Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1068 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#66047 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#3151 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10615047 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10602553 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10602077 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #10602063 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10600650 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C2×C5×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 9 | C5×#532 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C5×#127 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#4082 | — | 2 | 2 | no | — |
| 12 | C2×#4081 | — | 2 | 2 | no | — |
| 13 | C2×D4×Dic5 | — | 2 | 2 | no | — |
| 14 | #66047 | — | 2 | 4 | no | — |
| 15 | #66047 | — | 2 | 2 | no | — |
| 16 | #66047 | C22 | 1 | 4 | yes | — |
| 17 | C2×C4×Dic10 | C22 | 1 | 2 | yes | — |
| 18 | C5×#496 | — | 2 | 2 | no | — |
| 19 | C5×#493 | C22 | 1 | 2 | yes | — |
| 20 | C5×#492 | C22 | 1 | 2 | yes | — |
| 21 | C4×C5×M16 | C22 | 1 | 1 | yes | characteristic |
| 22 | #65380 | C22 | 1 | 4 | yes | complemented |
| 23 | #65364 | C22 | 1 | 4 | yes | complemented |
| 24 | #65329 | — | 2 | 2 | no | — |
| 25 | #65322 | — | 2 | 2 | no | — |
| 26 | #65307 | — | 2 | 4 | no | — |
| 27 | #65215 | C22 | 1 | 2 | yes | — |
| 28 | #64764 | C22 | 1 | 4 | yes | — |
| 29 | #64676 | C22 | 1 | 8 | yes | complemented |
| 30 | #18526 | — | 5 | 1 | no | maximal, Sylow |
| 31 | C2×C5×#41 | — | 2 | 1 | no | — |
| 32 | C22×C5×D4 | — | 2 | 2 | no | — |
| 33 | C22×C5×D4 | D4 | 1 | 2 | yes | — |
| 34 | C23×Dic5 | — | 2 | 2 | no | — |
| 35 | C22×Dic10 | D4 | 1 | 2 | yes | — |
| 36 | C5×AGL(1,ℤ/8) | — | 4 | 8 | no | — |
| 37 | C2×C5×SD16 | — | 2 | 4 | no | — |
| 38 | C2×C5×D8 | — | 2 | 4 | no | — |
| 39 | C2×C5×M16 | — | 2 | 1 | no | — |
| 40 | C5×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 41 | C5×#127 | — | 2 | 2 | no | — |
| 42 | C5×#127 | D4 | 1 | 2 | yes | — |
| 43 | C5×#127 | C23 | 1 | 1 | yes | characteristic |
| 44 | C5×#127 | C23 | 1 | 2 | yes | — |
| 45 | C5×#124 | — | 2 | 1 | no | — |
| 46 | C5×#121 | — | 2 | 2 | no | — |
| 47 | C4×C5×D4 | — | 2 | 2 | no | — |
| 48 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C22⋊Dic5 | — | 2 | 4 | no | — |
| 50 | C2×C22⋊Dic5 | — | 2 | 2 | no | — |
| 51 | #4082 | — | 2 | 8 | no | — |
| 52 | #4082 | — | 2 | 8 | no | — |
| 53 | #4081 | — | 2 | 8 | no | — |
| 54 | #4081 | — | 2 | 8 | no | — |
| 55 | D4×Dic5 | — | 2 | 8 | no | — |
| 56 | D4×Dic5 | — | 2 | 8 | no | — |
| 57 | C2×#687 | — | 2 | 4 | no | — |
| 58 | C2×C4⋊Dic5 | D4 | 1 | 2 | yes | — |
| 59 | C2×#683 | — | 2 | 2 | no | — |
| 60 | C2×C4×Dic5 | — | 2 | 2 | no | — |
| 61 | C4×Dic10 | — | 2 | 2 | no | — |
| 62 | C4×Dic10 | — | 2 | 2 | no | — |
| 63 | C4×Dic10 | D4 | 1 | 4 | yes | — |
| 64 | C4×Dic10 | C23 | 1 | 4 | yes | — |
| 65 | C5×#97 | — | 2 | 1 | no | — |
| 66 | C4×C5×C8 | C23 | 1 | 2 | yes | — |
| 67 | #3964 | — | 2 | 4 | no | — |
| 68 | C22⋊#168 | — | 2 | 2 | no | — |
| 69 | #3942 | — | 2 | 4 | no | — |
| 70 | #3939 | — | 2 | 4 | no | — |
| 71 | C4⋊#168 | C23 | 1 | 4 | yes | — |
| 72 | #3935 | — | 2 | 2 | no | — |
| 73 | C2×#559 | — | 5 | 1 | no | — |
| 74 | #3151 | D5 | 1 | 1 | yes | characteristic |
| 75 | #3112 | — | 5 | 2 | no | — |
| 76 | #1630 | — | 5 | 4 | no | — |
| 77 | #1620 | — | 5 | 4 | no | — |
| 78 | #1555 | — | 5 | 1 | no | — |
| 79 | #1491 | — | 5 | 2 | no | — |
| 80 | C24×C5 | — | 2 | 2 | no | — |
| 81 | C5×#41 | — | 4 | 4 | no | — |
| 82 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 83 | C2×C5×D4 | — | 2 | 4 | no | — |
| 84 | C2×C5×D4 | — | 2 | 2 | no | — |
| 85 | C2×C5×D4 | — | 2 | 4 | no | — |
| 86 | C2×C5×D4 | — | 2 | 2 | no | — |
| 87 | C2×C5×D4 | — | 2 | 1 | no | — |
| 88 | C2×C5×D4 | — | 4 | 8 | no | — |
| 89 | C2×C5×D4 | — | 4 | 4 | no | — |
| 90 | C2×C5×D4 | — | 4 | 1 | no | — |
| 91 | C2×C5×D4 | C2×D4 | 1 | 4 | yes | — |
| 92 | C22×C4×C5 | — | 4 | 1 | no | — |
| 93 | C22×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 94 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C22×Dic5 | — | 2 | 2 | no | — |
| 96 | C22×Dic5 | — | 2 | 2 | no | — |
| 97 | C22×Dic5 | — | 2 | 8 | no | — |
| 98 | C22×Dic5 | — | 8 | 4 | no | — |
| 99 | C2×Dic10 | — | 2 | 2 | no | — |
| 100 | C2×Dic10 | — | 4 | 4 | no | — |
| 101 | C2×Dic10 | C2×D4 | 1 | 4 | yes | — |
| 102 | C5×SD16 | — | 4 | 8 | no | — |
| 103 | C5×D8 | — | 4 | 8 | no | — |
| 104 | C5×M16 | — | 2 | 4 | no | — |
| 105 | C2×C5×C8 | — | 2 | 2 | no | — |
| 106 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 107 | C5×C22⋊C4 | — | 4 | 2 | no | — |
| 108 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 109 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 111 | C22⋊Dic5 | — | 2 | 16 | no | — |
| 112 | C22⋊Dic5 | — | 2 | 8 | no | — |
| 113 | #687 | — | 4 | 8 | no | — |
| 114 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 115 | C4⋊Dic5 | C2×D4 | 1 | 4 | yes | — |
| 116 | #683 | — | 2 | 4 | no | — |
| 117 | #683 | — | 2 | 4 | no | — |
| 118 | C4×Dic5 | — | 2 | 4 | no | — |
| 119 | C4×Dic5 | — | 2 | 4 | no | — |
| 120 | C2×#168 | — | 2 | 4 | no | — |
| 121 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 122 | #559 | — | 5 | 4 | no | — |
| 123 | #559 | — | 10 | 4 | no | — |
| 124 | #559 | — | 10 | 2 | no | — |
| 125 | #532 | C2×D5 | 1 | 1 | yes | characteristic |
| 126 | C2×#127 | C2×D5 | 1 | 1 | yes | characteristic |
| 127 | C2×#124 | — | 10 | 2 | no | — |
| 128 | C2×#123 | — | 10 | 2 | no | — |
| 129 | C2×C4×Q8 | — | 5 | 2 | no | — |
| 130 | C2×C4×D4 | — | 10 | 2 | no | — |
| 131 | #496 | — | 2 | 2 | no | — |
| 132 | #493 | C2×D5 | 1 | 2 | yes | — |
| 133 | #492 | C2×D5 | 1 | 2 | yes | — |
| 134 | #480 | — | 10 | 2 | no | — |
| 135 | #477 | — | 5 | 4 | no | — |
| 136 | #460 | — | 5 | 4 | no | — |
| 137 | #448 | — | 10 | 2 | no | — |
| 138 | #440 | — | 10 | 4 | no | — |
| 139 | #432 | — | 5 | 2 | no | — |
| 140 | C4×M16 | C2×D5 | 1 | 1 | yes | characteristic |
| 141 | #331 | — | 5 | 4 | no | — |
| 142 | Q8⋊C8 | — | 5 | 8 | no | — |
| 143 | C23×C5 | — | 2 | 4 | no | — |
| 144 | C23×C5 | — | 4 | 8 | no | — |
| 145 | C23×C5 | — | 4 | 4 | no | — |
| 146 | C23×C5 | — | 4 | 1 | no | — |
| 147 | C23×C5 | #120 | 1 | 1 | yes | characteristic |
| 148 | C5×Q8 | — | 4 | 2 | no | — |
| 149 | C5×D4 | — | 4 | 4 | no | — |
| 150 | C5×D4 | — | 4 | 8 | no | — |
| 151 | C5×D4 | — | 4 | 2 | no | — |
| 152 | C5×D4 | — | 4 | 8 | no | — |
| 153 | C5×D4 | — | 4 | 4 | no | — |
| 154 | C5×D4 | — | 8 | 2 | no | — |
| 155 | C5×D4 | — | 8 | 2 | no | — |
| 156 | C2×C4×C5 | — | 2 | 2 | no | — |
| 157 | C2×C4×C5 | — | 2 | 4 | no | — |
| 158 | C2×C4×C5 | — | 4 | 1 | no | — |
| 159 | C2×C4×C5 | — | 8 | 2 | no | — |
| 160 | C2×C4×C5 | #120 | 1 | 2 | yes | — |
| 161 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 162 | C2×C4×C5 | C22×D4 | 1 | 2 | yes | — |
| 163 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 164 | C2×Dic5 | — | 2 | 4 | no | — |
| 165 | C2×Dic5 | — | 2 | 4 | no | — |
| 166 | C2×Dic5 | — | 4 | 4 | no | — |
| 167 | C2×Dic5 | — | 4 | 4 | no | — |
| 168 | C2×Dic5 | — | 8 | 8 | no | — |
| 169 | Dic10 | — | 2 | 4 | no | — |
| 170 | Dic10 | — | 2 | 4 | no | — |
| 171 | Dic10 | — | 4 | 4 | no | — |
| 172 | C5×C8 | — | 2 | 4 | no | — |
| 173 | #168 | — | 4 | 4 | no | — |
| 174 | C2×#41 | — | 2 | 1 | no | — |
| 175 | C22×Q8 | — | 5 | 2 | no | — |
| 176 | C22×D4 | — | 2 | 2 | no | — |
| 177 | C22×D4 | C22⋊D5 | 1 | 2 | yes | — |
| 178 | C23×C4 | — | 10 | 2 | no | — |
| 179 | AGL(1,ℤ/8) | — | 4 | 8 | no | — |
| 180 | C2×SD16 | — | 2 | 4 | no | — |
| 181 | C2×SD16 | — | 10 | 4 | no | — |
| 182 | C2×D8 | — | 2 | 4 | no | — |
| 183 | C2×M16 | — | 2 | 1 | no | — |
| 184 | C4⋊Q8 | C22×D5 | 1 | 1 | yes | characteristic |
| 185 | #127 | — | 2 | 2 | no | — |
| 186 | #127 | C22⋊D5 | 1 | 2 | yes | — |
| 187 | #127 | C22×D5 | 1 | 1 | yes | characteristic |
| 188 | #127 | C22×D5 | 1 | 2 | yes | — |
| 189 | #124 | — | 2 | 1 | no | — |
| 190 | #124 | — | 10 | 8 | no | — |
| 191 | #124 | — | 10 | 8 | no | — |
| 192 | #123 | — | 10 | 8 | no | — |
| 193 | #123 | — | 10 | 8 | no | — |
| 194 | #121 | — | 2 | 2 | no | — |
| 195 | C4×Q8 | — | 5 | 4 | no | — |
| 196 | C4×Q8 | — | 5 | 4 | no | — |
| 197 | C4×Q8 | — | 10 | 2 | no | — |
| 198 | C4×Q8 | — | 10 | 2 | no | — |
| 199 | C4×D4 | — | 2 | 2 | no | — |
| 200 | C4×D4 | — | 10 | 8 | no | — |
| 201 | C4×D4 | — | 10 | 8 | no | — |
| 202 | C2×C4⋊C4 | — | 5 | 2 | no | — |
| 203 | C2×C4⋊C4 | — | 10 | 2 | no | — |
| 204 | C2×C22⋊C4 | — | 10 | 4 | no | — |
| 205 | C2×C22⋊C4 | — | 10 | 2 | no | — |
| 206 | C2×C42 | — | 10 | 2 | no | — |
| 207 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 208 | #107 | — | 10 | 4 | no | — |
| 209 | C4⋊C8 | — | 5 | 4 | no | — |
| 210 | #103 | — | 10 | 4 | no | — |
| 211 | #102 | — | 10 | 4 | no | — |
| 212 | C22⋊C8 | — | 10 | 2 | no | — |
| 213 | #97 | — | 2 | 1 | no | — |
| 214 | #97 | — | 10 | 2 | no | — |
| 215 | C4×C8 | C22×D5 | 1 | 2 | yes | — |
| 216 | C22×C5 | — | 2 | 2 | no | — |
| 217 | C22×C5 | — | 4 | 8 | no | — |
| 218 | C22×C5 | — | 4 | 4 | no | — |
| 219 | C22×C5 | — | 8 | 8 | no | — |
| 220 | C22×C5 | — | 8 | 2 | no | — |
| 221 | C22×C5 | — | 8 | 1 | no | — |
| 222 | C22×C5 | #453 | 1 | 2 | yes | — |
| 223 | C22×C5 | C2×#120 | 1 | 1 | yes | characteristic |
| 224 | C4×C5 | — | 2 | 2 | no | — |
| 225 | C4×C5 | — | 2 | 1 | no | — |
| 226 | C4×C5 | — | 2 | 1 | no | — |
| 227 | C4×C5 | — | 8 | 1 | no | — |
| 228 | C4×C5 | #575 | 1 | 4 | yes | — |
| 229 | Dic5 | — | 4 | 4 | no | — |
| 230 | Dic5 | — | 4 | 4 | no | — |
| 231 | C24 | — | 2 | 2 | no | — |
| 232 | #41 | — | 4 | 4 | no | — |
| 233 | C2×Q8 | — | 2 | 1 | no | — |
| 234 | C2×Q8 | — | 5 | 4 | no | — |
| 235 | C2×Q8 | — | 10 | 2 | no | — |
| 236 | C2×Q8 | — | 20 | 4 | no | — |
| 237 | C2×D4 | — | 2 | 4 | no | — |
| 238 | C2×D4 | — | 2 | 2 | no | — |
| 239 | C2×D4 | — | 2 | 1 | no | — |
| 240 | C2×D4 | — | 2 | 4 | no | — |
| 241 | C2×D4 | — | 2 | 2 | no | — |
| 242 | C2×D4 | — | 4 | 8 | no | — |
| 243 | C2×D4 | — | 4 | 4 | no | — |
| 244 | C2×D4 | — | 4 | 1 | no | — |
| 245 | C2×D4 | C2×C22⋊D5 | 1 | 4 | yes | — |
| 246 | C22×C4 | — | 4 | 1 | no | — |
| 247 | C22×C4 | — | 10 | 2 | no | — |
| 248 | C22×C4 | — | 10 | 2 | no | — |
| 249 | C22×C4 | — | 10 | 8 | no | — |
| 250 | C22×C4 | — | 40 | 4 | no | — |
| 251 | C22×C4 | D4×D5 | 1 | 2 | yes | — |
| 252 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 253 | SD16 | — | 4 | 8 | no | — |
| 254 | SD16 | — | 20 | 8 | no | — |
| 255 | D8 | — | 4 | 8 | no | — |
| 256 | M16 | — | 2 | 4 | no | — |
| 257 | C2×C8 | — | 2 | 2 | no | — |
| 258 | C2×C8 | — | 10 | 4 | no | — |
| 259 | C4⋊C4 | — | 2 | 2 | no | — |
| 260 | C4⋊C4 | — | 5 | 4 | no | — |
| 261 | C4⋊C4 | — | 10 | 4 | no | — |
| 262 | C4⋊C4 | — | 10 | 2 | no | — |
| 263 | C4⋊C4 | — | 10 | 4 | no | — |
| 264 | C22⋊C4 | — | 4 | 2 | no | — |
| 265 | C22⋊C4 | — | 10 | 16 | no | — |
| 266 | C22⋊C4 | — | 10 | 8 | no | — |
| 267 | C42 | — | 10 | 4 | no | — |
| 268 | C42 | — | 10 | 4 | no | — |
| 269 | C42 | D4×D5 | 1 | 2 | yes | — |
| 270 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 271 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 272 | C2×C5 | — | 2 | 2 | no | — |
| 273 | C2×C5 | — | 8 | 4 | no | — |
| 274 | C2×C5 | — | 16 | 1 | no | — |
| 275 | C2×C5 | #3005 | 1 | 2 | yes | — |
| 276 | C2×C5 | C2×#453 | 1 | 1 | yes | characteristic |
| 277 | C23 | — | 2 | 4 | no | — |
| 278 | C23 | — | 4 | 8 | no | — |
| 279 | C23 | — | 4 | 4 | no | — |
| 280 | C23 | — | 4 | 1 | no | — |
| 281 | C23 | #4083 | 1 | 1 | yes | characteristic |
| 282 | Q8 | — | 4 | 2 | no | — |
| 283 | Q8 | — | 10 | 4 | no | — |
| 284 | Q8 | — | 10 | 4 | no | — |
| 285 | Q8 | — | 20 | 4 | no | — |
| 286 | D4 | — | 4 | 4 | no | — |
| 287 | D4 | — | 4 | 8 | no | — |
| 288 | D4 | — | 4 | 2 | no | — |
| 289 | D4 | — | 4 | 8 | no | — |
| 290 | D4 | — | 4 | 4 | no | — |
| 291 | D4 | — | 8 | 2 | no | — |
| 292 | D4 | — | 8 | 2 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 2 | 4 | no | — |
| 295 | C2×C4 | — | 4 | 1 | no | — |
| 296 | C2×C4 | — | 8 | 2 | no | — |
| 297 | C2×C4 | — | 10 | 4 | no | — |
| 298 | C2×C4 | — | 10 | 4 | no | — |
| 299 | C2×C4 | — | 20 | 4 | no | — |
| 300 | C2×C4 | — | 20 | 4 | no | — |
| 301 | C2×C4 | — | 40 | 8 | no | — |
| 302 | C2×C4 | #4083 | 1 | 2 | yes | — |
| 303 | C2×C4 | #4083 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×D4×D5 | 1 | 2 | yes | — |
| 305 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 306 | C8 | — | 2 | 4 | no | — |
| 307 | C8 | — | 20 | 4 | no | — |
| 308 | C5 | #18526 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 309 | C22 | — | 2 | 2 | no | — |
| 310 | C22 | — | 4 | 4 | no | — |
| 311 | C22 | — | 4 | 8 | no | — |
| 312 | C22 | — | 8 | 8 | no | — |
| 313 | C22 | — | 8 | 1 | no | — |
| 314 | C22 | — | 8 | 2 | no | — |
| 315 | C22 | #65366 | 1 | 2 | yes | — |
| 316 | C22 | C2×#4083 | 1 | 1 | yes | characteristic |
| 317 | C4 | — | 2 | 1 | no | — |
| 318 | C4 | — | 2 | 1 | no | — |
| 319 | C4 | — | 2 | 2 | no | — |
| 320 | C4 | — | 8 | 1 | no | — |
| 321 | C4 | — | 20 | 4 | no | — |
| 322 | C4 | — | 20 | 4 | no | — |
| 323 | C4 | #66107 | 1 | 4 | yes | — |
| 324 | C2 | — | 2 | 2 | no | — |
| 325 | C2 | — | 8 | 4 | no | — |
| 326 | C2 | — | 16 | 1 | no | — |
| 327 | C2 | C2×#65366 | 1 | 1 | yes | characteristic |
| 328 | C2 | #10615676 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#66047 | C2 | semidirect |
| 3. #10615047 | C2 | semidirect |
| 4. #10602553 | C2 | semidirect |
| 5. #10602077 | C2 | semidirect |
| 6. #10602063 | C2 | semidirect |
| 7. #10600650 | C2 | semidirect |
| 22. #65380 | C22 | semidirect |
| 23. #65364 | C22 | semidirect |
| 29. #64676 | C22 | semidirect |
| 308. C5 | #18526 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#66047 | 2 |
| C5×#3151 | 2 |
| #10615047 | 2 |
| #10602553 | 2 |
| #10602077 | 2 |
| #10602063 | 2 |
| #10600650 | 2 |
| #18526 | 5 |