← #30754500
Subgroups — #30754500
Order 17152.
370 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 | #22560973 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22560970 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22560365 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #22560079 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#16863148 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #22558093 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C67×#1296 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#14523975 | — | 2 | 1 | no | — |
| 9 | C2×#14523974 | — | 2 | 1 | no | — |
| 10 | C2×#14523974 | — | 2 | 1 | no | — |
| 11 | C2×#14523974 | — | 2 | 1 | no | — |
| 12 | C2×C4⋊(C67⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#14523953 | — | 2 | 1 | no | — |
| 14 | C2×C4×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | #16863266 | C22 | 1 | 1 | yes | characteristic |
| 16 | C4×C4⋊Dic67 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16863159 | — | 2 | 1 | no | — |
| 18 | #16863158 | — | 2 | 1 | no | — |
| 19 | #16863150 | — | 2 | 1 | no | — |
| 20 | #16863148 | C4 | 1 | 4 | yes | complemented |
| 21 | C2×C4⋊C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4⋊C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 23 | #384×C67 | C22 | 1 | 1 | yes | characteristic |
| 24 | C22×C67⋊C8 | — | 2 | 1 | no | — |
| 25 | C22×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 26 | C22×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×C4⋊Dic67 | — | 2 | 1 | no | — |
| 28 | C2×C4⋊Dic67 | — | 2 | 1 | no | — |
| 29 | C2×C4⋊Dic67 | — | 2 | 1 | no | — |
| 30 | C2×C4⋊Dic67 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C4⋊Dic67 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×C4×Dic67 | — | 2 | 1 | no | — |
| 33 | #14523975 | — | 4 | 2 | no | — |
| 34 | #14523974 | — | 4 | 2 | no | — |
| 35 | #14523974 | — | 4 | 2 | no | — |
| 36 | #14523974 | — | 4 | 2 | no | — |
| 37 | C4⋊(C67⋊C8) | — | 2 | 1 | no | — |
| 38 | C4⋊(C67⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 39 | #14523953 | — | 4 | 2 | no | — |
| 40 | C4×C67⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 41 | C4×C67⋊C8 | Q8 | 1 | 2 | yes | — |
| 42 | C67⋊#95 | — | 2 | 1 | no | — |
| 43 | C22×C8×C67 | — | 2 | 1 | no | — |
| 44 | C22×C8×C67 | — | 2 | 1 | no | — |
| 45 | C2×C4⋊C4×C67 | — | 2 | 1 | no | — |
| 46 | C2×C4⋊C4×C67 | — | 2 | 1 | no | — |
| 47 | C2×C4⋊C4×C67 | — | 2 | 1 | no | — |
| 48 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 49 | C4⋊C8×C67 | — | 2 | 1 | no | — |
| 50 | C4⋊C8×C67 | — | 2 | 1 | no | — |
| 51 | C4⋊C8×C67 | C2×C4 | 1 | 2 | yes | — |
| 52 | C4⋊C8×C67 | Q8 | 1 | 2 | yes | — |
| 53 | C22×Dic67 | — | 2 | 1 | no | — |
| 54 | C22×Dic67 | — | 2 | 1 | no | — |
| 55 | C22×Dic67 | — | 4 | 1 | no | — |
| 56 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 57 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 58 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 59 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 60 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 61 | C2×C67⋊C8 | C4⋊C4 | 1 | 2 | yes | — |
| 62 | C2×C67⋊C8 | C4⋊C4 | 1 | 2 | yes | — |
| 63 | C2×C67⋊C8 | SD16 | 1 | 2 | yes | — |
| 64 | C2×C67⋊C8 | SD16 | 1 | 2 | yes | — |
| 65 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 66 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 67 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 68 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 69 | C4⋊Dic67 | — | 4 | 1 | no | — |
| 70 | C4⋊Dic67 | — | 4 | 1 | no | — |
| 71 | C4⋊Dic67 | — | 4 | 1 | no | — |
| 72 | C4⋊Dic67 | — | 4 | 1 | no | — |
| 73 | C4⋊Dic67 | — | 8 | 1 | no | — |
| 74 | C4×Dic67 | — | 4 | 1 | no | — |
| 75 | C4×Dic67 | — | 4 | 1 | no | — |
| 76 | C22×C4×C67 | — | 4 | 1 | no | — |
| 77 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C22×C4×C67 | #41 | 1 | 1 | yes | characteristic |
| 80 | C2×C8×C67 | — | 2 | 2 | no | — |
| 81 | C2×C8×C67 | — | 2 | 2 | no | — |
| 82 | C2×C8×C67 | — | 4 | 1 | no | — |
| 83 | C2×C8×C67 | — | 4 | 1 | no | — |
| 84 | C2×C8×C67 | — | 4 | 1 | no | — |
| 85 | C2×C8×C67 | — | 4 | 1 | no | — |
| 86 | C4⋊C4×C67 | — | 4 | 1 | no | — |
| 87 | C4⋊C4×C67 | — | 4 | 1 | no | — |
| 88 | C4⋊C4×C67 | — | 4 | 1 | no | — |
| 89 | C4⋊C4×C67 | — | 4 | 1 | no | — |
| 90 | C4⋊C4×C67 | — | 8 | 1 | no | — |
| 91 | C42×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C42×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 93 | C42×C67 | C2×Q8 | 1 | 1 | yes | characteristic |
| 94 | C42×C67 | #41 | 1 | 1 | yes | characteristic |
| 95 | C23×C67 | #121 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C67 | — | 2 | 1 | no | — |
| 97 | C2×C4×C67 | — | 2 | 1 | no | — |
| 98 | C2×C4×C67 | — | 8 | 1 | no | — |
| 99 | C2×C4×C67 | — | 8 | 1 | no | — |
| 100 | C2×C4×C67 | — | 8 | 1 | no | — |
| 101 | C2×C4×C67 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×C4×C67 | C4×Q8 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C67 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C67 | #125 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C67 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C67 | C2×SD16 | 1 | 1 | yes | characteristic |
| 108 | C2×C4×C67 | C2×SD16 | 1 | 1 | yes | characteristic |
| 109 | C2×C4×C67 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C67 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 111 | C2×C4×C67 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C67 | #137 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C67 | #137 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C67 | #137 | 1 | 1 | yes | characteristic |
| 115 | C2×Dic67 | — | 4 | 1 | no | — |
| 116 | C2×Dic67 | — | 4 | 1 | no | — |
| 117 | C2×Dic67 | — | 4 | 1 | no | — |
| 118 | C2×Dic67 | — | 4 | 1 | no | — |
| 119 | C2×Dic67 | — | 4 | 1 | no | — |
| 120 | C2×Dic67 | — | 4 | 1 | no | — |
| 121 | C2×Dic67 | — | 8 | 1 | no | — |
| 122 | C2×Dic67 | — | 8 | 1 | no | — |
| 123 | C2×Dic67 | — | 8 | 1 | no | — |
| 124 | C8×C67 | — | 4 | 2 | no | — |
| 125 | C8×C67 | — | 4 | 2 | no | — |
| 126 | C67⋊C8 | — | 2 | 2 | no | — |
| 127 | C67⋊C8 | — | 4 | 2 | no | — |
| 128 | C67⋊C8 | #106 | 1 | 4 | yes | complemented |
| 129 | C22×C67 | #389 | 1 | 1 | yes | characteristic |
| 130 | C22×C67 | #460 | 1 | 1 | yes | characteristic |
| 131 | C22×C67 | #461 | 1 | 1 | yes | characteristic |
| 132 | C22×C67 | #468 | 1 | 1 | yes | characteristic |
| 133 | C22×C67 | #468 | 1 | 1 | yes | characteristic |
| 134 | C22×C67 | #471 | 1 | 1 | yes | characteristic |
| 135 | C22×C67 | #471 | 1 | 1 | yes | characteristic |
| 136 | C4×C67 | — | 2 | 1 | no | — |
| 137 | C4×C67 | — | 2 | 1 | no | — |
| 138 | C4×C67 | — | 16 | 1 | no | — |
| 139 | C4×C67 | C2×#106 | 1 | 1 | yes | characteristic |
| 140 | C4×C67 | #428 | 1 | 1 | yes | characteristic |
| 141 | C4×C67 | #441 | 1 | 1 | yes | characteristic |
| 142 | C4×C67 | #442 | 1 | 1 | yes | characteristic |
| 143 | C4×C67 | #475 | 1 | 1 | yes | characteristic |
| 144 | C4×C67 | #476 | 1 | 1 | yes | characteristic |
| 145 | C4×C67 | #501 | 1 | 1 | yes | characteristic |
| 146 | C4×C67 | #501 | 1 | 1 | yes | characteristic |
| 147 | Dic67 | — | 8 | 1 | no | — |
| 148 | Dic67 | — | 8 | 1 | no | — |
| 149 | Dic67 | — | 16 | 1 | no | — |
| 150 | #9618 | — | 67 | 1 | no | maximal, Sylow |
| 151 | C2×C67 | #1686 | 1 | 1 | yes | characteristic |
| 152 | C2×C67 | #1686 | 1 | 1 | yes | characteristic |
| 153 | C2×C67 | #1691 | 1 | 1 | yes | characteristic |
| 154 | C2×C67 | #1691 | 1 | 1 | yes | characteristic |
| 155 | C2×C67 | #1943 | 1 | 1 | yes | characteristic |
| 156 | C2×C67 | #1948 | 1 | 1 | yes | characteristic |
| 157 | C2×C67 | #1951 | 1 | 1 | yes | characteristic |
| 158 | #1948 | — | 67 | 1 | no | — |
| 159 | #1948 | — | 67 | 1 | no | — |
| 160 | #1844 | — | 67 | 1 | no | — |
| 161 | #1770 | — | 67 | 1 | no | — |
| 162 | C2×#335 | — | 67 | 1 | no | — |
| 163 | #1296 | — | 67 | 1 | no | — |
| 164 | #1296 | D67 | 1 | 1 | yes | characteristic |
| 165 | C67 | #9618 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 166 | C2×#106 | — | 134 | 1 | no | — |
| 167 | C2×#106 | — | 134 | 1 | no | — |
| 168 | C2×#106 | — | 134 | 1 | no | — |
| 169 | C2×#106 | — | 134 | 1 | no | — |
| 170 | C2×C4⋊C8 | — | 67 | 1 | no | — |
| 171 | C2×C4⋊C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 172 | C2×C4⋊C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 173 | C2×#97 | — | 134 | 1 | no | — |
| 174 | C2×C4×C8 | — | 67 | 1 | no | — |
| 175 | #384 | — | 67 | 1 | no | — |
| 176 | #384 | C2×D67 | 1 | 1 | yes | characteristic |
| 177 | C4×C4⋊C4 | — | 67 | 1 | no | — |
| 178 | #340 | — | 134 | 1 | no | — |
| 179 | #340 | — | 134 | 1 | no | — |
| 180 | #336 | — | 134 | 1 | no | — |
| 181 | #335 | — | 67 | 4 | no | — |
| 182 | C22×C8 | — | 2 | 1 | no | — |
| 183 | C22×C8 | — | 2 | 1 | no | — |
| 184 | C22×C8 | — | 67 | 1 | no | — |
| 185 | C22×C8 | — | 67 | 1 | no | — |
| 186 | C22×C8 | — | 134 | 1 | no | — |
| 187 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 188 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 189 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 190 | C2×C4⋊C4 | — | 67 | 1 | no | — |
| 191 | C2×C4⋊C4 | — | 67 | 1 | no | — |
| 192 | C2×C4⋊C4 | — | 134 | 1 | no | — |
| 193 | C2×C4⋊C4 | — | 134 | 1 | no | — |
| 194 | C2×C4⋊C4 | — | 134 | 1 | no | — |
| 195 | C2×C42 | — | 134 | 1 | no | — |
| 196 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 197 | #106 | — | 268 | 2 | no | — |
| 198 | #106 | — | 268 | 2 | no | — |
| 199 | #106 | — | 268 | 2 | no | — |
| 200 | #106 | — | 268 | 2 | no | — |
| 201 | C4⋊C8 | — | 2 | 1 | no | — |
| 202 | C4⋊C8 | — | 2 | 1 | no | — |
| 203 | C4⋊C8 | — | 67 | 2 | no | — |
| 204 | C4⋊C8 | — | 134 | 1 | no | — |
| 205 | C4⋊C8 | Dic134 | 1 | 2 | yes | — |
| 206 | C4⋊C8 | C4×D67 | 1 | 2 | yes | — |
| 207 | #97 | — | 268 | 2 | no | — |
| 208 | C4×C8 | — | 67 | 2 | no | — |
| 209 | C4×C8 | — | 67 | 2 | no | — |
| 210 | #95 | — | 134 | 1 | no | — |
| 211 | C22×C4 | — | 4 | 1 | no | — |
| 212 | C22×C4 | — | 268 | 1 | no | — |
| 213 | C22×C4 | — | 134 | 1 | no | — |
| 214 | C22×C4 | — | 134 | 1 | no | — |
| 215 | C22×C4 | D4×D67 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | D4×D67 | 1 | 1 | yes | characteristic |
| 217 | C22×C4 | #11761628 | 1 | 1 | yes | characteristic |
| 218 | C2×C8 | — | 2 | 2 | no | — |
| 219 | C2×C8 | — | 2 | 2 | no | — |
| 220 | C2×C8 | — | 4 | 1 | no | — |
| 221 | C2×C8 | — | 4 | 1 | no | — |
| 222 | C2×C8 | — | 4 | 1 | no | — |
| 223 | C2×C8 | — | 4 | 1 | no | — |
| 224 | C2×C8 | — | 268 | 1 | no | — |
| 225 | C2×C8 | — | 268 | 1 | no | — |
| 226 | C2×C8 | — | 67 | 2 | no | — |
| 227 | C2×C8 | — | 67 | 2 | no | — |
| 228 | C2×C8 | — | 67 | 2 | no | — |
| 229 | C2×C8 | — | 67 | 2 | no | — |
| 230 | C2×C8 | — | 134 | 2 | no | — |
| 231 | C2×C8 | — | 134 | 1 | no | — |
| 232 | C2×C8 | — | 134 | 1 | no | — |
| 233 | C4⋊C4 | — | 4 | 1 | no | — |
| 234 | C4⋊C4 | — | 4 | 1 | no | — |
| 235 | C4⋊C4 | — | 4 | 1 | no | — |
| 236 | C4⋊C4 | — | 4 | 1 | no | — |
| 237 | C4⋊C4 | — | 8 | 1 | no | — |
| 238 | C4⋊C4 | — | 268 | 1 | no | — |
| 239 | C4⋊C4 | — | 268 | 1 | no | — |
| 240 | C4⋊C4 | — | 268 | 1 | no | — |
| 241 | C4⋊C4 | — | 268 | 1 | no | — |
| 242 | C4⋊C4 | — | 536 | 1 | no | — |
| 243 | C4⋊C4 | — | 134 | 1 | no | — |
| 244 | C4⋊C4 | — | 134 | 1 | no | — |
| 245 | C4⋊C4 | — | 134 | 1 | no | — |
| 246 | C4⋊C4 | — | 134 | 1 | no | — |
| 247 | C42 | — | 268 | 1 | no | — |
| 248 | C42 | — | 268 | 1 | no | — |
| 249 | C42 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 250 | C42 | Q8×D67 | 1 | 1 | yes | characteristic |
| 251 | C42 | C2×Dic134 | 1 | 1 | yes | characteristic |
| 252 | C42 | #11761627 | 1 | 1 | yes | characteristic |
| 253 | C23 | #14524016 | 1 | 1 | yes | characteristic |
| 254 | C2×C4 | — | 2 | 1 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 8 | 1 | no | — |
| 257 | C2×C4 | — | 8 | 1 | no | — |
| 258 | C2×C4 | — | 8 | 1 | no | — |
| 259 | C2×C4 | — | 268 | 1 | no | — |
| 260 | C2×C4 | — | 268 | 1 | no | — |
| 261 | C2×C4 | — | 268 | 1 | no | — |
| 262 | C2×C4 | — | 268 | 1 | no | — |
| 263 | C2×C4 | — | 268 | 1 | no | — |
| 264 | C2×C4 | — | 268 | 1 | no | — |
| 265 | C2×C4 | — | 536 | 1 | no | — |
| 266 | C2×C4 | — | 536 | 1 | no | — |
| 267 | C2×C4 | — | 536 | 1 | no | — |
| 268 | C2×C4 | C4⋊C4×D67 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | (C22⋊D67)⋊C4 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | C4×Dic134 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | #14524020 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | #14524033 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | #14524042 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | SD16×D67 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | SD16×D67 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | #14524067 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | #14524067 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | #14524070 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | #14524072 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | #14524075 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | #14524075 | 1 | 1 | yes | characteristic |
| 282 | C8 | — | 4 | 2 | no | — |
| 283 | C8 | — | 4 | 2 | no | — |
| 284 | C8 | — | 268 | 2 | no | — |
| 285 | C8 | — | 67 | 4 | no | — |
| 286 | C8 | — | 134 | 2 | no | — |
| 287 | C22 | #16863293 | 1 | 1 | yes | characteristic |
| 288 | C22 | #16863595 | 1 | 1 | yes | characteristic |
| 289 | C22 | #16863600 | 1 | 1 | yes | characteristic |
| 290 | C22 | #16863641 | 1 | 1 | yes | characteristic |
| 291 | C22 | #16863645 | 1 | 1 | yes | characteristic |
| 292 | C22 | #16863658 | 1 | 1 | yes | characteristic |
| 293 | C22 | #16863662 | 1 | 1 | yes | characteristic |
| 294 | C4 | — | 2 | 1 | no | — |
| 295 | C4 | — | 2 | 1 | no | — |
| 296 | C4 | — | 16 | 1 | no | — |
| 297 | C4 | — | 536 | 1 | no | — |
| 298 | C4 | — | 536 | 1 | no | — |
| 299 | C4 | — | 1072 | 1 | no | — |
| 300 | C4 | #106×D67 | 1 | 1 | yes | characteristic |
| 301 | C4 | #16863437 | 1 | 1 | yes | characteristic |
| 302 | C4 | #16863500 | 1 | 1 | yes | characteristic |
| 303 | C4 | #16863505 | 1 | 1 | yes | characteristic |
| 304 | C4 | #16863679 | 1 | 1 | yes | characteristic |
| 305 | C4 | #16863684 | 1 | 1 | yes | characteristic |
| 306 | C4 | #16863803 | 1 | 1 | yes | characteristic |
| 307 | C4 | #16863804 | 1 | 1 | yes | characteristic |
| 308 | C2 | #22559680 | 1 | 1 | yes | characteristic |
| 309 | C2 | #22559684 | 1 | 1 | yes | characteristic |
| 310 | C2 | #22559707 | 1 | 1 | yes | characteristic |
| 311 | C2 | #22559711 | 1 | 1 | yes | characteristic |
| 312 | C2 | #22560946 | 1 | 1 | yes | characteristic |
| 313 | C2 | #22560971 | 1 | 1 | yes | characteristic |
| 314 | C2 | #22560988 | 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