← #30955706
Subgroups — #30955706
Order 17152.
742 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22568572 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4×#14524031 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22561361 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22561356 | C2 | 1 | 2 | yes | maximal |
| 5 | #22561025 | C2 | 1 | 2 | yes | maximal |
| 6 | #22560343 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22560225 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #22560220 | C2 | 1 | 2 | yes | maximal |
| 9 | C67×#1767 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C2×#14524035 | C22 | 1 | 2 | yes | — |
| 11 | C2×#14524031 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×Dic134 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×D268 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#14523968 | C22 | 1 | 4 | yes | — |
| 15 | C2×#14523965 | C22 | 1 | 4 | yes | — |
| 16 | #16863302 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16863298 | C22 | 1 | 1 | yes | characteristic |
| 18 | #16863294 | — | 2 | 2 | no | — |
| 19 | #16863294 | — | 2 | 2 | no | — |
| 20 | #16863287 | — | 2 | 2 | no | — |
| 21 | #16863266 | C22 | 1 | 2 | yes | — |
| 22 | #16863262 | C22 | 1 | 1 | yes | characteristic |
| 23 | #16863262 | C22 | 1 | 2 | yes | — |
| 24 | C4×(C2×D67)⋊C4 | — | 2 | 2 | no | — |
| 25 | #16863159 | C22 | 1 | 4 | yes | — |
| 26 | #16863159 | C22 | 1 | 4 | yes | — |
| 27 | C2×C4⋊C8×C67 | C22 | 1 | 2 | yes | — |
| 28 | C2×#97×C67 | C22 | 1 | 2 | yes | — |
| 29 | C43×C67 | C22 | 1 | 1 | yes | characteristic |
| 30 | #336×C67 | C22 | 1 | 2 | yes | — |
| 31 | C22×Dic134 | C23 | 1 | 1 | yes | characteristic |
| 32 | C22×D268 | C23 | 1 | 1 | yes | characteristic |
| 33 | C22×C4×D67 | — | 2 | 1 | no | — |
| 34 | #14524035 | — | 4 | 4 | no | — |
| 35 | #14524031 | — | 2 | 2 | no | — |
| 36 | #14524031 | Q8 | 1 | 4 | yes | — |
| 37 | C4×Dic134 | — | 2 | 4 | no | — |
| 38 | C4×D268 | — | 2 | 4 | no | — |
| 39 | C2×C4⋊Dic67 | — | 2 | 2 | no | — |
| 40 | C2×C4⋊Dic67 | — | 2 | 2 | no | — |
| 41 | C2×C4⋊Dic67 | C23 | 1 | 4 | yes | — |
| 42 | C2×C4⋊Dic67 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×C4⋊Dic67 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×Dic67⋊C4 | — | 2 | 1 | no | — |
| 45 | C2×Dic67⋊C4 | — | 2 | 4 | no | — |
| 46 | C2×(C2×D67)⋊C4 | — | 2 | 2 | no | — |
| 47 | C2×(C2×D67)⋊C4 | — | 2 | 2 | no | — |
| 48 | C2×(C2×D67)⋊C4 | — | 2 | 1 | no | — |
| 49 | C2×C4×Dic67 | — | 2 | 1 | no | — |
| 50 | #14523968 | — | 4 | 4 | no | — |
| 51 | #14523965 | — | 4 | 4 | no | — |
| 52 | C67⋊#95 | — | 2 | 2 | no | — |
| 53 | C67⋊#95 | — | 2 | 2 | no | — |
| 54 | C67⋊#95 | — | 2 | 2 | no | — |
| 55 | C67⋊#95 | — | 2 | 2 | no | — |
| 56 | C22×C8×C67 | C23 | 1 | 4 | yes | — |
| 57 | C2×C42×C67 | D4 | 1 | 2 | yes | — |
| 58 | C2×C42×C67 | D4 | 1 | 2 | yes | — |
| 59 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 62 | C4⋊C8×C67 | — | 2 | 4 | no | — |
| 63 | #97×C67 | — | 2 | 4 | no | — |
| 64 | C23×D67 | — | 2 | 1 | no | — |
| 65 | C2×Dic134 | — | 2 | 2 | no | — |
| 66 | C2×Dic134 | — | 2 | 2 | no | — |
| 67 | C2×Dic134 | — | 2 | 1 | no | — |
| 68 | C2×Dic134 | C2×Q8 | 1 | 2 | yes | — |
| 69 | C2×D268 | — | 2 | 2 | no | — |
| 70 | C2×D268 | — | 2 | 2 | no | — |
| 71 | C2×D268 | — | 2 | 1 | no | — |
| 72 | C2×D268 | C2×Q8 | 1 | 2 | yes | — |
| 73 | C22×Dic67 | — | 2 | 1 | no | — |
| 74 | C22×Dic67 | — | 2 | 1 | no | — |
| 75 | C22×Dic67 | — | 2 | 1 | no | — |
| 76 | C22×Dic67 | — | 2 | 4 | no | — |
| 77 | C2×C4×D67 | — | 4 | 4 | no | — |
| 78 | C2×C4×D67 | — | 4 | 2 | no | — |
| 79 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 80 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 81 | C4⋊Dic67 | — | 2 | 4 | no | — |
| 82 | C4⋊Dic67 | — | 2 | 4 | no | — |
| 83 | C4⋊Dic67 | — | 4 | 4 | no | — |
| 84 | C4⋊Dic67 | — | 4 | 4 | no | — |
| 85 | Dic67⋊C4 | — | 4 | 8 | no | — |
| 86 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 87 | (C2×D67)⋊C4 | — | 2 | 8 | no | — |
| 88 | (C2×D67)⋊C4 | — | 4 | 4 | no | — |
| 89 | (C2×D67)⋊C4 | — | 4 | 2 | no | — |
| 90 | C4×Dic67 | — | 4 | 2 | no | — |
| 91 | C22×C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 92 | C22×C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 93 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C22×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 96 | C2×C8×C67 | — | 2 | 4 | no | — |
| 97 | C2×C8×C67 | — | 2 | 4 | no | — |
| 98 | C2×C8×C67 | — | 2 | 4 | no | — |
| 99 | C42×C67 | — | 2 | 2 | no | — |
| 100 | C42×C67 | — | 4 | 2 | no | — |
| 101 | C42×C67 | — | 4 | 2 | no | — |
| 102 | C42×C67 | C2×Q8 | 1 | 2 | yes | — |
| 103 | C42×C67 | #41 | 1 | 2 | yes | — |
| 104 | C42×C67 | #41 | 1 | 4 | yes | — |
| 105 | C23×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 106 | C22×D67 | — | 2 | 2 | no | — |
| 107 | C22×D67 | — | 4 | 1 | no | — |
| 108 | C22×D67 | — | 4 | 1 | no | — |
| 109 | C22×D67 | — | 4 | 2 | no | — |
| 110 | C22×D67 | — | 4 | 2 | no | — |
| 111 | C2×C4×C67 | — | 2 | 2 | no | — |
| 112 | C2×C4×C67 | — | 2 | 2 | no | — |
| 113 | C2×C4×C67 | — | 2 | 2 | no | — |
| 114 | C2×C4×C67 | — | 2 | 2 | no | — |
| 115 | C2×C4×C67 | — | 2 | 2 | no | — |
| 116 | C2×C4×C67 | — | 2 | 2 | no | — |
| 117 | C2×C4×C67 | — | 2 | 2 | no | — |
| 118 | C2×C4×C67 | C22⋊Q8 | 1 | 4 | yes | — |
| 119 | C2×C4×C67 | #124 | 1 | 2 | yes | — |
| 120 | C2×C4×C67 | #124 | 1 | 2 | yes | — |
| 121 | C2×C4×C67 | C22×Q8 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C67 | C2×#41 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C67 | C2×#41 | 1 | 2 | yes | — |
| 124 | C2×C4×C67 | H(5,𝔽2) | 1 | 2 | yes | — |
| 125 | C2×Dic67 | — | 2 | 2 | no | — |
| 126 | C2×Dic67 | — | 4 | 1 | no | — |
| 127 | C2×Dic67 | — | 4 | 1 | no | — |
| 128 | C2×Dic67 | — | 4 | 1 | no | — |
| 129 | C2×Dic67 | — | 4 | 1 | no | — |
| 130 | C2×Dic67 | — | 4 | 4 | no | — |
| 131 | C2×Dic67 | — | 4 | 2 | no | — |
| 132 | C2×Dic67 | — | 4 | 4 | no | — |
| 133 | C2×Dic67 | — | 4 | 4 | no | — |
| 134 | C2×Dic67 | — | 4 | 2 | no | — |
| 135 | D268 | — | 2 | 4 | no | — |
| 136 | D268 | — | 4 | 2 | no | — |
| 137 | C4×D67 | — | 8 | 4 | no | — |
| 138 | Dic134 | — | 2 | 4 | no | — |
| 139 | Dic134 | — | 4 | 2 | no | — |
| 140 | C8×C67 | — | 4 | 4 | no | — |
| 141 | C22×C67 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 142 | C22×C67 | C2×#124 | 1 | 1 | yes | characteristic |
| 143 | C22×C67 | #535 | 1 | 1 | yes | characteristic |
| 144 | C22×C67 | #575 | 1 | 2 | yes | — |
| 145 | C22×C67 | #575 | 1 | 2 | yes | — |
| 146 | C2×D67 | — | 4 | 2 | no | — |
| 147 | C2×D67 | — | 4 | 4 | no | — |
| 148 | C2×D67 | — | 8 | 1 | no | — |
| 149 | C2×D67 | — | 8 | 1 | no | — |
| 150 | C2×D67 | — | 8 | 2 | no | — |
| 151 | C4×C67 | — | 2 | 4 | no | — |
| 152 | C4×C67 | — | 2 | 2 | no | — |
| 153 | C4×C67 | — | 2 | 2 | no | — |
| 154 | C4×C67 | — | 4 | 2 | no | — |
| 155 | C4×C67 | #554 | 1 | 2 | yes | — |
| 156 | C4×C67 | #555 | 1 | 2 | yes | — |
| 157 | Dic67 | — | 4 | 2 | no | — |
| 158 | Dic67 | — | 8 | 1 | no | — |
| 159 | Dic67 | — | 8 | 1 | no | — |
| 160 | Dic67 | — | 8 | 4 | no | — |
| 161 | #29146 | — | 67 | 1 | no | maximal, Sylow |
| 162 | C2×C67 | #2563 | 1 | 1 | yes | characteristic |
| 163 | C2×C67 | #3080 | 1 | 2 | yes | — |
| 164 | C2×C67 | #3135 | 1 | 1 | yes | characteristic |
| 165 | C2×C67 | #3140 | 1 | 1 | yes | characteristic |
| 166 | C2×C67 | #3219 | 1 | 2 | yes | — |
| 167 | D67 | — | 8 | 2 | no | — |
| 168 | #2859 | — | 67 | 1 | no | — |
| 169 | C4×#124 | — | 67 | 1 | no | — |
| 170 | #2026 | — | 67 | 2 | no | — |
| 171 | #2025 | — | 67 | 2 | no | — |
| 172 | #1959 | — | 67 | 2 | no | — |
| 173 | #1836 | — | 67 | 2 | no | — |
| 174 | #1805 | — | 67 | 2 | no | — |
| 175 | #1804 | — | 67 | 2 | no | — |
| 176 | #1767 | D67 | 1 | 1 | yes | characteristic, complemented |
| 177 | C67 | #29146 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 178 | C2×#125 | — | 67 | 2 | no | — |
| 179 | C2×#124 | — | 67 | 1 | no | — |
| 180 | C2×C4×Q8 | — | 67 | 1 | no | — |
| 181 | C2×C4×D4 | — | 67 | 1 | no | — |
| 182 | C2×C4⋊C8 | C2×D67 | 1 | 2 | yes | — |
| 183 | C2×#103 | — | 67 | 4 | no | — |
| 184 | C2×#102 | — | 67 | 4 | no | — |
| 185 | C2×#97 | C2×D67 | 1 | 2 | yes | — |
| 186 | #391 | — | 67 | 1 | no | — |
| 187 | #390 | — | 67 | 1 | no | — |
| 188 | #389 | — | 134 | 2 | no | — |
| 189 | #389 | — | 134 | 2 | no | — |
| 190 | #388 | — | 134 | 2 | no | — |
| 191 | #384 | — | 67 | 2 | no | — |
| 192 | #382 | — | 67 | 2 | no | — |
| 193 | #382 | — | 67 | 1 | no | — |
| 194 | C4×C22⋊C4 | — | 134 | 2 | no | — |
| 195 | C43 | C2×D67 | 1 | 1 | yes | characteristic |
| 196 | #340 | — | 67 | 4 | no | — |
| 197 | #340 | — | 67 | 4 | no | — |
| 198 | #336 | C2×D67 | 1 | 2 | yes | — |
| 199 | C22×Q8 | — | 67 | 1 | no | — |
| 200 | C22×D4 | — | 67 | 1 | no | — |
| 201 | C23×C4 | — | 134 | 1 | no | — |
| 202 | C22×C8 | C22×D67 | 1 | 4 | yes | — |
| 203 | #125 | — | 268 | 4 | no | — |
| 204 | #124 | — | 67 | 4 | no | — |
| 205 | #124 | — | 134 | 2 | no | — |
| 206 | C4×Q8 | — | 134 | 4 | no | — |
| 207 | C4×D4 | — | 134 | 4 | no | — |
| 208 | C2×C4⋊C4 | — | 67 | 1 | no | — |
| 209 | C2×C4⋊C4 | — | 67 | 1 | no | — |
| 210 | C2×C4⋊C4 | — | 67 | 4 | no | — |
| 211 | C2×C4⋊C4 | — | 134 | 1 | no | — |
| 212 | C2×C4⋊C4 | — | 134 | 4 | no | — |
| 213 | C2×C4⋊C4 | — | 134 | 2 | no | — |
| 214 | C2×C4⋊C4 | — | 134 | 2 | no | — |
| 215 | C2×C22⋊C4 | — | 134 | 1 | no | — |
| 216 | C2×C22⋊C4 | — | 134 | 2 | no | — |
| 217 | C2×C22⋊C4 | — | 134 | 2 | no | — |
| 218 | C2×C42 | — | 134 | 1 | no | — |
| 219 | C2×C42 | D268 | 1 | 2 | yes | — |
| 220 | C2×C42 | D268 | 1 | 2 | yes | — |
| 221 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 222 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 223 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 224 | C4⋊C8 | — | 2 | 4 | no | — |
| 225 | #103 | — | 268 | 4 | no | — |
| 226 | #102 | — | 268 | 4 | no | — |
| 227 | #97 | — | 2 | 4 | no | — |
| 228 | #95 | — | 134 | 2 | no | — |
| 229 | #95 | — | 134 | 2 | no | — |
| 230 | #95 | — | 134 | 2 | no | — |
| 231 | #95 | — | 134 | 2 | no | — |
| 232 | C24 | — | 134 | 1 | no | — |
| 233 | C2×Q8 | — | 67 | 2 | no | — |
| 234 | C2×Q8 | — | 134 | 2 | no | — |
| 235 | C2×Q8 | — | 134 | 2 | no | — |
| 236 | C2×Q8 | — | 134 | 1 | no | — |
| 237 | C2×D4 | — | 67 | 2 | no | — |
| 238 | C2×D4 | — | 134 | 1 | no | — |
| 239 | C2×D4 | — | 134 | 2 | no | — |
| 240 | C2×D4 | — | 134 | 2 | no | — |
| 241 | C22×C4 | — | 268 | 4 | no | — |
| 242 | C22×C4 | — | 268 | 2 | no | — |
| 243 | C22×C4 | — | 134 | 1 | no | — |
| 244 | C22×C4 | — | 134 | 1 | no | — |
| 245 | C22×C4 | — | 134 | 4 | no | — |
| 246 | C22×C4 | — | 134 | 1 | no | — |
| 247 | C22×C4 | C2×D268 | 1 | 2 | yes | — |
| 248 | C22×C4 | C2×D268 | 1 | 2 | yes | — |
| 249 | C22×C4 | C2×D268 | 1 | 1 | yes | characteristic |
| 250 | C22×C4 | C2×D268 | 1 | 1 | yes | characteristic |
| 251 | C22×C4 | C23×D67 | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | — | 2 | 4 | no | — |
| 253 | C2×C8 | — | 2 | 4 | no | — |
| 254 | C2×C8 | — | 2 | 4 | no | — |
| 255 | C4⋊C4 | — | 268 | 8 | no | — |
| 256 | C4⋊C4 | — | 268 | 4 | no | — |
| 257 | C4⋊C4 | — | 268 | 4 | no | — |
| 258 | C4⋊C4 | — | 268 | 2 | no | — |
| 259 | C4⋊C4 | — | 134 | 2 | no | — |
| 260 | C4⋊C4 | — | 134 | 2 | no | — |
| 261 | C4⋊C4 | — | 134 | 4 | no | — |
| 262 | C4⋊C4 | — | 134 | 4 | no | — |
| 263 | C22⋊C4 | — | 268 | 4 | no | — |
| 264 | C22⋊C4 | — | 268 | 2 | no | — |
| 265 | C22⋊C4 | — | 134 | 8 | no | — |
| 266 | C42 | — | 2 | 2 | no | — |
| 267 | C42 | — | 4 | 2 | no | — |
| 268 | C42 | — | 4 | 2 | no | — |
| 269 | C42 | — | 268 | 2 | no | — |
| 270 | C42 | Q8×D67 | 1 | 2 | yes | — |
| 271 | C42 | #11761627 | 1 | 2 | yes | — |
| 272 | C42 | #11761628 | 1 | 4 | yes | — |
| 273 | C23 | — | 268 | 1 | no | — |
| 274 | C23 | — | 268 | 2 | no | — |
| 275 | C23 | — | 268 | 2 | no | — |
| 276 | C23 | — | 268 | 1 | no | — |
| 277 | C23 | — | 134 | 2 | no | — |
| 278 | C23 | C22×D268 | 1 | 1 | yes | characteristic |
| 279 | Q8 | — | 268 | 2 | no | — |
| 280 | Q8 | — | 134 | 4 | no | — |
| 281 | D4 | — | 268 | 2 | no | — |
| 282 | D4 | — | 134 | 4 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 2 | 2 | no | — |
| 285 | C2×C4 | — | 2 | 2 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 2 | 2 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 268 | 1 | no | — |
| 291 | C2×C4 | — | 268 | 1 | no | — |
| 292 | C2×C4 | — | 268 | 1 | no | — |
| 293 | C2×C4 | — | 268 | 1 | no | — |
| 294 | C2×C4 | — | 268 | 2 | no | — |
| 295 | C2×C4 | — | 268 | 4 | no | — |
| 296 | C2×C4 | — | 268 | 4 | no | — |
| 297 | C2×C4 | — | 268 | 2 | no | — |
| 298 | C2×C4 | — | 268 | 4 | no | — |
| 299 | C2×C4 | — | 536 | 4 | no | — |
| 300 | C2×C4 | — | 134 | 2 | no | — |
| 301 | C2×C4 | #14524023 | 1 | 4 | yes | — |
| 302 | C2×C4 | #14524031 | 1 | 2 | yes | — |
| 303 | C2×C4 | #14524031 | 1 | 2 | yes | — |
| 304 | C2×C4 | C2×Q8×D67 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | C2×#11761627 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | C2×#11761628 | 1 | 2 | yes | — |
| 307 | C2×C4 | #14524089 | 1 | 2 | yes | — |
| 308 | C8 | — | 4 | 4 | no | — |
| 309 | C22 | — | 268 | 2 | no | — |
| 310 | C22 | — | 268 | 4 | no | — |
| 311 | C22 | — | 536 | 1 | no | — |
| 312 | C22 | — | 536 | 1 | no | — |
| 313 | C22 | — | 536 | 2 | no | — |
| 314 | C22 | C2×#14524023 | 1 | 1 | yes | characteristic |
| 315 | C22 | C2×#14524031 | 1 | 1 | yes | characteristic |
| 316 | C22 | #16863965 | 1 | 1 | yes | characteristic |
| 317 | C22 | #16864196 | 1 | 2 | yes | — |
| 318 | C22 | #16864196 | 1 | 2 | yes | — |
| 319 | C4 | — | 2 | 4 | no | — |
| 320 | C4 | — | 2 | 2 | no | — |
| 321 | C4 | — | 2 | 2 | no | — |
| 322 | C4 | — | 4 | 2 | no | — |
| 323 | C4 | — | 268 | 2 | no | — |
| 324 | C4 | — | 536 | 1 | no | — |
| 325 | C4 | — | 536 | 1 | no | — |
| 326 | C4 | — | 536 | 4 | no | — |
| 327 | C4 | #16864106 | 1 | 2 | yes | — |
| 328 | C4 | #16864111 | 1 | 2 | yes | — |
| 329 | C2 | — | 536 | 2 | no | — |
| 330 | C2 | #22565378 | 1 | 1 | yes | characteristic |
| 331 | C2 | #22570032 | 1 | 2 | yes | — |
| 332 | C2 | #22570555 | 1 | 1 | yes | characteristic |
| 333 | C2 | #22570596 | 1 | 1 | yes | characteristic |
| 334 | C2 | #22571248 | 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