← #31725221
Subgroups — #31725221
Order 17152.
1998 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22574035 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22570609 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #22570589 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #22570574 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #22570554 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×#16863764 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #22568742 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #22568637 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C67×#3499 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C2×#14524050 | C22 | 1 | 2 | yes | — |
| 11 | #16864122 | — | 2 | 2 | no | — |
| 12 | #16864105 | — | 2 | 6 | no | — |
| 13 | #16864024 | C22 | 1 | 6 | yes | — |
| 14 | #16863953 | C22 | 1 | 3 | yes | — |
| 15 | C2×#14524035 | C22 | 1 | 3 | yes | — |
| 16 | #16863879 | C22 | 1 | 2 | yes | — |
| 17 | #16863764 | C22 | 1 | 3 | yes | — |
| 18 | #16863764 | C22 | 1 | 12 | yes | — |
| 19 | #16863764 | C22 | 1 | 6 | yes | — |
| 20 | #16863764 | C22 | 1 | 3 | yes | — |
| 21 | #16863760 | C22 | 1 | 1 | yes | characteristic |
| 22 | #16863760 | C22 | 1 | 3 | yes | — |
| 23 | #16863755 | C22 | 1 | 12 | yes | — |
| 24 | #16863750 | C22 | 1 | 12 | yes | — |
| 25 | #16863746 | C22 | 1 | 12 | yes | — |
| 26 | #16863405 | C22 | 1 | 12 | yes | — |
| 27 | #16863397 | C22 | 1 | 12 | yes | — |
| 28 | #16863392 | C22 | 1 | 12 | yes | — |
| 29 | C2×#14523967 | C22 | 1 | 12 | yes | — |
| 30 | #16863352 | C22 | 1 | 6 | yes | — |
| 31 | C4×#11761612 | C22 | 1 | 3 | yes | — |
| 32 | C2×#14523954 | C22 | 1 | 3 | yes | — |
| 33 | C2×2−1+4×C67 | C22 | 1 | 2 | yes | — |
| 34 | D4×Q8×C67 | — | 2 | 2 | no | — |
| 35 | #547×C67 | — | 2 | 6 | no | — |
| 36 | #536×C67 | C22 | 1 | 6 | yes | — |
| 37 | #532×C67 | C22 | 1 | 3 | yes | — |
| 38 | C2×C4⋊Q8×C67 | C22 | 1 | 3 | yes | — |
| 39 | C4×#41×C67 | C22 | 1 | 1 | yes | characteristic |
| 40 | #14524050 | — | 2 | 8 | no | — |
| 41 | C2×#11761612 | C23 | 1 | 6 | yes | — |
| 42 | C22×C67⋊C8 | C23 | 1 | 6 | yes | — |
| 43 | #14524043 | C23 | 1 | 6 | yes | — |
| 44 | #14524042 | — | 2 | 6 | no | — |
| 45 | #14524035 | — | 2 | 6 | no | — |
| 46 | #14524035 | C23 | 1 | 6 | yes | — |
| 47 | #14524035 | C23 | 1 | 3 | yes | — |
| 48 | #14524035 | C23 | 1 | 1 | yes | characteristic |
| 49 | #14524034 | — | 2 | 6 | no | — |
| 50 | #14524022 | — | 2 | 6 | no | — |
| 51 | #14524020 | — | 2 | 12 | no | — |
| 52 | Q8×Dic67 | — | 2 | 2 | no | — |
| 53 | C4×Dic134 | — | 2 | 6 | no | — |
| 54 | D4×Dic67 | — | 2 | 6 | no | — |
| 55 | #14524002 | C23 | 1 | 6 | yes | — |
| 56 | C2×C4⋊Dic67 | C23 | 1 | 6 | yes | — |
| 57 | C2×Dic67⋊C4 | — | 2 | 6 | no | — |
| 58 | #14523967 | C23 | 1 | 12 | yes | — |
| 59 | #14523967 | C23 | 1 | 24 | yes | — |
| 60 | #14523967 | C23 | 1 | 12 | yes | — |
| 61 | #14523964 | C23 | 1 | 12 | yes | — |
| 62 | #14523964 | C23 | 1 | 4 | yes | — |
| 63 | #14523954 | C23 | 1 | 6 | yes | — |
| 64 | #14523954 | C23 | 1 | 1 | yes | characteristic |
| 65 | #14523954 | C23 | 1 | 3 | yes | — |
| 66 | C4×C67⋊C8 | C23 | 1 | 6 | yes | — |
| 67 | 2−1+4×C67 | — | 2 | 16 | no | — |
| 68 | C2×#41×C67 | — | 2 | 6 | no | — |
| 69 | C2×#41×C67 | C23 | 1 | 6 | yes | — |
| 70 | C2×#41×C67 | C23 | 1 | 1 | yes | characteristic |
| 71 | C22×Q8×C67 | — | 2 | 2 | no | — |
| 72 | C22×Q8×C67 | C23 | 1 | 6 | yes | — |
| 73 | C4⋊Q8×C67 | — | 2 | 6 | no | — |
| 74 | C4⋊Q8×C67 | C23 | 1 | 6 | yes | — |
| 75 | C4⋊Q8×C67 | C23 | 1 | 3 | yes | — |
| 76 | #127×C67 | C23 | 1 | 1 | yes | characteristic |
| 77 | #124×C67 | C23 | 1 | 6 | yes | — |
| 78 | #123×C67 | — | 2 | 12 | no | — |
| 79 | C22⋊Q8×C67 | — | 2 | 12 | no | — |
| 80 | #121×C67 | — | 2 | 6 | no | — |
| 81 | C4×Q8×C67 | D4 | 1 | 2 | yes | — |
| 82 | C4×D4×C67 | — | 2 | 6 | no | — |
| 83 | C4×D4×C67 | D4 | 1 | 6 | yes | — |
| 84 | #117×C67 | C23 | 1 | 3 | yes | — |
| 85 | C2×C4⋊C4×C67 | — | 2 | 6 | no | — |
| 86 | C2×C42×C67 | C23 | 1 | 3 | yes | — |
| 87 | C2×Dic134 | — | 2 | 6 | no | — |
| 88 | C22×Dic67 | — | 2 | 6 | no | — |
| 89 | #11761612 | — | 2 | 12 | no | — |
| 90 | C2×C67⋊C8 | — | 2 | 12 | no | — |
| 91 | C2×C67⋊C8 | C24 | 1 | 6 | yes | — |
| 92 | C2×C67⋊C8 | C24 | 1 | 2 | yes | — |
| 93 | C4⋊Dic67 | — | 2 | 6 | no | — |
| 94 | C4⋊Dic67 | — | 2 | 6 | no | — |
| 95 | C4⋊Dic67 | C24 | 1 | 6 | yes | — |
| 96 | C4⋊Dic67 | C24 | 1 | 2 | yes | — |
| 97 | Dic67⋊C4 | — | 2 | 6 | no | — |
| 98 | Dic67⋊C4 | — | 2 | 12 | no | — |
| 99 | Dic67⋊C4 | — | 2 | 2 | no | — |
| 100 | C22⋊Dic67 | — | 2 | 6 | no | — |
| 101 | C4×Dic67 | — | 2 | 6 | no | — |
| 102 | #41×C67 | — | 2 | 24 | no | — |
| 103 | #41×C67 | — | 4 | 24 | no | — |
| 104 | #41×C67 | #41 | 1 | 8 | yes | — |
| 105 | C2×Q8×C67 | — | 2 | 6 | no | — |
| 106 | C2×Q8×C67 | — | 2 | 24 | no | — |
| 107 | C2×Q8×C67 | — | 4 | 8 | no | — |
| 108 | C2×Q8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C2×Q8×C67 | C24 | 1 | 6 | yes | — |
| 110 | C2×D4×C67 | — | 2 | 6 | no | — |
| 111 | C2×D4×C67 | — | 2 | 2 | no | — |
| 112 | C2×D4×C67 | C2×D4 | 1 | 3 | yes | — |
| 113 | C2×D4×C67 | C24 | 1 | 2 | yes | — |
| 114 | C22×C4×C67 | — | 2 | 6 | no | — |
| 115 | C22×C4×C67 | — | 2 | 6 | no | — |
| 116 | C22×C4×C67 | C2×D4 | 1 | 6 | yes | — |
| 117 | C22×C4×C67 | C24 | 1 | 3 | yes | — |
| 118 | C4⋊C4×C67 | — | 2 | 6 | no | — |
| 119 | C4⋊C4×C67 | — | 2 | 12 | no | — |
| 120 | C4⋊C4×C67 | C2×D4 | 1 | 6 | yes | — |
| 121 | C22⋊C4×C67 | — | 2 | 12 | no | — |
| 122 | C22⋊C4×C67 | C2×D4 | 1 | 6 | yes | — |
| 123 | C42×C67 | C2×D4 | 1 | 6 | yes | — |
| 124 | C42×C67 | C24 | 1 | 1 | yes | characteristic |
| 125 | C42×C67 | C24 | 1 | 3 | yes | — |
| 126 | C23×C67 | — | 2 | 2 | no | — |
| 127 | C23×C67 | C22×D4 | 1 | 3 | yes | — |
| 128 | Q8×C67 | — | 2 | 12 | no | — |
| 129 | Q8×C67 | — | 4 | 12 | no | — |
| 130 | Q8×C67 | #124 | 1 | 4 | yes | — |
| 131 | D4×C67 | — | 2 | 4 | no | — |
| 132 | D4×C67 | — | 4 | 4 | no | — |
| 133 | D4×C67 | — | 4 | 12 | no | — |
| 134 | D4×C67 | #124 | 1 | 12 | yes | — |
| 135 | C2×C4×C67 | — | 2 | 6 | no | — |
| 136 | C2×C4×C67 | — | 4 | 6 | no | — |
| 137 | C2×C4×C67 | — | 4 | 12 | no | — |
| 138 | C2×C4×C67 | — | 4 | 12 | no | — |
| 139 | C2×C4×C67 | C22×D4 | 1 | 2 | yes | — |
| 140 | C2×C4×C67 | C22×D4 | 1 | 6 | yes | — |
| 141 | C2×C4×C67 | C22×D4 | 1 | 3 | yes | — |
| 142 | C2×C4×C67 | C2×#41 | 1 | 12 | yes | — |
| 143 | C2×C4×C67 | C25 | 1 | 1 | yes | characteristic |
| 144 | C2×Dic67 | — | 2 | 2 | no | — |
| 145 | C2×Dic67 | — | 2 | 6 | no | — |
| 146 | C2×Dic67 | — | 8 | 6 | no | — |
| 147 | Dic134 | — | 8 | 6 | no | — |
| 148 | C67⋊C8 | — | 2 | 2 | no | — |
| 149 | C67⋊C8 | — | 2 | 6 | no | — |
| 150 | C22×C67 | — | 2 | 6 | no | — |
| 151 | C22×C67 | — | 4 | 4 | no | — |
| 152 | C22×C67 | — | 4 | 2 | no | — |
| 153 | C22×C67 | C2×#124 | 1 | 6 | yes | — |
| 154 | C22×C67 | C23×D4 | 1 | 1 | yes | characteristic |
| 155 | C4×C67 | — | 2 | 6 | no | — |
| 156 | C4×C67 | — | 2 | 2 | no | — |
| 157 | C4×C67 | — | 4 | 6 | no | — |
| 158 | C4×C67 | C2×#124 | 1 | 6 | yes | — |
| 159 | C4×C67 | C22×#41 | 1 | 2 | yes | — |
| 160 | Dic67 | — | 4 | 2 | no | — |
| 161 | Dic67 | — | 4 | 6 | no | — |
| 162 | #60627 | — | 67 | 1 | no | maximal, Sylow |
| 163 | C2×C67 | — | 2 | 6 | no | — |
| 164 | C2×C67 | — | 8 | 2 | no | — |
| 165 | C2×C67 | C22×#124 | 1 | 1 | yes | characteristic |
| 166 | C2×C67 | #3586 | 1 | 2 | yes | — |
| 167 | #3504 | — | 67 | 1 | no | — |
| 168 | #3499 | D67 | 1 | 1 | yes | characteristic |
| 169 | #3142 | — | 67 | 3 | no | — |
| 170 | #3139 | — | 67 | 3 | no | — |
| 171 | #3137 | — | 67 | 12 | no | — |
| 172 | #3135 | — | 67 | 3 | no | — |
| 173 | C2×#491 | — | 67 | 3 | no | — |
| 174 | #2898 | — | 67 | 4 | no | — |
| 175 | #2875 | — | 67 | 1 | no | — |
| 176 | C67 | #60627 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 177 | C2×2−1+4 | C2×D67 | 1 | 2 | yes | — |
| 178 | C2×#131 | — | 67 | 2 | no | — |
| 179 | #557 | — | 134 | 2 | no | — |
| 180 | #554 | — | 134 | 6 | no | — |
| 181 | D4×Q8 | — | 2 | 2 | no | — |
| 182 | #547 | — | 2 | 6 | no | — |
| 183 | #544 | — | 67 | 6 | no | — |
| 184 | #536 | C2×D67 | 1 | 6 | yes | — |
| 185 | #533 | — | 67 | 3 | no | — |
| 186 | #532 | C2×D67 | 1 | 3 | yes | — |
| 187 | C2×C4⋊Q8 | C2×D67 | 1 | 3 | yes | — |
| 188 | C2×#125 | — | 67 | 3 | no | — |
| 189 | #520 | — | 67 | 2 | no | — |
| 190 | C4×#41 | C2×D67 | 1 | 1 | yes | characteristic |
| 191 | #491 | — | 67 | 3 | no | — |
| 192 | #491 | — | 67 | 12 | no | — |
| 193 | #491 | — | 67 | 6 | no | — |
| 194 | #491 | — | 67 | 3 | no | — |
| 195 | #490 | — | 67 | 1 | no | — |
| 196 | #490 | — | 67 | 3 | no | — |
| 197 | #489 | — | 67 | 12 | no | — |
| 198 | #488 | — | 67 | 12 | no | — |
| 199 | #487 | — | 67 | 12 | no | — |
| 200 | #419 | — | 67 | 12 | no | — |
| 201 | #417 | — | 67 | 12 | no | — |
| 202 | #416 | — | 67 | 12 | no | — |
| 203 | C2×#103 | — | 67 | 12 | no | — |
| 204 | #405 | — | 67 | 6 | no | — |
| 205 | C4×M16 | — | 67 | 3 | no | — |
| 206 | C2×#97 | — | 67 | 3 | no | — |
| 207 | 2−1+4 | — | 2 | 16 | no | — |
| 208 | C2×#41 | — | 2 | 6 | no | — |
| 209 | C2×#41 | C22×D67 | 1 | 6 | yes | — |
| 210 | C2×#41 | C22×D67 | 1 | 1 | yes | characteristic |
| 211 | C22×Q8 | — | 2 | 2 | no | — |
| 212 | C22×Q8 | C22×D67 | 1 | 6 | yes | — |
| 213 | #131 | — | 134 | 8 | no | — |
| 214 | C2×M16 | — | 67 | 6 | no | — |
| 215 | C22×C8 | — | 67 | 6 | no | — |
| 216 | C4⋊Q8 | — | 2 | 6 | no | — |
| 217 | C4⋊Q8 | — | 67 | 6 | no | — |
| 218 | C4⋊Q8 | — | 134 | 6 | no | — |
| 219 | C4⋊Q8 | C22×D67 | 1 | 3 | yes | — |
| 220 | C4⋊Q8 | C22×D67 | 1 | 6 | yes | — |
| 221 | #127 | C22×D67 | 1 | 1 | yes | characteristic |
| 222 | #125 | — | 67 | 3 | no | — |
| 223 | #125 | — | 67 | 6 | no | — |
| 224 | #125 | — | 67 | 1 | no | — |
| 225 | #125 | — | 134 | 6 | no | — |
| 226 | #125 | — | 134 | 6 | no | — |
| 227 | #124 | C22×D67 | 1 | 6 | yes | — |
| 228 | #123 | — | 2 | 12 | no | — |
| 229 | C22⋊Q8 | — | 2 | 12 | no | — |
| 230 | C22⋊Q8 | — | 134 | 6 | no | — |
| 231 | C22⋊Q8 | — | 134 | 12 | no | — |
| 232 | #121 | — | 2 | 6 | no | — |
| 233 | C4×Q8 | — | 134 | 2 | no | — |
| 234 | C4×Q8 | — | 134 | 6 | no | — |
| 235 | C4×Q8 | C22⋊D67 | 1 | 2 | yes | — |
| 236 | C4×D4 | — | 2 | 6 | no | — |
| 237 | C4×D4 | — | 134 | 6 | no | — |
| 238 | C4×D4 | C22⋊D67 | 1 | 6 | yes | — |
| 239 | #117 | — | 67 | 6 | no | — |
| 240 | #117 | C22×D67 | 1 | 3 | yes | — |
| 241 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 242 | C2×C4⋊C4 | — | 67 | 6 | no | — |
| 243 | C2×C4⋊C4 | — | 134 | 6 | no | — |
| 244 | C2×C42 | C22×D67 | 1 | 3 | yes | — |
| 245 | #103 | — | 67 | 12 | no | — |
| 246 | #103 | — | 67 | 24 | no | — |
| 247 | #103 | — | 67 | 12 | no | — |
| 248 | #102 | — | 67 | 4 | no | — |
| 249 | #102 | — | 67 | 12 | no | — |
| 250 | #97 | — | 67 | 1 | no | — |
| 251 | #97 | — | 67 | 3 | no | — |
| 252 | #97 | — | 67 | 6 | no | — |
| 253 | C4×C8 | — | 67 | 6 | no | — |
| 254 | #41 | — | 2 | 24 | no | — |
| 255 | #41 | — | 4 | 24 | no | — |
| 256 | #41 | #11761627 | 1 | 8 | yes | — |
| 257 | C2×Q8 | — | 2 | 6 | no | — |
| 258 | C2×Q8 | — | 2 | 24 | no | — |
| 259 | C2×Q8 | — | 4 | 8 | no | — |
| 260 | C2×Q8 | — | 134 | 6 | no | — |
| 261 | C2×Q8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 262 | C2×Q8 | C23×D67 | 1 | 6 | yes | — |
| 263 | C2×D4 | — | 2 | 6 | no | — |
| 264 | C2×D4 | — | 2 | 2 | no | — |
| 265 | C2×D4 | C2×C22⋊D67 | 1 | 3 | yes | — |
| 266 | C2×D4 | C23×D67 | 1 | 2 | yes | — |
| 267 | C22×C4 | — | 2 | 6 | no | — |
| 268 | C22×C4 | — | 2 | 6 | no | — |
| 269 | C22×C4 | — | 134 | 6 | no | — |
| 270 | C22×C4 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 271 | C22×C4 | C23×D67 | 1 | 3 | yes | — |
| 272 | M16 | — | 134 | 12 | no | — |
| 273 | C2×C8 | — | 67 | 6 | no | — |
| 274 | C2×C8 | — | 67 | 2 | no | — |
| 275 | C2×C8 | — | 134 | 12 | no | — |
| 276 | C4⋊C4 | — | 2 | 6 | no | — |
| 277 | C4⋊C4 | — | 2 | 12 | no | — |
| 278 | C4⋊C4 | — | 67 | 6 | no | — |
| 279 | C4⋊C4 | — | 67 | 2 | no | — |
| 280 | C4⋊C4 | — | 134 | 6 | no | — |
| 281 | C4⋊C4 | — | 134 | 6 | no | — |
| 282 | C4⋊C4 | — | 134 | 12 | no | — |
| 283 | C4⋊C4 | — | 134 | 2 | no | — |
| 284 | C4⋊C4 | — | 134 | 6 | no | — |
| 285 | C4⋊C4 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 286 | C22⋊C4 | — | 2 | 12 | no | — |
| 287 | C22⋊C4 | — | 134 | 6 | no | — |
| 288 | C22⋊C4 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 289 | C42 | — | 134 | 6 | no | — |
| 290 | C42 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 291 | C42 | C23×D67 | 1 | 3 | yes | — |
| 292 | C42 | C23×D67 | 1 | 1 | yes | characteristic |
| 293 | C23 | — | 2 | 2 | no | — |
| 294 | C23 | C22×C22⋊D67 | 1 | 3 | yes | — |
| 295 | Q8 | — | 2 | 12 | no | — |
| 296 | Q8 | — | 4 | 12 | no | — |
| 297 | Q8 | — | 536 | 6 | no | — |
| 298 | Q8 | #14524030 | 1 | 4 | yes | — |
| 299 | D4 | — | 2 | 4 | no | — |
| 300 | D4 | — | 4 | 4 | no | — |
| 301 | D4 | — | 4 | 12 | no | — |
| 302 | D4 | #14524030 | 1 | 12 | yes | — |
| 303 | C2×C4 | — | 2 | 6 | no | — |
| 304 | C2×C4 | — | 4 | 6 | no | — |
| 305 | C2×C4 | — | 4 | 12 | no | — |
| 306 | C2×C4 | — | 4 | 12 | no | — |
| 307 | C2×C4 | — | 536 | 6 | no | — |
| 308 | C2×C4 | — | 134 | 2 | no | — |
| 309 | C2×C4 | — | 134 | 6 | no | — |
| 310 | C2×C4 | C22×C22⋊D67 | 1 | 2 | yes | — |
| 311 | C2×C4 | C22×C22⋊D67 | 1 | 6 | yes | — |
| 312 | C2×C4 | C22×C22⋊D67 | 1 | 3 | yes | — |
| 313 | C2×C4 | C2×#11761627 | 1 | 12 | yes | — |
| 314 | C2×C4 | C24×D67 | 1 | 1 | yes | characteristic |
| 315 | C8 | — | 134 | 2 | no | — |
| 316 | C8 | — | 134 | 6 | no | — |
| 317 | C22 | — | 2 | 6 | no | — |
| 318 | C22 | — | 4 | 2 | no | — |
| 319 | C22 | — | 4 | 4 | no | — |
| 320 | C22 | C2×#14524030 | 1 | 6 | yes | — |
| 321 | C22 | C23×C22⋊D67 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 2 | 6 | no | — |
| 324 | C4 | — | 4 | 6 | no | — |
| 325 | C4 | — | 268 | 6 | no | — |
| 326 | C4 | — | 268 | 2 | no | — |
| 327 | C4 | C2×#14524030 | 1 | 6 | yes | — |
| 328 | C4 | C22×#11761627 | 1 | 2 | yes | — |
| 329 | C2 | — | 2 | 6 | no | — |
| 330 | C2 | — | 8 | 2 | no | — |
| 331 | C2 | C22×#14524030 | 1 | 1 | yes | characteristic |
| 332 | C2 | #22574996 | 1 | 2 | yes | — |
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