← #30856194
Subgroups — #30856194
Order 17152.
616 conjugacy classes of subgroups.
The subgroup lattice has 318 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #131×Dic67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22560356 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22560332 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22560310 | C2 | 1 | 2 | yes | maximal |
| 5 | #22560259 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #22560090 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22560011 | C2 | 1 | 2 | yes | maximal |
| 8 | #22559963 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C67×#1812 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C2×#14524050 | C22 | 1 | 1 | yes | characteristic |
| 11 | #41×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#14523979 | C22 | 1 | 2 | yes | — |
| 13 | #16863380 | C22 | 1 | 2 | yes | — |
| 14 | #16863354 | C4 | 1 | 2 | yes | complemented |
| 15 | #16863354 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16863352 | C22 | 1 | 4 | yes | — |
| 17 | #16863351 | C4 | 1 | 2 | yes | complemented |
| 18 | #16863351 | C22 | 1 | 1 | yes | characteristic |
| 19 | M16×Dic67 | C4 | 1 | 2 | yes | complemented |
| 20 | M16×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C8×Dic67 | C4 | 1 | 2 | yes | complemented |
| 22 | C2×C8×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 23 | #16863166 | C22 | 1 | 4 | yes | — |
| 24 | #16863155 | C22 | 1 | 4 | yes | — |
| 25 | C67⋊#338 | C22 | 1 | 2 | yes | — |
| 26 | #16863152 | C22 | 1 | 4 | yes | — |
| 27 | C2×#131×C67 | C22 | 1 | 1 | yes | characteristic |
| 28 | #427×C67 | C22 | 1 | 2 | yes | — |
| 29 | #410×C67 | C22 | 1 | 2 | yes | — |
| 30 | H(3,ℤ/4)×C67 | C22 | 1 | 2 | yes | — |
| 31 | #14524050 | — | 2 | 2 | no | — |
| 32 | #14524050 | — | 2 | 2 | no | — |
| 33 | C2×#11761612 | C2×C4 | 1 | 2 | yes | — |
| 34 | C2×#11761612 | C23 | 1 | 4 | yes | — |
| 35 | C2×#11761612 | C23 | 1 | 1 | yes | characteristic |
| 36 | C22×C67⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 37 | C22×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 38 | Q8×Dic67 | — | 2 | 1 | no | — |
| 39 | D4×Dic67 | — | 2 | 2 | no | — |
| 40 | D4×Dic67 | — | 2 | 1 | no | — |
| 41 | #14524002 | C2×C4 | 1 | 2 | yes | — |
| 42 | #14524002 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×C4×Dic67 | C2×C4 | 1 | 2 | yes | — |
| 44 | C2×C4×Dic67 | C23 | 1 | 1 | yes | characteristic |
| 45 | #14523979 | — | 2 | 2 | no | — |
| 46 | #14523979 | — | 2 | 2 | no | — |
| 47 | #14523962 | — | 2 | 2 | no | — |
| 48 | #14523960 | — | 2 | 2 | no | — |
| 49 | #14523954 | — | 2 | 4 | no | — |
| 50 | #14523953 | C2×C4 | 1 | 2 | yes | — |
| 51 | #14523953 | C2×C4 | 1 | 2 | yes | — |
| 52 | #14523953 | C2×C4 | 1 | 2 | yes | — |
| 53 | C4×C67⋊C8 | — | 2 | 4 | no | — |
| 54 | C8×Dic67 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | C8×Dic67 | C2×C4 | 1 | 2 | yes | — |
| 56 | C8×Dic67 | C2×C4 | 1 | 2 | yes | — |
| 57 | C8×Dic67 | C2×C4 | 1 | 2 | yes | — |
| 58 | C8×Dic67 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C8×Dic67 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | C8×Dic67 | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | C2×#41×C67 | C23 | 1 | 1 | yes | characteristic |
| 62 | #131×C67 | — | 2 | 2 | no | — |
| 63 | #131×C67 | — | 2 | 2 | no | — |
| 64 | C2×M16×C67 | C2×C4 | 1 | 2 | yes | — |
| 65 | C2×M16×C67 | C23 | 1 | 1 | yes | characteristic |
| 66 | C22×C8×C67 | C2×C4 | 1 | 2 | yes | — |
| 67 | C22×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 68 | #117×C67 | C23 | 1 | 4 | yes | — |
| 69 | #107×C67 | — | 2 | 2 | no | — |
| 70 | #106×C67 | — | 2 | 2 | no | — |
| 71 | #99×C67 | — | 2 | 4 | no | — |
| 72 | C22×Dic67 | — | 2 | 2 | no | — |
| 73 | C22×Dic67 | — | 2 | 1 | no | — |
| 74 | #11761612 | — | 2 | 2 | no | — |
| 75 | #11761612 | — | 2 | 4 | no | — |
| 76 | #11761612 | — | 2 | 4 | no | — |
| 77 | #11761612 | — | 2 | 4 | no | — |
| 78 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 79 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 80 | C2×C67⋊C8 | — | 2 | 4 | no | — |
| 81 | C2×C67⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 82 | C2×C67⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C67⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 84 | C2×C67⋊C8 | #41 | 1 | 2 | yes | — |
| 85 | C2×C67⋊C8 | #41 | 1 | 2 | yes | — |
| 86 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 87 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 88 | C22⋊Dic67 | — | 2 | 2 | no | — |
| 89 | C22⋊Dic67 | — | 2 | 1 | no | — |
| 90 | C4×Dic67 | — | 2 | 2 | no | — |
| 91 | C4×Dic67 | — | 2 | 1 | no | — |
| 92 | C4×Dic67 | C22×C4 | 1 | 1 | yes | characteristic |
| 93 | C4×Dic67 | C22×C4 | 1 | 2 | yes | — |
| 94 | C4×Dic67 | C22×C4 | 1 | 1 | yes | characteristic |
| 95 | #41×C67 | — | 2 | 2 | no | — |
| 96 | #41×C67 | — | 2 | 2 | no | — |
| 97 | C2×Q8×C67 | #41 | 1 | 1 | yes | characteristic |
| 98 | C2×D4×C67 | #41 | 1 | 2 | yes | — |
| 99 | C2×D4×C67 | #41 | 1 | 1 | yes | characteristic |
| 100 | C22×C4×C67 | C22×C4 | 1 | 2 | yes | — |
| 101 | C22×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 102 | M16×C67 | — | 2 | 2 | no | — |
| 103 | M16×C67 | — | 2 | 4 | no | — |
| 104 | C2×C8×C67 | — | 2 | 2 | no | — |
| 105 | C2×C8×C67 | — | 2 | 2 | no | — |
| 106 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 107 | C2×C8×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 108 | C2×C8×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C2×C8×C67 | #41 | 1 | 2 | yes | — |
| 110 | C2×C8×C67 | #41 | 1 | 2 | yes | — |
| 111 | C4⋊C4×C67 | — | 2 | 4 | no | — |
| 112 | C22⋊C4×C67 | — | 2 | 4 | no | — |
| 113 | C42×C67 | — | 2 | 4 | no | — |
| 114 | C23×C67 | #117 | 1 | 2 | yes | — |
| 115 | C23×C67 | C2×#41 | 1 | 1 | yes | characteristic |
| 116 | Q8×C67 | — | 2 | 2 | no | — |
| 117 | D4×C67 | — | 2 | 4 | no | — |
| 118 | D4×C67 | — | 2 | 2 | no | — |
| 119 | C2×C4×C67 | — | 2 | 2 | no | — |
| 120 | C2×C4×C67 | — | 2 | 2 | no | — |
| 121 | C2×C4×C67 | — | 4 | 4 | no | — |
| 122 | C2×C4×C67 | #117 | 1 | 2 | yes | — |
| 123 | C2×C4×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C67 | C2×#41 | 1 | 2 | yes | — |
| 125 | C2×C4×C67 | C2×#41 | 1 | 2 | yes | — |
| 126 | C2×C4×C67 | C2×#41 | 1 | 1 | yes | characteristic |
| 127 | C2×Dic67 | — | 2 | 1 | no | — |
| 128 | C2×Dic67 | — | 2 | 2 | no | — |
| 129 | C2×Dic67 | — | 2 | 1 | no | — |
| 130 | C2×Dic67 | — | 4 | 2 | no | — |
| 131 | C2×Dic67 | — | 4 | 4 | no | — |
| 132 | C8×C67 | — | 2 | 2 | no | — |
| 133 | C8×C67 | #117 | 1 | 1 | yes | characteristic |
| 134 | C8×C67 | #117 | 1 | 1 | yes | characteristic |
| 135 | C8×C67 | #117 | 1 | 2 | yes | — |
| 136 | C67⋊C8 | — | 2 | 2 | no | — |
| 137 | C67⋊C8 | — | 4 | 4 | no | — |
| 138 | C67⋊C8 | #117 | 1 | 1 | yes | characteristic |
| 139 | C67⋊C8 | #117 | 1 | 1 | yes | characteristic |
| 140 | C67⋊C8 | #117 | 1 | 2 | yes | — |
| 141 | C22×C67 | — | 2 | 2 | no | — |
| 142 | C22×C67 | — | 4 | 1 | no | — |
| 143 | C22×C67 | — | 4 | 2 | no | — |
| 144 | C22×C67 | C2×#117 | 1 | 1 | yes | characteristic |
| 145 | C22×C67 | #529 | 1 | 2 | yes | — |
| 146 | C4×C67 | — | 2 | 2 | no | — |
| 147 | C4×C67 | — | 8 | 4 | no | — |
| 148 | C4×C67 | C2×#117 | 1 | 1 | yes | characteristic |
| 149 | C4×C67 | C2×#117 | 1 | 1 | yes | characteristic |
| 150 | C4×C67 | #529 | 1 | 2 | yes | — |
| 151 | Dic67 | — | 2 | 2 | no | — |
| 152 | Dic67 | — | 4 | 2 | no | — |
| 153 | Dic67 | — | 4 | 1 | no | — |
| 154 | #20787 | — | 67 | 1 | no | maximal, Sylow |
| 155 | C2×C67 | — | 2 | 1 | no | — |
| 156 | C2×C67 | — | 2 | 2 | no | — |
| 157 | C2×C67 | — | 4 | 2 | no | — |
| 158 | C2×C67 | #2283 | 1 | 1 | yes | characteristic |
| 159 | C4×#131 | — | 67 | 1 | no | — |
| 160 | #1841 | — | 67 | 2 | no | — |
| 161 | #1833 | — | 67 | 2 | no | — |
| 162 | #1827 | — | 67 | 2 | no | — |
| 163 | #1813 | — | 67 | 1 | no | — |
| 164 | #1812 | D67 | 1 | 1 | yes | characteristic |
| 165 | #1772 | — | 67 | 2 | no | — |
| 166 | #1757 | — | 67 | 2 | no | — |
| 167 | #1747 | — | 67 | 2 | no | — |
| 168 | C67 | #20787 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 169 | C2×#131 | — | 67 | 1 | no | — |
| 170 | C2×#131 | C2×D67 | 1 | 1 | yes | characteristic |
| 171 | C4×#41 | — | 67 | 1 | no | — |
| 172 | C2×#108 | — | 67 | 2 | no | — |
| 173 | #427 | C2×D67 | 1 | 2 | yes | — |
| 174 | #413 | — | 67 | 2 | no | — |
| 175 | #410 | C2×D67 | 1 | 2 | yes | — |
| 176 | #405 | — | 67 | 2 | no | — |
| 177 | #405 | — | 67 | 1 | no | — |
| 178 | #405 | — | 67 | 2 | no | — |
| 179 | #405 | — | 67 | 1 | no | — |
| 180 | #405 | — | 67 | 4 | no | — |
| 181 | C4×M16 | — | 67 | 2 | no | — |
| 182 | C4×M16 | — | 67 | 1 | no | — |
| 183 | C2×C4×C8 | — | 67 | 2 | no | — |
| 184 | C2×C4×C8 | — | 67 | 1 | no | — |
| 185 | #344 | — | 67 | 4 | no | — |
| 186 | #339 | — | 67 | 4 | no | — |
| 187 | #338 | — | 67 | 2 | no | — |
| 188 | H(3,ℤ/4) | — | 67 | 4 | no | — |
| 189 | H(3,ℤ/4) | C2×D67 | 1 | 2 | yes | — |
| 190 | C2×#41 | C22×D67 | 1 | 1 | yes | characteristic |
| 191 | #131 | — | 2 | 2 | no | — |
| 192 | #131 | — | 2 | 2 | no | — |
| 193 | #131 | — | 134 | 2 | no | — |
| 194 | #131 | — | 134 | 2 | no | — |
| 195 | C2×M16 | — | 67 | 2 | no | — |
| 196 | C2×M16 | — | 67 | 1 | no | — |
| 197 | C2×M16 | — | 67 | 4 | no | — |
| 198 | C2×M16 | C4×D67 | 1 | 2 | yes | — |
| 199 | C2×M16 | C22×D67 | 1 | 1 | yes | characteristic |
| 200 | C22×C8 | — | 67 | 1 | no | — |
| 201 | C22×C8 | — | 67 | 2 | no | — |
| 202 | C22×C8 | C4×D67 | 1 | 2 | yes | — |
| 203 | C22×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 204 | C4×Q8 | — | 134 | 1 | no | — |
| 205 | C4×D4 | — | 134 | 2 | no | — |
| 206 | C4×D4 | — | 134 | 1 | no | — |
| 207 | #117 | — | 67 | 2 | no | — |
| 208 | #117 | — | 67 | 1 | no | — |
| 209 | #117 | C22×D67 | 1 | 4 | yes | — |
| 210 | C2×C42 | — | 67 | 2 | no | — |
| 211 | C2×C42 | — | 67 | 1 | no | — |
| 212 | #108 | — | 134 | 2 | no | — |
| 213 | #108 | — | 134 | 2 | no | — |
| 214 | #107 | — | 2 | 2 | no | — |
| 215 | #106 | — | 2 | 2 | no | — |
| 216 | #101 | — | 134 | 2 | no | — |
| 217 | #100 | — | 134 | 2 | no | — |
| 218 | #99 | — | 2 | 4 | no | — |
| 219 | #97 | — | 67 | 2 | no | — |
| 220 | #97 | — | 67 | 2 | no | — |
| 221 | #97 | — | 67 | 2 | no | — |
| 222 | #97 | — | 134 | 4 | no | — |
| 223 | C4×C8 | — | 67 | 1 | no | — |
| 224 | C4×C8 | — | 67 | 1 | no | — |
| 225 | C4×C8 | — | 67 | 2 | no | — |
| 226 | C4×C8 | — | 67 | 2 | no | — |
| 227 | C4×C8 | — | 67 | 1 | no | — |
| 228 | C4×C8 | — | 67 | 1 | no | — |
| 229 | C4×C8 | — | 67 | 2 | no | — |
| 230 | C4×C8 | — | 134 | 4 | no | — |
| 231 | #41 | — | 2 | 2 | no | — |
| 232 | #41 | — | 2 | 2 | no | — |
| 233 | C2×Q8 | #11761629 | 1 | 1 | yes | characteristic |
| 234 | C2×D4 | #11761627 | 1 | 2 | yes | — |
| 235 | C2×D4 | #11761629 | 1 | 1 | yes | characteristic |
| 236 | C22×C4 | — | 134 | 2 | no | — |
| 237 | C22×C4 | — | 134 | 1 | no | — |
| 238 | C22×C4 | C2×C4×D67 | 1 | 2 | yes | — |
| 239 | C22×C4 | C23×D67 | 1 | 1 | yes | characteristic |
| 240 | M16 | — | 2 | 2 | no | — |
| 241 | M16 | — | 2 | 4 | no | — |
| 242 | M16 | — | 134 | 4 | no | — |
| 243 | M16 | — | 134 | 4 | no | — |
| 244 | M16 | — | 134 | 4 | no | — |
| 245 | M16 | — | 134 | 2 | no | — |
| 246 | C2×C8 | — | 2 | 2 | no | — |
| 247 | C2×C8 | — | 2 | 2 | no | — |
| 248 | C2×C8 | — | 67 | 1 | no | — |
| 249 | C2×C8 | — | 67 | 1 | no | — |
| 250 | C2×C8 | — | 67 | 2 | no | — |
| 251 | C2×C8 | — | 67 | 2 | no | — |
| 252 | C2×C8 | — | 67 | 2 | no | — |
| 253 | C2×C8 | — | 134 | 4 | no | — |
| 254 | C2×C8 | — | 134 | 2 | no | — |
| 255 | C2×C8 | — | 134 | 2 | no | — |
| 256 | C2×C8 | C2×C4×D67 | 1 | 2 | yes | — |
| 257 | C2×C8 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | #11761627 | 1 | 2 | yes | — |
| 260 | C2×C8 | #11761627 | 1 | 2 | yes | — |
| 261 | C4⋊C4 | — | 2 | 4 | no | — |
| 262 | C4⋊C4 | — | 134 | 2 | no | — |
| 263 | C4⋊C4 | — | 134 | 1 | no | — |
| 264 | C22⋊C4 | — | 2 | 4 | no | — |
| 265 | C22⋊C4 | — | 134 | 2 | no | — |
| 266 | C22⋊C4 | — | 134 | 1 | no | — |
| 267 | C42 | — | 2 | 4 | no | — |
| 268 | C42 | — | 67 | 1 | no | — |
| 269 | C42 | — | 67 | 2 | no | — |
| 270 | C42 | — | 67 | 1 | no | — |
| 271 | C42 | — | 134 | 2 | no | — |
| 272 | C42 | — | 134 | 1 | no | — |
| 273 | C23 | #14524001 | 1 | 2 | yes | — |
| 274 | C23 | C2×#11761627 | 1 | 1 | yes | characteristic |
| 275 | Q8 | — | 2 | 2 | no | — |
| 276 | D4 | — | 2 | 4 | no | — |
| 277 | D4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 2 | 2 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 268 | 2 | no | — |
| 282 | C2×C4 | — | 268 | 4 | no | — |
| 283 | C2×C4 | — | 134 | 2 | no | — |
| 284 | C2×C4 | — | 134 | 1 | no | — |
| 285 | C2×C4 | — | 134 | 1 | no | — |
| 286 | C2×C4 | #14524001 | 1 | 2 | yes | — |
| 287 | C2×C4 | C22×C4×D67 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | #41×D67 | 1 | 2 | yes | — |
| 289 | C2×C4 | C2×#11761627 | 1 | 2 | yes | — |
| 290 | C2×C4 | C2×#11761629 | 1 | 1 | yes | characteristic |
| 291 | C8 | — | 2 | 2 | no | — |
| 292 | C8 | — | 268 | 4 | no | — |
| 293 | C8 | — | 67 | 1 | no | — |
| 294 | C8 | — | 67 | 1 | no | — |
| 295 | C8 | — | 67 | 2 | no | — |
| 296 | C8 | — | 134 | 2 | no | — |
| 297 | C8 | #14523999 | 1 | 1 | yes | characteristic |
| 298 | C8 | #14523999 | 1 | 1 | yes | characteristic |
| 299 | C8 | #14523999 | 1 | 2 | yes | — |
| 300 | C22 | — | 2 | 2 | no | — |
| 301 | C22 | — | 4 | 1 | no | — |
| 302 | C22 | — | 4 | 2 | no | — |
| 303 | C22 | C2×#14524001 | 1 | 1 | yes | characteristic |
| 304 | C22 | #16863927 | 1 | 2 | yes | — |
| 305 | C4 | — | 2 | 2 | no | — |
| 306 | C4 | — | 8 | 4 | no | — |
| 307 | C4 | — | 268 | 1 | no | — |
| 308 | C4 | — | 268 | 2 | no | — |
| 309 | C4 | — | 134 | 2 | no | — |
| 310 | C4 | #117×D67 | 1 | 1 | yes | characteristic |
| 311 | C4 | C2×#14523999 | 1 | 1 | yes | characteristic |
| 312 | C4 | #16863932 | 1 | 2 | yes | — |
| 313 | C2 | — | 2 | 2 | no | — |
| 314 | C2 | — | 2 | 1 | no | — |
| 315 | C2 | — | 4 | 2 | no | — |
| 316 | C2 | #22562604 | 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