← #10693713
Subgroups — #10693713
Order 768.
318 conjugacy classes of subgroups.
The subgroup lattice has 306 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×#1986 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #71828 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #71826 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #69875 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #69847 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #67620 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #67619 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #9591 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×#533 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C3×C4×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#822 | — | 2 | 1 | no | — |
| 13 | #4925 | C22 | 1 | 1 | yes | characteristic |
| 14 | #4924 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 17 | C3×#339 | — | 2 | 1 | no | — |
| 18 | C3×#339 | — | 2 | 1 | no | — |
| 19 | #4558 | — | 2 | 1 | no | — |
| 20 | #4547 | C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | #4547 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | #4546 | C4 | 1 | 1 | yes | characteristic, complemented |
| 23 | #4546 | C4 | 1 | 1 | yes | characteristic, complemented |
| 24 | #4535 | — | 2 | 1 | no | — |
| 25 | #4535 | — | 2 | 1 | no | — |
| 26 | #4454 | — | 2 | 1 | no | — |
| 27 | #4454 | — | 2 | 1 | no | — |
| 28 | #1986 | D3 | 1 | 1 | yes | characteristic |
| 29 | #1950 | — | 3 | 1 | no | — |
| 30 | #1542 | — | 3 | 1 | no | — |
| 31 | #1493 | — | 3 | 1 | no | — |
| 32 | #1451 | — | 3 | 1 | no | — |
| 33 | #1303 | — | 3 | 1 | no | — |
| 34 | #1279 | — | 3 | 1 | no | — |
| 35 | C2×C3×M16 | D4 | 1 | 1 | yes | characteristic |
| 36 | C2×C3×M16 | D4 | 1 | 1 | yes | characteristic |
| 37 | C3×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C3×C4⋊Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C3×#125 | — | 2 | 1 | no | — |
| 40 | C3×C22⋊Q8 | — | 2 | 1 | no | — |
| 41 | C3×C22⋊Q8 | — | 2 | 1 | no | — |
| 42 | C3×C4×Q8 | — | 2 | 1 | no | — |
| 43 | C3×C4×Q8 | — | 2 | 1 | no | — |
| 44 | C3×#117 | — | 2 | 1 | no | — |
| 45 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 46 | C2×C3×C42 | — | 2 | 1 | no | — |
| 47 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 48 | C2×#208 | — | 2 | 1 | no | — |
| 49 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 50 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 51 | C3×#97 | — | 2 | 1 | no | — |
| 52 | C3×C4×C8 | — | 2 | 1 | no | — |
| 53 | C3×#95 | — | 2 | 1 | no | — |
| 54 | #830 | — | 2 | 1 | no | — |
| 55 | #822 | — | 4 | 2 | no | — |
| 56 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 57 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 58 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 59 | C4⋊(C3⋊C8) | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | C4⋊(C3⋊C8) | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | C4⋊(C3⋊C8) | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C4⋊(C3⋊C8) | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 64 | C4×C3⋊C8 | — | 2 | 1 | no | — |
| 65 | C4×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 66 | C4×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 67 | #533 | C2×D3 | 1 | 1 | yes | characteristic |
| 68 | #431 | — | 3 | 1 | no | — |
| 69 | C2×#108 | — | 6 | 1 | no | — |
| 70 | #423 | — | 3 | 1 | no | — |
| 71 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 72 | C4×M16 | C2×D3 | 1 | 1 | yes | characteristic |
| 73 | C2×C4×C8 | — | 3 | 1 | no | — |
| 74 | C4×C4⋊C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 75 | #341 | — | 6 | 1 | no | — |
| 76 | #339 | — | 2 | 1 | no | — |
| 77 | #339 | — | 2 | 1 | no | — |
| 78 | #333 | — | 3 | 1 | no | — |
| 79 | #333 | — | 3 | 1 | no | — |
| 80 | #332 | — | 3 | 1 | no | — |
| 81 | #332 | — | 3 | 1 | no | — |
| 82 | Q8⋊C8 | — | 6 | 1 | no | — |
| 83 | Q8⋊C8 | — | 6 | 1 | no | — |
| 84 | #322 | — | 6 | 1 | no | — |
| 85 | #322 | — | 6 | 1 | no | — |
| 86 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 87 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 88 | C22×C3×C4 | — | 4 | 1 | no | — |
| 89 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 90 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 91 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 92 | C3×M16 | — | 2 | 1 | no | — |
| 93 | C3×M16 | — | 2 | 1 | no | — |
| 94 | C3×M16 | — | 2 | 1 | no | — |
| 95 | C3×M16 | — | 2 | 1 | no | — |
| 96 | C2×C3×C8 | — | 2 | 1 | no | — |
| 97 | C2×C3×C8 | — | 2 | 1 | no | — |
| 98 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 99 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 100 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 101 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 102 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 103 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 104 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 105 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 106 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 107 | C3×C42 | — | 4 | 1 | no | — |
| 108 | C3×C42 | — | 4 | 1 | no | — |
| 109 | C3×C42 | — | 4 | 1 | no | — |
| 110 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 111 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 113 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 114 | #208 | — | 4 | 1 | no | — |
| 115 | #208 | — | 4 | 1 | no | — |
| 116 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 117 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 118 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 119 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 120 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 121 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 122 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 123 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 124 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 125 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 126 | C2×M16 | — | 6 | 1 | no | — |
| 127 | C2×M16 | #67 | 1 | 1 | yes | characteristic |
| 128 | C2×M16 | #67 | 1 | 1 | yes | characteristic |
| 129 | C22×C8 | — | 6 | 1 | no | — |
| 130 | C22×C8 | — | 6 | 1 | no | — |
| 131 | C4⋊Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 132 | C4⋊Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 133 | #125 | — | 2 | 1 | no | — |
| 134 | C22⋊Q8 | — | 2 | 1 | no | — |
| 135 | C22⋊Q8 | — | 2 | 1 | no | — |
| 136 | C4×Q8 | — | 2 | 1 | no | — |
| 137 | C4×Q8 | — | 2 | 1 | no | — |
| 138 | #117 | — | 2 | 1 | no | — |
| 139 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 140 | C2×C42 | — | 2 | 1 | no | — |
| 141 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 142 | #108 | — | 12 | 2 | no | — |
| 143 | C4⋊C8 | — | 3 | 1 | no | — |
| 144 | C4⋊C8 | — | 3 | 1 | no | — |
| 145 | C4⋊C8 | — | 3 | 1 | no | — |
| 146 | C4⋊C8 | — | 3 | 1 | no | — |
| 147 | C4⋊C8 | — | 6 | 1 | no | — |
| 148 | C4⋊C8 | — | 6 | 1 | no | — |
| 149 | C4⋊C8 | — | 6 | 1 | no | — |
| 150 | C22⋊C8 | — | 6 | 1 | no | — |
| 151 | #97 | — | 2 | 1 | no | — |
| 152 | C4×C8 | — | 2 | 1 | no | — |
| 153 | C4×C8 | — | 3 | 1 | no | — |
| 154 | C4×C8 | — | 3 | 1 | no | — |
| 155 | C4×C8 | — | 6 | 1 | no | — |
| 156 | C4×C8 | — | 6 | 1 | no | — |
| 157 | #95 | — | 2 | 1 | no | — |
| 158 | C23×C3 | #124 | 1 | 1 | yes | characteristic |
| 159 | C3×Q8 | — | 4 | 1 | no | — |
| 160 | C3×Q8 | — | 4 | 1 | no | — |
| 161 | C3×Q8 | — | 8 | 1 | no | — |
| 162 | C2×C3×C4 | — | 2 | 1 | no | — |
| 163 | C2×C3×C4 | — | 2 | 1 | no | — |
| 164 | C2×C3×C4 | — | 4 | 1 | no | — |
| 165 | C2×C3×C4 | — | 4 | 2 | no | — |
| 166 | C2×C3×C4 | — | 4 | 1 | no | — |
| 167 | C2×C3×C4 | — | 4 | 1 | no | — |
| 168 | C2×C3×C4 | — | 8 | 1 | no | — |
| 169 | C2×C3×C4 | — | 8 | 1 | no | — |
| 170 | C2×C3×C4 | #117 | 1 | 1 | yes | characteristic |
| 171 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 172 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 173 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 174 | C2×C3×C4 | #123 | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | #126 | 1 | 1 | yes | characteristic |
| 176 | C2×C3×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 177 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 179 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 180 | C3×C8 | — | 4 | 1 | no | — |
| 181 | C3×C8 | — | 4 | 1 | no | — |
| 182 | C3⋊C8 | — | 2 | 2 | no | — |
| 183 | C3⋊C8 | — | 4 | 1 | no | — |
| 184 | C3⋊C8 | — | 4 | 1 | no | — |
| 185 | C3⋊C8 | — | 4 | 1 | no | — |
| 186 | C3⋊C8 | — | 4 | 1 | no | — |
| 187 | C3⋊C8 | — | 4 | 1 | no | — |
| 188 | C3⋊C8 | — | 8 | 1 | no | — |
| 189 | C2×Q8 | — | 2 | 1 | no | — |
| 190 | C2×Q8 | — | 2 | 1 | no | — |
| 191 | C22×C4 | — | 4 | 1 | no | — |
| 192 | C22×C4 | #237 | 1 | 1 | yes | characteristic |
| 193 | C22×C4 | #237 | 1 | 1 | yes | characteristic |
| 194 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 195 | M16 | — | 2 | 1 | no | — |
| 196 | M16 | — | 2 | 1 | no | — |
| 197 | M16 | — | 2 | 1 | no | — |
| 198 | M16 | — | 2 | 1 | no | — |
| 199 | M16 | — | 12 | 1 | no | — |
| 200 | M16 | — | 12 | 1 | no | — |
| 201 | C2×C8 | — | 2 | 1 | no | — |
| 202 | C2×C8 | — | 2 | 1 | no | — |
| 203 | C2×C8 | — | 6 | 1 | no | — |
| 204 | C2×C8 | — | 6 | 1 | no | — |
| 205 | C2×C8 | — | 6 | 1 | no | — |
| 206 | C2×C8 | — | 6 | 1 | no | — |
| 207 | C2×C8 | — | 6 | 1 | no | — |
| 208 | C2×C8 | — | 6 | 1 | no | — |
| 209 | C2×C8 | — | 12 | 2 | no | — |
| 210 | C2×C8 | — | 12 | 1 | no | — |
| 211 | C2×C8 | — | 12 | 1 | no | — |
| 212 | C2×C8 | — | 12 | 1 | no | — |
| 213 | C4⋊C4 | — | 2 | 2 | no | — |
| 214 | C4⋊C4 | — | 2 | 1 | no | — |
| 215 | C4⋊C4 | — | 2 | 1 | no | — |
| 216 | C4⋊C4 | — | 2 | 1 | no | — |
| 217 | C4⋊C4 | — | 2 | 1 | no | — |
| 218 | C4⋊C4 | — | 4 | 1 | no | — |
| 219 | C4⋊C4 | — | 4 | 1 | no | — |
| 220 | C4⋊C4 | — | 4 | 1 | no | — |
| 221 | C22⋊C4 | — | 4 | 1 | no | — |
| 222 | C42 | — | 4 | 1 | no | — |
| 223 | C42 | — | 4 | 1 | no | — |
| 224 | C42 | — | 4 | 1 | no | — |
| 225 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 226 | C42 | #235 | 1 | 1 | yes | characteristic |
| 227 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 228 | C42 | #237 | 1 | 1 | yes | characteristic |
| 229 | C22×C3 | — | 2 | 1 | no | — |
| 230 | C22×C3 | — | 2 | 1 | no | — |
| 231 | C22×C3 | #388 | 1 | 1 | yes | characteristic |
| 232 | C22×C3 | #489 | 1 | 1 | yes | characteristic |
| 233 | C22×C3 | #491 | 1 | 1 | yes | characteristic |
| 234 | C3×C4 | — | 2 | 1 | no | — |
| 235 | C3×C4 | — | 2 | 1 | no | — |
| 236 | C3×C4 | — | 2 | 1 | no | — |
| 237 | C3×C4 | — | 8 | 2 | no | — |
| 238 | C3×C4 | — | 8 | 1 | no | — |
| 239 | C3×C4 | — | 8 | 1 | no | — |
| 240 | C3×C4 | #440 | 1 | 1 | yes | characteristic |
| 241 | C3×C4 | #441 | 1 | 1 | yes | characteristic |
| 242 | C3×C4 | #443 | 1 | 1 | yes | characteristic |
| 243 | C3×C4 | #444 | 1 | 1 | yes | characteristic |
| 244 | C3×C4 | #468 | 1 | 1 | yes | characteristic |
| 245 | C3×C4 | #470 | 1 | 1 | yes | characteristic |
| 246 | C23 | #936 | 1 | 1 | yes | characteristic |
| 247 | Q8 | — | 4 | 1 | no | — |
| 248 | Q8 | — | 4 | 1 | no | — |
| 249 | Q8 | — | 8 | 1 | no | — |
| 250 | C2×C4 | — | 2 | 1 | no | — |
| 251 | C2×C4 | — | 2 | 1 | no | — |
| 252 | C2×C4 | — | 4 | 1 | no | — |
| 253 | C2×C4 | — | 4 | 2 | no | — |
| 254 | C2×C4 | — | 4 | 1 | no | — |
| 255 | C2×C4 | — | 4 | 1 | no | — |
| 256 | C2×C4 | — | 8 | 1 | no | — |
| 257 | C2×C4 | — | 8 | 1 | no | — |
| 258 | C2×C4 | #877 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | #879 | 1 | 1 | yes | characteristic |
| 260 | C2×C4 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 261 | C2×C4 | #883 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | C4×#67 | 1 | 1 | yes | characteristic |
| 263 | C2×C4 | #939 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | #942 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | #942 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | #949 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | #951 | 1 | 1 | yes | characteristic |
| 268 | C8 | — | 4 | 1 | no | — |
| 269 | C8 | — | 4 | 1 | no | — |
| 270 | C8 | — | 6 | 2 | no | — |
| 271 | C8 | — | 12 | 1 | no | — |
| 272 | C8 | — | 12 | 1 | no | — |
| 273 | C8 | — | 12 | 1 | no | — |
| 274 | C8 | — | 12 | 1 | no | — |
| 275 | C8 | — | 12 | 1 | no | — |
| 276 | C8 | — | 24 | 1 | no | — |
| 277 | C2×C3 | — | 2 | 1 | no | — |
| 278 | C2×C3 | — | 2 | 1 | no | — |
| 279 | C2×C3 | #1929 | 1 | 1 | yes | characteristic |
| 280 | C2×C3 | #1932 | 1 | 1 | yes | characteristic |
| 281 | C2×C3 | #1935 | 1 | 1 | yes | characteristic |
| 282 | C22 | — | 2 | 1 | no | — |
| 283 | C22 | — | 2 | 1 | no | — |
| 284 | C22 | #4945 | 1 | 1 | yes | characteristic |
| 285 | C22 | #5055 | 1 | 1 | yes | characteristic |
| 286 | C22 | #5082 | 1 | 1 | yes | characteristic |
| 287 | C4 | — | 2 | 1 | no | — |
| 288 | C4 | — | 2 | 1 | no | — |
| 289 | C4 | — | 2 | 1 | no | — |
| 290 | C4 | — | 8 | 2 | no | — |
| 291 | C4 | — | 8 | 1 | no | — |
| 292 | C4 | — | 8 | 1 | no | — |
| 293 | C4 | #4823 | 1 | 1 | yes | characteristic |
| 294 | C4 | #4824 | 1 | 1 | yes | characteristic |
| 295 | C4 | #5026 | 1 | 1 | yes | characteristic |
| 296 | C4 | #5030 | 1 | 1 | yes | characteristic |
| 297 | C4 | #5049 | 1 | 1 | yes | characteristic |
| 298 | C4 | #5052 | 1 | 1 | yes | characteristic |
| 299 | C3 | #9591 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 300 | C2 | — | 2 | 1 | no | — |
| 301 | C2 | — | 2 | 1 | no | — |
| 302 | C2 | #69675 | 1 | 1 | yes | characteristic |
| 303 | C2 | #70367 | 1 | 1 | yes | characteristic |
| 304 | C2 | #70401 | 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