← #12082413
Subgroups — #12082413
Order 1280.
650 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×C5×#500 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10614947 | C2 | 1 | 2 | yes | maximal |
| 3 | #10610189 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10602567 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10602167 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #10602165 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10600522 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C5×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 9 | C2×#4047 | — | 2 | 1 | no | — |
| 10 | #65811 | C22 | 1 | 1 | yes | characteristic |
| 11 | C5×#500 | — | 2 | 1 | no | — |
| 12 | C5×#500 | — | 2 | 1 | no | — |
| 13 | C5×#500 | C22 | 1 | 2 | yes | — |
| 14 | C5×#500 | C22 | 1 | 2 | yes | — |
| 15 | C2×C5×#107 | — | 2 | 1 | no | — |
| 16 | C2×C5×#107 | — | 2 | 1 | no | — |
| 17 | C2×C4×C5×C8 | C22 | 1 | 1 | yes | characteristic |
| 18 | #65399 | — | 2 | 1 | no | — |
| 19 | #65377 | — | 2 | 2 | no | — |
| 20 | #65373 | C22 | 1 | 8 | yes | — |
| 21 | #65368 | — | 2 | 2 | no | — |
| 22 | #65257 | — | 2 | 4 | no | — |
| 23 | #65214 | C22 | 1 | 2 | yes | — |
| 24 | #65213 | C22 | 1 | 1 | yes | characteristic |
| 25 | #64977 | — | 2 | 1 | no | — |
| 26 | #64977 | C22 | 1 | 2 | yes | — |
| 27 | C8×D20 | — | 2 | 1 | no | — |
| 28 | C8×D20 | C22 | 1 | 2 | yes | — |
| 29 | #64766 | C22 | 1 | 8 | yes | — |
| 30 | #64678 | C22 | 1 | 2 | yes | — |
| 31 | #64677 | C22 | 1 | 2 | yes | — |
| 32 | #19193 | — | 5 | 1 | no | maximal, Sylow |
| 33 | C22×C5×Q8 | — | 2 | 2 | no | — |
| 34 | C2×#709 | — | 2 | 1 | no | — |
| 35 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 37 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 38 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 39 | C5×C4⋊Q8 | — | 2 | 2 | no | — |
| 40 | C5×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 41 | C2×C5×C4⋊C4 | — | 2 | 2 | no | — |
| 42 | C2×C5×C4⋊C4 | — | 2 | 2 | no | — |
| 43 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 44 | #4076 | — | 2 | 1 | no | — |
| 45 | #4076 | D4 | 1 | 2 | yes | — |
| 46 | C2×#681 | — | 2 | 1 | no | — |
| 47 | #4047 | — | 4 | 4 | no | — |
| 48 | C2×#676 | — | 2 | 2 | no | — |
| 49 | C2×C8×D5 | — | 2 | 2 | no | — |
| 50 | #4021 | — | 2 | 1 | no | — |
| 51 | #4020 | D4 | 1 | 1 | yes | characteristic |
| 52 | C4×D20 | — | 2 | 1 | no | — |
| 53 | C4×D20 | C23 | 1 | 2 | yes | — |
| 54 | #4015 | D4 | 1 | 1 | yes | characteristic |
| 55 | C5×#107 | — | 2 | 2 | no | — |
| 56 | C5×#107 | — | 2 | 2 | no | — |
| 57 | C5×#107 | — | 2 | 2 | no | — |
| 58 | C5×#107 | — | 2 | 2 | no | — |
| 59 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 60 | C4×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 61 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 62 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 63 | C22⋊#168 | — | 2 | 2 | no | — |
| 64 | #3952 | — | 2 | 2 | no | — |
| 65 | #3952 | — | 2 | 2 | no | — |
| 66 | #3941 | — | 2 | 8 | no | — |
| 67 | #3940 | — | 2 | 8 | no | — |
| 68 | C4⋊#168 | — | 2 | 1 | no | — |
| 69 | C4⋊#168 | C23 | 1 | 4 | yes | — |
| 70 | C4⋊#168 | C23 | 1 | 2 | yes | — |
| 71 | C4×#168 | — | 2 | 2 | no | — |
| 72 | #3227 | — | 5 | 2 | no | — |
| 73 | C2×#500 | D5 | 1 | 1 | yes | characteristic, complemented |
| 74 | #2931 | — | 5 | 1 | no | — |
| 75 | #1695 | — | 5 | 4 | no | — |
| 76 | #1675 | — | 5 | 4 | no | — |
| 77 | #1564 | — | 5 | 1 | no | — |
| 78 | #1548 | — | 5 | 2 | no | — |
| 79 | C2×C5×Q8 | — | 2 | 4 | no | — |
| 80 | C2×C5×Q8 | — | 4 | 2 | no | — |
| 81 | C2×C5×Q8 | — | 8 | 4 | no | — |
| 82 | C22×C4×C5 | — | 4 | 2 | no | — |
| 83 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C22×C4×C5 | #41 | 1 | 1 | yes | characteristic |
| 86 | C2×C22⋊D5 | — | 2 | 2 | no | — |
| 87 | #709 | — | 4 | 2 | no | — |
| 88 | #709 | — | 4 | 2 | no | — |
| 89 | C2×D20 | — | 2 | 1 | no | — |
| 90 | C2×D20 | — | 2 | 1 | no | — |
| 91 | C2×D20 | C2×D4 | 1 | 2 | yes | — |
| 92 | C2×C4×D5 | — | 2 | 2 | no | — |
| 93 | C2×Dic10 | — | 2 | 1 | no | — |
| 94 | C2×C5×C8 | — | 2 | 2 | no | — |
| 95 | C2×C5×C8 | SD16 | 1 | 4 | yes | — |
| 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×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 101 | C5×C4⋊C4 | — | 2 | 4 | no | — |
| 102 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 103 | C5×C4⋊C4 | — | 4 | 2 | no | — |
| 104 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 106 | C42×C5 | #41 | 1 | 1 | yes | characteristic |
| 107 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 108 | #685 | — | 2 | 2 | no | — |
| 109 | #685 | — | 2 | 2 | no | — |
| 110 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 111 | C4⋊Dic5 | C2×D4 | 1 | 2 | yes | — |
| 112 | #681 | — | 4 | 2 | no | — |
| 113 | C2×#168 | — | 2 | 4 | no | — |
| 114 | C2×#168 | — | 2 | 2 | no | — |
| 115 | #676 | — | 4 | 4 | no | — |
| 116 | C8×D5 | — | 4 | 4 | no | — |
| 117 | C2×#131 | — | 10 | 1 | no | — |
| 118 | #532 | — | 5 | 1 | no | — |
| 119 | C2×C4⋊Q8 | C2×D5 | 1 | 2 | yes | — |
| 120 | #500 | — | 2 | 1 | no | — |
| 121 | #500 | — | 2 | 1 | no | — |
| 122 | #500 | C2×D5 | 1 | 2 | yes | — |
| 123 | #500 | C2×D5 | 1 | 2 | yes | — |
| 124 | #498 | — | 10 | 2 | no | — |
| 125 | #486 | — | 10 | 2 | no | — |
| 126 | #481 | — | 10 | 4 | no | — |
| 127 | #476 | — | 5 | 8 | no | — |
| 128 | #436 | — | 5 | 2 | no | — |
| 129 | #436 | — | 10 | 1 | no | — |
| 130 | C8×D4 | — | 5 | 2 | no | — |
| 131 | C8×D4 | — | 10 | 1 | no | — |
| 132 | #431 | — | 5 | 2 | no | — |
| 133 | C2×#107 | — | 2 | 1 | no | — |
| 134 | C2×#107 | — | 2 | 1 | no | — |
| 135 | #423 | — | 5 | 1 | no | — |
| 136 | #408 | — | 10 | 1 | no | — |
| 137 | C2×C4×C8 | C2×D5 | 1 | 1 | yes | characteristic |
| 138 | #335 | — | 5 | 2 | no | — |
| 139 | #334 | — | 5 | 2 | no | — |
| 140 | #333 | — | 5 | 8 | no | — |
| 141 | C23×C5 | #121 | 1 | 1 | yes | characteristic |
| 142 | C22×D5 | — | 2 | 2 | no | — |
| 143 | C5×Q8 | — | 8 | 4 | no | — |
| 144 | C5×Q8 | — | 8 | 4 | no | — |
| 145 | C2×C4×C5 | — | 2 | 2 | no | — |
| 146 | C2×C4×C5 | — | 2 | 2 | no | — |
| 147 | C2×C4×C5 | — | 4 | 4 | no | — |
| 148 | C2×C4×C5 | — | 8 | 4 | no | — |
| 149 | C2×C4×C5 | #121 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×C5 | #121 | 1 | 1 | yes | characteristic |
| 151 | C2×C4×C5 | #121 | 1 | 1 | yes | characteristic |
| 152 | C2×C4×C5 | C2×SD16 | 1 | 2 | yes | — |
| 153 | C2×C4×C5 | #137 | 1 | 2 | yes | — |
| 154 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 155 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 156 | C2×C4×C5 | C2×#41 | 1 | 1 | yes | characteristic |
| 157 | C22⋊D5 | — | 4 | 4 | no | — |
| 158 | C2×Dic5 | — | 2 | 2 | no | — |
| 159 | D20 | — | 2 | 4 | no | — |
| 160 | D20 | — | 4 | 2 | no | — |
| 161 | D20 | — | 4 | 2 | no | — |
| 162 | C4×D5 | — | 4 | 2 | no | — |
| 163 | C4×D5 | — | 4 | 2 | no | — |
| 164 | Dic10 | — | 4 | 2 | no | — |
| 165 | C5×C8 | — | 2 | 1 | no | — |
| 166 | C5×C8 | — | 2 | 1 | no | — |
| 167 | C5×C8 | C2×SD16 | 1 | 2 | yes | — |
| 168 | C5×C8 | C2×SD16 | 1 | 2 | yes | — |
| 169 | #168 | — | 4 | 2 | no | — |
| 170 | #168 | — | 4 | 4 | no | — |
| 171 | C2×#41 | — | 10 | 1 | no | — |
| 172 | C22×Q8 | — | 2 | 2 | no | — |
| 173 | #131 | — | 20 | 4 | no | — |
| 174 | C2×M16 | — | 10 | 1 | no | — |
| 175 | C2×M16 | — | 10 | 2 | no | — |
| 176 | C22×C8 | — | 10 | 2 | no | — |
| 177 | C22×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 178 | C22×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 179 | C4⋊Q8 | — | 2 | 2 | no | — |
| 180 | C4⋊Q8 | — | 2 | 2 | no | — |
| 181 | C4⋊Q8 | — | 2 | 2 | no | — |
| 182 | C4⋊Q8 | — | 5 | 1 | no | — |
| 183 | C4⋊Q8 | C22×D5 | 1 | 4 | yes | — |
| 184 | #127 | — | 5 | 1 | no | — |
| 185 | #124 | — | 10 | 1 | no | — |
| 186 | #121 | — | 5 | 2 | no | — |
| 187 | #121 | — | 10 | 1 | no | — |
| 188 | C4×D4 | — | 5 | 2 | no | — |
| 189 | C4×D4 | — | 10 | 1 | no | — |
| 190 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 191 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 192 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 193 | #107 | — | 2 | 2 | no | — |
| 194 | #107 | — | 2 | 2 | no | — |
| 195 | #107 | — | 2 | 2 | no | — |
| 196 | #107 | — | 2 | 2 | no | — |
| 197 | #106 | — | 10 | 8 | no | — |
| 198 | C4⋊C8 | — | 5 | 2 | no | — |
| 199 | C4⋊C8 | — | 5 | 4 | no | — |
| 200 | C4⋊C8 | — | 10 | 1 | no | — |
| 201 | #102 | — | 10 | 8 | no | — |
| 202 | C22⋊C8 | — | 10 | 2 | no | — |
| 203 | C22⋊C8 | — | 10 | 2 | no | — |
| 204 | C22⋊C8 | — | 10 | 2 | no | — |
| 205 | C4×C8 | — | 10 | 2 | no | — |
| 206 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 207 | C4×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 208 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 209 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 210 | C22×C5 | — | 2 | 1 | no | — |
| 211 | C22×C5 | — | 2 | 1 | no | — |
| 212 | C22×C5 | #461 | 1 | 2 | yes | — |
| 213 | C22×C5 | C2×#121 | 1 | 1 | yes | characteristic |
| 214 | C2×D5 | — | 4 | 2 | no | — |
| 215 | C2×D5 | — | 4 | 2 | no | — |
| 216 | C2×D5 | — | 4 | 2 | no | — |
| 217 | C4×C5 | — | 2 | 1 | no | — |
| 218 | C4×C5 | — | 2 | 1 | no | — |
| 219 | C4×C5 | — | 8 | 4 | no | — |
| 220 | C4×C5 | #461 | 1 | 2 | yes | — |
| 221 | C4×C5 | #472 | 1 | 2 | yes | — |
| 222 | C4×C5 | C22×SD16 | 1 | 1 | yes | characteristic |
| 223 | C4×C5 | C2×#137 | 1 | 1 | yes | characteristic |
| 224 | C4×C5 | #578 | 1 | 2 | yes | — |
| 225 | Dic5 | — | 4 | 2 | no | — |
| 226 | #41 | — | 20 | 2 | no | — |
| 227 | #41 | — | 20 | 2 | no | — |
| 228 | C2×Q8 | — | 2 | 4 | no | — |
| 229 | C2×Q8 | — | 4 | 2 | no | — |
| 230 | C2×Q8 | — | 8 | 4 | no | — |
| 231 | C2×Q8 | — | 10 | 1 | no | — |
| 232 | C2×D4 | — | 5 | 2 | no | — |
| 233 | C2×D4 | — | 10 | 2 | no | — |
| 234 | C2×D4 | — | 10 | 1 | no | — |
| 235 | C2×D4 | — | 10 | 1 | no | — |
| 236 | C22×C4 | — | 4 | 2 | no | — |
| 237 | C22×C4 | — | 10 | 2 | no | — |
| 238 | C22×C4 | D4×D5 | 1 | 1 | yes | characteristic |
| 239 | C22×C4 | #711 | 1 | 1 | yes | characteristic |
| 240 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 241 | M16 | — | 20 | 2 | no | — |
| 242 | M16 | — | 20 | 4 | no | — |
| 243 | C2×C8 | — | 2 | 2 | no | — |
| 244 | C2×C8 | — | 10 | 4 | no | — |
| 245 | C2×C8 | — | 10 | 2 | no | — |
| 246 | C2×C8 | — | 20 | 4 | no | — |
| 247 | C2×C8 | #688 | 1 | 4 | yes | — |
| 248 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 252 | C4⋊C4 | — | 2 | 4 | no | — |
| 253 | C4⋊C4 | — | 2 | 4 | no | — |
| 254 | C4⋊C4 | — | 4 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 2 | no | — |
| 256 | C4⋊C4 | — | 5 | 2 | no | — |
| 257 | C4⋊C4 | — | 10 | 1 | no | — |
| 258 | C22⋊C4 | — | 10 | 2 | no | — |
| 259 | C22⋊C4 | — | 10 | 2 | no | — |
| 260 | C42 | D4×D5 | 1 | 1 | yes | characteristic |
| 261 | C42 | #711 | 1 | 1 | yes | characteristic |
| 262 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 263 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 264 | C2×C5 | — | 2 | 2 | no | — |
| 265 | C2×C5 | C2×#461 | 1 | 1 | yes | characteristic |
| 266 | C2×C5 | #3059 | 1 | 1 | yes | characteristic |
| 267 | C2×C5 | C2×#472 | 1 | 1 | yes | characteristic |
| 268 | D5 | — | 8 | 2 | no | — |
| 269 | C23 | — | 10 | 2 | no | — |
| 270 | C23 | #4084 | 1 | 1 | yes | characteristic |
| 271 | Q8 | — | 8 | 4 | no | — |
| 272 | Q8 | — | 8 | 4 | no | — |
| 273 | Q8 | — | 20 | 2 | no | — |
| 274 | D4 | — | 10 | 4 | no | — |
| 275 | D4 | — | 20 | 4 | no | — |
| 276 | D4 | — | 20 | 2 | no | — |
| 277 | D4 | — | 20 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 2 | 2 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 8 | 4 | no | — |
| 282 | C2×C4 | — | 10 | 2 | no | — |
| 283 | C2×C4 | — | 20 | 2 | no | — |
| 284 | C2×C4 | — | 20 | 2 | no | — |
| 285 | C2×C4 | #4084 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #4084 | 1 | 1 | yes | characteristic |
| 287 | C2×C4 | #4084 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C2×#688 | 1 | 2 | yes | — |
| 289 | C2×C4 | #4088 | 1 | 2 | yes | — |
| 290 | C2×C4 | C2×D4×D5 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C2×#711 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 293 | C8 | — | 2 | 1 | no | — |
| 294 | C8 | — | 2 | 1 | no | — |
| 295 | C8 | — | 20 | 4 | no | — |
| 296 | C8 | — | 20 | 2 | no | — |
| 297 | C8 | C2×#688 | 1 | 2 | yes | — |
| 298 | C8 | C2×#688 | 1 | 2 | yes | — |
| 299 | C5 | #19193 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 300 | C22 | — | 2 | 1 | no | — |
| 301 | C22 | — | 2 | 1 | no | — |
| 302 | C22 | — | 20 | 2 | no | — |
| 303 | C22 | — | 20 | 2 | no | — |
| 304 | C22 | — | 20 | 2 | no | — |
| 305 | C22 | #65372 | 1 | 2 | yes | — |
| 306 | C22 | C2×#4084 | 1 | 1 | yes | characteristic |
| 307 | C4 | — | 2 | 1 | no | — |
| 308 | C4 | — | 2 | 1 | no | — |
| 309 | C4 | — | 8 | 4 | no | — |
| 310 | C4 | — | 20 | 2 | no | — |
| 311 | C4 | #65372 | 1 | 2 | yes | — |
| 312 | C4 | #65449 | 1 | 2 | yes | — |
| 313 | C4 | #66100 | 1 | 2 | yes | — |
| 314 | C4 | C22×#688 | 1 | 1 | yes | characteristic |
| 315 | C4 | C2×#4088 | 1 | 1 | yes | characteristic |
| 316 | C2 | — | 2 | 2 | no | — |
| 317 | C2 | — | 40 | 2 | no | — |
| 318 | C2 | C2×#65372 | 1 | 1 | yes | characteristic |
| 319 | C2 | C2×#65449 | 1 | 1 | yes | characteristic |
| 320 | C2 | #10615618 | 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