← #12254656
Subgroups — #12254656
Order 1280.
578 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#65470 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10610901 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C5×#2141 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10602854 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10602238 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | Dic5×M32 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #10601447 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #10600011 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #10600008 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | C22×C5×Q16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#4090 | — | 2 | 1 | no | — |
| 12 | C22×#689 | — | 2 | 1 | no | — |
| 13 | C22×Dic20 | C22 | 1 | 1 | yes | characteristic |
| 14 | #65851 | — | 2 | 1 | no | — |
| 15 | C2×C5×M32 | C22 | 1 | 1 | yes | characteristic |
| 16 | C5×#427 | C22 | 1 | 1 | yes | characteristic |
| 17 | #65470 | — | 2 | 1 | no | — |
| 18 | #65470 | — | 2 | 1 | no | — |
| 19 | #65470 | C22 | 1 | 2 | yes | — |
| 20 | #65470 | C22 | 1 | 2 | yes | — |
| 21 | #65395 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C8×Dic5 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×#3944 | C22 | 1 | 1 | yes | characteristic |
| 24 | #65287 | — | 2 | 1 | no | — |
| 25 | #65167 | C22 | 1 | 2 | yes | — |
| 26 | #65163 | C22 | 1 | 2 | yes | — |
| 27 | #65147 | — | 2 | 1 | no | — |
| 28 | #65144 | — | 2 | 1 | no | — |
| 29 | C5×#358 | C22 | 1 | 4 | yes | — |
| 30 | #64784 | C22 | 1 | 4 | yes | — |
| 31 | #64723 | C22 | 1 | 4 | yes | — |
| 32 | #64722 | C22 | 1 | 2 | yes | — |
| 33 | C16×Dic5 | C22 | 1 | 2 | yes | — |
| 34 | #64712 | C22 | 1 | 4 | yes | — |
| 35 | #33705 | — | 5 | 1 | no | maximal, Sylow |
| 36 | C22×C5×Q8 | — | 2 | 1 | no | — |
| 37 | C22×Dic10 | — | 2 | 1 | no | — |
| 38 | C2×C5×Q16 | — | 2 | 1 | no | — |
| 39 | C2×C5×Q16 | — | 2 | 2 | no | — |
| 40 | C2×C5×Q16 | — | 2 | 2 | no | — |
| 41 | C2×C5×Q16 | C23 | 1 | 2 | yes | — |
| 42 | C22×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C5×#117 | — | 2 | 1 | no | — |
| 44 | #4090 | — | 2 | 2 | no | — |
| 45 | #4090 | — | 2 | 2 | no | — |
| 46 | #4090 | — | 4 | 2 | no | — |
| 47 | C2×#689 | — | 2 | 2 | no | — |
| 48 | C2×#689 | — | 2 | 2 | no | — |
| 49 | C2×#689 | — | 4 | 4 | no | — |
| 50 | #4072 | — | 2 | 1 | no | — |
| 51 | C2×#683 | — | 4 | 2 | no | — |
| 52 | C2×C4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 53 | C22×#168 | D4 | 1 | 1 | yes | characteristic |
| 54 | C2×Dic20 | — | 2 | 2 | no | — |
| 55 | C2×Dic20 | — | 2 | 2 | no | — |
| 56 | C2×Dic20 | — | 2 | 1 | no | — |
| 57 | C2×Dic20 | C23 | 1 | 2 | yes | — |
| 58 | #4036 | — | 2 | 2 | no | — |
| 59 | #4034 | — | 2 | 2 | no | — |
| 60 | #4024 | — | 4 | 2 | no | — |
| 61 | C4×Dic10 | — | 4 | 2 | no | — |
| 62 | C5×M32 | — | 2 | 2 | no | — |
| 63 | C2×C5×C16 | C23 | 1 | 2 | yes | — |
| 64 | C5×#107 | C23 | 1 | 2 | yes | — |
| 65 | C5×#106 | — | 2 | 1 | no | — |
| 66 | #3950 | C23 | 1 | 2 | yes | — |
| 67 | #3949 | — | 2 | 1 | no | — |
| 68 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 70 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 71 | C8×Dic5 | C23 | 1 | 1 | yes | characteristic |
| 72 | #3944 | — | 2 | 2 | no | — |
| 73 | C2×#672 | C23 | 1 | 2 | yes | — |
| 74 | #3940 | — | 2 | 2 | no | — |
| 75 | #3939 | — | 2 | 2 | no | — |
| 76 | #3160 | — | 5 | 1 | no | — |
| 77 | C2×#494 | — | 5 | 1 | no | — |
| 78 | #2245 | — | 5 | 4 | no | — |
| 79 | #2241 | — | 5 | 4 | no | — |
| 80 | #2141 | — | 5 | 1 | no | — |
| 81 | #2141 | — | 5 | 1 | no | — |
| 82 | #2141 | — | 5 | 1 | no | — |
| 83 | #2141 | D5 | 1 | 1 | yes | characteristic |
| 84 | C4×M32 | — | 5 | 1 | no | — |
| 85 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 86 | C2×C5×Q8 | — | 4 | 1 | no | — |
| 87 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 88 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 89 | C22×C4×C5 | — | 4 | 1 | no | — |
| 90 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 91 | C22×Dic5 | — | 4 | 1 | no | — |
| 92 | C22×Dic5 | D8 | 1 | 2 | yes | — |
| 93 | C2×Dic10 | — | 2 | 2 | no | — |
| 94 | C2×Dic10 | — | 4 | 2 | no | — |
| 95 | C2×Dic10 | — | 4 | 2 | no | — |
| 96 | C2×Dic10 | — | 4 | 1 | no | — |
| 97 | C2×Dic10 | — | 4 | 2 | no | — |
| 98 | C5×Q16 | — | 2 | 2 | no | — |
| 99 | C5×Q16 | — | 2 | 2 | no | — |
| 100 | C5×Q16 | — | 4 | 2 | no | — |
| 101 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 103 | C2×C5×C8 | #41 | 1 | 2 | yes | — |
| 104 | C2×C5×C8 | C24 | 1 | 1 | yes | characteristic |
| 105 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 106 | C5×C22⋊C4 | — | 4 | 1 | no | — |
| 107 | C42×C5 | — | 4 | 1 | no | — |
| 108 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 109 | #689 | — | 4 | 4 | no | — |
| 110 | #689 | — | 4 | 4 | no | — |
| 111 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 112 | C4⋊Dic5 | — | 4 | 2 | no | — |
| 113 | #683 | — | 4 | 2 | no | — |
| 114 | #683 | — | 4 | 2 | no | — |
| 115 | #683 | — | 4 | 4 | no | — |
| 116 | #683 | — | 4 | 4 | no | — |
| 117 | C4×Dic5 | — | 4 | 1 | no | — |
| 118 | C4×Dic5 | D8 | 1 | 2 | yes | — |
| 119 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 120 | C4×Dic5 | C2×D4 | 1 | 1 | yes | characteristic |
| 121 | C2×#168 | SD16 | 1 | 4 | yes | — |
| 122 | C2×#168 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C2×#168 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | Dic20 | — | 2 | 2 | no | — |
| 125 | Dic20 | — | 2 | 2 | no | — |
| 126 | Dic20 | — | 4 | 2 | no | — |
| 127 | C5×C16 | — | 2 | 2 | no | — |
| 128 | #672 | — | 2 | 2 | no | — |
| 129 | C22×Q16 | — | 5 | 1 | no | — |
| 130 | C22×Q16 | — | 10 | 1 | no | — |
| 131 | C22×Q16 | C2×D5 | 1 | 1 | yes | characteristic |
| 132 | #533 | — | 10 | 1 | no | — |
| 133 | C2×C4⋊Q8 | — | 10 | 1 | no | — |
| 134 | C2×M32 | — | 5 | 1 | no | — |
| 135 | C2×M32 | C2×D5 | 1 | 1 | yes | characteristic |
| 136 | #500 | — | 5 | 2 | no | — |
| 137 | #499 | — | 5 | 2 | no | — |
| 138 | #499 | — | 10 | 1 | no | — |
| 139 | #498 | — | 10 | 1 | no | — |
| 140 | #494 | — | 5 | 2 | no | — |
| 141 | #494 | — | 5 | 2 | no | — |
| 142 | #494 | — | 10 | 1 | no | — |
| 143 | #494 | — | 10 | 1 | no | — |
| 144 | #427 | — | 5 | 1 | no | — |
| 145 | #427 | — | 10 | 1 | no | — |
| 146 | #427 | C2×D5 | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C8 | — | 5 | 1 | no | — |
| 148 | #358 | — | 5 | 4 | no | — |
| 149 | #358 | — | 5 | 4 | no | — |
| 150 | #358 | — | 5 | 4 | no | — |
| 151 | #358 | C2×D5 | 1 | 4 | yes | — |
| 152 | #346 | — | 5 | 2 | no | — |
| 153 | C4×C16 | — | 5 | 2 | no | — |
| 154 | C23×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 155 | C5×Q8 | — | 4 | 2 | no | — |
| 156 | C5×Q8 | — | 4 | 2 | no | — |
| 157 | C5×Q8 | — | 8 | 2 | no | — |
| 158 | C2×C4×C5 | — | 4 | 2 | no | — |
| 159 | C2×C4×C5 | — | 4 | 2 | no | — |
| 160 | C2×C4×C5 | — | 8 | 2 | no | — |
| 161 | C2×C4×C5 | #124 | 1 | 2 | yes | — |
| 162 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 163 | C2×C4×C5 | C2×SD16 | 1 | 2 | yes | — |
| 164 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 165 | C2×Dic5 | — | 2 | 4 | no | — |
| 166 | C2×Dic5 | — | 4 | 2 | no | — |
| 167 | C2×Dic5 | — | 4 | 2 | no | — |
| 168 | C2×Dic5 | — | 8 | 2 | no | — |
| 169 | C2×Dic5 | C2×D8 | 1 | 2 | yes | — |
| 170 | C2×Dic5 | C2×D8 | 1 | 2 | yes | — |
| 171 | Dic10 | — | 4 | 2 | no | — |
| 172 | Dic10 | — | 4 | 2 | no | — |
| 173 | Dic10 | — | 8 | 2 | no | — |
| 174 | Dic10 | — | 8 | 4 | no | — |
| 175 | C5×C8 | #124 | 1 | 2 | yes | — |
| 176 | C5×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 177 | C5×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 178 | #168 | C2×SD16 | 1 | 2 | yes | — |
| 179 | #168 | C2×SD16 | 1 | 2 | yes | — |
| 180 | C22×Q8 | — | 2 | 1 | no | — |
| 181 | C22×Q8 | — | 10 | 1 | no | — |
| 182 | C2×Q16 | — | 2 | 2 | no | — |
| 183 | C2×Q16 | — | 2 | 1 | no | — |
| 184 | C2×Q16 | — | 2 | 2 | no | — |
| 185 | C2×Q16 | — | 5 | 2 | no | — |
| 186 | C2×Q16 | — | 10 | 2 | no | — |
| 187 | C2×Q16 | — | 10 | 2 | no | — |
| 188 | C2×Q16 | — | 10 | 2 | no | — |
| 189 | C2×Q16 | — | 10 | 1 | no | — |
| 190 | C2×Q16 | — | 10 | 2 | no | — |
| 191 | C2×Q16 | — | 20 | 4 | no | — |
| 192 | C2×Q16 | C22×D5 | 1 | 2 | yes | — |
| 193 | C22×C8 | — | 5 | 1 | no | — |
| 194 | C22×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 195 | C4⋊Q8 | — | 10 | 2 | no | — |
| 196 | C4⋊Q8 | — | 10 | 2 | no | — |
| 197 | C4⋊Q8 | — | 10 | 2 | no | — |
| 198 | C4⋊Q8 | — | 20 | 2 | no | — |
| 199 | #125 | — | 10 | 2 | no | — |
| 200 | C22⋊Q8 | — | 20 | 2 | no | — |
| 201 | C4×Q8 | — | 20 | 2 | no | — |
| 202 | #117 | — | 2 | 1 | no | — |
| 203 | #117 | — | 10 | 1 | no | — |
| 204 | C2×C4⋊C4 | — | 20 | 2 | no | — |
| 205 | C2×C42 | — | 5 | 1 | no | — |
| 206 | M32 | — | 2 | 2 | no | — |
| 207 | M32 | — | 10 | 2 | no | — |
| 208 | C2×C16 | — | 5 | 2 | no | — |
| 209 | C2×C16 | C22×D5 | 1 | 2 | yes | — |
| 210 | #107 | — | 5 | 2 | no | — |
| 211 | #107 | — | 10 | 2 | no | — |
| 212 | #107 | C22×D5 | 1 | 2 | yes | — |
| 213 | #106 | — | 2 | 1 | no | — |
| 214 | #106 | — | 10 | 2 | no | — |
| 215 | #106 | — | 10 | 1 | no | — |
| 216 | C4×C8 | — | 5 | 1 | no | — |
| 217 | C4×C8 | — | 5 | 1 | no | — |
| 218 | C4×C8 | — | 5 | 1 | no | — |
| 219 | C4×C8 | — | 5 | 1 | no | — |
| 220 | C22×C5 | — | 2 | 2 | no | — |
| 221 | C22×C5 | #486 | 1 | 2 | yes | — |
| 222 | C22×C5 | C22×D8 | 1 | 1 | yes | characteristic |
| 223 | C4×C5 | — | 8 | 2 | no | — |
| 224 | C4×C5 | — | 8 | 2 | no | — |
| 225 | C4×C5 | #486 | 1 | 2 | yes | — |
| 226 | C4×C5 | C2×#124 | 1 | 1 | yes | characteristic |
| 227 | C4×C5 | C22×SD16 | 1 | 1 | yes | characteristic |
| 228 | Dic5 | — | 2 | 2 | no | — |
| 229 | Dic5 | — | 8 | 2 | no | — |
| 230 | Dic5 | — | 8 | 2 | no | — |
| 231 | Dic5 | #510 | 1 | 4 | yes | — |
| 232 | C2×Q8 | — | 2 | 2 | no | — |
| 233 | C2×Q8 | — | 4 | 1 | no | — |
| 234 | C2×Q8 | — | 4 | 2 | no | — |
| 235 | C2×Q8 | — | 4 | 2 | no | — |
| 236 | C2×Q8 | — | 10 | 2 | no | — |
| 237 | C2×Q8 | — | 20 | 2 | no | — |
| 238 | C2×Q8 | — | 20 | 1 | no | — |
| 239 | C2×Q8 | — | 20 | 2 | no | — |
| 240 | C2×Q8 | — | 20 | 2 | no | — |
| 241 | C22×C4 | — | 4 | 1 | no | — |
| 242 | C22×C4 | — | 5 | 2 | no | — |
| 243 | C22×C4 | — | 20 | 1 | no | — |
| 244 | C22×C4 | D4×D5 | 1 | 1 | yes | characteristic |
| 245 | Q16 | — | 2 | 2 | no | — |
| 246 | Q16 | — | 2 | 2 | no | — |
| 247 | Q16 | — | 4 | 2 | no | — |
| 248 | Q16 | — | 10 | 2 | no | — |
| 249 | Q16 | — | 10 | 2 | no | — |
| 250 | Q16 | — | 20 | 4 | no | — |
| 251 | Q16 | — | 20 | 4 | no | — |
| 252 | Q16 | — | 20 | 2 | no | — |
| 253 | C2×C8 | — | 5 | 1 | no | — |
| 254 | C2×C8 | — | 5 | 1 | no | — |
| 255 | C2×C8 | — | 5 | 4 | no | — |
| 256 | C2×C8 | #709 | 1 | 2 | yes | — |
| 257 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | #711 | 1 | 2 | yes | — |
| 259 | C2×C8 | C23×D5 | 1 | 1 | yes | characteristic |
| 260 | C4⋊C4 | — | 2 | 2 | no | — |
| 261 | C4⋊C4 | — | 10 | 2 | no | — |
| 262 | C4⋊C4 | — | 20 | 4 | no | — |
| 263 | C4⋊C4 | — | 20 | 4 | no | — |
| 264 | C4⋊C4 | — | 20 | 2 | no | — |
| 265 | C4⋊C4 | — | 20 | 2 | no | — |
| 266 | C4⋊C4 | — | 20 | 2 | no | — |
| 267 | C22⋊C4 | — | 4 | 1 | no | — |
| 268 | C22⋊C4 | — | 20 | 1 | no | — |
| 269 | C42 | — | 4 | 1 | no | — |
| 270 | C42 | — | 5 | 1 | no | — |
| 271 | C42 | — | 5 | 1 | no | — |
| 272 | C42 | — | 5 | 2 | no | — |
| 273 | C42 | — | 20 | 1 | no | — |
| 274 | C16 | — | 2 | 2 | no | — |
| 275 | C16 | — | 10 | 2 | no | — |
| 276 | C2×C5 | — | 2 | 2 | no | — |
| 277 | C2×C5 | C2×#486 | 1 | 1 | yes | characteristic |
| 278 | C2×C5 | C2×#510 | 1 | 2 | yes | — |
| 279 | C23 | D5×D8 | 1 | 1 | yes | characteristic |
| 280 | Q8 | — | 4 | 2 | no | — |
| 281 | Q8 | — | 4 | 2 | no | — |
| 282 | Q8 | — | 8 | 2 | no | — |
| 283 | Q8 | — | 20 | 2 | no | — |
| 284 | Q8 | — | 20 | 2 | no | — |
| 285 | Q8 | — | 40 | 4 | no | — |
| 286 | Q8 | — | 40 | 2 | no | — |
| 287 | C2×C4 | — | 4 | 2 | no | — |
| 288 | C2×C4 | — | 4 | 2 | no | — |
| 289 | C2×C4 | — | 5 | 2 | no | — |
| 290 | C2×C4 | — | 5 | 2 | no | — |
| 291 | C2×C4 | — | 8 | 2 | no | — |
| 292 | C2×C4 | — | 10 | 4 | no | — |
| 293 | C2×C4 | — | 20 | 2 | no | — |
| 294 | C2×C4 | — | 20 | 2 | no | — |
| 295 | C2×C4 | — | 40 | 2 | no | — |
| 296 | C2×C4 | #4031 | 1 | 2 | yes | — |
| 297 | C2×C4 | D5×D8 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | D5×SD16 | 1 | 2 | yes | — |
| 299 | C2×C4 | C2×D4×D5 | 1 | 1 | yes | characteristic |
| 300 | C8 | — | 5 | 2 | no | — |
| 301 | C8 | — | 5 | 2 | no | — |
| 302 | C8 | #4031 | 1 | 2 | yes | — |
| 303 | C8 | C2×#709 | 1 | 1 | yes | characteristic |
| 304 | C8 | C2×#711 | 1 | 1 | yes | characteristic |
| 305 | C5 | #33705 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 306 | C22 | — | 2 | 2 | no | — |
| 307 | C22 | #65049 | 1 | 2 | yes | — |
| 308 | C22 | C2×D5×D8 | 1 | 1 | yes | characteristic |
| 309 | C4 | — | 5 | 4 | no | — |
| 310 | C4 | — | 8 | 2 | no | — |
| 311 | C4 | — | 8 | 2 | no | — |
| 312 | C4 | — | 10 | 2 | no | — |
| 313 | C4 | — | 40 | 2 | no | — |
| 314 | C4 | — | 40 | 2 | no | — |
| 315 | C4 | #65049 | 1 | 2 | yes | — |
| 316 | C4 | C2×#4031 | 1 | 1 | yes | characteristic |
| 317 | C4 | C2×D5×SD16 | 1 | 1 | yes | characteristic |
| 318 | C2 | — | 2 | 2 | no | — |
| 319 | C2 | C2×#65049 | 1 | 1 | yes | characteristic |
| 320 | C2 | D5×#510 | 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