← #31725216
Subgroups — #31725216
Order 17152.
2134 conjugacy classes of subgroups.
The subgroup lattice has 342 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22573987 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22570612 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #22570585 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #22570568 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #22570557 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×#16863762 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #22568732 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #22568637 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C67×#3504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C2×#14524089 | C22 | 1 | 2 | yes | — |
| 11 | C2×#14524050 | C22 | 1 | 2 | yes | — |
| 12 | #16864068 | — | 2 | 2 | no | — |
| 13 | D4×D268 | — | 2 | 6 | no | — |
| 14 | #16863965 | C22 | 1 | 6 | yes | — |
| 15 | #16863946 | C22 | 1 | 3 | yes | — |
| 16 | C2×#14524040 | C22 | 1 | 3 | yes | — |
| 17 | #16863766 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | #16863766 | C22 | 1 | 3 | yes | complemented |
| 19 | #16863762 | C22 | 1 | 3 | yes | complemented |
| 20 | #16863762 | C22 | 1 | 12 | yes | complemented |
| 21 | #16863762 | C22 | 1 | 6 | yes | complemented |
| 22 | #16863762 | C22 | 1 | 3 | yes | complemented |
| 23 | #16863758 | C22 | 1 | 12 | yes | complemented |
| 24 | #16863753 | C22 | 1 | 12 | yes | complemented |
| 25 | #16863744 | C22 | 1 | 12 | yes | complemented |
| 26 | #16863400 | C22 | 1 | 12 | yes | complemented |
| 27 | #16863395 | C22 | 1 | 12 | yes | complemented |
| 28 | #16863390 | C22 | 1 | 12 | yes | complemented |
| 29 | C2×#14523963 | C22 | 1 | 12 | yes | complemented |
| 30 | #16863352 | C22 | 1 | 6 | yes | complemented |
| 31 | C4×#11761612 | C22 | 1 | 3 | yes | complemented |
| 32 | C2×#14523954 | C22 | 1 | 3 | yes | complemented |
| 33 | #557×C67 | — | 2 | 2 | no | — |
| 34 | #554×C67 | — | 2 | 6 | no | — |
| 35 | #544×C67 | C22 | 1 | 6 | yes | complemented |
| 36 | #533×C67 | C22 | 1 | 3 | yes | complemented |
| 37 | C2×#125×C67 | C22 | 1 | 3 | yes | complemented |
| 38 | #520×C67 | C22 | 1 | 2 | yes | — |
| 39 | C4×#41×C67 | C22 | 1 | 1 | yes | characteristic |
| 40 | #14524089 | — | 2 | 16 | no | — |
| 41 | C2×#11761629 | — | 2 | 2 | no | — |
| 42 | C2×#11761628 | C23 | 1 | 6 | yes | — |
| 43 | C22×D268 | — | 2 | 6 | no | — |
| 44 | C22×D268 | C23 | 1 | 6 | yes | — |
| 45 | C2×D4×D67 | — | 2 | 6 | no | — |
| 46 | #14524050 | — | 2 | 8 | no | — |
| 47 | C2×#11761612 | C23 | 1 | 6 | yes | — |
| 48 | C22×C67⋊C8 | C23 | 1 | 6 | yes | — |
| 49 | #14524043 | C23 | 1 | 1 | yes | characteristic |
| 50 | #14524040 | — | 2 | 6 | no | — |
| 51 | #14524040 | C23 | 1 | 6 | yes | — |
| 52 | #14524040 | C23 | 1 | 3 | yes | — |
| 53 | #14524031 | C23 | 1 | 6 | yes | — |
| 54 | #14524018 | — | 2 | 12 | no | — |
| 55 | #14524017 | — | 2 | 6 | no | — |
| 56 | #14524013 | — | 2 | 12 | no | — |
| 57 | C4×D268 | — | 2 | 6 | no | — |
| 58 | #14523966 | C23 | 1 | 12 | yes | — |
| 59 | #14523966 | C23 | 1 | 4 | yes | — |
| 60 | #14523963 | C23 | 1 | 12 | yes | — |
| 61 | #14523963 | C23 | 1 | 24 | yes | — |
| 62 | #14523963 | C23 | 1 | 12 | 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 | C2×#41×C67 | C23 | 1 | 1 | yes | characteristic |
| 68 | C4⋊Q8×C67 | — | 2 | 6 | no | — |
| 69 | C4⋊Q8×C67 | C23 | 1 | 6 | yes | — |
| 70 | #125×C67 | — | 2 | 6 | no | — |
| 71 | #125×C67 | — | 2 | 6 | no | — |
| 72 | #125×C67 | C23 | 1 | 6 | yes | — |
| 73 | #125×C67 | C23 | 1 | 3 | yes | — |
| 74 | #125×C67 | C23 | 1 | 1 | yes | characteristic |
| 75 | C22⋊Q8×C67 | — | 2 | 6 | no | — |
| 76 | C22⋊Q8×C67 | — | 2 | 12 | no | — |
| 77 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 78 | C4×Q8×C67 | — | 2 | 6 | no | — |
| 79 | C4×Q8×C67 | D4 | 1 | 2 | yes | — |
| 80 | C4×D4×C67 | — | 2 | 6 | no | — |
| 81 | C4×D4×C67 | D4 | 1 | 6 | yes | — |
| 82 | #117×C67 | C23 | 1 | 6 | yes | — |
| 83 | #117×C67 | C23 | 1 | 3 | yes | — |
| 84 | C2×C4⋊C4×C67 | — | 2 | 6 | no | — |
| 85 | C2×C4⋊C4×C67 | C23 | 1 | 6 | yes | — |
| 86 | C2×C42×C67 | C23 | 1 | 3 | yes | — |
| 87 | C23×D67 | — | 2 | 6 | no | — |
| 88 | #11761629 | — | 4 | 8 | no | — |
| 89 | #11761628 | — | 2 | 24 | no | — |
| 90 | C2×Dic134 | C24 | 1 | 2 | yes | — |
| 91 | C2×D268 | — | 2 | 24 | no | — |
| 92 | C2×D268 | — | 2 | 6 | no | — |
| 93 | C2×D268 | — | 2 | 6 | no | — |
| 94 | C2×D268 | — | 2 | 12 | no | — |
| 95 | C2×D268 | — | 8 | 12 | no | — |
| 96 | C2×D268 | C24 | 1 | 6 | yes | — |
| 97 | C2×C22⋊D67 | — | 2 | 6 | no | — |
| 98 | D4×D67 | — | 4 | 24 | no | — |
| 99 | C2×C4×D67 | — | 2 | 6 | no | — |
| 100 | #11761612 | — | 2 | 12 | no | — |
| 101 | C2×C67⋊C8 | — | 2 | 12 | no | — |
| 102 | C2×C67⋊C8 | C24 | 1 | 6 | yes | — |
| 103 | C2×C67⋊C8 | C24 | 1 | 2 | yes | — |
| 104 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 105 | (C2×D67)⋊C4 | — | 2 | 12 | no | — |
| 106 | #41×C67 | #41 | 1 | 8 | yes | — |
| 107 | C2×Q8×C67 | — | 2 | 6 | no | — |
| 108 | C2×Q8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C2×D4×C67 | C2×D4 | 1 | 3 | yes | — |
| 110 | C22×C4×C67 | — | 2 | 6 | no | — |
| 111 | C22×C4×C67 | C2×D4 | 1 | 6 | yes | — |
| 112 | C22×C4×C67 | C24 | 1 | 3 | yes | — |
| 113 | C4⋊C4×C67 | — | 2 | 6 | no | — |
| 114 | C4⋊C4×C67 | — | 2 | 12 | no | — |
| 115 | C4⋊C4×C67 | — | 2 | 6 | no | — |
| 116 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 117 | C4⋊C4×C67 | — | 2 | 6 | no | — |
| 118 | C4⋊C4×C67 | C2×D4 | 1 | 6 | yes | — |
| 119 | C4⋊C4×C67 | C24 | 1 | 6 | yes | — |
| 120 | C4⋊C4×C67 | C24 | 1 | 2 | yes | — |
| 121 | C22⋊C4×C67 | — | 2 | 6 | no | — |
| 122 | C22⋊C4×C67 | C2×D4 | 1 | 6 | yes | — |
| 123 | C42×C67 | — | 2 | 6 | no | — |
| 124 | C42×C67 | C2×D4 | 1 | 6 | yes | — |
| 125 | C42×C67 | C24 | 1 | 1 | yes | characteristic |
| 126 | C42×C67 | C24 | 1 | 3 | yes | — |
| 127 | C23×C67 | C22×D4 | 1 | 3 | yes | — |
| 128 | C22×D67 | — | 2 | 6 | no | — |
| 129 | C22×D67 | — | 4 | 12 | no | — |
| 130 | C22×D67 | — | 8 | 12 | no | — |
| 131 | Q8×C67 | — | 8 | 6 | no | — |
| 132 | Q8×C67 | #124 | 1 | 4 | yes | — |
| 133 | D4×C67 | #124 | 1 | 12 | yes | — |
| 134 | C2×C4×C67 | — | 2 | 2 | no | — |
| 135 | C2×C4×C67 | — | 2 | 6 | no | — |
| 136 | C2×C4×C67 | — | 4 | 6 | no | — |
| 137 | C2×C4×C67 | — | 8 | 6 | no | — |
| 138 | C2×C4×C67 | C22×D4 | 1 | 2 | yes | — |
| 139 | C2×C4×C67 | C22×D4 | 1 | 6 | yes | — |
| 140 | C2×C4×C67 | C22×D4 | 1 | 3 | yes | — |
| 141 | C2×C4×C67 | C2×#41 | 1 | 12 | yes | — |
| 142 | C2×C4×C67 | C25 | 1 | 1 | yes | characteristic |
| 143 | C22⋊D67 | — | 4 | 12 | no | — |
| 144 | C2×Dic67 | — | 2 | 2 | no | — |
| 145 | D268 | — | 2 | 12 | no | — |
| 146 | D268 | — | 4 | 12 | no | — |
| 147 | D268 | — | 4 | 12 | no | — |
| 148 | D268 | — | 8 | 12 | no | — |
| 149 | C4×D67 | — | 4 | 12 | no | — |
| 150 | Dic134 | — | 2 | 4 | no | — |
| 151 | C67⋊C8 | — | 2 | 2 | no | — |
| 152 | C67⋊C8 | — | 2 | 6 | no | — |
| 153 | C22×C67 | — | 2 | 6 | no | — |
| 154 | C22×C67 | C2×#124 | 1 | 6 | yes | — |
| 155 | C22×C67 | C23×D4 | 1 | 1 | yes | characteristic |
| 156 | C2×D67 | — | 4 | 12 | no | — |
| 157 | C2×D67 | — | 4 | 6 | no | — |
| 158 | C2×D67 | — | 16 | 12 | no | — |
| 159 | C4×C67 | — | 2 | 2 | no | — |
| 160 | C4×C67 | — | 2 | 6 | no | — |
| 161 | C4×C67 | — | 4 | 6 | no | — |
| 162 | C4×C67 | — | 4 | 2 | no | — |
| 163 | C4×C67 | C2×#124 | 1 | 6 | yes | — |
| 164 | C4×C67 | C22×#41 | 1 | 2 | yes | — |
| 165 | Dic67 | — | 4 | 2 | no | — |
| 166 | #60626 | — | 67 | 1 | no | maximal, Sylow |
| 167 | C2×C67 | — | 2 | 6 | no | — |
| 168 | C2×C67 | C22×#124 | 1 | 1 | yes | characteristic |
| 169 | C2×C67 | #3585 | 1 | 2 | yes | — |
| 170 | D67 | — | 8 | 6 | no | — |
| 171 | #3504 | D67 | 1 | 1 | yes | characteristic, complemented |
| 172 | #3498 | — | 67 | 1 | no | — |
| 173 | #3142 | — | 67 | 3 | no | — |
| 174 | #3138 | — | 67 | 3 | no | — |
| 175 | #3136 | — | 67 | 12 | no | — |
| 176 | #3135 | — | 67 | 3 | no | — |
| 177 | C2×#490 | — | 67 | 3 | no | — |
| 178 | #2897 | — | 67 | 4 | no | — |
| 179 | #2875 | — | 67 | 1 | no | — |
| 180 | C67 | #60626 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 181 | C2×H(5,𝔽2) | — | 67 | 2 | no | — |
| 182 | C2×#131 | — | 67 | 2 | no | — |
| 183 | #557 | — | 2 | 2 | no | — |
| 184 | #554 | — | 2 | 6 | no | — |
| 185 | #550 | — | 134 | 2 | no | — |
| 186 | D42 | — | 134 | 6 | no | — |
| 187 | #544 | C2×D67 | 1 | 6 | yes | complemented |
| 188 | #535 | — | 67 | 6 | no | — |
| 189 | #533 | C2×D67 | 1 | 3 | yes | complemented |
| 190 | #532 | — | 67 | 3 | no | — |
| 191 | C2×#127 | — | 67 | 3 | no | — |
| 192 | C2×#125 | C2×D67 | 1 | 3 | yes | complemented |
| 193 | #520 | C2×D67 | 1 | 2 | yes | — |
| 194 | C4×#41 | C2×D67 | 1 | 1 | yes | characteristic |
| 195 | #491 | — | 67 | 1 | no | — |
| 196 | #491 | — | 67 | 3 | no | — |
| 197 | #490 | — | 67 | 3 | no | — |
| 198 | #490 | — | 67 | 12 | no | — |
| 199 | #490 | — | 67 | 6 | no | — |
| 200 | #490 | — | 67 | 3 | no | — |
| 201 | #489 | — | 67 | 12 | no | — |
| 202 | #488 | — | 67 | 12 | no | — |
| 203 | #486 | — | 67 | 12 | no | — |
| 204 | #418 | — | 67 | 12 | no | — |
| 205 | #417 | — | 67 | 12 | no | — |
| 206 | #416 | — | 67 | 12 | no | — |
| 207 | C2×#102 | — | 67 | 12 | no | — |
| 208 | #405 | — | 67 | 6 | no | — |
| 209 | C4×M16 | — | 67 | 3 | no | — |
| 210 | C2×#97 | — | 67 | 3 | no | — |
| 211 | H(5,𝔽2) | — | 134 | 16 | no | — |
| 212 | C2×#41 | — | 67 | 6 | no | — |
| 213 | C2×#41 | — | 134 | 2 | no | — |
| 214 | C2×#41 | C22×D67 | 1 | 1 | yes | characteristic |
| 215 | C22×D4 | — | 67 | 6 | no | — |
| 216 | C22×D4 | — | 134 | 6 | no | — |
| 217 | C22×D4 | — | 134 | 6 | no | — |
| 218 | #131 | — | 134 | 8 | no | — |
| 219 | C2×M16 | — | 67 | 6 | no | — |
| 220 | C22×C8 | — | 67 | 6 | no | — |
| 221 | C4⋊Q8 | — | 2 | 6 | no | — |
| 222 | C4⋊Q8 | — | 67 | 1 | no | — |
| 223 | C4⋊Q8 | C22×D67 | 1 | 6 | yes | — |
| 224 | #127 | — | 67 | 3 | no | — |
| 225 | #127 | — | 67 | 6 | no | — |
| 226 | #127 | — | 134 | 6 | no | — |
| 227 | #125 | — | 2 | 6 | no | — |
| 228 | #125 | — | 2 | 6 | no | — |
| 229 | #125 | C22×D67 | 1 | 3 | yes | — |
| 230 | #125 | C22×D67 | 1 | 6 | yes | — |
| 231 | #125 | C22×D67 | 1 | 1 | yes | characteristic |
| 232 | #124 | — | 67 | 6 | no | — |
| 233 | C22⋊Q8 | — | 2 | 6 | no | — |
| 234 | C22⋊Q8 | — | 2 | 12 | no | — |
| 235 | #121 | — | 134 | 6 | no | — |
| 236 | #121 | — | 134 | 12 | no | — |
| 237 | #120 | — | 134 | 12 | no | — |
| 238 | C4×Q8 | — | 2 | 2 | no | — |
| 239 | C4×Q8 | — | 2 | 6 | no | — |
| 240 | C4×Q8 | C22⋊D67 | 1 | 2 | yes | — |
| 241 | C4×D4 | — | 2 | 6 | no | — |
| 242 | C4×D4 | — | 134 | 6 | no | — |
| 243 | C4×D4 | C22⋊D67 | 1 | 6 | yes | — |
| 244 | #117 | C22×D67 | 1 | 6 | yes | — |
| 245 | #117 | C22×D67 | 1 | 3 | yes | — |
| 246 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 247 | C2×C4⋊C4 | C22×D67 | 1 | 6 | yes | — |
| 248 | C2×C42 | C22×D67 | 1 | 3 | yes | — |
| 249 | #103 | — | 67 | 4 | no | — |
| 250 | #103 | — | 67 | 12 | no | — |
| 251 | #102 | — | 67 | 12 | no | — |
| 252 | #102 | — | 67 | 24 | no | — |
| 253 | #102 | — | 67 | 12 | no | — |
| 254 | #97 | — | 67 | 1 | no | — |
| 255 | #97 | — | 67 | 3 | no | — |
| 256 | #97 | — | 67 | 6 | no | — |
| 257 | C4×C8 | — | 67 | 6 | no | — |
| 258 | C24 | — | 134 | 6 | no | — |
| 259 | #41 | — | 268 | 8 | no | — |
| 260 | #41 | — | 134 | 24 | no | — |
| 261 | #41 | #11761629 | 1 | 8 | yes | — |
| 262 | C2×Q8 | — | 2 | 6 | no | — |
| 263 | C2×Q8 | — | 67 | 2 | no | — |
| 264 | C2×Q8 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 265 | C2×D4 | — | 268 | 24 | no | — |
| 266 | C2×D4 | — | 536 | 12 | no | — |
| 267 | C2×D4 | — | 67 | 6 | no | — |
| 268 | C2×D4 | — | 134 | 6 | no | — |
| 269 | C2×D4 | — | 134 | 6 | no | — |
| 270 | C2×D4 | — | 134 | 6 | no | — |
| 271 | C2×D4 | — | 134 | 12 | no | — |
| 272 | C2×D4 | — | 134 | 24 | no | — |
| 273 | C2×D4 | C2×C22⋊D67 | 1 | 3 | yes | — |
| 274 | C22×C4 | — | 2 | 6 | no | — |
| 275 | C22×C4 | — | 134 | 6 | no | — |
| 276 | C22×C4 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 277 | C22×C4 | C23×D67 | 1 | 3 | yes | — |
| 278 | M16 | — | 134 | 12 | no | — |
| 279 | C2×C8 | — | 67 | 6 | no | — |
| 280 | C2×C8 | — | 67 | 2 | no | — |
| 281 | C2×C8 | — | 134 | 12 | no | — |
| 282 | C4⋊C4 | — | 2 | 6 | no | — |
| 283 | C4⋊C4 | — | 2 | 12 | no | — |
| 284 | C4⋊C4 | — | 2 | 6 | no | — |
| 285 | C4⋊C4 | — | 2 | 2 | no | — |
| 286 | C4⋊C4 | — | 2 | 6 | no | — |
| 287 | C4⋊C4 | — | 134 | 2 | no | — |
| 288 | C4⋊C4 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 289 | C4⋊C4 | C23×D67 | 1 | 6 | yes | — |
| 290 | C4⋊C4 | C23×D67 | 1 | 2 | yes | — |
| 291 | C22⋊C4 | — | 2 | 6 | no | — |
| 292 | C22⋊C4 | — | 134 | 12 | no | — |
| 293 | C22⋊C4 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 294 | C42 | — | 2 | 6 | no | — |
| 295 | C42 | C2×C22⋊D67 | 1 | 6 | yes | — |
| 296 | C42 | C23×D67 | 1 | 3 | yes | — |
| 297 | C42 | C23×D67 | 1 | 1 | yes | characteristic |
| 298 | C23 | — | 268 | 12 | no | — |
| 299 | C23 | — | 536 | 12 | no | — |
| 300 | C23 | — | 134 | 6 | no | — |
| 301 | C23 | C22×C22⋊D67 | 1 | 3 | yes | — |
| 302 | Q8 | — | 8 | 6 | no | — |
| 303 | Q8 | — | 134 | 4 | no | — |
| 304 | Q8 | #14524032 | 1 | 4 | yes | — |
| 305 | D4 | — | 268 | 12 | no | — |
| 306 | D4 | — | 268 | 12 | no | — |
| 307 | D4 | — | 268 | 12 | no | — |
| 308 | D4 | — | 536 | 12 | no | — |
| 309 | D4 | — | 134 | 12 | no | — |
| 310 | D4 | #14524032 | 1 | 12 | yes | — |
| 311 | C2×C4 | — | 2 | 2 | no | — |
| 312 | C2×C4 | — | 2 | 6 | no | — |
| 313 | C2×C4 | — | 4 | 6 | no | — |
| 314 | C2×C4 | — | 8 | 6 | no | — |
| 315 | C2×C4 | — | 268 | 12 | no | — |
| 316 | C2×C4 | — | 134 | 2 | no | — |
| 317 | C2×C4 | C22×C22⋊D67 | 1 | 2 | yes | — |
| 318 | C2×C4 | C22×C22⋊D67 | 1 | 6 | yes | — |
| 319 | C2×C4 | C22×C22⋊D67 | 1 | 3 | yes | — |
| 320 | C2×C4 | C2×#11761629 | 1 | 12 | yes | — |
| 321 | C2×C4 | C24×D67 | 1 | 1 | yes | characteristic |
| 322 | C8 | — | 134 | 2 | no | — |
| 323 | C8 | — | 134 | 6 | no | — |
| 324 | C22 | — | 2 | 6 | no | — |
| 325 | C22 | — | 268 | 12 | no | — |
| 326 | C22 | — | 268 | 6 | no | — |
| 327 | C22 | — | 1072 | 12 | no | — |
| 328 | C22 | C2×#14524032 | 1 | 6 | yes | — |
| 329 | C22 | C23×C22⋊D67 | 1 | 1 | yes | characteristic |
| 330 | C4 | — | 2 | 2 | no | — |
| 331 | C4 | — | 2 | 6 | no | — |
| 332 | C4 | — | 4 | 6 | no | — |
| 333 | C4 | — | 4 | 2 | no | — |
| 334 | C4 | — | 268 | 2 | no | — |
| 335 | C4 | C2×#14524032 | 1 | 6 | yes | — |
| 336 | C4 | C22×#11761629 | 1 | 2 | yes | — |
| 337 | C2 | — | 2 | 6 | no | — |
| 338 | C2 | — | 536 | 6 | no | — |
| 339 | C2 | C22×#14524032 | 1 | 1 | yes | characteristic |
| 340 | C2 | #22574988 | 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