← #12015411
Subgroups — #12015411
Order 1280.
400 conjugacy classes of subgroups.
The subgroup lattice has 298 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C5×#443 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10602874 | C2 | 1 | 2 | yes | maximal |
| 3 | #10600585 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#64683 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10598222 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C2×C5×#135 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×C5×#131 | — | 2 | 1 | no | — |
| 8 | C5×#443 | — | 2 | 1 | no | — |
| 9 | C5×#443 | — | 2 | 1 | no | — |
| 10 | C5×#443 | C4 | 1 | 2 | yes | — |
| 11 | C5×#443 | C4 | 1 | 2 | yes | — |
| 12 | C2×C5×#108 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C5×#104 | — | 2 | 1 | no | — |
| 14 | C2×C4×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#3951 | — | 2 | 1 | no | — |
| 16 | C2×#3951 | C22 | 1 | 2 | yes | — |
| 17 | C2×#3944 | C22 | 1 | 2 | yes | — |
| 18 | #65213 | — | 2 | 1 | no | — |
| 19 | #64785 | — | 2 | 2 | no | — |
| 20 | #64783 | — | 2 | 2 | no | — |
| 21 | #64683 | — | 2 | 1 | no | — |
| 22 | #64683 | — | 2 | 1 | no | — |
| 23 | #12825 | — | 5 | 1 | no | maximal, Sylow |
| 24 | C2×C5×#41 | — | 2 | 1 | no | — |
| 25 | C5×#135 | — | 2 | 1 | no | — |
| 26 | C5×#135 | — | 2 | 1 | no | — |
| 27 | C5×#135 | C2×C4 | 1 | 2 | yes | — |
| 28 | C5×#135 | Q8 | 1 | 2 | yes | — |
| 29 | C2×C5×Q16 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×C5×SD16 | — | 2 | 1 | no | — |
| 31 | C2×C5×D8 | D4 | 1 | 1 | yes | characteristic |
| 32 | C5×#131 | — | 2 | 2 | no | — |
| 33 | C5×#131 | — | 2 | 2 | no | — |
| 34 | C5×#131 | — | 4 | 1 | no | — |
| 35 | C5×#131 | — | 4 | 1 | no | — |
| 36 | C2×C5×M16 | — | 2 | 1 | no | — |
| 37 | C2×C5×M16 | — | 4 | 1 | no | — |
| 38 | C22×C5×C8 | — | 4 | 1 | no | — |
| 39 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 40 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×C42×C5 | D4 | 1 | 1 | yes | characteristic |
| 42 | C2×#681 | — | 2 | 2 | no | — |
| 43 | C5×#108 | — | 2 | 1 | no | — |
| 44 | C5×#108 | C2×C4 | 1 | 2 | yes | — |
| 45 | C5×#104 | — | 2 | 2 | no | — |
| 46 | C5×#104 | — | 2 | 2 | no | — |
| 47 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C4×C5×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 50 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 51 | #3951 | — | 2 | 2 | no | — |
| 52 | #3951 | — | 2 | 2 | no | — |
| 53 | #3951 | — | 4 | 1 | no | — |
| 54 | #3951 | — | 4 | 1 | no | — |
| 55 | #3944 | — | 2 | 2 | no | — |
| 56 | #3944 | — | 2 | 2 | no | — |
| 57 | C2×#672 | — | 2 | 2 | no | — |
| 58 | C4⋊#168 | — | 4 | 2 | no | — |
| 59 | C2×#443 | D5 | 1 | 1 | yes | characteristic |
| 60 | C2×#364 | — | 5 | 1 | no | — |
| 61 | #2148 | — | 5 | 2 | no | — |
| 62 | #1848 | — | 5 | 1 | no | — |
| 63 | #1385 | — | 5 | 2 | no | — |
| 64 | C5×#41 | — | 2 | 2 | no | — |
| 65 | C5×#41 | — | 2 | 2 | no | — |
| 66 | C5×#41 | — | 4 | 1 | no | — |
| 67 | C5×#41 | — | 4 | 1 | no | — |
| 68 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 69 | C2×C5×D4 | — | 2 | 1 | no | — |
| 70 | C2×C5×D4 | — | 4 | 1 | no | — |
| 71 | C22×C4×C5 | — | 2 | 1 | no | — |
| 72 | C22×C4×C5 | — | 4 | 1 | no | — |
| 73 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C5×Q16 | — | 2 | 1 | no | — |
| 75 | C5×Q16 | C4⋊C4 | 1 | 2 | yes | — |
| 76 | C5×SD16 | — | 2 | 2 | no | — |
| 77 | C5×SD16 | — | 2 | 2 | no | — |
| 78 | C5×D8 | — | 2 | 1 | no | — |
| 79 | C5×D8 | C4⋊C4 | 1 | 2 | yes | — |
| 80 | C5×M16 | — | 2 | 2 | no | — |
| 81 | C5×M16 | — | 4 | 1 | no | — |
| 82 | C5×M16 | — | 4 | 2 | no | — |
| 83 | C5×M16 | — | 4 | 2 | no | — |
| 84 | C2×C5×C8 | — | 4 | 1 | no | — |
| 85 | C2×C5×C8 | — | 4 | 2 | no | — |
| 86 | C2×C5×C8 | — | 4 | 2 | no | — |
| 87 | C2×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 88 | C2×C5×C8 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C5×C8 | D8 | 1 | 1 | yes | characteristic |
| 90 | C2×C5×C8 | D8 | 1 | 1 | yes | characteristic |
| 91 | C2×C5×C8 | SD16 | 1 | 1 | yes | characteristic |
| 92 | C2×C5×C8 | SD16 | 1 | 1 | yes | characteristic |
| 93 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 94 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C5×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 97 | C2×C5×C8 | #41 | 1 | 1 | yes | characteristic |
| 98 | C2×C5×C8 | #41 | 1 | 1 | yes | characteristic |
| 99 | C42×C5 | — | 2 | 1 | no | — |
| 100 | C42×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 101 | C42×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 102 | #681 | — | 4 | 2 | no | — |
| 103 | #681 | — | 4 | 2 | no | — |
| 104 | C2×#168 | — | 4 | 2 | no | — |
| 105 | #672 | — | 4 | 2 | no | — |
| 106 | C2×#135 | C2×D5 | 1 | 1 | yes | characteristic |
| 107 | C2×#131 | — | 2 | 1 | no | — |
| 108 | C2×M32 | — | 5 | 2 | no | — |
| 109 | #443 | — | 2 | 1 | no | — |
| 110 | #443 | — | 2 | 1 | no | — |
| 111 | #443 | Dic5 | 1 | 2 | yes | — |
| 112 | #443 | Dic5 | 1 | 2 | yes | — |
| 113 | C2×#108 | — | 5 | 2 | no | — |
| 114 | C2×#108 | — | 10 | 1 | no | — |
| 115 | C2×#108 | C2×D5 | 1 | 1 | yes | characteristic |
| 116 | #423 | — | 10 | 1 | no | — |
| 117 | C2×#104 | — | 2 | 1 | no | — |
| 118 | C2×C4×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 119 | #364 | — | 10 | 1 | no | — |
| 120 | #364 | — | 10 | 1 | no | — |
| 121 | #362 | — | 10 | 2 | no | — |
| 122 | #361 | — | 10 | 2 | no | — |
| 123 | C23×C5 | — | 4 | 1 | no | — |
| 124 | C23×C5 | #137 | 1 | 1 | yes | characteristic |
| 125 | C5×Q8 | — | 2 | 2 | no | — |
| 126 | C5×Q8 | — | 4 | 1 | no | — |
| 127 | C5×D4 | — | 2 | 2 | no | — |
| 128 | C5×D4 | — | 4 | 2 | no | — |
| 129 | C5×D4 | — | 4 | 2 | no | — |
| 130 | C5×D4 | — | 4 | 1 | no | — |
| 131 | C2×C4×C5 | — | 2 | 1 | no | — |
| 132 | C2×C4×C5 | — | 2 | 1 | no | — |
| 133 | C2×C4×C5 | — | 4 | 2 | no | — |
| 134 | C2×C4×C5 | — | 4 | 1 | no | — |
| 135 | C2×C4×C5 | — | 4 | 2 | no | — |
| 136 | C2×C4×C5 | — | 4 | 2 | no | — |
| 137 | C2×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 138 | C2×C4×C5 | #121 | 1 | 1 | yes | characteristic |
| 139 | C2×C4×C5 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 140 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 141 | C2×C4×C5 | C2×SD16 | 1 | 1 | yes | characteristic |
| 142 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 143 | C5×C8 | — | 4 | 2 | no | — |
| 144 | C5×C8 | #102 | 1 | 1 | yes | characteristic |
| 145 | C5×C8 | #102 | 1 | 1 | yes | characteristic |
| 146 | C5×C8 | #102 | 1 | 1 | yes | characteristic |
| 147 | C5×C8 | #102 | 1 | 1 | yes | characteristic |
| 148 | C5×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 149 | C5×C8 | #117 | 1 | 1 | yes | characteristic |
| 150 | C5×C8 | #121 | 1 | 1 | yes | characteristic |
| 151 | C5×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 152 | #168 | — | 8 | 2 | no | — |
| 153 | C2×#41 | — | 2 | 1 | no | — |
| 154 | #135 | — | 2 | 1 | no | — |
| 155 | #135 | — | 2 | 1 | no | — |
| 156 | #135 | Dic10 | 1 | 2 | yes | — |
| 157 | #135 | C2×Dic5 | 1 | 2 | yes | — |
| 158 | C2×Q16 | D20 | 1 | 1 | yes | characteristic |
| 159 | C2×SD16 | — | 2 | 1 | no | — |
| 160 | C2×D8 | D20 | 1 | 1 | yes | characteristic |
| 161 | #131 | — | 2 | 2 | no | — |
| 162 | #131 | — | 2 | 2 | no | — |
| 163 | #131 | — | 4 | 1 | no | — |
| 164 | #131 | — | 4 | 1 | no | — |
| 165 | C2×M16 | — | 2 | 1 | no | — |
| 166 | C2×M16 | — | 4 | 1 | no | — |
| 167 | C2×M16 | — | 10 | 2 | no | — |
| 168 | C22×C8 | — | 4 | 1 | no | — |
| 169 | C22×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 170 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 171 | C2×C42 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 172 | M32 | — | 10 | 2 | no | — |
| 173 | M32 | — | 10 | 2 | no | — |
| 174 | C2×C16 | — | 10 | 2 | no | — |
| 175 | #108 | — | 2 | 1 | no | — |
| 176 | #108 | — | 10 | 2 | no | — |
| 177 | #108 | — | 10 | 2 | no | — |
| 178 | #108 | — | 20 | 1 | no | — |
| 179 | #108 | — | 20 | 1 | no | — |
| 180 | #108 | C2×Dic5 | 1 | 2 | yes | — |
| 181 | C4⋊C8 | — | 20 | 2 | no | — |
| 182 | #104 | — | 2 | 2 | no | — |
| 183 | #104 | — | 2 | 2 | no | — |
| 184 | C4×C8 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 185 | C4×C8 | C2×Dic5 | 1 | 1 | yes | characteristic |
| 186 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 187 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 188 | C22×C5 | — | 4 | 1 | no | — |
| 189 | C22×C5 | — | 4 | 2 | no | — |
| 190 | C22×C5 | — | 8 | 2 | no | — |
| 191 | C22×C5 | #417 | 1 | 1 | yes | characteristic |
| 192 | C22×C5 | #461 | 1 | 1 | yes | characteristic |
| 193 | C22×C5 | #474 | 1 | 1 | yes | characteristic |
| 194 | C4×C5 | — | 2 | 2 | no | — |
| 195 | C4×C5 | — | 4 | 1 | no | — |
| 196 | C4×C5 | — | 4 | 2 | no | — |
| 197 | C4×C5 | #380 | 1 | 1 | yes | characteristic |
| 198 | C4×C5 | C2×#102 | 1 | 1 | yes | characteristic |
| 199 | C4×C5 | #459 | 1 | 1 | yes | characteristic |
| 200 | C4×C5 | #476 | 1 | 1 | yes | characteristic |
| 201 | #41 | — | 2 | 2 | no | — |
| 202 | #41 | — | 2 | 2 | no | — |
| 203 | #41 | — | 4 | 1 | no | — |
| 204 | #41 | — | 4 | 1 | no | — |
| 205 | C2×Q8 | — | 2 | 1 | no | — |
| 206 | C2×D4 | — | 2 | 1 | no | — |
| 207 | C2×D4 | — | 4 | 1 | no | — |
| 208 | C22×C4 | — | 2 | 1 | no | — |
| 209 | C22×C4 | — | 4 | 1 | no | — |
| 210 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 211 | Q16 | — | 2 | 1 | no | — |
| 212 | Q16 | C4⋊Dic5 | 1 | 2 | yes | — |
| 213 | SD16 | — | 2 | 2 | no | — |
| 214 | SD16 | — | 2 | 2 | no | — |
| 215 | D8 | — | 2 | 1 | no | — |
| 216 | D8 | C4⋊Dic5 | 1 | 2 | yes | — |
| 217 | M16 | — | 2 | 2 | no | — |
| 218 | M16 | — | 4 | 1 | no | — |
| 219 | M16 | — | 4 | 2 | no | — |
| 220 | M16 | — | 4 | 2 | no | — |
| 221 | M16 | — | 20 | 2 | no | — |
| 222 | M16 | — | 20 | 2 | no | — |
| 223 | C2×C8 | — | 4 | 1 | no | — |
| 224 | C2×C8 | — | 4 | 2 | no | — |
| 225 | C2×C8 | — | 4 | 2 | no | — |
| 226 | C2×C8 | — | 20 | 2 | no | — |
| 227 | C2×C8 | #686 | 1 | 1 | yes | characteristic |
| 228 | C2×C8 | #686 | 1 | 1 | yes | characteristic |
| 229 | C2×C8 | #687 | 1 | 1 | yes | characteristic |
| 230 | C2×C8 | #687 | 1 | 1 | yes | characteristic |
| 231 | C2×C8 | C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 232 | C2×C8 | C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 233 | C2×C8 | C2×Dic10 | 1 | 1 | yes | characteristic |
| 234 | C2×C8 | C2×D20 | 1 | 1 | yes | characteristic |
| 235 | C2×C8 | #709 | 1 | 1 | yes | characteristic |
| 236 | C2×C8 | #709 | 1 | 1 | yes | characteristic |
| 237 | C2×C8 | C22×Dic5 | 1 | 1 | yes | characteristic |
| 238 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 239 | C42 | — | 2 | 1 | no | — |
| 240 | C42 | C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 241 | C42 | C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 242 | C16 | — | 20 | 2 | no | — |
| 243 | C2×C5 | — | 2 | 1 | no | — |
| 244 | C2×C5 | — | 2 | 1 | no | — |
| 245 | C2×C5 | — | 2 | 1 | no | — |
| 246 | C2×C5 | — | 8 | 2 | no | — |
| 247 | C2×C5 | #1802 | 1 | 1 | yes | characteristic |
| 248 | C23 | — | 4 | 1 | no | — |
| 249 | C23 | #4097 | 1 | 1 | yes | characteristic |
| 250 | Q8 | — | 2 | 2 | no | — |
| 251 | Q8 | — | 4 | 1 | no | — |
| 252 | D4 | — | 2 | 2 | no | — |
| 253 | D4 | — | 4 | 2 | no | — |
| 254 | D4 | — | 4 | 2 | no | — |
| 255 | D4 | — | 4 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 2 | no | — |
| 259 | C2×C4 | — | 4 | 1 | no | — |
| 260 | C2×C4 | — | 4 | 2 | no | — |
| 261 | C2×C4 | — | 4 | 2 | no | — |
| 262 | C2×C4 | #4070 | 1 | 1 | yes | characteristic |
| 263 | C2×C4 | #4076 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | C2×#687 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | #4095 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | C2×C22⋊Dic5 | 1 | 1 | yes | characteristic |
| 268 | C8 | — | 4 | 2 | no | — |
| 269 | C8 | — | 40 | 2 | no | — |
| 270 | C8 | #3964 | 1 | 1 | yes | characteristic |
| 271 | C8 | #3964 | 1 | 1 | yes | characteristic |
| 272 | C8 | #3964 | 1 | 1 | yes | characteristic |
| 273 | C8 | #3964 | 1 | 1 | yes | characteristic |
| 274 | C8 | #4070 | 1 | 1 | yes | characteristic |
| 275 | C8 | C2×C4⋊Dic5 | 1 | 1 | yes | characteristic |
| 276 | C8 | #4072 | 1 | 1 | yes | characteristic |
| 277 | C8 | #4076 | 1 | 1 | yes | characteristic |
| 278 | C5 | #12825 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 279 | C22 | — | 4 | 1 | no | — |
| 280 | C22 | — | 4 | 2 | no | — |
| 281 | C22 | — | 8 | 2 | no | — |
| 282 | C22 | #65296 | 1 | 1 | yes | characteristic |
| 283 | C22 | #65308 | 1 | 1 | yes | characteristic |
| 284 | C22 | #65521 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 2 | no | — |
| 286 | C4 | — | 4 | 1 | no | — |
| 287 | C4 | — | 4 | 2 | no | — |
| 288 | C4 | #65297 | 1 | 1 | yes | characteristic |
| 289 | C4 | #65304 | 1 | 1 | yes | characteristic |
| 290 | C4 | #65501 | 1 | 1 | yes | characteristic |
| 291 | C4 | C2×#3964 | 1 | 1 | yes | characteristic |
| 292 | C2 | — | 2 | 1 | no | — |
| 293 | C2 | — | 2 | 1 | no | — |
| 294 | C2 | — | 2 | 1 | no | — |
| 295 | C2 | — | 8 | 2 | no | — |
| 296 | C2 | #10602397 | 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