Q8 | — | 83 | 1 | no | — |
| 178 | C4⋊Q8 | — | 83 | 1 | no | — |
| 179 | C4⋊Q8 | — | 83 | 4 | no | — |
| 180 | C4⋊Q8 | C2×D83 | 1 | 1 | yes | characteristic |
| 181 | C4⋊Q8 | C2×D83 | 1 | 1 | yes | characteristic |
| 182 | #127 | — | 83 | 1 | no | — |
| 183 | #126 | — | 83 | 8 | no | — |
| 184 | #126 | — | 83 | 8 | no | — |
| 185 | #125 | — | 83 | 2 | no | — |
| 186 | #125 | C2×D83 | 1 | 2 | yes | — |
| 187 | #124 | — | 83 | 4 | no | — |
| 188 | #124 | — | 83 | 8 | no | — |
| 189 | #124 | — | 83 | 2 | no | — |
| 190 | #124 | — | 83 | 4 | no | — |
| 191 | #123 | — | 83 | 8 | no | — |
| 192 | #123 | — | 83 | 8 | no | — |
| 193 | C22⋊Q8 | — | 83 | 4 | no | — |
| 194 | C22⋊Q8 | — | 83 | 2 | no | — |
| 195 | C22⋊Q8 | — | 83 | 2 | no | — |
| 196 | C22⋊Q8 | — | 83 | 8 | no | — |
| 197 | C22⋊Q8 | — | 83 | 4 | no | — |
| 198 | C22⋊Q8 | — | 83 | 8 | no | — |
| 199 | C22⋊Q8 | C2×D83 | 1 | 2 | yes | — |
| 200 | C22⋊Q8 | C2×D83 | 1 | 2 | yes | — |
| 201 | #121 | — | 83 | 2 | no | — |
| 202 | #121 | — | 83 | 2 | no | — |
| 203 | C4×Q8 | — | 83 | 4 | no | — |
| 204 | C4×Q8 | — | 83 | 2 | no | — |
| 205 | C4×Q8 | — | 83 | 2 | no | — |
| 206 | C4×Q8 | — | 83 | 4 | no | — |
| 207 | C4×Q8 | C2×D83 | 1 | 2 | yes | — |
| 208 | C4×Q8 | C2×D83 | 1 | 2 | yes | — |
| 209 | C4×D4 | — | 83 | 4 | no | — |
| 210 | C4×D4 | — | 83 | 2 | no | — |
| 211 | C4×D4 | — | 83 | 2 | no | — |
| 212 | C4×D4 | — | 83 | 4 | no | — |
| 213 | #117 | — | 83 | 2 | no | — |
| 214 | #117 | — | 83 | 4 | no | — |
| 215 | #117 | C2×D83 | 1 | 2 | yes | — |
| 216 | C2×C42 | C2×D83 | 1 | 1 | yes | characteristic |
| 217 | #41 | — | 332 | 4 | no | — |
| 218 | C2×Q8 | — | 332 | 8 | no | — |
| 219 | C2×Q8 | — | 83 | 8 | no | — |
| 220 | C2×Q8 | — | 83 | 2 | no | — |
| 221 | C2×Q8 | — | 83 | 2 | no | — |
| 222 | C2×Q8 | — | 83 | 2 | no | — |
| 223 | C2×Q8 | C22×D83 | 1 | 2 | yes | — |
| 224 | C2×Q8 | C22×D83 | 1 | 2 | yes | — |
| 225 | C2×D4 | — | 83 | 4 | no | — |
| 226 | C2×D4 | — | 83 | 2 | no | — |
| 227 | C2×D4 | — | 83 | 2 | no | — |
| 228 | C2×D4 | — | 83 | 2 | no | — |
| 229 | C22×C4 | — | 83 | 8 | no | — |
| 230 | C22×C4 | — | 83 | 4 | no | — |
| 231 | C22×C4 | C22×D83 | 1 | 1 | yes | characteristic |
| 232 | C22×C4 | C22×D83 | 1 | 1 | yes | characteristic |
| 233 | C22×C4 | C22×D83 | 1 | 1 | yes | characteristic |
| 234 | C4⋊C4 | — | 83 | 2 | no | — |
| 235 | C4⋊C4 | — | 83 | 2 | no | — |
| 236 | C4⋊C4 | — | 83 | 2 | no | — |
| 237 | C4⋊C4 | — | 83 | 2 | no | — |
| 238 | C4⋊C4 | — | 83 | 2 | no | — |
| 239 | C4⋊C4 | — | 83 | 2 | no | — |
| 240 | C4⋊C4 | — | 83 | 4 | no | — |
| 241 | C4⋊C4 | — | 83 | 8 | no | — |
| 242 | C4⋊C4 | — | 83 | 8 | no | — |
| 243 | C4⋊C4 | — | 83 | 4 | no | — |
| 244 | C4⋊C4 | C22×D83 | 1 | 2 | yes | — |
| 245 | C4⋊C4 | C22×D83 | 1 | 2 | yes | — |
| 246 | C4⋊C4 | C22×D83 | 1 | 2 | yes | — |
| 247 | C4⋊C4 | C22×D83 | 1 | 2 | yes | — |
| 248 | C4⋊C4 | C22×D83 | 1 | 4 | yes | — |
| 249 | C4⋊C4 | C22×D83 | 1 | 4 | yes | — |
| 250 | C22⋊C4 | — | 83 | 4 | no | — |
| 251 | C22⋊C4 | — | 83 | 8 | no | — |
| 252 | C22⋊C4 | — | 83 | 4 | no | — |
| 253 | C22⋊C4 | — | 83 | 4 | no | — |
| 254 | C22⋊C4 | — | 83 | 8 | no | — |
| 255 | C22⋊C4 | — | 83 | 8 | no | — |
| 256 | C22⋊C4 | C22×D83 | 1 | 4 | yes | — |
| 257 | C42 | — | 83 | 4 | no | — |
| 258 | C42 | — | 83 | 8 | no | — |
| 259 | C42 | C22×D83 | 1 | 1 | yes | characteristic |
| 260 | C42 | C22×D83 | 1 | 1 | yes | characteristic |
| 261 | C42 | C22×D83 | 1 | 1 | yes | characteristic |
| 262 | C42 | C22×D83 | 1 | 1 | yes | characteristic |
| 263 | C42 | C22×D83 | 1 | 4 | yes | — |
| 264 | C23 | — | 83 | 4 | no | — |
| 265 | C23 | C23×D83 | 1 | 1 | yes | characteristic |
| 266 | Q8 | — | 2 | 4 | no | — |
| 267 | Q8 | — | 2 | 4 | no | — |
| 268 | Q8 | — | 332 | 8 | no | — |
| 269 | Q8 | — | 332 | 2 | no | — |
| 270 | Q8 | — | 166 | 4 | no | — |
| 271 | Q8 | — | 166 | 4 | no | — |
| 272 | D4 | — | 332 | 2 | no | — |
| 273 | D4 | — | 332 | 4 | no | — |
| 274 | D4 | — | 166 | 4 | no | — |
| 275 | D4 | — | 166 | 4 | no | — |
| 276 | C2×C4 | — | 2 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 332 | 8 | no | — |
| 280 | C2×C4 | — | 83 | 4 | no | — |
| 281 | C2×C4 | — | 83 | 4 | no | — |
| 282 | C2×C4 | — | 83 | 4 | no | — |
| 283 | C2×C4 | — | 166 | 8 | no | — |
| 284 | C2×C4 | C23×D83 | 1 | 4 | yes | — |
| 285 | C2×C4 | C23×D83 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | C23×D83 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C23×D83 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C23×D83 | 1 | 4 | yes | — |
| 289 | C2×C4 | C23×D83 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C23×D83 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C23×D83 | 1 | 1 | yes | characteristic |
| 292 | C22 | — | 2 | 1 | no | — |
| 293 | C22 | — | 2 | 1 | no | — |
| 294 | C22 | — | 2 | 1 | no | — |
| 295 | C22 | — | 166 | 4 | no | — |
| 296 | C22 | — | 166 | 4 | no | — |
| 297 | C22 | — | 166 | 4 | no | — |
| 298 | C22 | C24×D83 | 1 | 1 | yes | characteristic |
| 299 | C4 | — | 2 | 1 | no | — |
| 300 | C4 | — | 2 | 4 | no | — |
| 301 | C4 | — | 2 | 1 | no | — |
| 302 | C4 | — | 2 | 4 | no | — |
| 303 | C4 | — | 2 | 1 | no | — |
| 304 | C4 | — | 166 | 4 | no | — |
| 305 | C4 | — | 166 | 4 | no | — |
| 306 | C4 | — | 166 | 4 | no | — |
| 307 | C4 | #14581838 | 1 | 2 | yes | — |
| 308 | C4 | #14581839 | 1 | 2 | yes | — |
| 309 | C4 | #14581840 | 1 | 2 | yes | — |
| 310 | C2 | — | 4 | 1 | no | — |
| 311 | C2 | — | 332 | 4 | no | — |
| 312 | C2 | C2×#14581838 | 1 | 1 | yes | characteristic |
| 313 | C2 | C2×#14581839 | 1 | 1 | yes | characteristic |
| 314 | C2 | C2×#14581840 | 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