Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
384 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 65 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 | #10611290 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C5×#1687 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10602087 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10602085 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10598252 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10597958 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10597957 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#4055 | — | 2 | 1 | no | — |
| 9 | #65886 | — | 2 | 1 | no | — |
| 10 | C5×#493 | C22 | 1 | 1 | yes | characteristic |
| 11 | C5×#459 | — | 2 | 1 | no | — |
| 12 | C5×#436 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#3959 | — | 2 | 1 | no | — |
| 14 | C2×#3959 | — | 2 | 1 | no | — |
| 15 | C2×#3958 | — | 2 | 1 | no | — |
| 16 | #65423 | — | 2 | 1 | no | — |
| 17 | #65134 | — | 2 | 1 | no | — |
| 18 | #65028 | — | 2 | 1 | no | — |
| 19 | #65023 | — | 2 | 1 | no | — |
| 20 | C4×Dic20 | C22 | 1 | 1 | yes | characteristic |
| 21 | #64985 | C22 | 1 | 1 | yes | characteristic |
| 22 | C5×#334 | C22 | 1 | 1 | yes | characteristic |
| 23 | C5×#331 | — | 2 | 1 | no | — |
| 24 | #64782 | — | 2 | 1 | no | — |
| 25 | C4⋊#672 | C22 | 1 | 1 | yes | characteristic |
| 26 | #64681 | C22 | 1 | 1 | yes | characteristic |
| 27 | #12325 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C2×#709 | — | 2 | 1 | no | — |
| 29 | C22×Dic10 | — | 2 | 1 | no | — |
| 30 | C2×C5×D8 | — | 2 | 1 | no | — |
| 31 | C2×C5×D8 | — | 2 | 1 | no | — |
| 32 | C2×C5×D8 | — | 4 | 1 | no | — |
| 33 | C2×C5×M16 | — | 2 | 1 | no | — |
| 34 | C5×#127 | — | 2 | 1 | no | — |
| 35 | C4×C5×D4 | D4 | 1 | 1 | yes | characteristic |
| 36 | #4085 | — | 4 | 1 | no | — |
| 37 | #4076 | — | 2 | 1 | no | — |
| 38 | #4070 | — | 2 | 1 | no | — |
| 39 | #4055 | — | 4 | 2 | no | — |
| 40 | #4055 | — | 4 | 2 | no | — |
| 41 | C2×Dic20 | — | 2 | 1 | no | — |
| 42 | C2×Dic20 | D4 | 1 | 1 | yes | characteristic |
| 43 | C2×Dic20 | D4 | 1 | 1 | yes | characteristic |
| 44 | C2×D40 | D4 | 1 | 1 | yes | characteristic |
| 45 | C2×#677 | — | 2 | 1 | no | — |
| 46 | C2×#677 | — | 2 | 1 | no | — |
| 47 | #4042 | — | 4 | 1 | no | — |
| 48 | #4031 | — | 4 | 1 | no | — |
| 49 | #4027 | — | 4 | 1 | no | — |
| 50 | #4021 | — | 2 | 1 | no | — |
| 51 | C4×Dic10 | — | 2 | 1 | no | — |
| 52 | C5×C4⋊C8 | — | 2 | 1 | no | — |
| 53 | C5×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 54 | C5×#102 | — | 4 | 1 | no | — |
| 55 | C5×C22⋊C8 | — | 2 | 1 | no | — |
| 56 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 57 | #3959 | — | 4 | 2 | no | — |
| 58 | #3959 | — | 4 | 2 | no | — |
| 59 | #3958 | — | 4 | 2 | no | — |
| 60 | #3953 | — | 2 | 1 | no | — |
| 61 | #3950 | D4 | 1 | 1 | yes | characteristic |
| 62 | #3948 | — | 2 | 1 | no | — |
| 63 | #3948 | — | 2 | 1 | no | — |
| 64 | C2×#672 | — | 2 | 1 | no | — |
| 65 | C2×#672 | — | 2 | 1 | no | — |
| 66 | #3277 | — | 5 | 1 | no | — |
| 67 | #2250 | — | 5 | 1 | no | — |
| 68 | #2207 | — | 5 | 1 | no | — |
| 69 | #1687 | D5 | 1 | 1 | yes | characteristic, complemented |
| 70 | #1360 | — | 5 | 1 | no | — |
| 71 | #1334 | — | 5 | 1 | no | — |
| 72 | #1330 | — | 5 | 1 | no | — |
| 73 | C2×C5×D4 | — | 4 | 1 | no | — |
| 74 | C2×C5×D4 | — | 4 | 1 | no | — |
| 75 | C2×C5×D4 | — | 4 | 1 | no | — |
| 76 | C2×C5×D4 | D8 | 1 | 1 | yes | characteristic |
| 77 | C22×C4×C5 | — | 2 | 1 | no | — |
| 78 | C2×C22⋊D5 | — | 4 | 1 | no | — |
| 79 | C2×C22⋊D5 | — | 4 | 1 | no | — |
| 80 | C22×Dic5 | — | 4 | 1 | no | — |
| 81 | #709 | — | 4 | 2 | no | — |
| 82 | #709 | — | 8 | 1 | no | — |
| 83 | C2×D20 | — | 2 | 1 | no | — |
| 84 | C2×C4×D5 | — | 4 | 1 | no | — |
| 85 | C2×Dic10 | — | 2 | 1 | no | — |
| 86 | C2×Dic10 | — | 2 | 1 | no | — |
| 87 | C2×Dic10 | — | 4 | 2 | no | — |
| 88 | C2×Dic10 | — | 4 | 1 | no | — |
| 89 | C2×Dic10 | — | 8 | 1 | no | — |
| 90 | C5×D8 | — | 4 | 1 | no | — |
| 91 | C5×D8 | — | 4 | 1 | no | — |
| 92 | C5×D8 | — | 4 | 1 | no | — |
| 93 | C5×D8 | — | 4 | 1 | no | — |
| 94 | C5×D8 | — | 8 | 2 | no | — |
| 95 | C5×M16 | — | 4 | 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 | D8 | 1 | 1 | yes | characteristic |
| 102 | C5×C22⋊C4 | — | 2 | 1 | no | — |
| 103 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 105 | #685 | — | 4 | 1 | no | — |
| 106 | #685 | — | 4 | 1 | no | — |
| 107 | #685 | — | 4 | 1 | no | — |
| 108 | C4⋊Dic5 | — | 2 | 1 | no | — |
| 109 | #683 | — | 4 | 1 | no | — |
| 110 | #683 | — | 4 | 1 | no | — |
| 111 | C4×Dic5 | — | 4 | 1 | no | — |
| 112 | Dic20 | — | 2 | 1 | no | — |
| 113 | Dic20 | — | 4 | 1 | no | — |
| 114 | Dic20 | — | 4 | 2 | no | — |
| 115 | Dic20 | D8 | 1 | 2 | yes | complemented |
| 116 | D40 | — | 4 | 1 | no | — |
| 117 | #677 | — | 4 | 2 | no | — |
| 118 | #677 | — | 4 | 1 | no | — |
| 119 | #677 | — | 4 | 1 | no | — |
| 120 | #672 | — | 2 | 2 | no | — |
| 121 | #672 | — | 4 | 1 | no | — |
| 122 | C2×#137 | — | 10 | 1 | no | — |
| 123 | #548 | — | 10 | 1 | no | — |
| 124 | C2×SD32 | — | 10 | 1 | no | — |
| 125 | C2×SD32 | — | 10 | 1 | no | — |
| 126 | C2×D16 | — | 10 | 1 | no | — |
| 127 | #495 | — | 5 | 1 | no | — |
| 128 | #493 | C2×D5 | 1 | 1 | yes | characteristic |
| 129 | #468 | — | 10 | 1 | no | — |
| 130 | #464 | — | 10 | 1 | no | — |
| 131 | #459 | — | 2 | 1 | no | — |
| 132 | #451 | — | 10 | 1 | no | — |
| 133 | #449 | — | 10 | 1 | no | — |
| 134 | C4×Q16 | — | 5 | 1 | no | — |
| 135 | #436 | C2×D5 | 1 | 1 | yes | characteristic |
| 136 | C4⋊C16 | — | 5 | 1 | no | — |
| 137 | #357 | — | 10 | 1 | no | — |
| 138 | #346 | — | 5 | 1 | no | — |
| 139 | #334 | C2×D5 | 1 | 1 | yes | characteristic |
| 140 | #331 | — | 2 | 1 | no | — |
| 141 | C23×C5 | — | 2 | 1 | no | — |
| 142 | C23×C5 | — | 8 | 1 | no | — |
| 143 | C22×D5 | — | 4 | 1 | no | — |
| 144 | C5×D4 | — | 4 | 1 | no | — |
| 145 | C5×D4 | — | 8 | 1 | no | — |
| 146 | C5×D4 | — | 8 | 1 | no | — |
| 147 | C5×D4 | — | 8 | 1 | no | — |
| 148 | C5×D4 | — | 8 | 1 | no | — |
| 149 | C5×D4 | — | 16 | 1 | no | — |
| 150 | C2×C4×C5 | — | 2 | 1 | no | — |
| 151 | C2×C4×C5 | — | 4 | 1 | no | — |
| 152 | C2×C4×C5 | — | 4 | 1 | no | — |
| 153 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 155 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 156 | C22⋊D5 | — | 8 | 2 | no | — |
| 157 | C22⋊D5 | — | 8 | 2 | no | — |
| 158 | C2×Dic5 | — | 4 | 1 | no | — |
| 159 | C2×Dic5 | — | 4 | 1 | no | — |
| 160 | C2×Dic5 | — | 4 | 1 | no | — |
| 161 | C2×Dic5 | — | 8 | 2 | no | — |
| 162 | D20 | — | 4 | 1 | no | — |
| 163 | D20 | — | 4 | 1 | no | — |
| 164 | C4×D5 | — | 8 | 2 | no | — |
| 165 | Dic10 | — | 2 | 2 | no | — |
| 166 | Dic10 | — | 4 | 1 | no | — |
| 167 | Dic10 | — | 4 | 1 | no | — |
| 168 | Dic10 | — | 4 | 1 | no | — |
| 169 | Dic10 | — | 8 | 2 | no | — |
| 170 | C5×C8 | — | 2 | 1 | no | — |
| 171 | C5×C8 | — | 4 | 1 | no | — |
| 172 | C5×C8 | — | 8 | 1 | no | — |
| 173 | C5×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 174 | C5×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 175 | C2×#41 | — | 10 | 1 | no | — |
| 176 | C22×Q8 | — | 10 | 1 | no | — |
| 177 | #137 | — | 20 | 2 | no | — |
| 178 | #137 | — | 20 | 2 | no | — |
| 179 | C2×Q16 | — | 5 | 1 | no | — |
| 180 | C2×Q16 | — | 5 | 1 | no | — |
| 181 | C2×Q16 | — | 10 | 1 | no | — |
| 182 | C2×SD16 | — | 10 | 1 | no | — |
| 183 | C2×SD16 | — | 10 | 1 | no | — |
| 184 | C2×D8 | — | 2 | 1 | no | — |
| 185 | C2×D8 | — | 2 | 1 | no | — |
| 186 | C2×D8 | — | 4 | 1 | no | — |
| 187 | C2×D8 | — | 5 | 1 | no | — |
| 188 | C2×M16 | — | 2 | 1 | no | — |
| 189 | #127 | — | 2 | 1 | no | — |
| 190 | #124 | — | 10 | 1 | no | — |
| 191 | #124 | — | 20 | 1 | no | — |
| 192 | C22⋊Q8 | — | 10 | 1 | no | — |
| 193 | C22⋊Q8 | — | 20 | 1 | no | — |
| 194 | #121 | — | 10 | 1 | no | — |
| 195 | #121 | — | 20 | 1 | no | — |
| 196 | C4×Q8 | — | 10 | 1 | no | — |
| 197 | C4×D4 | — | 20 | 1 | no | — |
| 198 | C4×D4 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 199 | SD32 | — | 20 | 2 | no | — |
| 200 | SD32 | — | 20 | 2 | no | — |
| 201 | D16 | — | 20 | 2 | no | — |
| 202 | C2×C16 | — | 10 | 1 | no | — |
| 203 | C2×C16 | — | 10 | 1 | no | — |
| 204 | #107 | — | 5 | 1 | no | — |
| 205 | C4⋊C8 | — | 2 | 1 | no | — |
| 206 | C4⋊C8 | C22⋊D5 | 1 | 1 | yes | characteristic |
| 207 | #103 | — | 10 | 1 | no | — |
| 208 | #103 | — | 10 | 1 | no | — |
| 209 | #102 | — | 4 | 1 | no | — |
| 210 | #102 | — | 10 | 1 | no | — |
| 211 | C22⋊C8 | — | 2 | 1 | no | — |
| 212 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 213 | C22×C5 | — | 4 | 1 | no | — |
| 214 | C22×C5 | — | 4 | 1 | no | — |
| 215 | C22×C5 | — | 4 | 1 | no | — |
| 216 | C22×C5 | — | 16 | 1 | no | — |
| 217 | C22×C5 | — | 16 | 1 | no | — |
| 218 | C22×C5 | — | 16 | 1 | no | — |
| 219 | C22×C5 | #447 | 1 | 1 | yes | characteristic |
| 220 | C2×D5 | — | 8 | 1 | no | — |
| 221 | C2×D5 | — | 8 | 1 | no | — |
| 222 | C2×D5 | — | 8 | 1 | no | — |
| 223 | C4×C5 | — | 2 | 1 | no | — |
| 224 | C4×C5 | — | 2 | 1 | no | — |
| 225 | C4×C5 | — | 4 | 1 | no | — |
| 226 | C4×C5 | #447 | 1 | 1 | yes | characteristic |
| 227 | C4×C5 | #453 | 1 | 1 | yes | characteristic |
| 228 | Dic5 | — | 4 | 2 | no | — |
| 229 | Dic5 | — | 8 | 1 | no | — |
| 230 | Dic5 | — | 8 | 1 | no | — |
| 231 | #41 | — | 20 | 2 | no | — |
| 232 | #41 | — | 40 | 1 | no | — |
| 233 | C2×Q8 | — | 10 | 1 | no | — |
| 234 | C2×Q8 | — | 10 | 1 | no | — |
| 235 | C2×Q8 | — | 20 | 1 | no | — |
| 236 | C2×Q8 | — | 20 | 2 | no | — |
| 237 | C2×Q8 | — | 40 | 1 | no | — |
| 238 | C2×D4 | — | 4 | 1 | no | — |
| 239 | C2×D4 | — | 4 | 1 | no | — |
| 240 | C2×D4 | — | 4 | 1 | no | — |
| 241 | C2×D4 | — | 10 | 1 | no | — |
| 242 | C2×D4 | — | 20 | 1 | no | — |
| 243 | C2×D4 | — | 20 | 1 | no | — |
| 244 | C2×D4 | #686 | 1 | 1 | yes | characteristic |
| 245 | C22×C4 | — | 2 | 1 | no | — |
| 246 | C22×C4 | — | 20 | 1 | no | — |
| 247 | C22×C4 | — | 20 | 1 | no | — |
| 248 | Q16 | — | 5 | 2 | no | — |
| 249 | Q16 | — | 10 | 1 | no | — |
| 250 | Q16 | — | 20 | 1 | no | — |
| 251 | Q16 | — | 20 | 2 | no | — |
| 252 | SD16 | — | 20 | 2 | no | — |
| 253 | SD16 | — | 20 | 1 | no | — |
| 254 | SD16 | — | 20 | 1 | no | — |
| 255 | D8 | — | 4 | 1 | no | — |
| 256 | D8 | — | 4 | 1 | no | — |
| 257 | D8 | — | 4 | 1 | no | — |
| 258 | D8 | — | 4 | 1 | no | — |
| 259 | D8 | — | 8 | 2 | no | — |
| 260 | D8 | — | 20 | 1 | no | — |
| 261 | M16 | — | 4 | 1 | no | — |
| 262 | M16 | — | 4 | 1 | no | — |
| 263 | C2×C8 | — | 2 | 1 | no | — |
| 264 | C2×C8 | — | 4 | 1 | no | — |
| 265 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 266 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 267 | C4⋊C4 | — | 10 | 1 | no | — |
| 268 | C4⋊C4 | — | 20 | 1 | no | — |
| 269 | C4⋊C4 | — | 20 | 1 | no | — |
| 270 | C4⋊C4 | #686 | 1 | 1 | yes | characteristic |
| 271 | C22⋊C4 | — | 2 | 1 | no | — |
| 272 | C22⋊C4 | — | 20 | 1 | no | — |
| 273 | C22⋊C4 | — | 20 | 1 | no | — |
| 274 | C22⋊C4 | — | 20 | 1 | no | — |
| 275 | C22⋊C4 | — | 20 | 1 | no | — |
| 276 | C42 | — | 20 | 1 | no | — |
| 277 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 278 | C16 | — | 10 | 2 | no | — |
| 279 | C16 | — | 20 | 1 | no | — |
| 280 | C2×C5 | — | 8 | 1 | no | — |
| 281 | C2×C5 | — | 32 | 1 | no | — |
| 282 | C2×C5 | #1619 | 1 | 1 | yes | characteristic |
| 283 | C2×C5 | #2190 | 1 | 1 | yes | characteristic |
| 284 | C2×C5 | #2193 | 1 | 1 | yes | characteristic |
| 285 | D5 | — | 16 | 1 | no | — |
| 286 | C23 | — | 2 | 1 | no | — |
| 287 | C23 | — | 8 | 1 | no | — |
| 288 | C23 | — | 20 | 1 | no | — |
| 289 | Q8 | — | 10 | 2 | no | — |
| 290 | Q8 | — | 20 | 1 | no | — |
| 291 | Q8 | — | 20 | 1 | no | — |
| 292 | Q8 | — | 20 | 1 | no | — |
| 293 | Q8 | — | 40 | 2 | no | — |
| 294 | D4 | — | 4 | 1 | no | — |
| 295 | D4 | — | 8 | 1 | no | — |
| 296 | D4 | — | 8 | 1 | no | — |
| 297 | D4 | — | 8 | 1 | no | — |
| 298 | D4 | — | 8 | 1 | no | — |
| 299 | D4 | — | 16 | 1 | no | — |
| 300 | D4 | — | 20 | 1 | no | — |
| 301 | D4 | — | 20 | 1 | no | — |
| 302 | D4 | — | 40 | 2 | no | — |
| 303 | D4 | — | 40 | 2 | no | — |
| 304 | C2×C4 | — | 2 | 1 | no | — |
| 305 | C2×C4 | — | 4 | 1 | no | — |
| 306 | C2×C4 | — | 4 | 1 | no | — |
| 307 | C2×C4 | — | 20 | 1 | no | — |
| 308 | C2×C4 | — | 20 | 1 | no | — |
| 309 | C2×C4 | — | 20 | 1 | no | — |
| 310 | C2×C4 | — | 40 | 2 | no | — |
| 311 | C2×C4 | — | 40 | 2 | no | — |
| 312 | C2×C4 | C2×#686 | 1 | 1 | yes | characteristic |
| 313 | C2×C4 | #4083 | 1 | 1 | yes | characteristic |
| 314 | C2×C4 | #4095 | 1 | 1 | yes | characteristic |
| 315 | C8 | — | 2 | 1 | no | — |
| 316 | C8 | — | 4 | 1 | no | — |
| 317 | C8 | — | 8 | 1 | no | — |
| 318 | C8 | D5×D8 | 1 | 1 | yes | characteristic |
| 319 | C8 | #4057 | 1 | 1 | yes | characteristic |
| 320 | C5 | #12325 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 321 | C22 | — | 4 | 1 | no | — |
| 322 | C22 | — | 4 | 1 | no | — |
| 323 | C22 | — | 4 | 1 | no | — |
| 324 | C22 | — | 16 | 1 | no | — |
| 325 | C22 | — | 16 | 1 | no | — |
| 326 | C22 | — | 16 | 1 | no | — |
| 327 | C22 | — | 40 | 1 | no | — |
| 328 | C22 | — | 40 | 1 | no | — |
| 329 | C22 | — | 40 | 1 | no | — |
| 330 | C22 | #65325 | 1 | 1 | yes | characteristic |
| 331 | C4 | — | 2 | 1 | no | — |
| 332 | C4 | — | 2 | 1 | no | — |
| 333 | C4 | — | 4 | 1 | no | — |
| 334 | C4 | — | 20 | 2 | no | — |
| 335 | C4 | — | 40 | 1 | no | — |
| 336 | C4 | — | 40 | 1 | no | — |
| 337 | C4 | #65445 | 1 | 1 | yes | characteristic |
| 338 | C4 | #65487 | 1 | 1 | yes | characteristic |
| 339 | C2 | — | 8 | 1 | no | — |
| 340 | C2 | — | 32 | 1 | no | — |
| 341 | C2 | — | 80 | 1 | no | — |
| 342 | C2 | #10601808 | 1 | 1 | yes | characteristic |
| 343 | C2 | #10601882 | 1 | 1 | yes | characteristic |
| 344 | C2 | #10601888 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10611290 | C2 | semidirect |
| 2. C5×#1687 | C2 | semidirect |
| 3. #10602087 | C2 | semidirect |
| 4. #10602085 | C2 | semidirect |
| 5. #10598252 | C2 | semidirect |
| 6. #10597958 | C2 | semidirect |
| 7. #10597957 | C2 | semidirect |
| 69. #1687 | D5 | semidirect |
| 115. Dic20 | D8 | semidirect |
| 320. C5 | #12325 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10611290 | 2 |
| C5×#1687 | 2 |
| #10602087 | 2 |
| #10602085 | 2 |
| #10598252 | 2 |
| #10597958 | 2 |
| #10597957 | 2 |
| #12325 | 5 |