Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
344 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 55 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 | #10610829 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#2053 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10602906 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10602905 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #10598210 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10597973 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10597972 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C5×#575 | C22 | 1 | 1 | yes | characteristic |
| 9 | #65868 | — | 2 | 1 | no | — |
| 10 | C5×#424 | C22 | 1 | 1 | yes | characteristic |
| 11 | C5×#416 | — | 2 | 1 | no | — |
| 12 | C5×#410 | — | 2 | 1 | no | — |
| 13 | #65484 | — | 2 | 1 | no | — |
| 14 | #65482 | — | 2 | 1 | no | — |
| 15 | #65426 | C22 | 1 | 1 | yes | characteristic |
| 16 | #65424 | C22 | 1 | 1 | yes | characteristic |
| 17 | #65403 | — | 2 | 1 | no | — |
| 18 | #65140 | — | 2 | 1 | no | — |
| 19 | #65134 | — | 2 | 1 | no | — |
| 20 | #65133 | — | 2 | 1 | no | — |
| 21 | #65131 | — | 2 | 1 | no | — |
| 22 | C5×H(3,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 23 | #64786 | — | 2 | 1 | no | — |
| 24 | #64786 | — | 2 | 1 | no | — |
| 25 | #64773 | C22 | 1 | 1 | yes | characteristic |
| 26 | #64773 | C22 | 1 | 1 | yes | characteristic |
| 27 | #12172 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C2×C5×#41 | — | 2 | 1 | no | — |
| 29 | C5×#137 | — | 4 | 1 | no | — |
| 30 | C5×AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 31 | C5×#135 | — | 2 | 1 | no | — |
| 32 | C5×#135 | — | 2 | 1 | no | — |
| 33 | C2×C5×M16 | C23 | 1 | 1 | yes | characteristic |
| 34 | C22×C5×C8 | D4 | 1 | 1 | yes | characteristic |
| 35 | C5×#117 | — | 2 | 1 | no | — |
| 36 | C5×#117 | D4 | 1 | 1 | yes | characteristic |
| 37 | #4076 | — | 2 | 1 | no | — |
| 38 | #4070 | — | 2 | 1 | no | — |
| 39 | C2×Dic20 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×D40 | D4 | 1 | 1 | yes | characteristic |
| 41 | C2×#677 | — | 2 | 1 | no | — |
| 42 | #4044 | — | 4 | 1 | no | — |
| 43 | #4042 | — | 4 | 1 | no | — |
| 44 | #4031 | — | 4 | 1 | no | — |
| 45 | #4029 | — | 4 | 1 | no | — |
| 46 | #4021 | — | 2 | 1 | no | — |
| 47 | C4×D20 | — | 2 | 1 | no | — |
| 48 | #4015 | — | 2 | 1 | no | — |
| 49 | C4×Dic10 | — | 2 | 1 | no | — |
| 50 | C5×C4⋊C8 | — | 2 | 1 | no | — |
| 51 | C5×C4⋊C8 | — | 2 | 1 | no | — |
| 52 | C5×#103 | — | 4 | 1 | no | — |
| 53 | C5×#102 | — | 4 | 1 | no | — |
| 54 | C5×#99 | — | 4 | 1 | no | — |
| 55 | C5×#99 | — | 4 | 1 | no | — |
| 56 | #3961 | — | 4 | 1 | no | — |
| 57 | #3960 | — | 4 | 1 | no | — |
| 58 | #3959 | — | 4 | 1 | no | — |
| 59 | #3958 | — | 4 | 1 | no | — |
| 60 | #3953 | — | 2 | 1 | no | — |
| 61 | #3953 | — | 2 | 1 | no | — |
| 62 | #3949 | — | 2 | 1 | no | — |
| 63 | #3949 | D4 | 1 | 1 | yes | characteristic |
| 64 | #3949 | D4 | 1 | 1 | yes | characteristic |
| 65 | #3948 | — | 2 | 1 | no | — |
| 66 | #3948 | — | 2 | 1 | no | — |
| 67 | #3944 | — | 2 | 1 | no | — |
| 68 | #3944 | — | 2 | 1 | no | — |
| 69 | #3263 | — | 5 | 1 | no | — |
| 70 | #2191 | — | 5 | 1 | no | — |
| 71 | #2190 | — | 5 | 1 | no | — |
| 72 | #2053 | D5 | 1 | 1 | yes | characteristic, complemented |
| 73 | #1340 | — | 5 | 1 | no | — |
| 74 | #1339 | — | 5 | 1 | no | — |
| 75 | #1328 | — | 5 | 1 | no | — |
| 76 | C5×#41 | — | 4 | 1 | no | — |
| 77 | C5×#41 | — | 4 | 1 | no | — |
| 78 | C5×#41 | — | 4 | 1 | no | — |
| 79 | C5×#41 | — | 4 | 1 | no | — |
| 80 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 81 | C2×C5×D4 | — | 2 | 1 | no | — |
| 82 | C2×C5×D4 | — | 4 | 1 | no | — |
| 83 | C22×C4×C5 | — | 4 | 1 | no | — |
| 84 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C22⋊D5 | — | 4 | 1 | no | — |
| 86 | C2×D20 | — | 2 | 1 | no | — |
| 87 | C2×C4×D5 | — | 4 | 1 | no | — |
| 88 | C2×Dic10 | — | 2 | 1 | no | — |
| 89 | C2×Dic10 | — | 4 | 1 | no | — |
| 90 | C5×Q16 | — | 2 | 1 | no | — |
| 91 | C5×Q16 | — | 2 | 1 | no | — |
| 92 | C5×SD16 | — | 4 | 1 | no | — |
| 93 | C5×SD16 | — | 4 | 1 | no | — |
| 94 | C5×D8 | — | 2 | 1 | no | — |
| 95 | C5×D8 | — | 2 | 1 | no | — |
| 96 | C5×M16 | — | 4 | 1 | no | — |
| 97 | C2×C5×C8 | — | 2 | 1 | no | — |
| 98 | C2×C5×C8 | — | 4 | 1 | no | — |
| 99 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 102 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 103 | C5×C22⋊C4 | — | 4 | 1 | no | — |
| 104 | C5×C22⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 105 | C5×C22⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 106 | C42×C5 | — | 2 | 1 | no | — |
| 107 | C42×C5 | — | 4 | 1 | no | — |
| 108 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 109 | #685 | — | 4 | 1 | no | — |
| 110 | #685 | — | 4 | 1 | no | — |
| 111 | #685 | — | 4 | 1 | no | — |
| 112 | #685 | — | 4 | 1 | no | — |
| 113 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 114 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 115 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 116 | #683 | — | 4 | 1 | no | — |
| 117 | #683 | — | 4 | 1 | no | — |
| 118 | C4×Dic5 | — | 4 | 1 | no | — |
| 119 | Dic20 | — | 4 | 1 | no | — |
| 120 | D40 | — | 4 | 1 | no | — |
| 121 | #677 | — | 8 | 1 | no | — |
| 122 | #672 | — | 4 | 1 | no | — |
| 123 | #672 | — | 4 | 1 | no | — |
| 124 | #575 | C2×D5 | 1 | 1 | yes | characteristic |
| 125 | #542 | — | 10 | 1 | no | — |
| 126 | #510 | — | 10 | 1 | no | — |
| 127 | #509 | — | 10 | 1 | no | — |
| 128 | #476 | — | 10 | 1 | no | — |
| 129 | #475 | — | 10 | 1 | no | — |
| 130 | #470 | — | 5 | 1 | no | — |
| 131 | #469 | — | 5 | 1 | no | — |
| 132 | #465 | — | 10 | 1 | no | — |
| 133 | #464 | — | 10 | 1 | no | — |
| 134 | #463 | — | 10 | 1 | no | — |
| 135 | #424 | C2×D5 | 1 | 1 | yes | characteristic |
| 136 | #416 | — | 2 | 1 | no | — |
| 137 | #410 | — | 2 | 1 | no | — |
| 138 | #360 | — | 10 | 1 | no | — |
| 139 | #360 | — | 10 | 1 | no | — |
| 140 | #349 | — | 5 | 1 | no | — |
| 141 | #349 | — | 5 | 1 | no | — |
| 142 | H(3,ℤ/4) | C2×D5 | 1 | 1 | yes | characteristic |
| 143 | C23×C5 | — | 4 | 1 | no | — |
| 144 | C23×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 145 | C22×D5 | — | 4 | 1 | no | — |
| 146 | C5×Q8 | — | 4 | 1 | no | — |
| 147 | C5×Q8 | — | 4 | 1 | no | — |
| 148 | C5×D4 | — | 4 | 1 | no | — |
| 149 | C5×D4 | — | 4 | 1 | no | — |
| 150 | C5×D4 | — | 8 | 1 | no | — |
| 151 | C5×D4 | — | 8 | 1 | no | — |
| 152 | C2×C4×C5 | — | 2 | 1 | no | — |
| 153 | C2×C4×C5 | — | 2 | 1 | no | — |
| 154 | C2×C4×C5 | — | 2 | 1 | no | — |
| 155 | C2×C4×C5 | — | 2 | 1 | no | — |
| 156 | C2×C4×C5 | — | 4 | 1 | no | — |
| 157 | C2×C4×C5 | — | 8 | 1 | no | — |
| 158 | C2×C4×C5 | — | 8 | 1 | no | — |
| 159 | C2×C4×C5 | — | 8 | 1 | no | — |
| 160 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 161 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 162 | C22⋊D5 | — | 16 | 1 | no | — |
| 163 | C2×Dic5 | — | 4 | 1 | no | — |
| 164 | C2×Dic5 | — | 4 | 1 | no | — |
| 165 | C2×Dic5 | — | 4 | 1 | no | — |
| 166 | D20 | — | 4 | 1 | no | — |
| 167 | D20 | — | 4 | 1 | no | — |
| 168 | C4×D5 | — | 16 | 1 | no | — |
| 169 | Dic10 | — | 4 | 1 | no | — |
| 170 | Dic10 | — | 4 | 1 | no | — |
| 171 | Dic10 | — | 16 | 1 | no | — |
| 172 | C5×C8 | — | 2 | 1 | no | — |
| 173 | C5×C8 | — | 2 | 1 | no | — |
| 174 | C5×C8 | — | 4 | 1 | no | — |
| 175 | C2×#41 | — | 2 | 1 | no | — |
| 176 | #137 | — | 4 | 1 | no | — |
| 177 | AGL(1,ℤ/8) | — | 4 | 1 | no | — |
| 178 | #135 | — | 2 | 1 | no | — |
| 179 | #135 | — | 2 | 1 | no | — |
| 180 | C2×Q16 | — | 5 | 1 | no | — |
| 181 | C2×SD16 | — | 10 | 1 | no | — |
| 182 | C2×D8 | — | 5 | 1 | no | — |
| 183 | C2×M16 | C22×D5 | 1 | 1 | yes | characteristic |
| 184 | C22×C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 185 | C4⋊Q8 | — | 10 | 1 | no | — |
| 186 | #126 | — | 20 | 1 | no | — |
| 187 | #124 | — | 10 | 1 | no | — |
| 188 | #124 | — | 20 | 1 | no | — |
| 189 | #123 | — | 20 | 1 | no | — |
| 190 | C22⋊Q8 | — | 10 | 1 | no | — |
| 191 | C22⋊Q8 | — | 20 | 1 | no | — |
| 192 | #121 | — | 10 | 1 | no | — |
| 193 | C4×Q8 | — | 10 | 1 | no | — |
| 194 | C4×D4 | — | 10 | 1 | no | — |
| 195 | #117 | — | 2 | 1 | no | — |
| 196 | #117 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 197 | Q32 | — | 20 | 1 | no | — |
| 198 | SD32 | — | 20 | 1 | no | — |
| 199 | SD32 | — | 20 | 1 | no | — |
| 200 | D16 | — | 20 | 1 | no | — |
| 201 | M32 | — | 10 | 1 | no | — |
| 202 | M32 | — | 10 | 1 | no | — |
| 203 | #106 | — | 5 | 1 | no | — |
| 204 | #106 | — | 5 | 1 | no | — |
| 205 | #106 | — | 10 | 1 | no | — |
| 206 | C4⋊C8 | — | 2 | 1 | no | — |
| 207 | C4⋊C8 | — | 2 | 1 | no | — |
| 208 | #103 | — | 4 | 1 | no | — |
| 209 | #103 | — | 10 | 1 | no | — |
| 210 | #103 | — | 10 | 1 | no | — |
| 211 | #102 | — | 4 | 1 | no | — |
| 212 | #102 | — | 10 | 1 | no | — |
| 213 | #102 | — | 10 | 1 | no | — |
| 214 | #99 | — | 4 | 1 | no | — |
| 215 | #99 | — | 4 | 1 | no | — |
| 216 | C22×C5 | — | 2 | 1 | no | — |
| 217 | C22×C5 | — | 4 | 1 | no | — |
| 218 | C22×C5 | — | 8 | 1 | no | — |
| 219 | C22×C5 | — | 16 | 1 | no | — |
| 220 | C22×C5 | #447 | 1 | 1 | yes | characteristic |
| 221 | C2×D5 | — | 8 | 1 | no | — |
| 222 | C2×D5 | — | 8 | 1 | no | — |
| 223 | C2×D5 | — | 8 | 1 | no | — |
| 224 | C4×C5 | — | 2 | 1 | no | — |
| 225 | C4×C5 | — | 4 | 1 | no | — |
| 226 | C4×C5 | — | 4 | 1 | no | — |
| 227 | C4×C5 | — | 8 | 1 | no | — |
| 228 | C4×C5 | — | 16 | 1 | no | — |
| 229 | C4×C5 | #454 | 1 | 1 | yes | characteristic |
| 230 | C4×C5 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 231 | Dic5 | — | 8 | 1 | no | — |
| 232 | Dic5 | — | 8 | 1 | no | — |
| 233 | Dic5 | — | 8 | 1 | no | — |
| 234 | #41 | — | 4 | 1 | no | — |
| 235 | #41 | — | 4 | 1 | no | — |
| 236 | #41 | — | 4 | 1 | no | — |
| 237 | #41 | — | 4 | 1 | no | — |
| 238 | C2×Q8 | — | 2 | 1 | no | — |
| 239 | C2×Q8 | — | 10 | 1 | no | — |
| 240 | C2×Q8 | — | 20 | 1 | no | — |
| 241 | C2×D4 | — | 2 | 1 | no | — |
| 242 | C2×D4 | — | 4 | 1 | no | — |
| 243 | C2×D4 | — | 10 | 1 | no | — |
| 244 | C2×D4 | — | 20 | 1 | no | — |
| 245 | C22×C4 | — | 4 | 1 | no | — |
| 246 | C22×C4 | — | 20 | 1 | no | — |
| 247 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 248 | Q16 | — | 2 | 1 | no | — |
| 249 | Q16 | — | 2 | 1 | no | — |
| 250 | Q16 | — | 20 | 1 | no | — |
| 251 | SD16 | — | 4 | 1 | no | — |
| 252 | SD16 | — | 4 | 1 | no | — |
| 253 | SD16 | — | 40 | 1 | no | — |
| 254 | D8 | — | 2 | 1 | no | — |
| 255 | D8 | — | 2 | 1 | no | — |
| 256 | D8 | — | 20 | 1 | no | — |
| 257 | M16 | — | 4 | 1 | no | — |
| 258 | C2×C8 | — | 2 | 1 | no | — |
| 259 | C2×C8 | — | 4 | 1 | no | — |
| 260 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 261 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 262 | C4⋊C4 | — | 2 | 1 | no | — |
| 263 | C4⋊C4 | — | 4 | 1 | no | — |
| 264 | C4⋊C4 | — | 10 | 1 | no | — |
| 265 | C4⋊C4 | — | 10 | 1 | no | — |
| 266 | C4⋊C4 | — | 20 | 1 | no | — |
| 267 | C4⋊C4 | — | 20 | 1 | no | — |
| 268 | C4⋊C4 | — | 20 | 1 | no | — |
| 269 | C22⋊C4 | — | 4 | 1 | no | — |
| 270 | C22⋊C4 | — | 20 | 1 | no | — |
| 271 | C22⋊C4 | — | 20 | 1 | no | — |
| 272 | C22⋊C4 | — | 20 | 1 | no | — |
| 273 | C22⋊C4 | — | 20 | 1 | no | — |
| 274 | C22⋊C4 | — | 20 | 1 | no | — |
| 275 | C22⋊C4 | #686 | 1 | 1 | yes | characteristic |
| 276 | C22⋊C4 | #686 | 1 | 1 | yes | characteristic |
| 277 | C42 | — | 2 | 1 | no | — |
| 278 | C42 | — | 4 | 1 | no | — |
| 279 | C42 | — | 20 | 1 | no | — |
| 280 | C16 | — | 20 | 1 | no | — |
| 281 | C16 | — | 20 | 1 | no | — |
| 282 | C2×C5 | — | 2 | 1 | no | — |
| 283 | C2×C5 | — | 4 | 1 | no | — |
| 284 | C2×C5 | — | 16 | 1 | no | — |
| 285 | C2×C5 | #1598 | 1 | 1 | yes | characteristic |
| 286 | D5 | — | 16 | 1 | no | — |
| 287 | C23 | — | 4 | 1 | no | — |
| 288 | C23 | — | 20 | 1 | no | — |
| 289 | C23 | C2×#686 | 1 | 1 | yes | characteristic |
| 290 | Q8 | — | 4 | 1 | no | — |
| 291 | Q8 | — | 4 | 1 | no | — |
| 292 | Q8 | — | 20 | 1 | no | — |
| 293 | Q8 | — | 20 | 1 | no | — |
| 294 | Q8 | — | 80 | 1 | no | — |
| 295 | D4 | — | 4 | 1 | no | — |
| 296 | D4 | — | 4 | 1 | no | — |
| 297 | D4 | — | 8 | 1 | no | — |
| 298 | D4 | — | 8 | 1 | no | — |
| 299 | D4 | — | 20 | 1 | no | — |
| 300 | D4 | — | 20 | 1 | no | — |
| 301 | D4 | — | 80 | 1 | no | — |
| 302 | C2×C4 | — | 2 | 1 | no | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 1 | no | — |
| 305 | C2×C4 | — | 2 | 1 | no | — |
| 306 | C2×C4 | — | 4 | 1 | no | — |
| 307 | C2×C4 | — | 8 | 1 | no | — |
| 308 | C2×C4 | — | 8 | 1 | no | — |
| 309 | C2×C4 | — | 8 | 1 | no | — |
| 310 | C2×C4 | — | 20 | 1 | no | — |
| 311 | C2×C4 | — | 20 | 1 | no | — |
| 312 | C2×C4 | — | 20 | 1 | no | — |
| 313 | C2×C4 | — | 80 | 1 | no | — |
| 314 | C2×C4 | #4083 | 1 | 1 | yes | characteristic |
| 315 | C2×C4 | #4095 | 1 | 1 | yes | characteristic |
| 316 | C8 | — | 2 | 1 | no | — |
| 317 | C8 | — | 2 | 1 | no | — |
| 318 | C8 | — | 4 | 1 | no | — |
| 319 | C5 | #12172 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 320 | C22 | — | 2 | 1 | no | — |
| 321 | C22 | — | 4 | 1 | no | — |
| 322 | C22 | — | 8 | 1 | no | — |
| 323 | C22 | — | 16 | 1 | no | — |
| 324 | C22 | — | 40 | 1 | no | — |
| 325 | C22 | — | 40 | 1 | no | — |
| 326 | C22 | — | 40 | 1 | no | — |
| 327 | C22 | #65325 | 1 | 1 | yes | characteristic |
| 328 | C4 | — | 2 | 1 | no | — |
| 329 | C4 | — | 4 | 1 | no | — |
| 330 | C4 | — | 4 | 1 | no | — |
| 331 | C4 | — | 8 | 1 | no | — |
| 332 | C4 | — | 16 | 1 | no | — |
| 333 | C4 | — | 40 | 1 | no | — |
| 334 | C4 | — | 40 | 1 | no | — |
| 335 | C4 | — | 40 | 1 | no | — |
| 336 | C4 | #65321 | 1 | 1 | yes | characteristic |
| 337 | C4 | #65351 | 1 | 1 | yes | characteristic |
| 338 | C2 | — | 2 | 1 | no | — |
| 339 | C2 | — | 4 | 1 | no | — |
| 340 | C2 | — | 16 | 1 | no | — |
| 341 | C2 | — | 80 | 1 | no | — |
| 342 | C2 | #10600496 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10610829 | C2 | semidirect |
| 2. C5×#2053 | C2 | semidirect |
| 3. #10602906 | C2 | semidirect |
| 5. #10598210 | C2 | semidirect |
| 6. #10597973 | C2 | semidirect |
| 7. #10597972 | C2 | semidirect |
| 72. #2053 | D5 | semidirect |
| 319. C5 | #12172 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10610829 | 2 |
| C5×#2053 | 2 |
| #10602906 | 2 |
| #10602905 | 2 |
| #10598210 | 2 |
| #10597973 | 2 |
| #10597972 | 2 |
| #12172 | 5 |