← #30680372
Subgroups — #30680372
Order 16768.
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 | #22554066 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #22554047 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | Q8×Dic262 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22554036 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22554035 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | Q8×D524 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22553977 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #22553935 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #22553933 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #22553885 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #22553869 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #22553862 | C2 | 1 | 1 | yes | maximal, characteristic |
| 13 | #533×C131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | C2×#14523513 | C22 | 1 | 2 | yes | — |
| 15 | C2×Q8×D131 | C22 | 1 | 4 | yes | — |
| 16 | #16862246 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16862246 | C22 | 1 | 1 | yes | characteristic |
| 18 | #16862246 | C22 | 1 | 1 | yes | characteristic |
| 19 | #16862246 | C22 | 1 | 4 | yes | — |
| 20 | #16862245 | C22 | 1 | 8 | yes | — |
| 21 | #16862245 | C22 | 1 | 4 | yes | — |
| 22 | #16862243 | C22 | 1 | 1 | yes | characteristic |
| 23 | #16862241 | C22 | 1 | 8 | yes | — |
| 24 | #16862241 | C22 | 1 | 8 | yes | — |
| 25 | #16862238 | C22 | 1 | 2 | yes | — |
| 26 | #16862235 | C22 | 1 | 4 | yes | — |
| 27 | #16862234 | C22 | 1 | 2 | yes | — |
| 28 | #16862234 | C22 | 1 | 4 | yes | — |
| 29 | #16862232 | C22 | 1 | 8 | yes | — |
| 30 | #16862231 | C22 | 1 | 8 | yes | — |
| 31 | #16862228 | C22 | 1 | 8 | yes | — |
| 32 | #16862226 | C22 | 1 | 4 | yes | — |
| 33 | #16862226 | C22 | 1 | 8 | yes | — |
| 34 | #16862225 | C22 | 1 | 2 | yes | — |
| 35 | #16862225 | C22 | 1 | 2 | yes | — |
| 36 | #16862224 | C22 | 1 | 8 | yes | — |
| 37 | #16862222 | C22 | 1 | 4 | yes | — |
| 38 | #16862220 | C22 | 1 | 2 | yes | — |
| 39 | #16862220 | C22 | 1 | 2 | yes | — |
| 40 | Q8×Dic131 | C22 | 1 | 4 | yes | — |
| 41 | C4×Dic262 | C22 | 1 | 2 | yes | — |
| 42 | C4×Dic262 | C22 | 1 | 2 | yes | — |
| 43 | C4×Dic262 | C22 | 1 | 4 | yes | — |
| 44 | C4×D524 | C22 | 1 | 2 | yes | — |
| 45 | C4×D524 | C22 | 1 | 2 | yes | — |
| 46 | C4×D524 | C22 | 1 | 4 | yes | — |
| 47 | #16862208 | C22 | 1 | 4 | yes | — |
| 48 | #16862205 | C22 | 1 | 2 | yes | — |
| 49 | #16862204 | C22 | 1 | 4 | yes | — |
| 50 | C4⋊Q8×C131 | C22 | 1 | 1 | yes | characteristic |
| 51 | C4⋊Q8×C131 | C22 | 1 | 1 | yes | characteristic |
| 52 | #125×C131 | C22 | 1 | 2 | yes | — |
| 53 | C22⋊Q8×C131 | C22 | 1 | 2 | yes | — |
| 54 | C22⋊Q8×C131 | C22 | 1 | 2 | yes | — |
| 55 | C4×Q8×C131 | C22 | 1 | 2 | yes | — |
| 56 | C4×Q8×C131 | C22 | 1 | 2 | yes | — |
| 57 | #117×C131 | C22 | 1 | 2 | yes | — |
| 58 | C2×C42×C131 | C22 | 1 | 1 | yes | characteristic |
| 59 | #14523513 | — | 4 | 4 | no | — |
| 60 | C2×Dic262 | C23 | 1 | 2 | yes | — |
| 61 | C2×Dic262 | C23 | 1 | 2 | yes | — |
| 62 | C2×Dic262 | C23 | 1 | 8 | yes | — |
| 63 | C2×Dic262 | C23 | 1 | 2 | yes | — |
| 64 | Q8×D131 | — | 4 | 8 | no | — |
| 65 | C2×D524 | C23 | 1 | 2 | yes | — |
| 66 | C2×D524 | C23 | 1 | 2 | yes | — |
| 67 | C2×D524 | C23 | 1 | 2 | yes | — |
| 68 | C2×C22⋊D131 | C23 | 1 | 4 | yes | — |
| 69 | C2×C4×D131 | C23 | 1 | 4 | yes | — |
| 70 | C2×C4×D131 | C23 | 1 | 8 | yes | — |
| 71 | C4⋊Dic131 | C23 | 1 | 2 | yes | — |
| 72 | C4⋊Dic131 | C23 | 1 | 2 | yes | — |
| 73 | C4⋊Dic131 | C23 | 1 | 2 | yes | — |
| 74 | C4⋊Dic131 | C23 | 1 | 2 | yes | — |
| 75 | C4⋊Dic131 | C23 | 1 | 8 | yes | — |
| 76 | C4⋊Dic131 | C23 | 1 | 2 | yes | — |
| 77 | C4⋊Dic131 | C23 | 1 | 2 | yes | — |
| 78 | Dic131⋊C4 | C23 | 1 | 8 | yes | — |
| 79 | Dic131⋊C4 | C23 | 1 | 4 | yes | — |
| 80 | Dic131⋊C4 | C23 | 1 | 4 | yes | — |
| 81 | C22⋊Dic131 | C23 | 1 | 4 | yes | — |
| 82 | (C2×D131)⋊C4 | C23 | 1 | 8 | yes | — |
| 83 | (C2×D131)⋊C4 | C23 | 1 | 4 | yes | — |
| 84 | (C2×D131)⋊C4 | C23 | 1 | 4 | yes | — |
| 85 | (C2×D131)⋊C4 | C23 | 1 | 8 | yes | — |
| 86 | (C2×D131)⋊C4 | C23 | 1 | 8 | yes | — |
| 87 | C4×Dic131 | C23 | 1 | 4 | yes | — |
| 88 | C4×Dic131 | C23 | 1 | 8 | yes | — |
| 89 | C2×Q8×C131 | C23 | 1 | 2 | yes | — |
| 90 | C2×Q8×C131 | C23 | 1 | 2 | yes | — |
| 91 | C22×C4×C131 | C23 | 1 | 1 | yes | characteristic |
| 92 | C22×C4×C131 | C23 | 1 | 1 | yes | characteristic |
| 93 | C22×C4×C131 | C23 | 1 | 1 | yes | characteristic |
| 94 | C4⋊C4×C131 | C23 | 1 | 2 | yes | — |
| 95 | C4⋊C4×C131 | C23 | 1 | 2 | yes | — |
| 96 | C4⋊C4×C131 | C23 | 1 | 4 | yes | — |
| 97 | C4⋊C4×C131 | C23 | 1 | 2 | yes | — |
| 98 | C4⋊C4×C131 | C23 | 1 | 4 | yes | — |
| 99 | C4⋊C4×C131 | C23 | 1 | 2 | yes | — |
| 100 | C22⋊C4×C131 | C23 | 1 | 4 | yes | — |
| 101 | C42×C131 | C23 | 1 | 1 | yes | characteristic |
| 102 | C42×C131 | C23 | 1 | 1 | yes | characteristic |
| 103 | C42×C131 | C23 | 1 | 4 | yes | — |
| 104 | C42×C131 | C23 | 1 | 1 | yes | characteristic |
| 105 | C42×C131 | C23 | 1 | 1 | yes | characteristic |
| 106 | C22×D131 | C24 | 1 | 4 | yes | — |
| 107 | Dic262 | — | 2 | 4 | no | — |
| 108 | Dic262 | — | 2 | 4 | no | — |
| 109 | Dic262 | — | 4 | 8 | no | — |
| 110 | Dic262 | — | 4 | 2 | no | — |
| 111 | D524 | — | 2 | 4 | no | — |
| 112 | D524 | — | 2 | 4 | no | — |
| 113 | D524 | — | 4 | 2 | no | — |
| 114 | C22⋊D131 | — | 4 | 4 | no | — |
| 115 | C2×Dic131 | C24 | 1 | 4 | yes | — |
| 116 | C2×Dic131 | C24 | 1 | 4 | yes | — |
| 117 | C2×Dic131 | C24 | 1 | 4 | yes | — |
| 118 | C4×D131 | — | 2 | 8 | no | — |
| 119 | C4×D131 | — | 4 | 8 | no | — |
| 120 | C23×C131 | C24 | 1 | 1 | yes | characteristic |
| 121 | Q8×C131 | — | 2 | 4 | no | — |
| 122 | Q8×C131 | — | 2 | 4 | no | — |
| 123 | C2×C4×C131 | — | 2 | 2 | no | — |
| 124 | C2×C4×C131 | — | 2 | 2 | no | — |
| 125 | C2×C4×C131 | — | 2 | 2 | no | — |
| 126 | C2×C4×C131 | C24 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C131 | C24 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C131 | C24 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C131 | C24 | 1 | 4 | yes | — |
| 130 | C2×C4×C131 | C24 | 1 | 4 | yes | — |
| 131 | C2×C4×C131 | C24 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C131 | C24 | 1 | 1 | yes | characteristic |
| 133 | C2×C4×C131 | C24 | 1 | 1 | yes | characteristic |
| 134 | C22×C131 | — | 2 | 1 | no | — |
| 135 | C22×C131 | — | 2 | 1 | no | — |
| 136 | C22×C131 | — | 2 | 1 | no | — |
| 137 | C22×C131 | C25 | 1 | 1 | yes | characteristic |
| 138 | C2×D131 | — | 2 | 4 | no | — |
| 139 | C2×D131 | — | 2 | 4 | no | — |
| 140 | C2×D131 | — | 2 | 4 | no | — |
| 141 | C4×C131 | — | 2 | 1 | no | — |
| 142 | C4×C131 | — | 2 | 4 | no | — |
| 143 | C4×C131 | — | 2 | 1 | no | — |
| 144 | C4×C131 | — | 2 | 1 | no | — |
| 145 | C4×C131 | — | 2 | 4 | no | — |
| 146 | C4×C131 | H(5,𝔽2) | 1 | 2 | yes | — |
| 147 | C4×C131 | 2−1+4 | 1 | 2 | yes | — |
| 148 | C4×C131 | 2−1+4 | 1 | 2 | yes | — |
| 149 | Dic131 | — | 2 | 4 | no | — |
| 150 | Dic131 | — | 2 | 4 | no | — |
| 151 | Dic131 | — | 2 | 4 | no | — |
| 152 | C2×C131 | — | 4 | 1 | no | — |
| 153 | C2×C131 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 154 | C2×C131 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 155 | C2×C131 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 156 | D131 | — | 4 | 4 | no | — |
| 157 | C131 | #3550 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 158 | #3550 | — | 131 | 1 | no | maximal, Sylow |
| 159 | #563 | — | 131 | 4 | no | — |
| 160 | #559 | — | 131 | 2 | no | — |
| 161 | Q82 | — | 131 | 2 | no | — |
| 162 | #556 | — | 131 | 2 | no | — |
| 163 | #556 | — | 131 | 2 | no | — |
| 164 | D4×Q8 | — | 131 | 2 | no | — |
| 165 | D4×Q8 | — | 131 | 2 | no | — |
| 166 | #542 | — | 131 | 4 | no | — |
| 167 | #542 | — | 131 | 4 | no | — |
| 168 | #536 | — | 131 | 4 | no | — |
| 169 | #533 | — | 131 | 1 | no | — |
| 170 | #533 | D131 | 1 | 1 | yes | characteristic, complemented |
| 171 | #532 | — | 131 | 1 | no | — |
| 172 | C2×#41 | — | 131 | 2 | no | — |
| 173 | C22×Q8 | — | 131 | 4 | no | — |
| 174 | C4⋊Q8 | — | 131 | 8 | no | — |
| 175 | C4⋊Q8 | — | 131 | 4 | no | — |
| 176 | C4⋊Q8 | — | 131 | 1 | no | — |
| 177 | C4⋊Q8 | — | 131 | 1 | no | — |
| 178 | C4⋊Q8 | — | 131 | 1 | no | — |
| 179 | C4⋊Q8 | — | 131 | 4 | no | — |
| 180 | C4⋊Q8 | C2×D131 | 1 | 1 | yes | characteristic |
| 181 | C4⋊Q8 | C2×D131 | 1 | 1 | yes | characteristic |
| 182 | #127 | — | 131 | 1 | no | — |
| 183 | #126 | — | 131 | 8 | no | — |
| 184 | #126 | — | 131 | 8 | no | — |
| 185 | #125 | — | 131 | 2 | no | — |
| 186 | #125 | C2×D131 | 1 | 2 | yes | — |
| 187 | #124 | — | 131 | 4 | no | — |
| 188 | #124 | — | 131 | 8 | no | — |
| 189 | #124 | — | 131 | 2 | no | — |
| 190 | #124 | — | 131 | 4 | no | — |
| 191 | #123 | — | 131 | 8 | no | — |
| 192 | #123 | — | 131 | 8 | no | — |
| 193 | C22⋊Q8 | — | 131 | 4 | no | — |
| 194 | C22⋊Q8 | — | 131 | 2 | no | — |
| 195 | C22⋊Q8 | — | 131 | 2 | no | — |
| 196 | C22⋊Q8 | — | 131 | 8 | no | — |
| 197 | C22⋊Q8 | — | 131 | 4 | no | — |
| 198 | C22⋊Q8 | — | 131 | 8 | no | — |
| 199 | C22⋊Q8 | C2×D131 | 1 | 2 | yes | — |
| 200 | C22⋊Q8 | C2×D131 | 1 | 2 | yes | — |
| 201 | #121 | — | 131 | 2 | no | — |
| 202 | #121 | — | 131 | 2 | no | — |
| 203 | C4×Q8 | — | 131 | 4 | no | — |
| 204 | C4×Q8 | — | 131 | 2 | no | — |
| 205 | C4×Q8 | — | 131 | 2 | no | — |
| 206 | C4×Q8 | — | 131 | 4 | no | — |
| 207 | C4×Q8 | C2×D131 | 1 | 2 | yes | — |
| 208 | C4×Q8 | C2×D131 | 1 | 2 | yes | — |
| 209 | C4×D4 | — | 131 | 4 | no | — |
| 210 | C4×D4 | — | 131 | 2 | no | — |
| 211 | C4×D4 | — | 131 | 2 | no | — |
| 212 | C4×D4 | — | 131 | 4 | no | — |
| 213 | #117 | — | 131 | 2 | no | — |
| 214 | #117 | — | 131 | 4 | no | — |
| 215 | #117 | C2×D131 | 1 | 2 | yes | — |
| 216 | C2×C42 | C2×D131 | 1 | 1 | yes | characteristic |
| 217 | #41 | — | 524 | 4 | no | — |
| 218 | C2×Q8 | — | 524 | 8 | no | — |
| 219 | C2×Q8 | — | 131 | 2 | no | — |
| 220 | C2×Q8 | — | 131 | 2 | no | — |
| 221 | C2×Q8 | — | 131 | 8 | no | — |
| 222 | C2×Q8 | — | 131 | 2 | no | — |
| 223 | C2×Q8 | C22×D131 | 1 | 2 | yes | — |
| 224 | C2×Q8 | C22×D131 | 1 | 2 | yes | — |
| 225 | C2×D4 | — | 131 | 4 | no | — |
| 226 | C2×D4 | — | 131 | 2 | no | — |
| 227 | C2×D4 | — | 131 | 2 | no | — |
| 228 | C2×D4 | — | 131 | 2 | no | — |
| 229 | C22×C4 | — | 131 | 4 | no | — |
| 230 | C22×C4 | — | 131 | 8 | no | — |
| 231 | C22×C4 | C22×D131 | 1 | 1 | yes | characteristic |
| 232 | C22×C4 | C22×D131 | 1 | 1 | yes | characteristic |
| 233 | C22×C4 | C22×D131 | 1 | 1 | yes | characteristic |
| 234 | C4⋊C4 | — | 131 | 4 | no | — |
| 235 | C4⋊C4 | — | 131 | 8 | no | — |
| 236 | C4⋊C4 | — | 131 | 4 | no | — |
| 237 | C4⋊C4 | — | 131 | 2 | no | — |
| 238 | C4⋊C4 | — | 131 | 2 | no | — |
| 239 | C4⋊C4 | — | 131 | 2 | no | — |
| 240 | C4⋊C4 | — | 131 | 2 | no | — |
| 241 | C4⋊C4 | — | 131 | 2 | no | — |
| 242 | C4⋊C4 | — | 131 | 2 | no | — |
| 243 | C4⋊C4 | — | 131 | 8 | no | — |
| 244 | C4⋊C4 | C22×D131 | 1 | 2 | yes | — |
| 245 | C4⋊C4 | C22×D131 | 1 | 2 | yes | — |
| 246 | C4⋊C4 | C22×D131 | 1 | 2 | yes | — |
| 247 | C4⋊C4 | C22×D131 | 1 | 2 | yes | — |
| 248 | C4⋊C4 | C22×D131 | 1 | 4 | yes | — |
| 249 | C4⋊C4 | C22×D131 | 1 | 4 | yes | — |
| 250 | C22⋊C4 | — | 131 | 4 | no | — |
| 251 | C22⋊C4 | — | 131 | 8 | no | — |
| 252 | C22⋊C4 | — | 131 | 4 | no | — |
| 253 | C22⋊C4 | — | 131 | 4 | no | — |
| 254 | C22⋊C4 | — | 131 | 8 | no | — |
| 255 | C22⋊C4 | — | 131 | 8 | no | — |
| 256 | C22⋊C4 | C22×D131 | 1 | 4 | yes | — |
| 257 | C42 | — | 131 | 4 | no | — |
| 258 | C42 | — | 131 | 8 | no | — |
| 259 | C42 | C22×D131 | 1 | 1 | yes | characteristic |
| 260 | C42 | C22×D131 | 1 | 1 | yes | characteristic |
| 261 | C42 | C22×D131 | 1 | 1 | yes | characteristic |
| 262 | C42 | C22×D131 | 1 | 1 | yes | characteristic |
| 263 | C42 | C22×D131 | 1 | 4 | yes | — |
| 264 | C23 | — | 131 | 4 | no | — |
| 265 | C23 | C23×D131 | 1 | 1 | yes | characteristic |
| 266 | Q8 | — | 2 | 4 | no | — |
| 267 | Q8 | — | 2 | 4 | no | — |
| 268 | Q8 | — | 262 | 4 | no | — |
| 269 | Q8 | — | 262 | 4 | no | — |
| 270 | Q8 | — | 524 | 8 | no | — |
| 271 | Q8 | — | 524 | 2 | no | — |
| 272 | D4 | — | 262 | 4 | no | — |
| 273 | D4 | — | 262 | 4 | no | — |
| 274 | D4 | — | 524 | 2 | no | — |
| 275 | D4 | — | 524 | 4 | no | — |
| 276 | C2×C4 | — | 2 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 262 | 8 | no | — |
| 280 | C2×C4 | — | 524 | 8 | no | — |
| 281 | C2×C4 | — | 131 | 4 | no | — |
| 282 | C2×C4 | — | 131 | 4 | no | — |
| 283 | C2×C4 | — | 131 | 4 | no | — |
| 284 | C2×C4 | C23×D131 | 1 | 4 | yes | — |
| 285 | C2×C4 | C23×D131 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | C23×D131 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | C23×D131 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C23×D131 | 1 | 4 | yes | — |
| 289 | C2×C4 | C23×D131 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C23×D131 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C23×D131 | 1 | 1 | yes | characteristic |
| 292 | C22 | — | 2 | 1 | no | — |
| 293 | C22 | — | 2 | 1 | no | — |
| 294 | C22 | — | 2 | 1 | no | — |
| 295 | C22 | — | 262 | 4 | no | — |
| 296 | C22 | — | 262 | 4 | no | — |
| 297 | C22 | — | 262 | 4 | no | — |
| 298 | C22 | C24×D131 | 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 | — | 262 | 4 | no | — |
| 305 | C4 | — | 262 | 4 | no | — |
| 306 | C4 | — | 262 | 4 | no | — |
| 307 | C4 | #16862292 | 1 | 2 | yes | — |
| 308 | C4 | #16862293 | 1 | 2 | yes | — |
| 309 | C4 | #16862294 | 1 | 2 | yes | — |
| 310 | C2 | — | 4 | 1 | no | — |
| 311 | C2 | — | 524 | 4 | no | — |
| 312 | C2 | C2×#16862292 | 1 | 1 | yes | characteristic |
| 313 | C2 | C2×#16862293 | 1 | 1 | yes | characteristic |
| 314 | C2 | C2×#16862294 | 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