Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1998 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #10617730 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10617728 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C5×#2875 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10615537 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #10610306 | C2 | 1 | 12 | yes | maximal, complemented |
| 6 | #10610268 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #10610260 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×#64992 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #10610228 | C2 | 1 | 3 | yes | maximal, complemented |
| 10 | C2×#4150 | C22 | 1 | 2 | yes | — |
| 11 | C2×C5×#131 | C22 | 1 | 2 | yes | — |
| 12 | C4×C5×#41 | C22 | 1 | 1 | yes | characteristic |
| 13 | #66161 | C22 | 1 | 2 | yes | — |
| 14 | Q8×D20 | — | 2 | 2 | no | — |
| 15 | #65903 | — | 2 | 2 | no | — |
| 16 | #65889 | — | 2 | 6 | no | — |
| 17 | #65873 | — | 2 | 6 | no | — |
| 18 | #65860 | C22 | 1 | 6 | yes | — |
| 19 | #65853 | C22 | 1 | 6 | yes | — |
| 20 | #65811 | C22 | 1 | 3 | yes | — |
| 21 | #65804 | C22 | 1 | 3 | yes | — |
| 22 | C2×#4016 | C22 | 1 | 3 | yes | — |
| 23 | C2×#4015 | C22 | 1 | 3 | yes | — |
| 24 | C5×#405 | C22 | 1 | 6 | yes | — |
| 25 | C4×C5×M16 | C22 | 1 | 3 | yes | — |
| 26 | C2×C5×#97 | C22 | 1 | 3 | yes | — |
| 27 | #65422 | C22 | 1 | 12 | yes | — |
| 28 | #65409 | C22 | 1 | 12 | yes | — |
| 29 | #65402 | C22 | 1 | 12 | yes | — |
| 30 | C2×#3948 | C22 | 1 | 12 | yes | — |
| 31 | #65003 | C22 | 1 | 12 | yes | — |
| 32 | #65002 | C22 | 1 | 1 | yes | characteristic |
| 33 | #65002 | C22 | 1 | 3 | yes | — |
| 34 | #64992 | C22 | 1 | 3 | yes | — |
| 35 | #64992 | C22 | 1 | 3 | yes | — |
| 36 | #64992 | C22 | 1 | 6 | yes | — |
| 37 | #64992 | C22 | 1 | 12 | yes | — |
| 38 | #64986 | C22 | 1 | 12 | yes | — |
| 39 | #64970 | C22 | 1 | 12 | yes | — |
| 40 | #60627 | — | 5 | 1 | no | maximal, Sylow |
| 41 | C2×C5×#41 | C23 | 1 | 1 | yes | characteristic |
| 42 | #4150 | — | 2 | 16 | no | — |
| 43 | C2×Q8×D5 | — | 2 | 2 | no | — |
| 44 | C2×#711 | — | 2 | 6 | no | — |
| 45 | C2×#709 | C23 | 1 | 6 | yes | — |
| 46 | C22×Dic10 | C23 | 1 | 6 | yes | — |
| 47 | C5×#131 | — | 2 | 8 | no | — |
| 48 | C2×C5×M16 | C23 | 1 | 6 | yes | — |
| 49 | C22×C5×C8 | C23 | 1 | 6 | yes | — |
| 50 | C4×C5×Q8 | D4 | 1 | 2 | yes | — |
| 51 | C4×C5×D4 | D4 | 1 | 6 | yes | — |
| 52 | C5×#117 | C23 | 1 | 3 | yes | — |
| 53 | C2×C42×C5 | C23 | 1 | 3 | yes | — |
| 54 | Q8×Dic5 | — | 2 | 2 | no | — |
| 55 | D4×Dic5 | — | 2 | 6 | no | — |
| 56 | #4076 | — | 2 | 6 | no | — |
| 57 | #4072 | C23 | 1 | 6 | yes | — |
| 58 | C2×C4⋊Dic5 | — | 2 | 6 | no | — |
| 59 | C2×C4⋊Dic5 | C23 | 1 | 6 | yes | — |
| 60 | #4070 | — | 2 | 6 | no | — |
| 61 | C2×#683 | — | 2 | 6 | no | — |
| 62 | #4043 | — | 2 | 12 | no | — |
| 63 | #4036 | — | 2 | 6 | no | — |
| 64 | #4034 | — | 2 | 6 | no | — |
| 65 | #4032 | — | 2 | 12 | no | — |
| 66 | #4024 | — | 2 | 12 | no | — |
| 67 | #4021 | C23 | 1 | 6 | yes | — |
| 68 | #4020 | C23 | 1 | 1 | yes | characteristic |
| 69 | C4×D20 | — | 2 | 6 | no | — |
| 70 | #4016 | — | 2 | 6 | no | — |
| 71 | #4016 | C23 | 1 | 3 | yes | — |
| 72 | #4016 | C23 | 1 | 6 | yes | — |
| 73 | #4016 | C23 | 1 | 1 | yes | characteristic |
| 74 | #4015 | — | 2 | 6 | no | — |
| 75 | #4015 | C23 | 1 | 3 | yes | — |
| 76 | #4015 | C23 | 1 | 6 | yes | — |
| 77 | #4015 | C23 | 1 | 6 | yes | — |
| 78 | C4×Dic10 | — | 2 | 6 | no | — |
| 79 | C5×#97 | C23 | 1 | 1 | yes | characteristic |
| 80 | C5×#97 | C23 | 1 | 3 | yes | — |
| 81 | C5×#97 | C23 | 1 | 6 | yes | — |
| 82 | C4×C5×C8 | C23 | 1 | 6 | yes | — |
| 83 | #3953 | C23 | 1 | 4 | yes | — |
| 84 | #3953 | C23 | 1 | 12 | yes | — |
| 85 | #3948 | C23 | 1 | 12 | yes | — |
| 86 | #3948 | C23 | 1 | 24 | yes | — |
| 87 | #3948 | C23 | 1 | 12 | yes | — |
| 88 | #3504 | — | 5 | 1 | no | — |
| 89 | #3499 | — | 5 | 1 | no | — |
| 90 | #3142 | — | 5 | 3 | no | — |
| 91 | #3139 | — | 5 | 3 | no | — |
| 92 | #3137 | — | 5 | 12 | no | — |
| 93 | #3135 | — | 5 | 3 | no | — |
| 94 | C2×#491 | — | 5 | 3 | no | — |
| 95 | #2898 | — | 5 | 4 | no | — |
| 96 | #2875 | D5 | 1 | 1 | yes | characteristic, complemented |
| 97 | C5×#41 | #41 | 1 | 8 | yes | — |
| 98 | C2×C5×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C2×C5×D4 | C2×D4 | 1 | 3 | yes | — |
| 100 | C22×C4×C5 | C2×D4 | 1 | 6 | yes | — |
| 101 | C22×C4×C5 | C24 | 1 | 3 | yes | — |
| 102 | C2×C22⋊D5 | — | 2 | 6 | no | — |
| 103 | C22×Dic5 | — | 2 | 6 | no | — |
| 104 | C22×Dic5 | — | 2 | 6 | no | — |
| 105 | Q8×D5 | — | 4 | 8 | no | — |
| 106 | #711 | — | 4 | 24 | no | — |
| 107 | #709 | — | 2 | 24 | no | — |
| 108 | C2×D20 | — | 2 | 2 | no | — |
| 109 | C2×D20 | C24 | 1 | 2 | yes | — |
| 110 | C2×C4×D5 | — | 2 | 6 | no | — |
| 111 | C2×Dic10 | — | 2 | 24 | no | — |
| 112 | C2×Dic10 | — | 2 | 6 | no | — |
| 113 | C2×Dic10 | — | 2 | 6 | no | — |
| 114 | C2×Dic10 | C24 | 1 | 6 | yes | — |
| 115 | C5×M16 | — | 2 | 12 | no | — |
| 116 | C2×C5×C8 | — | 2 | 12 | no | — |
| 117 | C2×C5×C8 | C24 | 1 | 2 | yes | — |
| 118 | C2×C5×C8 | C24 | 1 | 6 | yes | — |
| 119 | C5×C4⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 120 | C5×C22⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 121 | C42×C5 | C2×D4 | 1 | 6 | yes | — |
| 122 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 123 | C42×C5 | C24 | 1 | 3 | yes | — |
| 124 | C22⋊Dic5 | — | 2 | 6 | no | — |
| 125 | #685 | — | 2 | 12 | no | — |
| 126 | C4⋊Dic5 | — | 2 | 6 | no | — |
| 127 | C4⋊Dic5 | — | 2 | 6 | no | — |
| 128 | C4⋊Dic5 | — | 2 | 12 | no | — |
| 129 | C4⋊Dic5 | — | 2 | 6 | no | — |
| 130 | C4⋊Dic5 | C24 | 1 | 2 | yes | — |
| 131 | C4⋊Dic5 | C24 | 1 | 6 | yes | — |
| 132 | #683 | — | 2 | 2 | no | — |
| 133 | #683 | — | 2 | 6 | no | — |
| 134 | #683 | — | 2 | 12 | no | — |
| 135 | C4×Dic5 | — | 2 | 6 | no | — |
| 136 | C2×2−1+4 | — | 5 | 2 | no | — |
| 137 | C2×#131 | C2×D5 | 1 | 2 | yes | — |
| 138 | #557 | — | 10 | 2 | no | — |
| 139 | #554 | — | 10 | 6 | no | — |
| 140 | D4×Q8 | — | 10 | 2 | no | — |
| 141 | #547 | — | 10 | 6 | no | — |
| 142 | #544 | — | 5 | 6 | no | — |
| 143 | #536 | — | 5 | 6 | no | — |
| 144 | #533 | — | 5 | 3 | no | — |
| 145 | #532 | — | 5 | 3 | no | — |
| 146 | C2×C4⋊Q8 | — | 5 | 3 | no | — |
| 147 | C2×#125 | — | 5 | 3 | no | — |
| 148 | #520 | — | 5 | 2 | no | — |
| 149 | C4×#41 | C2×D5 | 1 | 1 | yes | characteristic |
| 150 | #491 | — | 5 | 3 | no | — |
| 151 | #491 | — | 5 | 12 | no | — |
| 152 | #491 | — | 5 | 6 | no | — |
| 153 | #491 | — | 5 | 3 | no | — |
| 154 | #490 | — | 5 | 1 | no | — |
| 155 | #490 | — | 5 | 3 | no | — |
| 156 | #489 | — | 5 | 12 | no | — |
| 157 | #488 | — | 5 | 12 | no | — |
| 158 | #487 | — | 5 | 12 | no | — |
| 159 | #419 | — | 5 | 12 | no | — |
| 160 | #417 | — | 5 | 12 | no | — |
| 161 | #416 | — | 5 | 12 | no | — |
| 162 | C2×#103 | — | 5 | 12 | no | — |
| 163 | #405 | C2×D5 | 1 | 6 | yes | — |
| 164 | C4×M16 | C2×D5 | 1 | 3 | yes | — |
| 165 | C2×#97 | C2×D5 | 1 | 3 | yes | — |
| 166 | C23×C5 | C22×D4 | 1 | 3 | yes | — |
| 167 | C22×D5 | — | 2 | 2 | no | — |
| 168 | C5×Q8 | #124 | 1 | 4 | yes | — |
| 169 | C5×D4 | #124 | 1 | 12 | yes | — |
| 170 | C2×C4×C5 | — | 4 | 6 | no | — |
| 171 | C2×C4×C5 | C22×D4 | 1 | 2 | yes | — |
| 172 | C2×C4×C5 | C22×D4 | 1 | 6 | yes | — |
| 173 | C2×C4×C5 | C22×D4 | 1 | 3 | yes | — |
| 174 | C2×C4×C5 | C2×#41 | 1 | 12 | yes | — |
| 175 | C2×C4×C5 | C25 | 1 | 1 | yes | characteristic |
| 176 | C22⋊D5 | — | 4 | 12 | no | — |
| 177 | C2×Dic5 | — | 2 | 2 | no | — |
| 178 | C2×Dic5 | — | 2 | 6 | no | — |
| 179 | C2×Dic5 | — | 2 | 6 | no | — |
| 180 | C2×Dic5 | — | 4 | 12 | no | — |
| 181 | C2×Dic5 | — | 8 | 6 | no | — |
| 182 | D20 | — | 2 | 4 | no | — |
| 183 | D20 | — | 4 | 4 | no | — |
| 184 | C4×D5 | — | 4 | 12 | no | — |
| 185 | Dic10 | — | 2 | 12 | no | — |
| 186 | Dic10 | — | 4 | 12 | no | — |
| 187 | Dic10 | — | 8 | 6 | no | — |
| 188 | C5×C8 | — | 2 | 2 | no | — |
| 189 | C5×C8 | — | 2 | 6 | no | — |
| 190 | 2−1+4 | — | 10 | 16 | no | — |
| 191 | C2×#41 | — | 5 | 6 | no | — |
| 192 | C2×#41 | — | 10 | 6 | no | — |
| 193 | C2×#41 | C22×D5 | 1 | 1 | yes | characteristic |
| 194 | C22×Q8 | — | 5 | 6 | no | — |
| 195 | C22×Q8 | — | 10 | 2 | no | — |
| 196 | #131 | — | 2 | 8 | no | — |
| 197 | C2×M16 | C22×D5 | 1 | 6 | yes | — |
| 198 | C22×C8 | C22×D5 | 1 | 6 | yes | — |
| 199 | C4⋊Q8 | — | 5 | 6 | no | — |
| 200 | C4⋊Q8 | — | 5 | 3 | no | — |
| 201 | C4⋊Q8 | — | 5 | 6 | no | — |
| 202 | C4⋊Q8 | — | 10 | 6 | no | — |
| 203 | C4⋊Q8 | — | 10 | 6 | no | — |
| 204 | #127 | — | 5 | 1 | no | — |
| 205 | #125 | — | 5 | 3 | no | — |
| 206 | #125 | — | 5 | 6 | no | — |
| 207 | #125 | — | 5 | 1 | no | — |
| 208 | #125 | — | 10 | 6 | no | — |
| 209 | #125 | — | 10 | 6 | no | — |
| 210 | #124 | — | 5 | 6 | no | — |
| 211 | #123 | — | 10 | 12 | no | — |
| 212 | C22⋊Q8 | — | 10 | 6 | no | — |
| 213 | C22⋊Q8 | — | 10 | 12 | no | — |
| 214 | C22⋊Q8 | — | 10 | 12 | no | — |
| 215 | #121 | — | 10 | 6 | no | — |
| 216 | C4×Q8 | — | 10 | 2 | no | — |
| 217 | C4×Q8 | — | 10 | 6 | no | — |
| 218 | C4×Q8 | D20 | 1 | 2 | yes | — |
| 219 | C4×D4 | — | 10 | 6 | no | — |
| 220 | C4×D4 | — | 10 | 6 | no | — |
| 221 | C4×D4 | D20 | 1 | 6 | yes | — |
| 222 | #117 | — | 5 | 6 | no | — |
| 223 | #117 | C22×D5 | 1 | 3 | yes | — |
| 224 | C2×C4⋊C4 | — | 5 | 6 | no | — |
| 225 | C2×C4⋊C4 | — | 10 | 6 | no | — |
| 226 | C2×C4⋊C4 | — | 10 | 6 | no | — |
| 227 | C2×C42 | C22×D5 | 1 | 3 | yes | — |
| 228 | #103 | — | 5 | 12 | no | — |
| 229 | #103 | — | 5 | 24 | no | — |
| 230 | #103 | — | 5 | 12 | no | — |
| 231 | #102 | — | 5 | 4 | no | — |
| 232 | #102 | — | 5 | 12 | no | — |
| 233 | #97 | C22×D5 | 1 | 1 | yes | characteristic |
| 234 | #97 | C22×D5 | 1 | 3 | yes | — |
| 235 | #97 | C22×D5 | 1 | 6 | yes | — |
| 236 | C4×C8 | C22×D5 | 1 | 6 | yes | — |
| 237 | C22×C5 | — | 2 | 6 | no | — |
| 238 | C22×C5 | C2×#124 | 1 | 6 | yes | — |
| 239 | C22×C5 | C23×D4 | 1 | 1 | yes | characteristic |
| 240 | C2×D5 | — | 4 | 4 | no | — |
| 241 | C2×D5 | — | 4 | 2 | no | — |
| 242 | C4×C5 | — | 2 | 6 | no | — |
| 243 | C4×C5 | — | 2 | 2 | no | — |
| 244 | C4×C5 | C2×#124 | 1 | 6 | yes | — |
| 245 | C4×C5 | C22×#41 | 1 | 2 | yes | — |
| 246 | Dic5 | — | 4 | 2 | no | — |
| 247 | Dic5 | — | 4 | 6 | no | — |
| 248 | Dic5 | — | 4 | 6 | no | — |
| 249 | #41 | — | 10 | 24 | no | — |
| 250 | #41 | — | 20 | 24 | no | — |
| 251 | #41 | #709 | 1 | 8 | yes | — |
| 252 | C2×Q8 | — | 5 | 6 | no | — |
| 253 | C2×Q8 | — | 10 | 6 | no | — |
| 254 | C2×Q8 | — | 10 | 6 | no | — |
| 255 | C2×Q8 | — | 10 | 24 | no | — |
| 256 | C2×Q8 | — | 20 | 8 | no | — |
| 257 | C2×Q8 | C2×D20 | 1 | 1 | yes | characteristic |
| 258 | C2×D4 | — | 5 | 2 | no | — |
| 259 | C2×D4 | — | 10 | 6 | no | — |
| 260 | C2×D4 | — | 10 | 2 | no | — |
| 261 | C2×D4 | C2×D20 | 1 | 3 | yes | — |
| 262 | C22×C4 | — | 10 | 6 | no | — |
| 263 | C22×C4 | — | 10 | 6 | no | — |
| 264 | C22×C4 | — | 10 | 6 | no | — |
| 265 | C22×C4 | C2×D20 | 1 | 6 | yes | — |
| 266 | C22×C4 | C23×D5 | 1 | 3 | yes | — |
| 267 | M16 | — | 2 | 12 | no | — |
| 268 | C2×C8 | — | 2 | 12 | no | — |
| 269 | C2×C8 | C23×D5 | 1 | 6 | yes | — |
| 270 | C2×C8 | C23×D5 | 1 | 2 | yes | — |
| 271 | C4⋊C4 | — | 5 | 6 | no | — |
| 272 | C4⋊C4 | — | 5 | 2 | no | — |
| 273 | C4⋊C4 | — | 10 | 6 | no | — |
| 274 | C4⋊C4 | — | 10 | 6 | no | — |
| 275 | C4⋊C4 | — | 10 | 6 | no | — |
| 276 | C4⋊C4 | — | 10 | 12 | no | — |
| 277 | C4⋊C4 | — | 10 | 12 | no | — |
| 278 | C4⋊C4 | — | 10 | 2 | no | — |
| 279 | C4⋊C4 | — | 10 | 6 | no | — |
| 280 | C4⋊C4 | C2×D20 | 1 | 6 | yes | — |
| 281 | C22⋊C4 | — | 10 | 6 | no | — |
| 282 | C22⋊C4 | — | 10 | 12 | no | — |
| 283 | C22⋊C4 | C2×D20 | 1 | 6 | yes | — |
| 284 | C42 | — | 10 | 6 | no | — |
| 285 | C42 | C2×D20 | 1 | 6 | yes | — |
| 286 | C42 | C23×D5 | 1 | 3 | yes | — |
| 287 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 288 | C2×C5 | — | 2 | 6 | no | — |
| 289 | C2×C5 | C22×#124 | 1 | 1 | yes | characteristic |
| 290 | C2×C5 | #3586 | 1 | 2 | yes | — |
| 291 | D5 | — | 8 | 2 | no | — |
| 292 | C23 | — | 10 | 2 | no | — |
| 293 | C23 | C22×D20 | 1 | 3 | yes | — |
| 294 | Q8 | — | 10 | 12 | no | — |
| 295 | Q8 | — | 20 | 12 | no | — |
| 296 | Q8 | — | 40 | 6 | no | — |
| 297 | Q8 | #4021 | 1 | 4 | yes | — |
| 298 | D4 | — | 10 | 4 | no | — |
| 299 | D4 | — | 20 | 4 | no | — |
| 300 | D4 | — | 20 | 12 | no | — |
| 301 | D4 | #4021 | 1 | 12 | yes | — |
| 302 | C2×C4 | — | 4 | 6 | no | — |
| 303 | C2×C4 | — | 10 | 2 | no | — |
| 304 | C2×C4 | — | 10 | 6 | no | — |
| 305 | C2×C4 | — | 10 | 6 | no | — |
| 306 | C2×C4 | — | 20 | 12 | no | — |
| 307 | C2×C4 | — | 20 | 12 | no | — |
| 308 | C2×C4 | — | 40 | 6 | no | — |
| 309 | C2×C4 | C22×D20 | 1 | 2 | yes | — |
| 310 | C2×C4 | C22×D20 | 1 | 6 | yes | — |
| 311 | C2×C4 | C22×D20 | 1 | 3 | yes | — |
| 312 | C2×C4 | C2×#709 | 1 | 12 | yes | — |
| 313 | C2×C4 | C24×D5 | 1 | 1 | yes | characteristic |
| 314 | C8 | — | 2 | 2 | no | — |
| 315 | C8 | — | 2 | 6 | no | — |
| 316 | C5 | #60627 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 317 | C22 | — | 2 | 6 | no | — |
| 318 | C22 | — | 20 | 4 | no | — |
| 319 | C22 | — | 20 | 2 | no | — |
| 320 | C22 | C2×#4021 | 1 | 6 | yes | — |
| 321 | C22 | C23×D20 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 2 | no | — |
| 323 | C4 | — | 2 | 6 | no | — |
| 324 | C4 | — | 20 | 6 | no | — |
| 325 | C4 | — | 20 | 2 | no | — |
| 326 | C4 | — | 20 | 6 | no | — |
| 327 | C4 | C2×#4021 | 1 | 6 | yes | — |
| 328 | C4 | C22×#709 | 1 | 2 | yes | — |
| 329 | C2 | — | 2 | 6 | no | — |
| 330 | C2 | — | 40 | 2 | no | — |
| 331 | C2 | C22×#4021 | 1 | 1 | yes | characteristic |
| 332 | C2 | #10619069 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10617730 | C2 | semidirect |
| 3. C5×#2875 | C2 | semidirect |
| 4. #10615537 | C2 | semidirect |
| 5. #10610306 | C2 | semidirect |
| 6. #10610268 | C2 | semidirect |
| 7. #10610260 | C2 | semidirect |
| 8. C2×#64992 | C2 | semidirect |
| 9. #10610228 | C2 | semidirect |
| 96. #2875 | D5 | semidirect |
| 316. C5 | #60627 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10617730 | 2 |
| #10617728 | 2 |
| C5×#2875 | 2 |
| #10615537 | 2 |
| #10610306 | 2 |
| #10610268 | 2 |
| #10610260 | 2 |
| C2×#64992 | 2 |
| #10610228 | 2 |
| #60627 | 5 |