← #10713618
Subgroups — #10713618
Order 768.
422 conjugacy classes of subgroups.
The subgroup lattice has 320 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C3×SD32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #72069 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#4565 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72041 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#4563 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×#4506 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #71399 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13246 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×C3×Q16 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C3×D8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×SD32 | — | 2 | 1 | no | — |
| 12 | C2×C3×SD32 | — | 2 | 1 | no | — |
| 13 | C2×C3×SD32 | — | 2 | 2 | no | — |
| 14 | C2×C3×SD32 | C4 | 1 | 4 | yes | complemented |
| 15 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#836 | — | 2 | 1 | no | — |
| 17 | #5225 | — | 2 | 1 | no | — |
| 18 | C2×#833 | — | 2 | 1 | no | — |
| 19 | #5215 | — | 2 | 1 | no | — |
| 20 | C2×#819 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#819 | C22 | 1 | 1 | yes | characteristic |
| 22 | #5101 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 24 | #4565 | — | 2 | 2 | no | — |
| 25 | #4563 | — | 2 | 2 | no | — |
| 26 | #4506 | — | 2 | 2 | no | — |
| 27 | #4502 | — | 2 | 1 | no | — |
| 28 | Dic3×C16 | — | 2 | 1 | no | — |
| 29 | C22×SD32 | D3 | 1 | 1 | yes | characteristic |
| 30 | C2×#368 | — | 3 | 1 | no | — |
| 31 | C2×#362 | — | 3 | 1 | no | — |
| 32 | C2×#361 | — | 3 | 1 | no | — |
| 33 | #2108 | — | 3 | 1 | no | — |
| 34 | #1979 | — | 3 | 1 | no | — |
| 35 | #1978 | — | 3 | 1 | no | — |
| 36 | C22×C3×Q8 | — | 2 | 1 | no | — |
| 37 | C22×C3×D4 | — | 2 | 1 | no | — |
| 38 | C2×C3×Q16 | — | 2 | 1 | no | — |
| 39 | C2×C3×Q16 | — | 2 | 1 | no | — |
| 40 | C2×C3×Q16 | — | 2 | 1 | no | — |
| 41 | C2×C3×Q16 | — | 2 | 1 | no | — |
| 42 | C2×C3×Q16 | — | 2 | 1 | no | — |
| 43 | C2×C3×Q16 | C2×C4 | 1 | 2 | yes | — |
| 44 | C2×C3×D8 | — | 2 | 1 | no | — |
| 45 | C2×C3×D8 | — | 2 | 1 | no | — |
| 46 | C2×C3×D8 | — | 2 | 1 | no | — |
| 47 | C2×C3×D8 | — | 2 | 1 | no | — |
| 48 | C2×C3×D8 | — | 2 | 1 | no | — |
| 49 | C2×C3×D8 | C2×C4 | 1 | 2 | yes | — |
| 50 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 51 | #926 | D4 | 1 | 1 | yes | characteristic |
| 52 | C2×#208 | — | 2 | 1 | no | — |
| 53 | C2×#208 | — | 2 | 1 | no | — |
| 54 | C2×#208 | D4 | 1 | 1 | yes | characteristic |
| 55 | C3×SD32 | — | 2 | 4 | no | — |
| 56 | C3×SD32 | — | 2 | 4 | no | — |
| 57 | C2×C3×C16 | C2×C4 | 1 | 2 | yes | — |
| 58 | C2×C3×C16 | D4 | 1 | 1 | yes | characteristic |
| 59 | C2×C3×C16 | D4 | 1 | 1 | yes | characteristic |
| 60 | C2×C3×C16 | D4 | 1 | 2 | yes | — |
| 61 | #836 | — | 4 | 2 | no | — |
| 62 | #835 | — | 4 | 2 | no | — |
| 63 | #833 | — | 4 | 2 | no | — |
| 64 | #832 | — | 4 | 2 | no | — |
| 65 | #819 | — | 2 | 1 | no | — |
| 66 | #819 | — | 2 | 1 | no | — |
| 67 | #819 | — | 2 | 1 | no | — |
| 68 | #819 | — | 2 | 1 | no | — |
| 69 | #815 | — | 2 | 1 | no | — |
| 70 | C8×Dic3 | — | 2 | 1 | no | — |
| 71 | #812 | — | 2 | 1 | no | — |
| 72 | #812 | D4 | 1 | 2 | yes | — |
| 73 | C2×C3⋊C16 | — | 2 | 1 | no | — |
| 74 | C22×Q16 | C2×D3 | 1 | 1 | yes | characteristic |
| 75 | C22×D8 | C2×D3 | 1 | 1 | yes | characteristic |
| 76 | C2×SD32 | — | 2 | 1 | no | — |
| 77 | C2×SD32 | — | 2 | 1 | no | — |
| 78 | C2×SD32 | — | 2 | 2 | no | — |
| 79 | C2×SD32 | Dic3 | 1 | 4 | yes | complemented |
| 80 | C2×M32 | — | 3 | 1 | no | — |
| 81 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 82 | C2×#108 | — | 3 | 1 | no | — |
| 83 | C2×#108 | — | 3 | 1 | no | — |
| 84 | #419 | — | 6 | 1 | no | — |
| 85 | #418 | — | 6 | 1 | no | — |
| 86 | C2×#101 | — | 6 | 1 | no | — |
| 87 | C2×#100 | — | 6 | 1 | no | — |
| 88 | #405 | — | 3 | 1 | no | — |
| 89 | #368 | — | 6 | 2 | no | — |
| 90 | #362 | — | 6 | 2 | no | — |
| 91 | #361 | — | 6 | 2 | no | — |
| 92 | #346 | — | 6 | 1 | no | — |
| 93 | C4×C16 | — | 6 | 1 | no | — |
| 94 | C24×C3 | — | 4 | 1 | no | — |
| 95 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 96 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 97 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 98 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 99 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 100 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 101 | C2×C3×D4 | — | 2 | 2 | no | — |
| 102 | C2×C3×D4 | — | 4 | 1 | no | — |
| 103 | C2×C3×D4 | — | 4 | 1 | no | — |
| 104 | C2×C3×D4 | — | 4 | 1 | no | — |
| 105 | C2×C3×D4 | — | 4 | 1 | no | — |
| 106 | C2×C3×D4 | — | 4 | 1 | no | — |
| 107 | C22×C3×C4 | — | 4 | 1 | no | — |
| 108 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C3×Q16 | — | 2 | 2 | no | — |
| 110 | C3×Q16 | — | 2 | 2 | no | — |
| 111 | C3×Q16 | — | 4 | 1 | no | — |
| 112 | C3×Q16 | — | 4 | 1 | no | — |
| 113 | C3×D8 | — | 2 | 2 | no | — |
| 114 | C3×D8 | — | 2 | 2 | no | — |
| 115 | C3×D8 | — | 4 | 1 | no | — |
| 116 | C3×D8 | — | 4 | 1 | no | — |
| 117 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 118 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 120 | C2×C3×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 122 | C2×C3×C8 | #41 | 1 | 1 | yes | characteristic |
| 123 | #217 | — | 2 | 1 | no | — |
| 124 | C4⋊Dic3 | D8 | 1 | 2 | yes | — |
| 125 | C4×Dic3 | — | 2 | 1 | no | — |
| 126 | #208 | — | 2 | 1 | no | — |
| 127 | #208 | — | 4 | 1 | no | — |
| 128 | #208 | — | 4 | 1 | no | — |
| 129 | #208 | — | 4 | 1 | no | — |
| 130 | #208 | — | 4 | 1 | no | — |
| 131 | #208 | D8 | 1 | 2 | yes | — |
| 132 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 133 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 134 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 135 | C3×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 136 | C3×C16 | C22⋊C4 | 1 | 2 | yes | — |
| 137 | C3⋊C16 | — | 2 | 2 | no | — |
| 138 | C22×Q8 | — | 2 | 1 | no | — |
| 139 | C22×D4 | — | 2 | 1 | no | — |
| 140 | C2×Q16 | — | 2 | 1 | no | — |
| 141 | C2×Q16 | — | 2 | 1 | no | — |
| 142 | C2×Q16 | — | 2 | 1 | no | — |
| 143 | C2×Q16 | — | 2 | 1 | no | — |
| 144 | C2×Q16 | — | 2 | 1 | no | — |
| 145 | C2×Q16 | C2×Dic3 | 1 | 2 | yes | — |
| 146 | C2×D8 | — | 2 | 1 | no | — |
| 147 | C2×D8 | — | 2 | 1 | no | — |
| 148 | C2×D8 | — | 2 | 1 | no | — |
| 149 | C2×D8 | — | 2 | 1 | no | — |
| 150 | C2×D8 | — | 2 | 1 | no | — |
| 151 | C2×D8 | C2×Dic3 | 1 | 2 | yes | — |
| 152 | C2×M16 | — | 3 | 1 | no | — |
| 153 | C2×M16 | — | 6 | 1 | no | — |
| 154 | C2×M16 | — | 6 | 1 | no | — |
| 155 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 156 | #117 | — | 3 | 1 | no | — |
| 157 | SD32 | — | 2 | 4 | no | — |
| 158 | SD32 | — | 2 | 4 | no | — |
| 159 | M32 | — | 3 | 2 | no | — |
| 160 | M32 | — | 6 | 1 | no | — |
| 161 | C2×C16 | — | 6 | 1 | no | — |
| 162 | C2×C16 | C2×Dic3 | 1 | 2 | yes | — |
| 163 | C2×C16 | #67 | 1 | 1 | yes | characteristic |
| 164 | C2×C16 | #67 | 1 | 1 | yes | characteristic |
| 165 | C2×C16 | #67 | 1 | 2 | yes | — |
| 166 | #108 | — | 6 | 1 | no | — |
| 167 | #108 | — | 6 | 1 | no | — |
| 168 | #108 | — | 6 | 1 | no | — |
| 169 | #108 | — | 6 | 1 | no | — |
| 170 | #103 | — | 12 | 2 | no | — |
| 171 | #102 | — | 12 | 2 | no | — |
| 172 | #101 | — | 12 | 2 | no | — |
| 173 | #100 | — | 12 | 2 | no | — |
| 174 | #97 | — | 6 | 1 | no | — |
| 175 | C4×C8 | — | 6 | 1 | no | — |
| 176 | C23×C3 | — | 4 | 2 | no | — |
| 177 | C23×C3 | — | 8 | 1 | no | — |
| 178 | C23×C3 | — | 8 | 1 | no | — |
| 179 | C23×C3 | — | 16 | 2 | no | — |
| 180 | C23×C3 | #135 | 1 | 1 | yes | characteristic |
| 181 | C3×Q8 | — | 4 | 2 | no | — |
| 182 | C3×Q8 | — | 4 | 2 | no | — |
| 183 | C3×Q8 | — | 8 | 1 | no | — |
| 184 | C3×Q8 | — | 8 | 1 | no | — |
| 185 | C3×D4 | — | 4 | 2 | no | — |
| 186 | C3×D4 | — | 4 | 2 | no | — |
| 187 | C3×D4 | — | 8 | 1 | no | — |
| 188 | C3×D4 | — | 8 | 1 | no | — |
| 189 | C2×C3×C4 | — | 4 | 2 | no | — |
| 190 | C2×C3×C4 | — | 8 | 1 | no | — |
| 191 | C2×C3×C4 | — | 8 | 1 | no | — |
| 192 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 193 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 194 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 195 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 196 | C2×C3×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 197 | C2×C3×C4 | #135 | 1 | 1 | yes | characteristic |
| 198 | C2×Dic3 | — | 2 | 2 | no | — |
| 199 | C3×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 200 | C3×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 201 | C3×C8 | #121 | 1 | 1 | yes | characteristic |
| 202 | C3×C8 | #124 | 1 | 1 | yes | characteristic |
| 203 | C3⋊C8 | — | 2 | 2 | no | — |
| 204 | C3⋊C8 | — | 8 | 1 | no | — |
| 205 | C3⋊C8 | — | 8 | 1 | no | — |
| 206 | C24 | — | 4 | 1 | no | — |
| 207 | C2×Q8 | — | 2 | 2 | no | — |
| 208 | C2×Q8 | — | 4 | 1 | no | — |
| 209 | C2×Q8 | — | 4 | 1 | no | — |
| 210 | C2×Q8 | — | 4 | 1 | no | — |
| 211 | C2×Q8 | — | 4 | 1 | no | — |
| 212 | C2×Q8 | — | 4 | 1 | no | — |
| 213 | C2×D4 | — | 2 | 2 | no | — |
| 214 | C2×D4 | — | 4 | 1 | no | — |
| 215 | C2×D4 | — | 4 | 1 | no | — |
| 216 | C2×D4 | — | 4 | 1 | no | — |
| 217 | C2×D4 | — | 4 | 1 | no | — |
| 218 | C2×D4 | — | 4 | 1 | no | — |
| 219 | C22×C4 | — | 4 | 1 | no | — |
| 220 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 221 | Q16 | — | 2 | 2 | no | — |
| 222 | Q16 | — | 2 | 2 | no | — |
| 223 | Q16 | — | 4 | 1 | no | — |
| 224 | Q16 | — | 4 | 1 | no | — |
| 225 | D8 | — | 2 | 2 | no | — |
| 226 | D8 | — | 2 | 2 | no | — |
| 227 | D8 | — | 4 | 1 | no | — |
| 228 | D8 | — | 4 | 1 | no | — |
| 229 | M16 | — | 3 | 2 | no | — |
| 230 | M16 | — | 6 | 1 | no | — |
| 231 | M16 | — | 12 | 1 | no | — |
| 232 | M16 | — | 12 | 1 | no | — |
| 233 | M16 | — | 12 | 1 | no | — |
| 234 | M16 | — | 12 | 1 | no | — |
| 235 | C2×C8 | — | 6 | 1 | no | — |
| 236 | C2×C8 | — | 12 | 1 | no | — |
| 237 | C2×C8 | — | 12 | 1 | no | — |
| 238 | C2×C8 | D3×D4 | 1 | 1 | yes | characteristic |
| 239 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 240 | C2×C8 | #237 | 1 | 1 | yes | characteristic |
| 241 | C2×C8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 242 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 243 | C2×C8 | C2×#67 | 1 | 1 | yes | characteristic |
| 244 | C4⋊C4 | — | 3 | 2 | no | — |
| 245 | C22⋊C4 | — | 6 | 1 | no | — |
| 246 | C42 | — | 6 | 1 | no | — |
| 247 | C16 | — | 6 | 2 | no | — |
| 248 | C16 | #217 | 1 | 2 | yes | — |
| 249 | C16 | #217 | 1 | 2 | yes | — |
| 250 | C22×C3 | — | 2 | 2 | no | — |
| 251 | C22×C3 | — | 8 | 2 | no | — |
| 252 | C22×C3 | — | 16 | 2 | no | — |
| 253 | C22×C3 | — | 16 | 2 | no | — |
| 254 | C22×C3 | — | 16 | 2 | no | — |
| 255 | C22×C3 | C4×D8 | 1 | 1 | yes | characteristic |
| 256 | C22×C3 | #466 | 1 | 1 | yes | characteristic |
| 257 | C22×C3 | #495 | 1 | 1 | yes | characteristic |
| 258 | C3×C4 | — | 8 | 2 | no | — |
| 259 | C3×C4 | #390 | 1 | 1 | yes | characteristic |
| 260 | C3×C4 | C4×D8 | 1 | 1 | yes | characteristic |
| 261 | C3×C4 | #466 | 1 | 1 | yes | characteristic |
| 262 | C3×C4 | #493 | 1 | 1 | yes | characteristic |
| 263 | Dic3 | — | 4 | 2 | no | — |
| 264 | C23 | — | 4 | 2 | no | — |
| 265 | C23 | — | 8 | 1 | no | — |
| 266 | C23 | — | 8 | 1 | no | — |
| 267 | C23 | — | 16 | 2 | no | — |
| 268 | C23 | #912 | 1 | 1 | yes | characteristic |
| 269 | Q8 | — | 4 | 2 | no | — |
| 270 | Q8 | — | 4 | 2 | no | — |
| 271 | Q8 | — | 8 | 1 | no | — |
| 272 | Q8 | — | 8 | 1 | no | — |
| 273 | D4 | — | 4 | 2 | no | — |
| 274 | D4 | — | 4 | 2 | no | — |
| 275 | D4 | — | 8 | 1 | no | — |
| 276 | D4 | — | 8 | 1 | no | — |
| 277 | C2×C4 | — | 4 | 2 | no | — |
| 278 | C2×C4 | — | 6 | 2 | no | — |
| 279 | C2×C4 | — | 8 | 1 | no | — |
| 280 | C2×C4 | — | 8 | 1 | no | — |
| 281 | C2×C4 | D3×D8 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | D3×D8 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | #912 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | D4×Dic3 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #938 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #940 | 1 | 1 | yes | characteristic |
| 287 | C8 | — | 6 | 2 | no | — |
| 288 | C8 | — | 24 | 1 | no | — |
| 289 | C8 | — | 24 | 1 | no | — |
| 290 | C8 | D4×Dic3 | 1 | 1 | yes | characteristic |
| 291 | C8 | #936 | 1 | 1 | yes | characteristic |
| 292 | C8 | #938 | 1 | 1 | yes | characteristic |
| 293 | C8 | C2×#217 | 1 | 1 | yes | characteristic |
| 294 | C2×C3 | — | 2 | 1 | no | — |
| 295 | C2×C3 | — | 2 | 1 | no | — |
| 296 | C2×C3 | — | 16 | 2 | no | — |
| 297 | C2×C3 | #1970 | 1 | 1 | yes | characteristic |
| 298 | C2×C3 | #2223 | 1 | 2 | yes | — |
| 299 | C22 | — | 2 | 2 | no | — |
| 300 | C22 | — | 8 | 2 | no | — |
| 301 | C22 | — | 16 | 2 | no | — |
| 302 | C22 | — | 16 | 2 | no | — |
| 303 | C22 | — | 16 | 2 | no | — |
| 304 | C22 | Dic3×D8 | 1 | 1 | yes | characteristic |
| 305 | C22 | #5155 | 1 | 1 | yes | characteristic |
| 306 | C22 | #5157 | 1 | 1 | yes | characteristic |
| 307 | C4 | — | 8 | 2 | no | — |
| 308 | C4 | — | 12 | 2 | no | — |
| 309 | C4 | Dic3×D8 | 1 | 1 | yes | characteristic |
| 310 | C4 | #5151 | 1 | 1 | yes | characteristic |
| 311 | C4 | #5157 | 1 | 1 | yes | characteristic |
| 312 | C4 | #5221 | 1 | 1 | yes | characteristic |
| 313 | C3 | #13246 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 314 | C2 | — | 2 | 1 | no | — |
| 315 | C2 | — | 2 | 1 | no | — |
| 316 | C2 | — | 16 | 2 | no | — |
| 317 | C2 | #71491 | 1 | 2 | yes | — |
| 318 | C2 | #72038 | 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