← #12253638
Subgroups — #12253638
Order 1280.
680 conjugacy classes of subgroups.
The subgroup lattice has 344 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C5×#3156 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#65484 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C22×#3961 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10610304 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10602870 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #10602854 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10602117 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #10602109 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #10600675 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C2×C5×#137 | — | 2 | 1 | no | — |
| 11 | C2×C5×#135 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×C5×Q16 | C22 | 1 | 1 | yes | characteristic |
| 13 | C5×#536 | — | 2 | 1 | no | — |
| 14 | C22×Dic20 | C22 | 1 | 1 | yes | characteristic |
| 15 | #65850 | — | 2 | 1 | no | — |
| 16 | C5×#497 | — | 2 | 2 | no | — |
| 17 | C5×#495 | C22 | 1 | 2 | yes | — |
| 18 | C5×#494 | C22 | 1 | 2 | yes | — |
| 19 | C5×#405 | C22 | 1 | 1 | yes | characteristic |
| 20 | #65484 | — | 2 | 4 | no | — |
| 21 | C2×#3961 | — | 2 | 4 | no | — |
| 22 | C2×#3961 | C22 | 1 | 4 | yes | — |
| 23 | C2×#3961 | C22 | 1 | 2 | yes | — |
| 24 | C2×#3959 | C22 | 1 | 2 | yes | — |
| 25 | #65409 | — | 2 | 1 | no | — |
| 26 | #65395 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#3944 | C22 | 1 | 1 | yes | characteristic |
| 28 | C22×#672 | C22 | 1 | 1 | yes | characteristic |
| 29 | #65005 | — | 2 | 2 | no | — |
| 30 | C4×Dic20 | C22 | 1 | 4 | yes | — |
| 31 | #64784 | C22 | 1 | 4 | yes | — |
| 32 | #64784 | C22 | 1 | 2 | yes | — |
| 33 | #64782 | C22 | 1 | 2 | yes | — |
| 34 | C4⋊#672 | C22 | 1 | 4 | yes | — |
| 35 | #33588 | — | 5 | 1 | no | maximal, Sylow |
| 36 | C2×C5×#41 | — | 2 | 1 | no | — |
| 37 | C22×C5×Q8 | — | 2 | 1 | no | — |
| 38 | C22×Dic10 | — | 2 | 1 | no | — |
| 39 | C5×#137 | — | 4 | 4 | no | — |
| 40 | C5×#135 | — | 2 | 2 | no | — |
| 41 | C5×#135 | — | 2 | 2 | no | — |
| 42 | C2×C5×Q16 | — | 2 | 2 | no | — |
| 43 | C2×C5×Q16 | — | 2 | 2 | no | — |
| 44 | C2×C5×Q16 | — | 2 | 1 | no | — |
| 45 | C2×C5×Q16 | — | 2 | 2 | no | — |
| 46 | C2×C5×Q16 | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×C5×Q16 | C23 | 1 | 2 | yes | — |
| 48 | C2×C5×SD16 | — | 2 | 2 | no | — |
| 49 | C2×C5×SD16 | — | 2 | 1 | no | — |
| 50 | C2×C5×D8 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×C5×M16 | D4 | 1 | 1 | yes | characteristic |
| 52 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 53 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 54 | C5×#124 | — | 2 | 2 | no | — |
| 55 | C5×#123 | — | 4 | 2 | no | — |
| 56 | C5×C22⋊Q8 | — | 4 | 2 | no | — |
| 57 | C5×#117 | D4 | 1 | 1 | yes | characteristic |
| 58 | #4072 | — | 2 | 1 | no | — |
| 59 | C2×Dic20 | — | 2 | 1 | no | — |
| 60 | C2×Dic20 | — | 2 | 2 | no | — |
| 61 | C2×Dic20 | D4 | 1 | 4 | yes | — |
| 62 | C2×Dic20 | C23 | 1 | 2 | yes | — |
| 63 | #4033 | — | 4 | 2 | no | — |
| 64 | #4023 | — | 4 | 2 | no | — |
| 65 | C4×Dic10 | — | 2 | 4 | no | — |
| 66 | C5×#97 | D4 | 1 | 2 | yes | — |
| 67 | C4×C5×C8 | C23 | 1 | 2 | yes | — |
| 68 | #3961 | — | 2 | 8 | no | — |
| 69 | #3961 | — | 2 | 4 | no | — |
| 70 | #3959 | — | 2 | 4 | no | — |
| 71 | #3950 | C23 | 1 | 2 | yes | — |
| 72 | #3949 | — | 2 | 1 | no | — |
| 73 | #3948 | — | 2 | 4 | no | — |
| 74 | #3944 | — | 2 | 2 | no | — |
| 75 | C2×#672 | — | 2 | 2 | no | — |
| 76 | C2×#672 | C23 | 1 | 2 | yes | — |
| 77 | C2×#672 | C23 | 1 | 2 | yes | — |
| 78 | C2×#510 | — | 5 | 1 | no | — |
| 79 | C22×Q32 | — | 5 | 1 | no | — |
| 80 | #3156 | D5 | 1 | 1 | yes | characteristic |
| 81 | #2950 | — | 5 | 1 | no | — |
| 82 | #2211 | — | 5 | 4 | no | — |
| 83 | #2209 | — | 5 | 4 | no | — |
| 84 | #2151 | — | 5 | 1 | no | — |
| 85 | #2141 | — | 5 | 1 | no | — |
| 86 | #2138 | — | 5 | 1 | no | — |
| 87 | C5×#41 | — | 4 | 2 | no | — |
| 88 | C5×#41 | — | 4 | 2 | no | — |
| 89 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 90 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 91 | C2×C5×Q8 | — | 4 | 1 | no | — |
| 92 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 93 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 94 | C2×C5×D4 | — | 2 | 1 | no | — |
| 95 | C2×C5×D4 | — | 4 | 1 | no | — |
| 96 | C22×C4×C5 | — | 4 | 1 | no | — |
| 97 | C22×C4×C5 | — | 4 | 1 | no | — |
| 98 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C22×Dic5 | — | 4 | 1 | no | — |
| 100 | C2×Dic10 | — | 2 | 4 | no | — |
| 101 | C2×Dic10 | — | 2 | 2 | no | — |
| 102 | C2×Dic10 | — | 4 | 1 | no | — |
| 103 | C2×Dic10 | — | 4 | 2 | no | — |
| 104 | C5×Q16 | — | 2 | 2 | no | — |
| 105 | C5×Q16 | — | 2 | 1 | no | — |
| 106 | C5×Q16 | — | 2 | 2 | no | — |
| 107 | C5×Q16 | — | 2 | 1 | no | — |
| 108 | C5×Q16 | — | 4 | 4 | no | — |
| 109 | C5×Q16 | — | 4 | 2 | no | — |
| 110 | C5×SD16 | — | 4 | 4 | no | — |
| 111 | C5×SD16 | — | 4 | 2 | no | — |
| 112 | C5×D8 | — | 2 | 1 | no | — |
| 113 | C5×D8 | — | 2 | 1 | no | — |
| 114 | C5×M16 | — | 2 | 2 | no | — |
| 115 | C2×C5×C8 | C2×D4 | 1 | 2 | yes | — |
| 116 | C2×C5×C8 | C2×D4 | 1 | 2 | yes | — |
| 117 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 118 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 119 | C2×C5×C8 | C24 | 1 | 1 | yes | characteristic |
| 120 | C5×C4⋊C4 | — | 4 | 4 | no | — |
| 121 | C5×C4⋊C4 | D8 | 1 | 2 | yes | — |
| 122 | C5×C22⋊C4 | — | 4 | 4 | no | — |
| 123 | C5×C22⋊C4 | D8 | 1 | 2 | yes | — |
| 124 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 125 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 126 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 127 | #683 | — | 4 | 4 | no | — |
| 128 | C4×Dic5 | — | 4 | 4 | no | — |
| 129 | C4×Dic5 | — | 4 | 1 | no | — |
| 130 | Dic20 | — | 2 | 4 | no | — |
| 131 | Dic20 | — | 2 | 2 | no | — |
| 132 | Dic20 | C2×D4 | 1 | 4 | yes | — |
| 133 | #672 | — | 2 | 2 | no | — |
| 134 | #672 | — | 2 | 2 | no | — |
| 135 | C2×#137 | — | 2 | 1 | no | — |
| 136 | C2×#135 | C2×D5 | 1 | 1 | yes | characteristic |
| 137 | C22×Q16 | — | 5 | 1 | no | — |
| 138 | C22×Q16 | C2×D5 | 1 | 1 | yes | characteristic |
| 139 | #536 | — | 2 | 1 | no | — |
| 140 | #519 | — | 10 | 1 | no | — |
| 141 | #510 | — | 10 | 4 | no | — |
| 142 | C2×Q32 | — | 5 | 4 | no | — |
| 143 | C2×Q32 | — | 5 | 2 | no | — |
| 144 | C2×Q32 | — | 10 | 4 | no | — |
| 145 | C2×SD32 | — | 5 | 2 | no | — |
| 146 | C2×M32 | — | 5 | 1 | no | — |
| 147 | C22×C16 | — | 5 | 1 | no | — |
| 148 | #497 | — | 2 | 2 | no | — |
| 149 | #495 | C2×D5 | 1 | 2 | yes | — |
| 150 | #494 | C2×D5 | 1 | 2 | yes | — |
| 151 | #441 | — | 10 | 2 | no | — |
| 152 | C4×Q16 | — | 5 | 4 | no | — |
| 153 | #427 | — | 5 | 1 | no | — |
| 154 | #419 | — | 10 | 1 | no | — |
| 155 | #405 | C2×D5 | 1 | 1 | yes | characteristic |
| 156 | C4⋊C16 | — | 5 | 4 | no | — |
| 157 | #358 | — | 5 | 2 | no | — |
| 158 | #358 | — | 5 | 4 | no | — |
| 159 | #357 | — | 5 | 2 | no | — |
| 160 | C23×C5 | — | 4 | 1 | no | — |
| 161 | C23×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 162 | C5×Q8 | — | 4 | 1 | no | — |
| 163 | C5×Q8 | — | 4 | 2 | no | — |
| 164 | C5×Q8 | — | 4 | 2 | no | — |
| 165 | C5×Q8 | — | 4 | 1 | no | — |
| 166 | C5×Q8 | — | 8 | 2 | no | — |
| 167 | C5×D4 | — | 4 | 1 | no | — |
| 168 | C5×D4 | — | 4 | 1 | no | — |
| 169 | C5×D4 | — | 8 | 2 | no | — |
| 170 | C2×C4×C5 | — | 4 | 2 | no | — |
| 171 | C2×C4×C5 | — | 4 | 1 | no | — |
| 172 | C2×C4×C5 | — | 8 | 2 | no | — |
| 173 | C2×C4×C5 | — | 8 | 2 | no | — |
| 174 | C2×C4×C5 | #121 | 1 | 2 | yes | — |
| 175 | C2×C4×C5 | C2×D8 | 1 | 4 | yes | — |
| 176 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 177 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 178 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 179 | C2×Dic5 | — | 4 | 2 | no | — |
| 180 | C2×Dic5 | — | 4 | 4 | no | — |
| 181 | C2×Dic5 | — | 4 | 2 | no | — |
| 182 | Dic10 | — | 2 | 4 | no | — |
| 183 | Dic10 | — | 4 | 2 | no | — |
| 184 | Dic10 | — | 4 | 4 | no | — |
| 185 | C5×C8 | — | 2 | 2 | no | — |
| 186 | C5×C8 | #121 | 1 | 2 | yes | — |
| 187 | C5×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 188 | C5×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 189 | C2×#41 | — | 2 | 1 | no | — |
| 190 | C22×Q8 | — | 2 | 1 | no | — |
| 191 | C22×Q8 | — | 10 | 1 | no | — |
| 192 | #137 | — | 4 | 4 | no | — |
| 193 | #135 | — | 2 | 2 | no | — |
| 194 | #135 | — | 2 | 2 | no | — |
| 195 | C2×Q16 | — | 2 | 2 | no | — |
| 196 | C2×Q16 | — | 2 | 2 | no | — |
| 197 | C2×Q16 | — | 2 | 1 | no | — |
| 198 | C2×Q16 | — | 2 | 2 | no | — |
| 199 | C2×Q16 | — | 5 | 4 | no | — |
| 200 | C2×Q16 | — | 5 | 2 | no | — |
| 201 | C2×Q16 | — | 10 | 2 | no | — |
| 202 | C2×Q16 | — | 10 | 1 | no | — |
| 203 | C2×Q16 | C22×D5 | 1 | 2 | yes | — |
| 204 | C2×Q16 | C22×D5 | 1 | 1 | yes | characteristic |
| 205 | C2×SD16 | — | 2 | 1 | no | — |
| 206 | C2×SD16 | — | 2 | 2 | no | — |
| 207 | C2×D8 | C22×D5 | 1 | 1 | yes | characteristic |
| 208 | C2×M16 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 209 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 210 | C4⋊Q8 | — | 2 | 2 | no | — |
| 211 | #124 | — | 2 | 2 | no | — |
| 212 | #123 | — | 4 | 2 | no | — |
| 213 | C22⋊Q8 | — | 4 | 2 | no | — |
| 214 | C4×Q8 | — | 10 | 4 | no | — |
| 215 | C4×Q8 | — | 20 | 2 | no | — |
| 216 | #117 | — | 10 | 1 | no | — |
| 217 | #117 | — | 20 | 2 | no | — |
| 218 | #117 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 219 | Q32 | — | 10 | 8 | no | — |
| 220 | Q32 | — | 10 | 4 | no | — |
| 221 | SD32 | — | 10 | 4 | no | — |
| 222 | M32 | — | 10 | 2 | no | — |
| 223 | C2×C16 | — | 5 | 2 | no | — |
| 224 | C2×C16 | — | 5 | 2 | no | — |
| 225 | C2×C16 | — | 10 | 2 | no | — |
| 226 | #107 | — | 5 | 2 | no | — |
| 227 | #106 | — | 10 | 1 | no | — |
| 228 | #103 | — | 10 | 4 | no | — |
| 229 | #97 | C22⋊D5 | 1 | 2 | yes | — |
| 230 | C4×C8 | C22×D5 | 1 | 2 | yes | — |
| 231 | C22×C5 | — | 2 | 2 | no | — |
| 232 | C22×C5 | — | 8 | 2 | no | — |
| 233 | C22×C5 | — | 8 | 1 | no | — |
| 234 | C22×C5 | #459 | 1 | 2 | yes | — |
| 235 | C22×C5 | C22×D8 | 1 | 1 | yes | characteristic |
| 236 | C4×C5 | — | 2 | 4 | no | — |
| 237 | C4×C5 | — | 8 | 1 | no | — |
| 238 | C4×C5 | — | 8 | 2 | no | — |
| 239 | C4×C5 | #459 | 1 | 2 | yes | — |
| 240 | C4×C5 | C2×#121 | 1 | 1 | yes | characteristic |
| 241 | C4×C5 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 242 | Dic5 | — | 4 | 4 | no | — |
| 243 | Dic5 | — | 8 | 2 | no | — |
| 244 | #41 | — | 4 | 2 | no | — |
| 245 | #41 | — | 4 | 2 | no | — |
| 246 | C2×Q8 | — | 2 | 2 | no | — |
| 247 | C2×Q8 | — | 2 | 1 | no | — |
| 248 | C2×Q8 | — | 4 | 1 | no | — |
| 249 | C2×Q8 | — | 4 | 2 | no | — |
| 250 | C2×Q8 | — | 4 | 2 | no | — |
| 251 | C2×Q8 | — | 10 | 2 | no | — |
| 252 | C2×Q8 | — | 10 | 4 | no | — |
| 253 | C2×Q8 | — | 20 | 1 | no | — |
| 254 | C2×Q8 | — | 20 | 2 | no | — |
| 255 | C2×D4 | — | 2 | 1 | no | — |
| 256 | C2×D4 | — | 4 | 1 | no | — |
| 257 | C22×C4 | — | 4 | 1 | no | — |
| 258 | C22×C4 | — | 4 | 1 | no | — |
| 259 | C22×C4 | — | 20 | 1 | no | — |
| 260 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 261 | Q16 | — | 2 | 2 | no | — |
| 262 | Q16 | — | 2 | 1 | no | — |
| 263 | Q16 | — | 2 | 2 | no | — |
| 264 | Q16 | — | 2 | 1 | no | — |
| 265 | Q16 | — | 4 | 4 | no | — |
| 266 | Q16 | — | 4 | 2 | no | — |
| 267 | Q16 | — | 5 | 4 | no | — |
| 268 | Q16 | — | 10 | 2 | no | — |
| 269 | Q16 | — | 10 | 4 | no | — |
| 270 | SD16 | — | 4 | 4 | no | — |
| 271 | SD16 | — | 4 | 2 | no | — |
| 272 | D8 | — | 2 | 1 | no | — |
| 273 | D8 | — | 2 | 1 | no | — |
| 274 | M16 | — | 2 | 2 | no | — |
| 275 | C2×C8 | D4×D5 | 1 | 2 | yes | — |
| 276 | C2×C8 | #711 | 1 | 2 | yes | — |
| 277 | C2×C8 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 278 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 279 | C2×C8 | C23×D5 | 1 | 1 | yes | characteristic |
| 280 | C4⋊C4 | — | 4 | 4 | no | — |
| 281 | C4⋊C4 | — | 10 | 2 | no | — |
| 282 | C4⋊C4 | — | 20 | 4 | no | — |
| 283 | C4⋊C4 | #686 | 1 | 2 | yes | — |
| 284 | C22⋊C4 | — | 4 | 4 | no | — |
| 285 | C22⋊C4 | — | 20 | 1 | no | — |
| 286 | C22⋊C4 | #686 | 1 | 2 | yes | — |
| 287 | C42 | — | 20 | 1 | no | — |
| 288 | C42 | — | 20 | 4 | no | — |
| 289 | C42 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 290 | C16 | — | 10 | 2 | no | — |
| 291 | C16 | — | 10 | 2 | no | — |
| 292 | C2×C5 | — | 2 | 2 | no | — |
| 293 | C2×C5 | — | 16 | 1 | no | — |
| 294 | C2×C5 | C2×#459 | 1 | 1 | yes | characteristic |
| 295 | C2×C5 | #3417 | 1 | 2 | yes | — |
| 296 | C23 | — | 4 | 1 | no | — |
| 297 | C23 | C2×#686 | 1 | 1 | yes | characteristic |
| 298 | Q8 | — | 4 | 2 | no | — |
| 299 | Q8 | — | 4 | 1 | no | — |
| 300 | Q8 | — | 4 | 1 | no | — |
| 301 | Q8 | — | 4 | 2 | no | — |
| 302 | Q8 | — | 8 | 2 | no | — |
| 303 | Q8 | — | 10 | 4 | no | — |
| 304 | Q8 | — | 20 | 2 | no | — |
| 305 | Q8 | — | 20 | 4 | no | — |
| 306 | D4 | — | 4 | 1 | no | — |
| 307 | D4 | — | 4 | 1 | no | — |
| 308 | D4 | — | 8 | 2 | no | — |
| 309 | C2×C4 | — | 4 | 2 | no | — |
| 310 | C2×C4 | — | 4 | 1 | no | — |
| 311 | C2×C4 | — | 8 | 2 | no | — |
| 312 | C2×C4 | — | 8 | 2 | no | — |
| 313 | C2×C4 | — | 20 | 2 | no | — |
| 314 | C2×C4 | — | 20 | 4 | no | — |
| 315 | C2×C4 | — | 20 | 2 | no | — |
| 316 | C2×C4 | C2×#686 | 1 | 4 | yes | — |
| 317 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 318 | C2×C4 | #4078 | 1 | 2 | yes | — |
| 319 | C2×C4 | #4084 | 1 | 2 | yes | — |
| 320 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 321 | C8 | — | 2 | 2 | no | — |
| 322 | C8 | #4084 | 1 | 2 | yes | — |
| 323 | C8 | C2×D4×D5 | 1 | 1 | yes | characteristic |
| 324 | C8 | C2×#711 | 1 | 1 | yes | characteristic |
| 325 | C5 | #33588 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 326 | C22 | — | 2 | 2 | no | — |
| 327 | C22 | — | 8 | 1 | no | — |
| 328 | C22 | — | 8 | 2 | no | — |
| 329 | C22 | #65361 | 1 | 2 | yes | — |
| 330 | C22 | C22×#686 | 1 | 1 | yes | characteristic |
| 331 | C4 | — | 2 | 4 | no | — |
| 332 | C4 | — | 8 | 1 | no | — |
| 333 | C4 | — | 8 | 2 | no | — |
| 334 | C4 | — | 20 | 4 | no | — |
| 335 | C4 | — | 40 | 2 | no | — |
| 336 | C4 | #65361 | 1 | 2 | yes | — |
| 337 | C4 | C2×#4078 | 1 | 1 | yes | characteristic |
| 338 | C4 | C2×#4084 | 1 | 1 | yes | characteristic |
| 339 | C2 | — | 2 | 2 | no | — |
| 340 | C2 | — | 16 | 1 | no | — |
| 341 | C2 | C2×#65361 | 1 | 1 | yes | characteristic |
| 342 | C2 | #10615748 | 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