← #544×D167 #36337534
Subgroups — #544×D167 #36337534
Order 21376.
1080 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #24923900 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #24923899 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #24923898 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #24923897 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #24923896 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C4⋊Q8×D167 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #125×D167 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | C22⋊Q8×D167 | C2 | 1 | 4 | yes | maximal |
| 9 | #117×D167 | C2 | 1 | 2 | yes | maximal |
| 10 | C2×C4⋊C4×D167 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #544×C167 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×Q8×D167 | C22 | 1 | 4 | yes | — |
| 13 | C22×C4×D167 | C22 | 1 | 2 | yes | — |
| 14 | C22×C4×D167 | C22 | 1 | 1 | yes | characteristic |
| 15 | #19128832 | C22 | 1 | 4 | yes | complemented |
| 16 | #19128831 | C22 | 1 | 4 | yes | complemented |
| 17 | #19128831 | C22 | 1 | 8 | yes | complemented |
| 18 | #19128831 | C22 | 1 | 4 | yes | complemented |
| 19 | #19128830 | C22 | 1 | 8 | yes | complemented |
| 20 | #19128824 | C22 | 1 | 4 | yes | complemented |
| 21 | #19128823 | C22 | 1 | 4 | yes | complemented |
| 22 | #19128823 | C22 | 1 | 4 | yes | complemented |
| 23 | #19128822 | C22 | 1 | 8 | yes | complemented |
| 24 | #19128822 | C22 | 1 | 8 | yes | complemented |
| 25 | #19128812 | C22 | 1 | 4 | yes | — |
| 26 | #19128811 | C22 | 1 | 4 | yes | — |
| 27 | #19128810 | C22 | 1 | 8 | yes | — |
| 28 | #19128809 | C22 | 1 | 8 | yes | — |
| 29 | #19128808 | C22 | 1 | 4 | yes | — |
| 30 | #19128791 | C22 | 1 | 2 | yes | — |
| 31 | #19128790 | C22 | 1 | 2 | yes | — |
| 32 | #19128790 | C22 | 1 | 2 | yes | — |
| 33 | #19128789 | C22 | 1 | 4 | yes | — |
| 34 | #19128788 | C22 | 1 | 4 | yes | — |
| 35 | C2×C4⋊Dic167 | C22 | 1 | 1 | yes | characteristic |
| 36 | C2×Dic167⋊C4 | C22 | 1 | 2 | yes | — |
| 37 | C4⋊C4×D167 | C22 | 1 | 4 | yes | — |
| 38 | C4⋊C4×D167 | C22 | 1 | 4 | yes | — |
| 39 | C4⋊C4×D167 | C22 | 1 | 2 | yes | — |
| 40 | C4⋊C4×D167 | C22 | 1 | 2 | yes | — |
| 41 | C4⋊C4×D167 | C22 | 1 | 4 | yes | — |
| 42 | C4⋊C4×D167 | C22 | 1 | 8 | yes | — |
| 43 | C22⋊C4×D167 | C22 | 1 | 4 | yes | — |
| 44 | C42×D167 | C22 | 1 | 4 | yes | — |
| 45 | C4⋊Q8×C167 | C22 | 1 | 4 | yes | complemented |
| 46 | #125×C167 | C22 | 1 | 4 | yes | complemented |
| 47 | C22⋊Q8×C167 | C22 | 1 | 4 | yes | — |
| 48 | #117×C167 | C22 | 1 | 2 | yes | — |
| 49 | C2×C4⋊C4×C167 | C22 | 1 | 1 | yes | characteristic |
| 50 | C23×D167 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×Dic334 | C23 | 1 | 8 | yes | — |
| 52 | C2×Dic334 | C23 | 1 | 4 | yes | — |
| 53 | Q8×D167 | — | 4 | 8 | no | — |
| 54 | C22×Dic167 | C23 | 1 | 2 | yes | — |
| 55 | C22×Dic167 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C4×D167 | — | 4 | 4 | no | — |
| 57 | C2×C4×D167 | Q8 | 1 | 8 | yes | complemented |
| 58 | C2×C4×D167 | C23 | 1 | 4 | yes | — |
| 59 | C2×C4×D167 | C23 | 1 | 4 | yes | — |
| 60 | C2×C4×D167 | C23 | 1 | 4 | yes | — |
| 61 | C2×C4×D167 | C23 | 1 | 4 | yes | — |
| 62 | C2×C4×D167 | C23 | 1 | 2 | yes | — |
| 63 | C2×C4×D167 | C23 | 1 | 2 | yes | — |
| 64 | C4⋊Dic167 | C23 | 1 | 2 | yes | — |
| 65 | C4⋊Dic167 | C23 | 1 | 4 | yes | — |
| 66 | C4⋊Dic167 | C23 | 1 | 8 | yes | — |
| 67 | C4⋊Dic167 | C23 | 1 | 2 | yes | — |
| 68 | C4⋊Dic167 | C23 | 1 | 4 | yes | — |
| 69 | Dic167⋊C4 | C23 | 1 | 8 | yes | — |
| 70 | Dic167⋊C4 | C23 | 1 | 4 | yes | — |
| 71 | Dic167⋊C4 | C23 | 1 | 4 | yes | — |
| 72 | Dic167⋊C4 | C23 | 1 | 8 | yes | — |
| 73 | Dic167⋊C4 | C23 | 1 | 8 | yes | — |
| 74 | Dic167⋊C4 | C23 | 1 | 8 | yes | — |
| 75 | C22⋊Dic167 | C23 | 1 | 4 | yes | — |
| 76 | (C2×D167)⋊C4 | C23 | 1 | 4 | yes | — |
| 77 | (C2×D167)⋊C4 | C23 | 1 | 4 | yes | — |
| 78 | C4×Dic167 | C23 | 1 | 4 | yes | — |
| 79 | C4×Dic167 | C23 | 1 | 4 | yes | — |
| 80 | C4×Dic167 | C23 | 1 | 4 | yes | — |
| 81 | C2×Q8×C167 | C23 | 1 | 4 | yes | — |
| 82 | C22×C4×C167 | C23 | 1 | 2 | yes | — |
| 83 | C22×C4×C167 | C23 | 1 | 1 | yes | characteristic |
| 84 | C4⋊C4×C167 | C23 | 1 | 4 | yes | — |
| 85 | C4⋊C4×C167 | C23 | 1 | 2 | yes | — |
| 86 | C4⋊C4×C167 | C23 | 1 | 2 | yes | — |
| 87 | C4⋊C4×C167 | C23 | 1 | 4 | yes | — |
| 88 | C4⋊C4×C167 | C23 | 1 | 8 | yes | — |
| 89 | C22⋊C4×C167 | C23 | 1 | 4 | yes | — |
| 90 | C42×C167 | C23 | 1 | 4 | yes | — |
| 91 | C22×D167 | — | 2 | 2 | no | — |
| 92 | C22×D167 | — | 2 | 2 | no | — |
| 93 | C22×D167 | C2×Q8 | 1 | 4 | yes | — |
| 94 | C22×D167 | C24 | 1 | 1 | yes | characteristic |
| 95 | C22×D167 | C24 | 1 | 1 | yes | characteristic |
| 96 | Dic334 | — | 4 | 8 | no | — |
| 97 | Dic334 | — | 4 | 4 | no | — |
| 98 | C2×Dic167 | — | 4 | 2 | no | — |
| 99 | C2×Dic167 | C2×Q8 | 1 | 4 | yes | complemented |
| 100 | C2×Dic167 | C24 | 1 | 4 | yes | — |
| 101 | C2×Dic167 | C24 | 1 | 4 | yes | — |
| 102 | C2×Dic167 | C24 | 1 | 4 | yes | — |
| 103 | C2×Dic167 | C24 | 1 | 2 | yes | — |
| 104 | C4×D167 | — | 2 | 8 | no | — |
| 105 | C4×D167 | — | 2 | 8 | no | — |
| 106 | C4×D167 | — | 2 | 8 | no | — |
| 107 | C4×D167 | — | 4 | 4 | no | — |
| 108 | C4×D167 | C2×Q8 | 1 | 8 | yes | — |
| 109 | C4×D167 | C2×Q8 | 1 | 8 | yes | complemented |
| 110 | C23×C167 | C24 | 1 | 1 | yes | characteristic |
| 111 | Q8×C167 | — | 4 | 4 | no | — |
| 112 | C2×C4×C167 | — | 4 | 2 | no | — |
| 113 | C2×C4×C167 | C2×Q8 | 1 | 4 | yes | complemented |
| 114 | C2×C4×C167 | C24 | 1 | 4 | yes | — |
| 115 | C2×C4×C167 | C24 | 1 | 4 | yes | — |
| 116 | C2×C4×C167 | C24 | 1 | 4 | yes | — |
| 117 | C2×C4×C167 | C24 | 1 | 2 | yes | — |
| 118 | C22×C167 | — | 2 | 1 | no | — |
| 119 | C22×C167 | — | 2 | 1 | no | — |
| 120 | C22×C167 | C22×Q8 | 1 | 2 | yes | — |
| 121 | C22×C167 | C25 | 1 | 1 | yes | characteristic |
| 122 | C2×D167 | — | 2 | 1 | no | — |
| 123 | C2×D167 | — | 2 | 1 | no | — |
| 124 | C2×D167 | — | 2 | 8 | no | — |
| 125 | C2×D167 | C22×Q8 | 1 | 2 | yes | — |
| 126 | C2×D167 | C22×Q8 | 1 | 2 | yes | — |
| 127 | C2×D167 | H(5,𝔽2) | 1 | 2 | yes | — |
| 128 | C2×D167 | 2−1+4 | 1 | 2 | yes | — |
| 129 | C4×C167 | — | 2 | 4 | no | — |
| 130 | C4×C167 | — | 2 | 4 | no | — |
| 131 | C4×C167 | — | 2 | 4 | no | — |
| 132 | C4×C167 | C22×Q8 | 1 | 4 | yes | — |
| 133 | Dic167 | — | 2 | 4 | no | — |
| 134 | Dic167 | — | 2 | 4 | no | — |
| 135 | Dic167 | — | 2 | 4 | no | — |
| 136 | Dic167 | C22×Q8 | 1 | 4 | yes | — |
| 137 | C2×C167 | — | 2 | 2 | no | — |
| 138 | C2×C167 | C23×Q8 | 1 | 1 | yes | characteristic |
| 139 | C2×C167 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 140 | C2×C167 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 141 | D167 | — | 2 | 2 | no | — |
| 142 | D167 | #544 | 1 | 4 | yes | direct product factor |
| 143 | C167 | C2×#544 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 144 | C2×#544 | — | 167 | 1 | no | maximal, Sylow |
| 145 | #544 | — | 167 | 4 | no | — |
| 146 | #544 | — | 167 | 4 | no | — |
| 147 | #544 | — | 167 | 4 | no | — |
| 148 | #544 | — | 167 | 2 | no | — |
| 149 | #544 | — | 167 | 1 | no | — |
| 150 | #544 | D167 | 1 | 1 | yes | characteristic, direct product factor |
| 151 | C2×C4⋊Q8 | — | 167 | 4 | no | — |
| 152 | C2×#125 | — | 167 | 4 | no | — |
| 153 | C2×C22⋊Q8 | — | 167 | 4 | no | — |
| 154 | C2×#117 | — | 167 | 2 | no | — |
| 155 | C22×C4⋊C4 | — | 167 | 1 | no | — |
| 156 | C22×Q8 | — | 167 | 4 | no | — |
| 157 | C23×C4 | — | 167 | 2 | no | — |
| 158 | C23×C4 | — | 167 | 1 | no | — |
| 159 | C4⋊Q8 | — | 167 | 8 | no | — |
| 160 | C4⋊Q8 | — | 167 | 8 | no | — |
| 161 | C4⋊Q8 | — | 167 | 4 | no | — |
| 162 | C4⋊Q8 | — | 167 | 4 | no | — |
| 163 | C4⋊Q8 | — | 167 | 4 | no | — |
| 164 | C4⋊Q8 | C2×D167 | 1 | 4 | yes | complemented |
| 165 | #125 | — | 167 | 8 | no | — |
| 166 | #125 | — | 167 | 8 | no | — |
| 167 | #125 | — | 167 | 4 | no | — |
| 168 | #125 | — | 167 | 4 | no | — |
| 169 | #125 | — | 167 | 4 | no | — |
| 170 | #125 | C2×D167 | 1 | 4 | yes | complemented |
| 171 | C22⋊Q8 | — | 167 | 4 | no | — |
| 172 | C22⋊Q8 | — | 167 | 4 | no | — |
| 173 | C22⋊Q8 | — | 167 | 8 | no | — |
| 174 | C22⋊Q8 | — | 167 | 4 | no | — |
| 175 | C22⋊Q8 | — | 167 | 8 | no | — |
| 176 | C22⋊Q8 | C2×D167 | 1 | 4 | yes | — |
| 177 | #117 | — | 167 | 2 | no | — |
| 178 | #117 | — | 167 | 2 | no | — |
| 179 | #117 | — | 167 | 2 | no | — |
| 180 | #117 | — | 167 | 4 | no | — |
| 181 | #117 | — | 167 | 4 | no | — |
| 182 | #117 | C2×D167 | 1 | 2 | yes | — |
| 183 | C2×C4⋊C4 | — | 167 | 2 | no | — |
| 184 | C2×C4⋊C4 | — | 167 | 1 | no | — |
| 185 | C2×C4⋊C4 | — | 167 | 4 | no | — |
| 186 | C2×C4⋊C4 | — | 167 | 4 | no | — |
| 187 | C2×C4⋊C4 | — | 167 | 2 | no | — |
| 188 | C2×C4⋊C4 | — | 167 | 2 | no | — |
| 189 | C2×C4⋊C4 | — | 167 | 4 | no | — |
| 190 | C2×C4⋊C4 | — | 167 | 8 | no | — |
| 191 | C2×C4⋊C4 | C2×D167 | 1 | 1 | yes | characteristic |
| 192 | C2×C22⋊C4 | — | 167 | 4 | no | — |
| 193 | C2×C42 | — | 167 | 4 | no | — |
| 194 | C24 | — | 167 | 1 | no | — |
| 195 | C2×Q8 | — | 668 | 8 | no | — |
| 196 | C2×Q8 | — | 167 | 4 | no | — |
| 197 | C2×Q8 | — | 167 | 8 | no | — |
| 198 | C2×Q8 | C22×D167 | 1 | 4 | yes | — |
| 199 | C22×C4 | — | 668 | 4 | no | — |
| 200 | C22×C4 | — | 167 | 4 | no | — |
| 201 | C22×C4 | — | 167 | 4 | no | — |
| 202 | C22×C4 | — | 167 | 4 | no | — |
| 203 | C22×C4 | — | 167 | 2 | no | — |
| 204 | C22×C4 | — | 167 | 4 | no | — |
| 205 | C22×C4 | — | 167 | 1 | no | — |
| 206 | C22×C4 | — | 167 | 8 | no | — |
| 207 | C22×C4 | — | 167 | 2 | no | — |
| 208 | C22×C4 | — | 167 | 2 | no | — |
| 209 | C22×C4 | C22×D167 | 1 | 2 | yes | — |
| 210 | C22×C4 | C22×D167 | 1 | 1 | yes | characteristic |
| 211 | C4⋊C4 | — | 167 | 8 | no | — |
| 212 | C4⋊C4 | — | 167 | 4 | no | — |
| 213 | C4⋊C4 | — | 167 | 4 | no | — |
| 214 | C4⋊C4 | — | 167 | 2 | no | — |
| 215 | C4⋊C4 | — | 167 | 4 | no | — |
| 216 | C4⋊C4 | — | 167 | 2 | no | — |
| 217 | C4⋊C4 | — | 167 | 8 | no | — |
| 218 | C4⋊C4 | — | 167 | 4 | no | — |
| 219 | C4⋊C4 | — | 167 | 8 | no | — |
| 220 | C4⋊C4 | — | 167 | 8 | no | — |
| 221 | C4⋊C4 | — | 167 | 8 | no | — |
| 222 | C4⋊C4 | C22×D167 | 1 | 4 | yes | — |
| 223 | C4⋊C4 | C22×D167 | 1 | 2 | yes | — |
| 224 | C4⋊C4 | C22×D167 | 1 | 2 | yes | — |
| 225 | C4⋊C4 | C22×D167 | 1 | 4 | yes | — |
| 226 | C4⋊C4 | C22×D167 | 1 | 8 | yes | — |
| 227 | C22⋊C4 | — | 167 | 4 | no | — |
| 228 | C22⋊C4 | — | 167 | 4 | no | — |
| 229 | C22⋊C4 | — | 167 | 4 | no | — |
| 230 | C22⋊C4 | C22×D167 | 1 | 4 | yes | — |
| 231 | C42 | — | 167 | 4 | no | — |
| 232 | C42 | — | 167 | 4 | no | — |
| 233 | C42 | — | 167 | 4 | no | — |
| 234 | C42 | C22×D167 | 1 | 4 | yes | — |
| 235 | C23 | — | 334 | 2 | no | — |
| 236 | C23 | — | 334 | 2 | no | — |
| 237 | C23 | — | 167 | 4 | no | — |
| 238 | C23 | — | 167 | 1 | no | — |
| 239 | C23 | — | 167 | 1 | no | — |
| 240 | C23 | C23×D167 | 1 | 1 | yes | characteristic |
| 241 | Q8 | — | 4 | 4 | no | — |
| 242 | Q8 | — | 668 | 4 | no | — |
| 243 | Q8 | — | 668 | 8 | no | — |
| 244 | C2×C4 | — | 4 | 2 | no | — |
| 245 | C2×C4 | — | 334 | 8 | no | — |
| 246 | C2×C4 | — | 334 | 8 | no | — |
| 247 | C2×C4 | — | 334 | 8 | no | — |
| 248 | C2×C4 | — | 668 | 4 | no | — |
| 249 | C2×C4 | — | 668 | 2 | no | — |
| 250 | C2×C4 | — | 167 | 4 | no | — |
| 251 | C2×C4 | — | 167 | 4 | no | — |
| 252 | C2×C4 | — | 167 | 4 | no | — |
| 253 | C2×C4 | — | 167 | 4 | no | — |
| 254 | C2×C4 | — | 167 | 2 | no | — |
| 255 | C2×C4 | — | 167 | 8 | no | — |
| 256 | C2×C4 | — | 167 | 8 | no | — |
| 257 | C2×C4 | Q8×D167 | 1 | 4 | yes | complemented |
| 258 | C2×C4 | C23×D167 | 1 | 4 | yes | — |
| 259 | C2×C4 | C23×D167 | 1 | 4 | yes | — |
| 260 | C2×C4 | C23×D167 | 1 | 4 | yes | — |
| 261 | C2×C4 | C23×D167 | 1 | 2 | yes | — |
| 262 | C22 | — | 2 | 1 | no | — |
| 263 | C22 | — | 2 | 1 | no | — |
| 264 | C22 | — | 334 | 8 | no | — |
| 265 | C22 | — | 334 | 1 | no | — |
| 266 | C22 | — | 334 | 1 | no | — |
| 267 | C22 | — | 167 | 2 | no | — |
| 268 | C22 | — | 167 | 2 | no | — |
| 269 | C22 | — | 167 | 2 | no | — |
| 270 | C22 | — | 167 | 2 | no | — |
| 271 | C22 | C2×Q8×D167 | 1 | 2 | yes | — |
| 272 | C22 | C24×D167 | 1 | 1 | yes | characteristic |
| 273 | C4 | — | 2 | 4 | no | — |
| 274 | C4 | — | 2 | 4 | no | — |
| 275 | C4 | — | 2 | 4 | no | — |
| 276 | C4 | — | 334 | 4 | no | — |
| 277 | C4 | — | 334 | 4 | no | — |
| 278 | C4 | — | 334 | 4 | no | — |
| 279 | C4 | — | 167 | 4 | no | — |
| 280 | C4 | C2×Q8×D167 | 1 | 4 | yes | — |
| 281 | C2 | — | 2 | 2 | no | — |
| 282 | C2 | — | 334 | 2 | no | — |
| 283 | C2 | — | 167 | 4 | no | — |
| 284 | C2 | C22×Q8×D167 | 1 | 1 | yes | characteristic |
| 285 | C2 | H(5,𝔽2)×D167 | 1 | 1 | yes | characteristic |
| 286 | C2 | 2−1+4×D167 | 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