← #12253994
Subgroups — #12253994
Order 1280.
520 conjugacy classes of subgroups.
The subgroup lattice has 330 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#3164 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10610304 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10602183 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10602182 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10602150 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10602150 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10601447 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #10601441 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | C2×#64712 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #10601389 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #10600675 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C5×#544 | — | 2 | 1 | no | — |
| 13 | C22×Dic20 | C22 | 1 | 1 | yes | characteristic |
| 14 | #65850 | — | 2 | 1 | no | — |
| 15 | C5×#501 | — | 2 | 2 | no | — |
| 16 | C5×#500 | C22 | 1 | 2 | yes | — |
| 17 | C5×#499 | C22 | 1 | 2 | yes | — |
| 18 | C5×#428 | — | 2 | 1 | no | — |
| 19 | C5×#427 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C5×#107 | C22 | 1 | 1 | yes | characteristic |
| 21 | C5×#405 | C22 | 1 | 1 | yes | characteristic |
| 22 | #65409 | — | 2 | 1 | no | — |
| 23 | #65395 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#3944 | C22 | 1 | 1 | yes | characteristic |
| 25 | C22×#672 | C22 | 1 | 1 | yes | characteristic |
| 26 | #65005 | — | 2 | 2 | no | — |
| 27 | C4×Dic20 | C22 | 1 | 2 | yes | — |
| 28 | C4×Dic20 | C22 | 1 | 2 | yes | — |
| 29 | #64712 | C22 | 1 | 2 | yes | — |
| 30 | #64712 | C22 | 1 | 2 | yes | — |
| 31 | #64712 | C22 | 1 | 2 | yes | — |
| 32 | #64712 | C22 | 1 | 2 | yes | — |
| 33 | #64710 | C22 | 1 | 1 | yes | characteristic |
| 34 | #64710 | C22 | 1 | 2 | yes | — |
| 35 | #64710 | C22 | 1 | 1 | yes | characteristic |
| 36 | #64709 | C22 | 1 | 1 | yes | characteristic |
| 37 | #64709 | C22 | 1 | 2 | yes | — |
| 38 | #64709 | C22 | 1 | 1 | yes | characteristic |
| 39 | C4⋊#672 | C22 | 1 | 2 | yes | — |
| 40 | C4⋊#672 | C22 | 1 | 2 | yes | — |
| 41 | #33632 | — | 5 | 1 | no | maximal, Sylow |
| 42 | C22×Dic10 | — | 2 | 1 | no | — |
| 43 | C2×C5×M16 | D4 | 1 | 1 | yes | characteristic |
| 44 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 45 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 46 | C5×C4⋊Q8 | — | 4 | 1 | no | — |
| 47 | C5×#125 | — | 2 | 2 | no | — |
| 48 | C5×#125 | — | 4 | 1 | no | — |
| 49 | C5×C22⋊Q8 | — | 4 | 2 | no | — |
| 50 | C5×#117 | — | 2 | 1 | no | — |
| 51 | C5×#117 | D4 | 1 | 1 | yes | characteristic |
| 52 | C2×C5×C4⋊C4 | — | 2 | 1 | no | — |
| 53 | #4072 | — | 2 | 1 | no | — |
| 54 | C2×Dic20 | — | 2 | 1 | no | — |
| 55 | C2×Dic20 | — | 2 | 2 | no | — |
| 56 | C2×Dic20 | Q8 | 1 | 4 | yes | complemented |
| 57 | C2×Dic20 | C23 | 1 | 2 | yes | — |
| 58 | #4033 | — | 4 | 1 | no | — |
| 59 | #4033 | — | 4 | 1 | no | — |
| 60 | #4023 | — | 4 | 2 | no | — |
| 61 | C4×Dic10 | — | 2 | 2 | no | — |
| 62 | C4×Dic10 | — | 2 | 2 | no | — |
| 63 | C5×#107 | — | 2 | 1 | no | — |
| 64 | C5×#107 | — | 2 | 2 | no | — |
| 65 | C5×#107 | C23 | 1 | 2 | yes | — |
| 66 | C5×#107 | C23 | 1 | 1 | yes | characteristic |
| 67 | C5×#107 | C23 | 1 | 1 | yes | characteristic |
| 68 | C5×#106 | — | 2 | 1 | no | — |
| 69 | C5×#106 | — | 2 | 2 | no | — |
| 70 | C5×#97 | D4 | 1 | 2 | yes | — |
| 71 | C4×C5×C8 | C23 | 1 | 2 | yes | — |
| 72 | #3950 | C23 | 1 | 1 | yes | characteristic |
| 73 | #3950 | C23 | 1 | 1 | yes | characteristic |
| 74 | #3949 | — | 2 | 1 | no | — |
| 75 | #3948 | — | 2 | 2 | no | — |
| 76 | #3948 | — | 2 | 2 | no | — |
| 77 | #3944 | — | 2 | 2 | no | — |
| 78 | C2×#672 | — | 2 | 2 | no | — |
| 79 | C2×#672 | C23 | 1 | 1 | yes | characteristic |
| 80 | C2×#672 | C23 | 1 | 1 | yes | characteristic |
| 81 | C2×#672 | C23 | 1 | 2 | yes | — |
| 82 | #3164 | D5 | 1 | 1 | yes | characteristic |
| 83 | #2950 | — | 5 | 1 | no | — |
| 84 | #2229 | — | 5 | 2 | no | — |
| 85 | #2229 | — | 5 | 2 | no | — |
| 86 | #2227 | — | 5 | 2 | no | — |
| 87 | #2225 | — | 5 | 2 | no | — |
| 88 | #2159 | — | 5 | 1 | no | — |
| 89 | #2158 | — | 5 | 1 | no | — |
| 90 | #2151 | — | 5 | 1 | no | — |
| 91 | #2141 | — | 5 | 1 | no | — |
| 92 | C2×#358 | — | 5 | 1 | no | — |
| 93 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 94 | C22×C4×C5 | — | 4 | 1 | no | — |
| 95 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C22×Dic5 | — | 4 | 1 | no | — |
| 97 | C2×Dic10 | — | 2 | 4 | no | — |
| 98 | C2×Dic10 | — | 2 | 2 | no | — |
| 99 | C2×Dic10 | — | 4 | 1 | no | — |
| 100 | C2×Dic10 | — | 4 | 2 | no | — |
| 101 | C5×M16 | — | 2 | 2 | no | — |
| 102 | C2×C5×C8 | C2×D4 | 1 | 2 | yes | — |
| 103 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C2×C5×C8 | C2×Q8 | 1 | 2 | yes | — |
| 105 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 106 | C2×C5×C8 | C24 | 1 | 1 | yes | characteristic |
| 107 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 108 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 109 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 110 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 111 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 112 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 113 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 114 | C5×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 115 | C5×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 116 | C5×C22⋊C4 | — | 4 | 1 | no | — |
| 117 | C5×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 118 | C42×C5 | — | 4 | 1 | no | — |
| 119 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 120 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 121 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 122 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 123 | #683 | — | 4 | 2 | no | — |
| 124 | #683 | — | 4 | 2 | no | — |
| 125 | C4×Dic5 | — | 4 | 2 | no | — |
| 126 | C4×Dic5 | — | 4 | 2 | no | — |
| 127 | C4×Dic5 | — | 4 | 1 | no | — |
| 128 | Dic20 | — | 2 | 4 | no | — |
| 129 | Dic20 | — | 2 | 2 | no | — |
| 130 | Dic20 | C2×Q8 | 1 | 4 | yes | — |
| 131 | #672 | — | 2 | 1 | no | — |
| 132 | #672 | — | 2 | 1 | no | — |
| 133 | #672 | — | 2 | 2 | no | — |
| 134 | C22×Q16 | — | 5 | 1 | no | — |
| 135 | #544 | — | 2 | 1 | no | — |
| 136 | #519 | — | 10 | 1 | no | — |
| 137 | C2×M32 | — | 5 | 1 | no | — |
| 138 | C22×C16 | — | 5 | 1 | no | — |
| 139 | #501 | — | 2 | 2 | no | — |
| 140 | #500 | C2×D5 | 1 | 2 | yes | — |
| 141 | #499 | C2×D5 | 1 | 2 | yes | — |
| 142 | #441 | — | 10 | 2 | no | — |
| 143 | C4×Q16 | — | 5 | 2 | no | — |
| 144 | C4×Q16 | — | 5 | 2 | no | — |
| 145 | #428 | — | 2 | 1 | no | — |
| 146 | #427 | — | 5 | 1 | no | — |
| 147 | #427 | C2×D5 | 1 | 1 | yes | characteristic |
| 148 | C2×#107 | C2×D5 | 1 | 1 | yes | characteristic |
| 149 | #419 | — | 10 | 1 | no | — |
| 150 | #405 | C2×D5 | 1 | 1 | yes | characteristic |
| 151 | #367 | — | 5 | 1 | no | — |
| 152 | #367 | — | 5 | 1 | no | — |
| 153 | #367 | — | 5 | 2 | no | — |
| 154 | #366 | — | 5 | 1 | no | — |
| 155 | #366 | — | 5 | 1 | no | — |
| 156 | #366 | — | 5 | 2 | no | — |
| 157 | C4⋊C16 | — | 5 | 2 | no | — |
| 158 | C4⋊C16 | — | 5 | 2 | no | — |
| 159 | #358 | — | 5 | 2 | no | — |
| 160 | #358 | — | 5 | 2 | no | — |
| 161 | #358 | — | 5 | 2 | no | — |
| 162 | #358 | — | 5 | 2 | no | — |
| 163 | C23×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 164 | C5×Q8 | — | 16 | 2 | no | — |
| 165 | C2×C4×C5 | — | 4 | 1 | no | — |
| 166 | C2×C4×C5 | — | 4 | 1 | no | — |
| 167 | C2×C4×C5 | — | 4 | 2 | no | — |
| 168 | C2×C4×C5 | — | 16 | 1 | no | — |
| 169 | C2×C4×C5 | C22⋊Q8 | 1 | 2 | yes | — |
| 170 | C2×C4×C5 | C2×D8 | 1 | 2 | yes | — |
| 171 | C2×C4×C5 | C2×D8 | 1 | 2 | yes | — |
| 172 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 173 | C2×C4×C5 | #137 | 1 | 2 | yes | — |
| 174 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 175 | C2×Dic5 | — | 4 | 2 | no | — |
| 176 | C2×Dic5 | — | 4 | 4 | no | — |
| 177 | C2×Dic5 | — | 4 | 1 | no | — |
| 178 | C2×Dic5 | — | 4 | 1 | no | — |
| 179 | Dic10 | — | 2 | 4 | no | — |
| 180 | Dic10 | — | 4 | 2 | no | — |
| 181 | Dic10 | — | 4 | 4 | no | — |
| 182 | C5×C8 | — | 2 | 2 | no | — |
| 183 | C5×C8 | C22⋊Q8 | 1 | 2 | yes | — |
| 184 | C5×C8 | C22×Q8 | 1 | 1 | yes | characteristic |
| 185 | C5×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 186 | C22×Q8 | — | 10 | 1 | no | — |
| 187 | C2×Q16 | — | 5 | 4 | no | — |
| 188 | C2×Q16 | — | 5 | 2 | no | — |
| 189 | C2×Q16 | — | 10 | 2 | no | — |
| 190 | C2×Q16 | — | 10 | 1 | no | — |
| 191 | C2×M16 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 192 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 193 | C4⋊Q8 | — | 2 | 2 | no | — |
| 194 | C4⋊Q8 | — | 4 | 1 | no | — |
| 195 | #125 | — | 2 | 2 | no | — |
| 196 | #125 | — | 4 | 1 | no | — |
| 197 | C22⋊Q8 | — | 4 | 2 | no | — |
| 198 | C4×Q8 | — | 10 | 2 | no | — |
| 199 | C4×Q8 | — | 10 | 2 | no | — |
| 200 | C4×Q8 | — | 20 | 1 | no | — |
| 201 | C4×Q8 | — | 20 | 1 | no | — |
| 202 | #117 | — | 2 | 1 | no | — |
| 203 | #117 | — | 10 | 1 | no | — |
| 204 | #117 | — | 20 | 2 | no | — |
| 205 | #117 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 206 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 207 | M32 | — | 10 | 2 | no | — |
| 208 | C2×C16 | — | 5 | 1 | no | — |
| 209 | C2×C16 | — | 5 | 2 | no | — |
| 210 | C2×C16 | — | 5 | 1 | no | — |
| 211 | C2×C16 | — | 10 | 2 | no | — |
| 212 | #107 | — | 2 | 1 | no | — |
| 213 | #107 | — | 2 | 2 | no | — |
| 214 | #107 | — | 5 | 1 | no | — |
| 215 | #107 | — | 5 | 1 | no | — |
| 216 | #107 | C22×D5 | 1 | 2 | yes | — |
| 217 | #107 | C22×D5 | 1 | 1 | yes | characteristic |
| 218 | #107 | C22×D5 | 1 | 1 | yes | characteristic |
| 219 | #106 | — | 2 | 1 | no | — |
| 220 | #106 | — | 2 | 2 | no | — |
| 221 | #106 | — | 10 | 1 | no | — |
| 222 | #103 | — | 10 | 2 | no | — |
| 223 | #103 | — | 10 | 2 | no | — |
| 224 | #97 | C22⋊D5 | 1 | 2 | yes | — |
| 225 | C4×C8 | C22×D5 | 1 | 2 | yes | — |
| 226 | C22×C5 | — | 2 | 1 | no | — |
| 227 | C22×C5 | — | 2 | 1 | no | — |
| 228 | C22×C5 | #474 | 1 | 2 | yes | — |
| 229 | C22×C5 | C22×D8 | 1 | 1 | yes | characteristic |
| 230 | C4×C5 | — | 2 | 2 | no | — |
| 231 | C4×C5 | — | 2 | 2 | no | — |
| 232 | C4×C5 | — | 8 | 2 | no | — |
| 233 | C4×C5 | — | 8 | 1 | no | — |
| 234 | C4×C5 | — | 8 | 1 | no | — |
| 235 | C4×C5 | #474 | 1 | 2 | yes | — |
| 236 | C4×C5 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 237 | C4×C5 | C2×#137 | 1 | 1 | yes | characteristic |
| 238 | Dic5 | — | 4 | 4 | no | — |
| 239 | Dic5 | — | 8 | 1 | no | — |
| 240 | Dic5 | — | 8 | 1 | no | — |
| 241 | C2×Q8 | — | 4 | 2 | no | — |
| 242 | C2×Q8 | — | 10 | 2 | no | — |
| 243 | C2×Q8 | — | 10 | 4 | no | — |
| 244 | C2×Q8 | — | 20 | 1 | no | — |
| 245 | C2×Q8 | — | 20 | 2 | no | — |
| 246 | C22×C4 | — | 4 | 1 | no | — |
| 247 | C22×C4 | — | 20 | 1 | no | — |
| 248 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 249 | Q16 | — | 5 | 4 | no | — |
| 250 | Q16 | — | 10 | 2 | no | — |
| 251 | Q16 | — | 10 | 4 | no | — |
| 252 | M16 | — | 2 | 2 | no | — |
| 253 | C2×C8 | Q8×D5 | 1 | 2 | yes | — |
| 254 | C2×C8 | #713 | 1 | 2 | yes | — |
| 255 | C2×C8 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 256 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | C23×D5 | 1 | 1 | yes | characteristic |
| 258 | C4⋊C4 | — | 2 | 1 | no | — |
| 259 | C4⋊C4 | — | 2 | 2 | no | — |
| 260 | C4⋊C4 | — | 2 | 1 | no | — |
| 261 | C4⋊C4 | — | 4 | 1 | no | — |
| 262 | C4⋊C4 | — | 4 | 2 | no | — |
| 263 | C4⋊C4 | — | 4 | 2 | no | — |
| 264 | C4⋊C4 | — | 4 | 2 | no | — |
| 265 | C4⋊C4 | — | 10 | 1 | no | — |
| 266 | C4⋊C4 | — | 10 | 1 | no | — |
| 267 | C4⋊C4 | — | 20 | 2 | no | — |
| 268 | C4⋊C4 | — | 20 | 2 | no | — |
| 269 | C4⋊C4 | #686 | 1 | 1 | yes | characteristic |
| 270 | C4⋊C4 | #686 | 1 | 1 | yes | characteristic |
| 271 | C22⋊C4 | — | 4 | 1 | no | — |
| 272 | C22⋊C4 | — | 20 | 1 | no | — |
| 273 | C22⋊C4 | #686 | 1 | 2 | yes | — |
| 274 | C42 | — | 4 | 1 | no | — |
| 275 | C42 | — | 20 | 1 | no | — |
| 276 | C42 | — | 20 | 2 | no | — |
| 277 | C42 | — | 20 | 2 | no | — |
| 278 | C42 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 279 | C16 | — | 10 | 1 | no | — |
| 280 | C16 | — | 10 | 1 | no | — |
| 281 | C16 | — | 10 | 2 | no | — |
| 282 | C2×C5 | — | 2 | 2 | no | — |
| 283 | C2×C5 | C2×#474 | 1 | 1 | yes | characteristic |
| 284 | C2×C5 | #3416 | 1 | 1 | yes | characteristic |
| 285 | C2×C5 | #3417 | 1 | 1 | yes | characteristic |
| 286 | C23 | C2×#686 | 1 | 1 | yes | characteristic |
| 287 | Q8 | — | 10 | 4 | no | — |
| 288 | Q8 | — | 16 | 2 | no | — |
| 289 | Q8 | — | 20 | 2 | no | — |
| 290 | Q8 | — | 20 | 4 | no | — |
| 291 | C2×C4 | — | 4 | 1 | no | — |
| 292 | C2×C4 | — | 4 | 1 | no | — |
| 293 | C2×C4 | — | 4 | 2 | no | — |
| 294 | C2×C4 | — | 16 | 1 | no | — |
| 295 | C2×C4 | — | 20 | 2 | no | — |
| 296 | C2×C4 | — | 20 | 4 | no | — |
| 297 | C2×C4 | — | 20 | 1 | no | — |
| 298 | C2×C4 | — | 20 | 1 | no | — |
| 299 | C2×C4 | C2×#686 | 1 | 2 | yes | — |
| 300 | C2×C4 | C2×#686 | 1 | 2 | yes | — |
| 301 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | #4088 | 1 | 2 | yes | — |
| 303 | C2×C4 | #4092 | 1 | 2 | yes | — |
| 304 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 305 | C8 | — | 2 | 2 | no | — |
| 306 | C8 | #4092 | 1 | 2 | yes | — |
| 307 | C8 | C2×Q8×D5 | 1 | 1 | yes | characteristic |
| 308 | C8 | C2×#713 | 1 | 1 | yes | characteristic |
| 309 | C5 | #33632 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 310 | C22 | — | 2 | 1 | no | — |
| 311 | C22 | — | 2 | 1 | no | — |
| 312 | C22 | #65376 | 1 | 2 | yes | — |
| 313 | C22 | C22×#686 | 1 | 1 | yes | characteristic |
| 314 | C4 | — | 2 | 2 | no | — |
| 315 | C4 | — | 2 | 2 | no | — |
| 316 | C4 | — | 8 | 2 | no | — |
| 317 | C4 | — | 8 | 1 | no | — |
| 318 | C4 | — | 8 | 1 | no | — |
| 319 | C4 | — | 20 | 4 | no | — |
| 320 | C4 | — | 40 | 1 | no | — |
| 321 | C4 | — | 40 | 1 | no | — |
| 322 | C4 | #65376 | 1 | 2 | yes | — |
| 323 | C4 | C2×#4088 | 1 | 1 | yes | characteristic |
| 324 | C4 | C2×#4092 | 1 | 1 | yes | characteristic |
| 325 | C2 | — | 2 | 2 | no | — |
| 326 | C2 | C2×#65376 | 1 | 1 | yes | characteristic |
| 327 | C2 | #10615739 | 1 | 1 | yes | characteristic |
| 328 | C2 | #10615748 | 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