← #10804041
Subgroups — #10804041
Order 768.
654 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 | #79170 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #78543 | C2 | 1 | 2 | yes | maximal |
| 3 | C3×#1773 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #70705 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #70693 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #70663 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #69889 | C2 | 1 | 2 | yes | maximal |
| 8 | #69829 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #69804 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #68747 | C2 | 1 | 2 | yes | maximal |
| 11 | #68732 | C2 | 1 | 2 | yes | maximal |
| 12 | #21733 | — | 3 | 1 | no | maximal, Sylow |
| 13 | #5517 | C22 | 1 | 2 | yes | — |
| 14 | C2×C3×#108 | C22 | 1 | 1 | yes | characteristic |
| 15 | C3×#427 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×#405 | C22 | 1 | 2 | yes | — |
| 17 | C2×C3×#97 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#822 | C22 | 1 | 2 | yes | — |
| 19 | #5119 | C22 | 1 | 2 | yes | — |
| 20 | #5118 | C22 | 1 | 2 | yes | — |
| 21 | C2×#819 | C22 | 1 | 1 | yes | characteristic |
| 22 | #5106 | C22 | 1 | 1 | yes | characteristic |
| 23 | #5101 | C22 | 1 | 2 | yes | — |
| 24 | C2×#815 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#815 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 28 | #4999 | C22 | 1 | 2 | yes | — |
| 29 | #4998 | C22 | 1 | 2 | yes | — |
| 30 | #4932 | C22 | 1 | 1 | yes | characteristic |
| 31 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 32 | #4878 | C4 | 1 | 2 | yes | complemented |
| 33 | #4874 | C4 | 1 | 2 | yes | complemented |
| 34 | #4858 | C4 | 1 | 2 | yes | complemented |
| 35 | #4855 | C4 | 1 | 2 | yes | complemented |
| 36 | #4701 | — | 2 | 4 | no | — |
| 37 | #4679 | — | 2 | 4 | no | — |
| 38 | C3×#339 | C22 | 1 | 2 | yes | — |
| 39 | #4555 | C22 | 1 | 2 | yes | — |
| 40 | #4532 | C22 | 1 | 2 | yes | — |
| 41 | #4523 | C22 | 1 | 2 | yes | — |
| 42 | #3160 | — | 3 | 1 | no | — |
| 43 | #2961 | — | 3 | 2 | no | — |
| 44 | #1995 | — | 3 | 2 | no | — |
| 45 | #1946 | — | 3 | 1 | no | — |
| 46 | #1946 | — | 3 | 1 | no | — |
| 47 | #1945 | — | 3 | 2 | no | — |
| 48 | #1848 | — | 3 | 1 | no | — |
| 49 | #1773 | — | 3 | 1 | no | — |
| 50 | #1773 | D3 | 1 | 1 | yes | characteristic |
| 51 | #1772 | — | 3 | 2 | no | — |
| 52 | #1726 | — | 3 | 1 | no | — |
| 53 | C2×C3×M16 | C23 | 1 | 2 | yes | — |
| 54 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 55 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 56 | C3×#117 | C23 | 1 | 2 | yes | — |
| 57 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 58 | #926 | C23 | 1 | 2 | yes | — |
| 59 | C2×C4×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×C4×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 61 | C2×#208 | C23 | 1 | 2 | yes | — |
| 62 | C22×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 63 | C22×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 64 | #890 | C2×C4 | 1 | 4 | yes | — |
| 65 | #888 | C2×C4 | 1 | 4 | yes | — |
| 66 | #878 | — | 2 | 4 | no | — |
| 67 | C4×Dic6 | — | 2 | 4 | no | — |
| 68 | C3×#108 | — | 2 | 2 | no | — |
| 69 | C3×#107 | C2×C4 | 1 | 2 | yes | — |
| 70 | C3×#106 | C2×C4 | 1 | 2 | yes | — |
| 71 | C3×#97 | — | 2 | 2 | no | — |
| 72 | C3×#97 | Q8 | 1 | 4 | yes | complemented |
| 73 | C3×C4×C8 | — | 2 | 2 | no | — |
| 74 | C3⋊#95 | — | 2 | 2 | no | — |
| 75 | #822 | — | 2 | 4 | no | — |
| 76 | #819 | — | 2 | 2 | no | — |
| 77 | #818 | C2×C4 | 1 | 2 | yes | — |
| 78 | #817 | C2×C4 | 1 | 2 | yes | — |
| 79 | #815 | — | 2 | 2 | no | — |
| 80 | #815 | — | 2 | 2 | no | — |
| 81 | #815 | D4 | 1 | 4 | yes | — |
| 82 | Dic3⋊C8 | — | 2 | 4 | no | — |
| 83 | Dic3⋊C8 | — | 2 | 4 | no | — |
| 84 | C8×Dic3 | — | 2 | 2 | no | — |
| 85 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 86 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 87 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 88 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 89 | C8×Dic3 | Q8 | 1 | 4 | yes | complemented |
| 90 | #808 | C2×C4 | 1 | 4 | yes | — |
| 91 | #807 | C2×C4 | 1 | 4 | yes | — |
| 92 | C4⋊(C3⋊C8) | — | 2 | 4 | no | — |
| 93 | C4×C3⋊C8 | — | 2 | 2 | no | — |
| 94 | #533 | — | 3 | 2 | no | — |
| 95 | #500 | — | 3 | 2 | no | — |
| 96 | #499 | — | 3 | 2 | no | — |
| 97 | #499 | — | 3 | 2 | no | — |
| 98 | #498 | — | 3 | 2 | no | — |
| 99 | #446 | — | 6 | 4 | no | — |
| 100 | #446 | — | 6 | 4 | no | — |
| 101 | C2×#108 | — | 3 | 2 | no | — |
| 102 | C2×#108 | — | 3 | 1 | no | — |
| 103 | C2×#108 | C2×D3 | 1 | 1 | yes | characteristic |
| 104 | #427 | — | 3 | 1 | no | — |
| 105 | #427 | — | 3 | 2 | no | — |
| 106 | #427 | C2×D3 | 1 | 1 | yes | characteristic |
| 107 | #424 | — | 3 | 2 | no | — |
| 108 | #424 | — | 3 | 2 | no | — |
| 109 | #423 | — | 3 | 2 | no | — |
| 110 | #405 | — | 3 | 2 | no | — |
| 111 | #405 | C2×D3 | 1 | 2 | yes | — |
| 112 | C2×#97 | — | 3 | 1 | no | — |
| 113 | C2×#97 | — | 3 | 1 | no | — |
| 114 | C2×#97 | C2×D3 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C8 | — | 3 | 1 | no | — |
| 116 | C2×C4×C8 | — | 3 | 1 | no | — |
| 117 | C2×C4×C8 | — | 3 | 1 | no | — |
| 118 | #376 | — | 3 | 1 | no | — |
| 119 | #339 | — | 3 | 2 | no | — |
| 120 | #339 | — | 3 | 2 | no | — |
| 121 | #339 | — | 3 | 2 | no | — |
| 122 | #339 | C2×D3 | 1 | 2 | yes | — |
| 123 | C22×C3×C4 | — | 2 | 1 | no | — |
| 124 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 125 | C22×Dic3 | — | 2 | 1 | no | — |
| 126 | C22×Dic3 | #41 | 1 | 2 | yes | — |
| 127 | C2×Dic6 | — | 2 | 4 | no | — |
| 128 | C3×M16 | — | 2 | 2 | no | — |
| 129 | C3×M16 | — | 2 | 2 | no | — |
| 130 | C2×C3×C8 | — | 2 | 2 | no | — |
| 131 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 132 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 133 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 134 | C2×C3×C8 | C2×Q8 | 1 | 2 | yes | — |
| 135 | C2×C3×C8 | C2×Q8 | 1 | 2 | yes | — |
| 136 | C2×C3×C8 | C2×Q8 | 1 | 2 | yes | — |
| 137 | C2×C3×C8 | #41 | 1 | 2 | yes | — |
| 138 | C3×C4⋊C4 | C22×C4 | 1 | 4 | yes | — |
| 139 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 140 | C3×C42 | — | 2 | 2 | no | — |
| 141 | C3×C42 | — | 2 | 1 | no | — |
| 142 | C3×C42 | C2×Q8 | 1 | 2 | yes | — |
| 143 | #217 | — | 2 | 2 | no | — |
| 144 | C4⋊Dic3 | — | 2 | 4 | no | — |
| 145 | C4⋊Dic3 | C22×C4 | 1 | 4 | yes | — |
| 146 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 147 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 148 | C4×Dic3 | — | 2 | 2 | no | — |
| 149 | C4×Dic3 | — | 2 | 1 | no | — |
| 150 | C4×Dic3 | C22×C4 | 1 | 1 | yes | characteristic |
| 151 | C4×Dic3 | C22×C4 | 1 | 1 | yes | characteristic |
| 152 | C4×Dic3 | C2×D4 | 1 | 2 | yes | — |
| 153 | C4×Dic3 | C2×Q8 | 1 | 2 | yes | — |
| 154 | #208 | — | 2 | 2 | no | — |
| 155 | #208 | — | 2 | 2 | no | — |
| 156 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 157 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 158 | C2×C3⋊C8 | — | 2 | 2 | no | — |
| 159 | C2×C3⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 160 | C2×C3⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 161 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 162 | C2×C3⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 163 | C2×C3⋊C8 | #41 | 1 | 2 | yes | — |
| 164 | C2×M16 | — | 3 | 2 | no | — |
| 165 | C2×M16 | C22×D3 | 1 | 2 | yes | — |
| 166 | C22×C8 | — | 3 | 1 | no | — |
| 167 | C22×C8 | — | 3 | 1 | no | — |
| 168 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 169 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 170 | C4⋊Q8 | — | 3 | 4 | no | — |
| 171 | #125 | — | 3 | 4 | no | — |
| 172 | C22⋊Q8 | — | 6 | 4 | no | — |
| 173 | C4×Q8 | — | 6 | 4 | no | — |
| 174 | #117 | — | 3 | 2 | no | — |
| 175 | #117 | C22×D3 | 1 | 2 | yes | — |
| 176 | C2×C42 | — | 3 | 1 | no | — |
| 177 | C2×C42 | — | 3 | 1 | no | — |
| 178 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 179 | #108 | — | 2 | 2 | no | — |
| 180 | #108 | — | 6 | 4 | no | — |
| 181 | #108 | — | 6 | 2 | no | — |
| 182 | #107 | — | 3 | 2 | no | — |
| 183 | #107 | — | 3 | 4 | no | — |
| 184 | #107 | C4×D3 | 1 | 2 | yes | — |
| 185 | #106 | — | 3 | 2 | no | — |
| 186 | #106 | — | 3 | 4 | no | — |
| 187 | #106 | C4×D3 | 1 | 2 | yes | — |
| 188 | C4⋊C8 | — | 6 | 4 | no | — |
| 189 | C4⋊C8 | — | 6 | 4 | no | — |
| 190 | C4⋊C8 | — | 6 | 4 | no | — |
| 191 | #97 | — | 2 | 2 | no | — |
| 192 | #97 | — | 3 | 4 | no | — |
| 193 | #97 | — | 6 | 2 | no | — |
| 194 | #97 | — | 6 | 2 | no | — |
| 195 | #97 | Dic6 | 1 | 4 | yes | complemented |
| 196 | C4×C8 | — | 2 | 2 | no | — |
| 197 | C4×C8 | — | 3 | 1 | no | — |
| 198 | C4×C8 | — | 3 | 1 | no | — |
| 199 | C4×C8 | — | 3 | 1 | no | — |
| 200 | C4×C8 | — | 3 | 1 | no | — |
| 201 | C4×C8 | — | 3 | 4 | no | — |
| 202 | C4×C8 | — | 6 | 2 | no | — |
| 203 | C4×C8 | — | 6 | 2 | no | — |
| 204 | #95 | — | 6 | 2 | no | — |
| 205 | C23×C3 | C2×#41 | 1 | 1 | yes | characteristic |
| 206 | C2×C3×C4 | — | 2 | 4 | no | — |
| 207 | C2×C3×C4 | — | 2 | 2 | no | — |
| 208 | C2×C3×C4 | — | 4 | 2 | no | — |
| 209 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 210 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 211 | C2×C3×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 212 | C2×C3×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 213 | C2×C3×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 214 | C2×C3×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 215 | C2×Dic3 | — | 2 | 4 | no | — |
| 216 | C2×Dic3 | — | 2 | 2 | no | — |
| 217 | C2×Dic3 | — | 4 | 2 | no | — |
| 218 | C2×Dic3 | — | 4 | 2 | no | — |
| 219 | C2×Dic3 | C4×D4 | 1 | 2 | yes | — |
| 220 | C2×Dic3 | C4×D4 | 1 | 2 | yes | — |
| 221 | Dic6 | — | 8 | 4 | no | — |
| 222 | C3×C8 | — | 4 | 2 | no | — |
| 223 | C3×C8 | C4×Q8 | 1 | 2 | yes | — |
| 224 | C3×C8 | C4×Q8 | 1 | 2 | yes | — |
| 225 | C3×C8 | C4⋊Q8 | 1 | 4 | yes | — |
| 226 | C3⋊C8 | — | 2 | 2 | no | — |
| 227 | C3⋊C8 | — | 4 | 2 | no | — |
| 228 | C3⋊C8 | C4×Q8 | 1 | 2 | yes | — |
| 229 | C3⋊C8 | C4×Q8 | 1 | 2 | yes | — |
| 230 | C2×Q8 | — | 6 | 4 | no | — |
| 231 | C22×C4 | — | 2 | 1 | no | — |
| 232 | C22×C4 | — | 3 | 2 | no | — |
| 233 | C22×C4 | — | 6 | 1 | no | — |
| 234 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 235 | M16 | — | 2 | 2 | no | — |
| 236 | M16 | — | 2 | 2 | no | — |
| 237 | M16 | — | 6 | 2 | no | — |
| 238 | M16 | — | 6 | 2 | no | — |
| 239 | C2×C8 | — | 2 | 2 | no | — |
| 240 | C2×C8 | — | 3 | 1 | no | — |
| 241 | C2×C8 | — | 3 | 1 | no | — |
| 242 | C2×C8 | — | 3 | 2 | no | — |
| 243 | C2×C8 | — | 3 | 2 | no | — |
| 244 | C2×C8 | — | 3 | 2 | no | — |
| 245 | C2×C8 | — | 6 | 1 | no | — |
| 246 | C2×C8 | — | 6 | 1 | no | — |
| 247 | C2×C8 | — | 6 | 2 | no | — |
| 248 | C2×C8 | C2×Dic6 | 1 | 2 | yes | — |
| 249 | C2×C8 | C2×Dic6 | 1 | 2 | yes | — |
| 250 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | #235 | 1 | 2 | yes | — |
| 253 | C2×C8 | D3×D4 | 1 | 2 | yes | — |
| 254 | C2×C8 | D3×Q8 | 1 | 2 | yes | — |
| 255 | C4⋊C4 | — | 3 | 4 | no | — |
| 256 | C4⋊C4 | — | 6 | 4 | no | — |
| 257 | C4⋊C4 | — | 6 | 4 | no | — |
| 258 | C4⋊C4 | — | 6 | 4 | no | — |
| 259 | C4⋊C4 | C2×C4×D3 | 1 | 4 | yes | — |
| 260 | C22⋊C4 | — | 2 | 2 | no | — |
| 261 | C22⋊C4 | — | 6 | 2 | no | — |
| 262 | C42 | — | 2 | 2 | no | — |
| 263 | C42 | — | 2 | 1 | no | — |
| 264 | C42 | — | 3 | 1 | no | — |
| 265 | C42 | — | 3 | 1 | no | — |
| 266 | C42 | — | 3 | 2 | no | — |
| 267 | C42 | — | 3 | 2 | no | — |
| 268 | C42 | — | 6 | 1 | no | — |
| 269 | C42 | — | 6 | 2 | no | — |
| 270 | C42 | C2×Dic6 | 1 | 2 | yes | — |
| 271 | C22×C3 | — | 2 | 2 | no | — |
| 272 | C22×C3 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 273 | C22×C3 | #533 | 1 | 1 | yes | characteristic |
| 274 | C22×C3 | #533 | 1 | 1 | yes | characteristic |
| 275 | C3×C4 | — | 4 | 2 | no | — |
| 276 | C3×C4 | — | 4 | 4 | no | — |
| 277 | C3×C4 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 278 | C3×C4 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 279 | C3×C4 | C2×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 280 | C3×C4 | #532 | 1 | 1 | yes | characteristic |
| 281 | Dic3 | — | 2 | 2 | no | — |
| 282 | Dic3 | — | 2 | 2 | no | — |
| 283 | Dic3 | — | 4 | 2 | no | — |
| 284 | Dic3 | — | 4 | 4 | no | — |
| 285 | C23 | D3×#41 | 1 | 1 | yes | characteristic |
| 286 | Q8 | — | 24 | 4 | no | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 2 | 4 | no | — |
| 289 | C2×C4 | — | 3 | 2 | no | — |
| 290 | C2×C4 | — | 3 | 2 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 6 | 4 | no | — |
| 293 | C2×C4 | — | 6 | 2 | no | — |
| 294 | C2×C4 | — | 12 | 2 | no | — |
| 295 | C2×C4 | — | 12 | 2 | no | — |
| 296 | C2×C4 | C22×Dic6 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | C2×#235 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | C2×D3×Q8 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | D3×#41 | 1 | 1 | yes | characteristic |
| 302 | C8 | — | 3 | 2 | no | — |
| 303 | C8 | — | 3 | 2 | no | — |
| 304 | C8 | — | 4 | 2 | no | — |
| 305 | C8 | — | 6 | 2 | no | — |
| 306 | C8 | — | 12 | 2 | no | — |
| 307 | C8 | C4×Dic6 | 1 | 2 | yes | — |
| 308 | C8 | C4×Dic6 | 1 | 2 | yes | — |
| 309 | C8 | #888 | 1 | 4 | yes | — |
| 310 | C2×C3 | — | 2 | 1 | no | — |
| 311 | C2×C3 | — | 2 | 1 | no | — |
| 312 | C2×C3 | C4×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 313 | C2×C3 | #2957 | 1 | 2 | yes | — |
| 314 | C22 | — | 2 | 2 | no | — |
| 315 | C22 | #5517 | 1 | 1 | yes | characteristic |
| 316 | C22 | C4×D3×D4 | 1 | 1 | yes | characteristic |
| 317 | C22 | #5578 | 1 | 1 | yes | characteristic |
| 318 | C4 | — | 4 | 2 | no | — |
| 319 | C4 | — | 4 | 4 | no | — |
| 320 | C4 | — | 6 | 2 | no | — |
| 321 | C4 | — | 6 | 2 | no | — |
| 322 | C4 | — | 12 | 2 | no | — |
| 323 | C4 | — | 12 | 4 | no | — |
| 324 | C4 | C2×C4×Dic6 | 1 | 1 | yes | characteristic |
| 325 | C4 | C2×#888 | 1 | 1 | yes | characteristic |
| 326 | C4 | #5548 | 1 | 1 | yes | characteristic |
| 327 | C4 | C4×D3×Q8 | 1 | 1 | yes | characteristic |
| 328 | C3 | #21733 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | C4×#888 | 1 | 1 | yes | characteristic |
| 332 | C2 | #79899 | 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