← #12015394
Subgroups — #12015394
Order 1280.
400 conjugacy classes of subgroups.
The subgroup lattice has 328 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#65485 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C5×#2143 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10602873 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#64731 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10601486 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10598312 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10598224 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C5×#135 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#4094 | — | 2 | 1 | no | — |
| 10 | C2×C5×M32 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C5×#108 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#3969 | — | 2 | 1 | no | — |
| 13 | #65485 | — | 2 | 1 | no | — |
| 14 | #65485 | — | 2 | 1 | no | — |
| 15 | #65485 | C4 | 1 | 2 | yes | — |
| 16 | #65485 | C4 | 1 | 2 | yes | — |
| 17 | C2×#3954 | — | 2 | 1 | no | — |
| 18 | #65407 | — | 2 | 1 | no | — |
| 19 | C2×#3951 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#3951 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#3944 | C22 | 1 | 1 | yes | characteristic |
| 23 | C5×#359 | — | 2 | 2 | no | — |
| 24 | #64787 | — | 2 | 2 | no | — |
| 25 | #64731 | — | 2 | 1 | no | — |
| 26 | #64731 | — | 2 | 1 | no | — |
| 27 | #12822 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C2×C5×#41 | — | 2 | 1 | no | — |
| 29 | C5×#135 | — | 2 | 1 | no | — |
| 30 | C5×#135 | — | 2 | 1 | no | — |
| 31 | C5×#135 | C2×C4 | 1 | 2 | yes | — |
| 32 | C5×#135 | D4 | 1 | 2 | yes | — |
| 33 | C2×C5×Q16 | Q8 | 1 | 1 | yes | characteristic |
| 34 | C2×C5×SD16 | — | 2 | 1 | no | — |
| 35 | C2×C5×D8 | Q8 | 1 | 1 | yes | characteristic |
| 36 | C2×C5×M16 | — | 2 | 1 | no | — |
| 37 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 38 | #4094 | — | 2 | 2 | no | — |
| 39 | #4094 | — | 2 | 2 | no | — |
| 40 | #4094 | — | 4 | 1 | no | — |
| 41 | #4094 | — | 4 | 1 | no | — |
| 42 | C2×C4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×#681 | — | 2 | 1 | no | — |
| 44 | C2×#681 | — | 2 | 1 | no | — |
| 45 | C2×#681 | — | 4 | 1 | no | — |
| 46 | C22×#168 | — | 4 | 1 | no | — |
| 47 | C22×#168 | D4 | 1 | 1 | yes | characteristic |
| 48 | C5×M32 | — | 2 | 1 | no | — |
| 49 | C5×M32 | — | 2 | 1 | no | — |
| 50 | C2×C5×C16 | — | 2 | 1 | no | — |
| 51 | C5×#108 | — | 2 | 1 | no | — |
| 52 | C5×#108 | — | 2 | 1 | no | — |
| 53 | #3969 | — | 2 | 2 | no | — |
| 54 | #3969 | — | 2 | 2 | no | — |
| 55 | #3954 | — | 4 | 1 | no | — |
| 56 | #3954 | — | 4 | 1 | no | — |
| 57 | #3951 | — | 2 | 1 | no | — |
| 58 | #3951 | — | 2 | 1 | no | — |
| 59 | #3951 | — | 2 | 1 | no | — |
| 60 | #3951 | C2×C4 | 1 | 2 | yes | — |
| 61 | #3946 | — | 4 | 2 | no | — |
| 62 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 63 | C8×Dic5 | C2×C4 | 1 | 1 | yes | characteristic |
| 64 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 65 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 66 | #3944 | — | 2 | 1 | no | — |
| 67 | #3944 | — | 2 | 1 | no | — |
| 68 | C2×#672 | — | 2 | 1 | no | — |
| 69 | C2×#443 | — | 5 | 1 | no | — |
| 70 | C2×#364 | — | 5 | 1 | no | — |
| 71 | #2143 | — | 5 | 1 | no | — |
| 72 | #2143 | D5 | 1 | 1 | yes | characteristic |
| 73 | #1848 | — | 5 | 1 | no | — |
| 74 | #1388 | — | 5 | 1 | no | — |
| 75 | #1388 | — | 5 | 1 | no | — |
| 76 | C5×#41 | — | 2 | 2 | no | — |
| 77 | C5×#41 | — | 2 | 2 | no | — |
| 78 | C5×#41 | — | 4 | 1 | no | — |
| 79 | C5×#41 | — | 4 | 1 | no | — |
| 80 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 81 | C2×C5×D4 | — | 2 | 1 | no | — |
| 82 | C2×C5×D4 | — | 4 | 1 | no | — |
| 83 | C22×C4×C5 | — | 4 | 1 | no | — |
| 84 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C22×Dic5 | — | 2 | 1 | no | — |
| 86 | C5×Q16 | — | 2 | 1 | no | — |
| 87 | C5×Q16 | C4⋊C4 | 1 | 2 | yes | — |
| 88 | C5×SD16 | — | 2 | 2 | no | — |
| 89 | C5×SD16 | — | 2 | 2 | no | — |
| 90 | C5×D8 | — | 2 | 1 | no | — |
| 91 | C5×D8 | C4⋊C4 | 1 | 2 | yes | — |
| 92 | C5×M16 | — | 4 | 1 | no | — |
| 93 | C5×M16 | — | 4 | 1 | no | — |
| 94 | C2×C5×C8 | — | 4 | 1 | no | — |
| 95 | C2×C5×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 96 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C5×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 99 | C2×C5×C8 | #41 | 1 | 1 | yes | characteristic |
| 100 | C2×C5×C8 | #41 | 1 | 1 | yes | characteristic |
| 101 | C4×Dic5 | — | 2 | 1 | no | — |
| 102 | C4×Dic5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 103 | C4×Dic5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 104 | #681 | — | 2 | 2 | no | — |
| 105 | #681 | — | 4 | 1 | no | — |
| 106 | #681 | — | 4 | 2 | no | — |
| 107 | #681 | — | 4 | 2 | no | — |
| 108 | #681 | — | 4 | 1 | no | — |
| 109 | #681 | — | 4 | 1 | no | — |
| 110 | C2×#168 | — | 4 | 1 | no | — |
| 111 | C2×#168 | — | 4 | 2 | no | — |
| 112 | C2×#168 | — | 4 | 2 | no | — |
| 113 | C2×#168 | — | 4 | 1 | no | — |
| 114 | C2×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C2×#168 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C2×#168 | D8 | 1 | 1 | yes | characteristic |
| 117 | C2×#168 | D8 | 1 | 1 | yes | characteristic |
| 118 | C2×#168 | SD16 | 1 | 1 | yes | characteristic |
| 119 | C2×#168 | SD16 | 1 | 1 | yes | characteristic |
| 120 | C5×C16 | — | 4 | 1 | no | — |
| 121 | #672 | — | 4 | 1 | no | — |
| 122 | C2×#135 | C2×D5 | 1 | 1 | yes | characteristic |
| 123 | C2×#131 | — | 10 | 1 | no | — |
| 124 | C2×M32 | — | 5 | 1 | no | — |
| 125 | C2×M32 | C2×D5 | 1 | 1 | yes | characteristic |
| 126 | #443 | — | 5 | 2 | no | — |
| 127 | #443 | — | 5 | 2 | no | — |
| 128 | #443 | — | 10 | 1 | no | — |
| 129 | #443 | — | 10 | 1 | no | — |
| 130 | C2×#108 | — | 5 | 1 | no | — |
| 131 | C2×#108 | — | 5 | 1 | no | — |
| 132 | C2×#108 | — | 10 | 1 | no | — |
| 133 | C2×#108 | C2×D5 | 1 | 1 | yes | characteristic |
| 134 | #423 | — | 10 | 1 | no | — |
| 135 | C2×#104 | — | 10 | 1 | no | — |
| 136 | C2×C4×C8 | — | 5 | 1 | no | — |
| 137 | #364 | — | 10 | 1 | no | — |
| 138 | #364 | — | 10 | 1 | no | — |
| 139 | #359 | — | 2 | 2 | no | — |
| 140 | #359 | — | 10 | 2 | no | — |
| 141 | C23×C5 | — | 4 | 1 | no | — |
| 142 | C23×C5 | #137 | 1 | 1 | yes | characteristic |
| 143 | C5×Q8 | — | 2 | 2 | no | — |
| 144 | C5×Q8 | — | 4 | 1 | no | — |
| 145 | C5×D4 | — | 2 | 2 | no | — |
| 146 | C5×D4 | — | 4 | 2 | no | — |
| 147 | C5×D4 | — | 4 | 2 | no | — |
| 148 | C5×D4 | — | 4 | 1 | no | — |
| 149 | C2×C4×C5 | — | 4 | 1 | no | — |
| 150 | C2×C4×C5 | — | 4 | 2 | no | — |
| 151 | C2×C4×C5 | — | 4 | 2 | no | — |
| 152 | C2×C4×C5 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 153 | C2×C4×C5 | #121 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C5 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 155 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 156 | C2×C4×C5 | C2×SD16 | 1 | 1 | yes | characteristic |
| 157 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 158 | C2×Dic5 | — | 2 | 1 | no | — |
| 159 | C2×Dic5 | — | 2 | 1 | no | — |
| 160 | C2×Dic5 | — | 4 | 2 | no | — |
| 161 | C5×C8 | — | 8 | 1 | no | — |
| 162 | C5×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 163 | C5×C8 | #117 | 1 | 1 | yes | characteristic |
| 164 | C5×C8 | #121 | 1 | 1 | yes | characteristic |
| 165 | C5×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 166 | #168 | — | 4 | 2 | no | — |
| 167 | #168 | — | 8 | 1 | no | — |
| 168 | #168 | #102 | 1 | 1 | yes | characteristic |
| 169 | #168 | #102 | 1 | 1 | yes | characteristic |
| 170 | #168 | #102 | 1 | 1 | yes | characteristic |
| 171 | #168 | #102 | 1 | 1 | yes | characteristic |
| 172 | C2×#41 | — | 2 | 1 | no | — |
| 173 | #135 | — | 2 | 1 | no | — |
| 174 | #135 | — | 2 | 1 | no | — |
| 175 | #135 | C4×D5 | 1 | 2 | yes | — |
| 176 | #135 | C22⋊D5 | 1 | 2 | yes | — |
| 177 | C2×Q16 | Dic10 | 1 | 1 | yes | characteristic |
| 178 | C2×SD16 | — | 2 | 1 | no | — |
| 179 | C2×D8 | Dic10 | 1 | 1 | yes | characteristic |
| 180 | #131 | — | 10 | 2 | no | — |
| 181 | #131 | — | 10 | 2 | no | — |
| 182 | #131 | — | 20 | 1 | no | — |
| 183 | #131 | — | 20 | 1 | no | — |
| 184 | C2×M16 | — | 2 | 1 | no | — |
| 185 | C2×M16 | — | 10 | 1 | no | — |
| 186 | C2×M16 | — | 10 | 1 | no | — |
| 187 | C2×M16 | — | 20 | 1 | no | — |
| 188 | C22×C8 | — | 5 | 1 | no | — |
| 189 | C22×C8 | — | 20 | 1 | no | — |
| 190 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 191 | C2×C42 | — | 5 | 1 | no | — |
| 192 | M32 | — | 2 | 1 | no | — |
| 193 | M32 | — | 2 | 1 | no | — |
| 194 | M32 | — | 10 | 1 | no | — |
| 195 | M32 | — | 10 | 1 | no | — |
| 196 | C2×C16 | — | 2 | 1 | no | — |
| 197 | C2×C16 | — | 10 | 1 | no | — |
| 198 | #108 | — | 2 | 1 | no | — |
| 199 | #108 | — | 2 | 1 | no | — |
| 200 | #108 | — | 5 | 2 | no | — |
| 201 | #108 | — | 10 | 1 | no | — |
| 202 | #108 | — | 10 | 1 | no | — |
| 203 | #108 | — | 10 | 1 | no | — |
| 204 | #108 | — | 20 | 1 | no | — |
| 205 | #108 | — | 20 | 1 | no | — |
| 206 | C4⋊C8 | — | 20 | 2 | no | — |
| 207 | #104 | — | 10 | 2 | no | — |
| 208 | #104 | — | 10 | 2 | no | — |
| 209 | C4×C8 | — | 5 | 1 | no | — |
| 210 | C4×C8 | — | 5 | 1 | no | — |
| 211 | C4×C8 | — | 5 | 1 | no | — |
| 212 | C4×C8 | — | 5 | 1 | no | — |
| 213 | C22×C5 | — | 4 | 1 | no | — |
| 214 | C22×C5 | — | 4 | 2 | no | — |
| 215 | C22×C5 | — | 8 | 2 | no | — |
| 216 | C22×C5 | #417 | 1 | 1 | yes | characteristic |
| 217 | C22×C5 | #461 | 1 | 1 | yes | characteristic |
| 218 | C22×C5 | #474 | 1 | 1 | yes | characteristic |
| 219 | C4×C5 | — | 4 | 2 | no | — |
| 220 | C4×C5 | #380 | 1 | 1 | yes | characteristic |
| 221 | C4×C5 | C2×#102 | 1 | 1 | yes | characteristic |
| 222 | C4×C5 | #459 | 1 | 1 | yes | characteristic |
| 223 | C4×C5 | #476 | 1 | 1 | yes | characteristic |
| 224 | Dic5 | — | 2 | 2 | no | — |
| 225 | Dic5 | — | 4 | 1 | no | — |
| 226 | #41 | — | 2 | 2 | no | — |
| 227 | #41 | — | 2 | 2 | no | — |
| 228 | #41 | — | 4 | 1 | no | — |
| 229 | #41 | — | 4 | 1 | no | — |
| 230 | C2×Q8 | — | 2 | 1 | no | — |
| 231 | C2×D4 | — | 2 | 1 | no | — |
| 232 | C2×D4 | — | 4 | 1 | no | — |
| 233 | C22×C4 | — | 4 | 1 | no | — |
| 234 | C22×C4 | — | 10 | 1 | no | — |
| 235 | C22×C4 | D4×D5 | 1 | 1 | yes | characteristic |
| 236 | Q16 | — | 2 | 1 | no | — |
| 237 | Q16 | #683 | 1 | 2 | yes | — |
| 238 | SD16 | — | 2 | 2 | no | — |
| 239 | SD16 | — | 2 | 2 | no | — |
| 240 | D8 | — | 2 | 1 | no | — |
| 241 | D8 | #683 | 1 | 2 | yes | — |
| 242 | M16 | — | 4 | 1 | no | — |
| 243 | M16 | — | 4 | 1 | no | — |
| 244 | M16 | — | 10 | 2 | no | — |
| 245 | M16 | — | 20 | 2 | no | — |
| 246 | M16 | — | 20 | 2 | no | — |
| 247 | M16 | — | 20 | 1 | no | — |
| 248 | M16 | — | 20 | 1 | no | — |
| 249 | M16 | — | 20 | 1 | no | — |
| 250 | C2×C8 | — | 4 | 1 | no | — |
| 251 | C2×C8 | — | 5 | 1 | no | — |
| 252 | C2×C8 | — | 5 | 1 | no | — |
| 253 | C2×C8 | — | 5 | 1 | no | — |
| 254 | C2×C8 | — | 5 | 1 | no | — |
| 255 | C2×C8 | — | 5 | 1 | no | — |
| 256 | C2×C8 | — | 5 | 1 | no | — |
| 257 | C2×C8 | — | 20 | 2 | no | — |
| 258 | C2×C8 | — | 20 | 2 | no | — |
| 259 | C2×C8 | — | 20 | 1 | no | — |
| 260 | C2×C8 | — | 20 | 1 | no | — |
| 261 | C2×C8 | C2×Dic10 | 1 | 1 | yes | characteristic |
| 262 | C2×C8 | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 263 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 264 | C2×C8 | #711 | 1 | 1 | yes | characteristic |
| 265 | C2×C8 | #711 | 1 | 1 | yes | characteristic |
| 266 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 267 | C42 | — | 5 | 1 | no | — |
| 268 | C42 | — | 5 | 1 | no | — |
| 269 | C42 | — | 10 | 1 | no | — |
| 270 | C16 | — | 4 | 1 | no | — |
| 271 | C16 | — | 20 | 1 | no | — |
| 272 | C2×C5 | — | 2 | 1 | no | — |
| 273 | C2×C5 | — | 2 | 1 | no | — |
| 274 | C2×C5 | — | 2 | 1 | no | — |
| 275 | C2×C5 | — | 8 | 2 | no | — |
| 276 | C2×C5 | #1802 | 1 | 1 | yes | characteristic |
| 277 | C23 | — | 4 | 1 | no | — |
| 278 | C23 | #4061 | 1 | 1 | yes | characteristic |
| 279 | Q8 | — | 2 | 2 | no | — |
| 280 | Q8 | — | 4 | 1 | no | — |
| 281 | D4 | — | 2 | 2 | no | — |
| 282 | D4 | — | 4 | 2 | no | — |
| 283 | D4 | — | 4 | 2 | no | — |
| 284 | D4 | — | 4 | 1 | no | — |
| 285 | C2×C4 | — | 4 | 1 | no | — |
| 286 | C2×C4 | — | 4 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 2 | no | — |
| 288 | C2×C4 | — | 10 | 1 | no | — |
| 289 | C2×C4 | — | 10 | 1 | no | — |
| 290 | C2×C4 | — | 20 | 2 | no | — |
| 291 | C2×C4 | #4024 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | D5×C22⋊C4 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | D5×D8 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | #4057 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | D5×SD16 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | #4085 | 1 | 1 | yes | characteristic |
| 297 | C8 | — | 5 | 1 | no | — |
| 298 | C8 | — | 5 | 1 | no | — |
| 299 | C8 | — | 5 | 1 | no | — |
| 300 | C8 | — | 5 | 1 | no | — |
| 301 | C8 | — | 8 | 1 | no | — |
| 302 | C8 | — | 20 | 2 | no | — |
| 303 | C8 | — | 40 | 1 | no | — |
| 304 | C8 | #4023 | 1 | 1 | yes | characteristic |
| 305 | C8 | #4024 | 1 | 1 | yes | characteristic |
| 306 | C8 | C2×#683 | 1 | 1 | yes | characteristic |
| 307 | C8 | #4085 | 1 | 1 | yes | characteristic |
| 308 | C5 | #12822 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 309 | C22 | — | 4 | 1 | no | — |
| 310 | C22 | — | 4 | 2 | no | — |
| 311 | C22 | — | 8 | 2 | no | — |
| 312 | C22 | #65048 | 1 | 1 | yes | characteristic |
| 313 | C22 | #65060 | 1 | 1 | yes | characteristic |
| 314 | C22 | #65451 | 1 | 1 | yes | characteristic |
| 315 | C4 | — | 4 | 2 | no | — |
| 316 | C4 | — | 10 | 2 | no | — |
| 317 | C4 | — | 20 | 1 | no | — |
| 318 | C4 | #65050 | 1 | 1 | yes | characteristic |
| 319 | C4 | D5×#102 | 1 | 1 | yes | characteristic |
| 320 | C4 | #65231 | 1 | 1 | yes | characteristic |
| 321 | C4 | #65439 | 1 | 1 | yes | characteristic |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 2 | 1 | no | — |
| 324 | C2 | — | 2 | 1 | no | — |
| 325 | C2 | — | 8 | 2 | no | — |
| 326 | C2 | #10600946 | 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