Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
636 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 105 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 | C5×#3150 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10615029 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10602593 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10600630 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10598151 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×C5×#135 | — | 2 | 2 | no | — |
| 7 | C5×#532 | C22 | 1 | 2 | yes | — |
| 8 | C2×C4×D20 | C22 | 1 | 1 | yes | characteristic |
| 9 | C5×#497 | — | 2 | 2 | no | — |
| 10 | C5×#496 | — | 2 | 2 | no | — |
| 11 | C2×C5×#97 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#3952 | — | 2 | 2 | no | — |
| 13 | #65376 | — | 2 | 2 | no | — |
| 14 | #65361 | — | 2 | 2 | no | — |
| 15 | #65325 | — | 2 | 2 | no | — |
| 16 | #65322 | — | 2 | 2 | no | — |
| 17 | C4×#686 | — | 2 | 4 | no | — |
| 18 | #65214 | C22 | 1 | 2 | yes | — |
| 19 | C2×C4⋊#168 | C22 | 1 | 1 | yes | characteristic |
| 20 | D4⋊#168 | — | 2 | 4 | no | — |
| 21 | #10032 | — | 5 | 1 | no | maximal, Sylow |
| 22 | C2×C5×#41 | — | 2 | 2 | no | — |
| 23 | C22×D20 | D4 | 1 | 1 | yes | characteristic |
| 24 | C22×C4×D5 | — | 2 | 1 | no | — |
| 25 | C5×#135 | — | 4 | 4 | no | — |
| 26 | C5×#135 | — | 4 | 4 | no | — |
| 27 | C2×C5×Q16 | — | 2 | 2 | no | — |
| 28 | C2×C5×SD16 | — | 2 | 4 | no | — |
| 29 | C2×C5×D8 | — | 2 | 2 | no | — |
| 30 | C22×C5×C8 | D4 | 1 | 2 | yes | — |
| 31 | C5×C4⋊Q8 | D4 | 1 | 2 | yes | — |
| 32 | C5×#127 | D4 | 1 | 2 | yes | — |
| 33 | C5×#124 | — | 2 | 2 | no | — |
| 34 | C5×#121 | — | 2 | 2 | no | — |
| 35 | C5×#121 | — | 2 | 2 | no | — |
| 36 | C4×C5×D4 | — | 2 | 2 | no | — |
| 37 | C4×C5×D4 | — | 2 | 2 | no | — |
| 38 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 39 | C2×#686 | — | 2 | 4 | no | — |
| 40 | C2×#685 | — | 2 | 1 | no | — |
| 41 | C2×C4⋊Dic5 | D4 | 1 | 1 | yes | characteristic |
| 42 | C22×#168 | — | 2 | 1 | no | — |
| 43 | C4×D20 | — | 2 | 2 | no | — |
| 44 | C4×D20 | — | 2 | 2 | no | — |
| 45 | C5×#97 | — | 2 | 2 | no | — |
| 46 | #3964 | — | 2 | 4 | no | — |
| 47 | C22⋊#168 | — | 2 | 2 | no | — |
| 48 | #3952 | — | 4 | 4 | no | — |
| 49 | #3941 | — | 2 | 4 | no | — |
| 50 | #3939 | — | 2 | 4 | no | — |
| 51 | C4⋊#168 | — | 2 | 2 | no | — |
| 52 | C4⋊#168 | — | 2 | 2 | no | — |
| 53 | C4×#168 | D4 | 1 | 4 | yes | — |
| 54 | #3150 | D5 | 1 | 1 | yes | characteristic, complemented |
| 55 | #3100 | — | 5 | 2 | no | — |
| 56 | #1955 | — | 5 | 1 | no | — |
| 57 | #1480 | — | 5 | 2 | no | — |
| 58 | #1275 | — | 5 | 1 | no | — |
| 59 | C23×D5 | — | 2 | 1 | no | — |
| 60 | C5×#41 | — | 4 | 4 | no | — |
| 61 | C5×#41 | — | 4 | 4 | no | — |
| 62 | C2×C5×Q8 | — | 2 | 2 | no | — |
| 63 | C2×C5×D4 | — | 2 | 2 | no | — |
| 64 | C2×C5×D4 | — | 2 | 2 | no | — |
| 65 | C2×C5×D4 | — | 2 | 2 | no | — |
| 66 | C2×C5×D4 | — | 4 | 2 | no | — |
| 67 | C22×C4×C5 | — | 4 | 2 | no | — |
| 68 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 69 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C22×C4×C5 | #41 | 1 | 1 | yes | characteristic |
| 71 | C22×Dic5 | — | 2 | 1 | no | — |
| 72 | C2×D20 | — | 2 | 2 | no | — |
| 73 | C2×D20 | — | 4 | 2 | no | — |
| 74 | C2×C4×D5 | — | 4 | 1 | no | — |
| 75 | C2×C4×D5 | — | 4 | 1 | no | — |
| 76 | C2×C4×D5 | — | 4 | 2 | no | — |
| 77 | C2×C4×D5 | — | 4 | 2 | no | — |
| 78 | C5×Q16 | — | 4 | 4 | no | — |
| 79 | C5×SD16 | — | 4 | 8 | no | — |
| 80 | C5×D8 | — | 4 | 4 | no | — |
| 81 | C2×C5×C8 | — | 2 | 2 | no | — |
| 82 | C2×C5×C8 | — | 2 | 2 | no | — |
| 83 | C2×C5×C8 | D8 | 1 | 4 | yes | complemented |
| 84 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 85 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 86 | C5×C22⋊C4 | — | 4 | 2 | no | — |
| 87 | C5×C22⋊C4 | — | 4 | 2 | no | — |
| 88 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 89 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 90 | #686 | — | 4 | 8 | no | — |
| 91 | #685 | — | 4 | 2 | no | — |
| 92 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 93 | C2×#168 | — | 2 | 4 | no | — |
| 94 | C2×#168 | — | 4 | 1 | no | — |
| 95 | C2×#168 | — | 4 | 2 | no | — |
| 96 | C2×#135 | — | 2 | 2 | no | — |
| 97 | #532 | C2×D5 | 1 | 2 | yes | — |
| 98 | C2×C4×D4 | — | 5 | 1 | no | — |
| 99 | #497 | — | 2 | 2 | no | — |
| 100 | #496 | — | 2 | 2 | no | — |
| 101 | #480 | — | 10 | 2 | no | — |
| 102 | #474 | — | 10 | 2 | no | — |
| 103 | #459 | — | 10 | 2 | no | — |
| 104 | #447 | — | 10 | 2 | no | — |
| 105 | C4×D8 | — | 10 | 4 | no | — |
| 106 | #431 | — | 5 | 2 | no | — |
| 107 | C2×C4⋊C8 | — | 5 | 1 | no | — |
| 108 | C2×C22⋊C8 | — | 10 | 2 | no | — |
| 109 | C2×#97 | C2×D5 | 1 | 1 | yes | characteristic |
| 110 | D4⋊C8 | — | 10 | 4 | no | — |
| 111 | C23×C5 | — | 4 | 2 | no | — |
| 112 | C23×C5 | #121 | 1 | 1 | yes | characteristic |
| 113 | C22×D5 | — | 2 | 2 | no | — |
| 114 | C22×D5 | — | 4 | 1 | no | — |
| 115 | C22×D5 | — | 4 | 2 | no | — |
| 116 | C22×D5 | — | 4 | 1 | no | — |
| 117 | C22×D5 | — | 4 | 2 | no | — |
| 118 | C5×Q8 | — | 4 | 4 | no | — |
| 119 | C5×D4 | — | 4 | 4 | no | — |
| 120 | C5×D4 | — | 4 | 4 | no | — |
| 121 | C5×D4 | — | 4 | 4 | no | — |
| 122 | C5×D4 | — | 8 | 4 | no | — |
| 123 | C2×C4×C5 | — | 4 | 1 | no | — |
| 124 | C2×C4×C5 | — | 4 | 1 | no | — |
| 125 | C2×C4×C5 | — | 4 | 2 | no | — |
| 126 | C2×C4×C5 | — | 8 | 2 | no | — |
| 127 | C2×C4×C5 | — | 8 | 2 | no | — |
| 128 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C5 | #120 | 1 | 2 | yes | — |
| 130 | C2×C4×C5 | #121 | 1 | 2 | yes | — |
| 131 | C2×C4×C5 | #127 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C5 | C2×D8 | 1 | 2 | yes | — |
| 133 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 134 | C2×Dic5 | — | 4 | 1 | no | — |
| 135 | C2×Dic5 | — | 4 | 2 | no | — |
| 136 | D20 | — | 2 | 4 | no | — |
| 137 | D20 | — | 4 | 2 | no | — |
| 138 | C4×D5 | — | 8 | 2 | no | — |
| 139 | C4×D5 | — | 8 | 1 | no | — |
| 140 | C4×D5 | — | 8 | 1 | no | — |
| 141 | C5×C8 | — | 2 | 4 | no | — |
| 142 | #168 | — | 4 | 4 | no | — |
| 143 | #168 | — | 8 | 1 | no | — |
| 144 | C2×#41 | — | 2 | 2 | no | — |
| 145 | C22×D4 | — | 5 | 1 | no | — |
| 146 | C23×C4 | — | 10 | 1 | no | — |
| 147 | #135 | — | 4 | 4 | no | — |
| 148 | #135 | — | 4 | 4 | no | — |
| 149 | C2×Q16 | — | 2 | 2 | no | — |
| 150 | C2×SD16 | — | 2 | 4 | no | — |
| 151 | C2×D8 | — | 2 | 2 | no | — |
| 152 | C2×D8 | — | 10 | 4 | no | — |
| 153 | C22×C8 | — | 10 | 1 | no | — |
| 154 | C22×C8 | C22⋊D5 | 1 | 2 | yes | — |
| 155 | C4⋊Q8 | C22⋊D5 | 1 | 2 | yes | — |
| 156 | #127 | C22⋊D5 | 1 | 2 | yes | — |
| 157 | #124 | — | 2 | 2 | no | — |
| 158 | #121 | — | 2 | 2 | no | — |
| 159 | #121 | — | 2 | 2 | no | — |
| 160 | C4×D4 | — | 2 | 2 | no | — |
| 161 | C4×D4 | — | 2 | 2 | no | — |
| 162 | C4×D4 | — | 10 | 2 | no | — |
| 163 | C4×D4 | — | 10 | 2 | no | — |
| 164 | C2×C4⋊C4 | — | 5 | 1 | no | — |
| 165 | C2×C22⋊C4 | — | 10 | 1 | no | — |
| 166 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 167 | #107 | — | 10 | 4 | no | — |
| 168 | C4⋊C8 | — | 10 | 2 | no | — |
| 169 | C4⋊C8 | — | 10 | 2 | no | — |
| 170 | #102 | — | 10 | 4 | no | — |
| 171 | #102 | — | 10 | 4 | no | — |
| 172 | C22⋊C8 | — | 10 | 2 | no | — |
| 173 | C22⋊C8 | — | 20 | 4 | no | — |
| 174 | #97 | — | 2 | 2 | no | — |
| 175 | C4×C8 | — | 5 | 4 | no | — |
| 176 | C22×C5 | — | 2 | 1 | no | — |
| 177 | C22×C5 | — | 2 | 1 | no | — |
| 178 | C22×C5 | — | 8 | 2 | no | — |
| 179 | C22×C5 | — | 8 | 2 | no | — |
| 180 | C22×C5 | — | 8 | 2 | no | — |
| 181 | C22×C5 | #392 | 1 | 1 | yes | characteristic |
| 182 | C22×C5 | #459 | 1 | 2 | yes | — |
| 183 | C2×D5 | — | 4 | 4 | no | — |
| 184 | C2×D5 | — | 4 | 2 | no | — |
| 185 | C2×D5 | — | 8 | 2 | no | — |
| 186 | C2×D5 | — | 8 | 1 | no | — |
| 187 | C2×D5 | — | 8 | 1 | no | — |
| 188 | C4×C5 | — | 2 | 2 | no | — |
| 189 | C4×C5 | — | 2 | 2 | no | — |
| 190 | C4×C5 | — | 8 | 2 | no | — |
| 191 | C4×C5 | #447 | 1 | 2 | yes | — |
| 192 | C4×C5 | #493 | 1 | 1 | yes | characteristic |
| 193 | C4×C5 | #496 | 1 | 1 | yes | characteristic |
| 194 | Dic5 | — | 8 | 1 | no | — |
| 195 | C24 | — | 10 | 1 | no | — |
| 196 | #41 | — | 4 | 4 | no | — |
| 197 | #41 | — | 4 | 4 | no | — |
| 198 | C2×Q8 | — | 2 | 2 | no | — |
| 199 | C2×D4 | — | 2 | 2 | no | — |
| 200 | C2×D4 | — | 2 | 2 | no | — |
| 201 | C2×D4 | — | 2 | 2 | no | — |
| 202 | C2×D4 | — | 4 | 2 | no | — |
| 203 | C2×D4 | — | 10 | 2 | no | — |
| 204 | C2×D4 | — | 20 | 2 | no | — |
| 205 | C22×C4 | — | 4 | 2 | no | — |
| 206 | C22×C4 | — | 10 | 1 | no | — |
| 207 | C22×C4 | — | 20 | 1 | no | — |
| 208 | C22×C4 | — | 20 | 2 | no | — |
| 209 | C22×C4 | — | 20 | 2 | no | — |
| 210 | C22×C4 | — | 20 | 1 | no | — |
| 211 | C22×C4 | D4×D5 | 1 | 1 | yes | characteristic |
| 212 | C22×C4 | #711 | 1 | 1 | yes | characteristic |
| 213 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 214 | Q16 | — | 4 | 4 | no | — |
| 215 | SD16 | — | 4 | 8 | no | — |
| 216 | D8 | — | 4 | 4 | no | — |
| 217 | D8 | — | 20 | 8 | no | — |
| 218 | C2×C8 | — | 2 | 2 | no | — |
| 219 | C2×C8 | — | 2 | 2 | no | — |
| 220 | C2×C8 | — | 10 | 4 | no | — |
| 221 | C2×C8 | — | 20 | 2 | no | — |
| 222 | C2×C8 | — | 20 | 1 | no | — |
| 223 | C2×C8 | #686 | 1 | 4 | yes | complemented |
| 224 | C4⋊C4 | — | 2 | 2 | no | — |
| 225 | C4⋊C4 | — | 2 | 2 | no | — |
| 226 | C4⋊C4 | — | 10 | 2 | no | — |
| 227 | C22⋊C4 | — | 4 | 2 | no | — |
| 228 | C22⋊C4 | — | 4 | 2 | no | — |
| 229 | C22⋊C4 | — | 20 | 2 | no | — |
| 230 | C42 | D4×D5 | 1 | 2 | yes | — |
| 231 | C42 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 232 | C2×C5 | — | 2 | 2 | no | — |
| 233 | C2×C5 | — | 16 | 2 | no | — |
| 234 | C2×C5 | #2010 | 1 | 1 | yes | characteristic |
| 235 | C2×C5 | #2011 | 1 | 1 | yes | characteristic |
| 236 | C2×C5 | #2017 | 1 | 1 | yes | characteristic |
| 237 | D5 | — | 8 | 2 | no | — |
| 238 | C23 | — | 4 | 2 | no | — |
| 239 | C23 | — | 10 | 2 | no | — |
| 240 | C23 | — | 20 | 1 | no | — |
| 241 | C23 | — | 20 | 2 | no | — |
| 242 | C23 | — | 20 | 2 | no | — |
| 243 | C23 | — | 20 | 1 | no | — |
| 244 | C23 | #4084 | 1 | 1 | yes | characteristic |
| 245 | Q8 | — | 4 | 4 | no | — |
| 246 | D4 | — | 4 | 4 | no | — |
| 247 | D4 | — | 4 | 4 | no | — |
| 248 | D4 | — | 4 | 4 | no | — |
| 249 | D4 | — | 8 | 4 | no | — |
| 250 | D4 | — | 10 | 4 | no | — |
| 251 | D4 | — | 20 | 2 | no | — |
| 252 | C2×C4 | — | 4 | 1 | no | — |
| 253 | C2×C4 | — | 4 | 1 | no | — |
| 254 | C2×C4 | — | 4 | 2 | no | — |
| 255 | C2×C4 | — | 8 | 2 | no | — |
| 256 | C2×C4 | — | 8 | 2 | no | — |
| 257 | C2×C4 | — | 20 | 2 | no | — |
| 258 | C2×C4 | — | 20 | 1 | no | — |
| 259 | C2×C4 | — | 40 | 1 | no | — |
| 260 | C2×C4 | — | 40 | 1 | no | — |
| 261 | C2×C4 | — | 40 | 2 | no | — |
| 262 | C2×C4 | C2×#686 | 1 | 2 | yes | — |
| 263 | C2×C4 | #4078 | 1 | 2 | yes | — |
| 264 | C2×C4 | #4083 | 1 | 2 | yes | — |
| 265 | C2×C4 | #4085 | 1 | 2 | yes | — |
| 266 | C2×C4 | #4086 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | #4099 | 1 | 1 | yes | characteristic |
| 268 | C8 | — | 2 | 4 | no | — |
| 269 | C8 | — | 20 | 4 | no | — |
| 270 | C8 | — | 40 | 1 | no | — |
| 271 | C5 | #10032 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 272 | C22 | — | 2 | 1 | no | — |
| 273 | C22 | — | 2 | 1 | no | — |
| 274 | C22 | — | 8 | 2 | no | — |
| 275 | C22 | — | 8 | 2 | no | — |
| 276 | C22 | — | 8 | 2 | no | — |
| 277 | C22 | — | 20 | 4 | no | — |
| 278 | C22 | — | 20 | 2 | no | — |
| 279 | C22 | — | 40 | 2 | no | — |
| 280 | C22 | — | 40 | 1 | no | — |
| 281 | C22 | — | 40 | 1 | no | — |
| 282 | C22 | #65361 | 1 | 2 | yes | — |
| 283 | C22 | #65512 | 1 | 1 | yes | characteristic |
| 284 | C4 | — | 2 | 2 | no | — |
| 285 | C4 | — | 2 | 2 | no | — |
| 286 | C4 | — | 8 | 2 | no | — |
| 287 | C4 | — | 40 | 1 | no | — |
| 288 | C4 | #65362 | 1 | 1 | yes | characteristic |
| 289 | C4 | #65363 | 1 | 1 | yes | characteristic |
| 290 | C4 | #65506 | 1 | 2 | yes | — |
| 291 | C2 | — | 2 | 2 | no | — |
| 292 | C2 | — | 16 | 2 | no | — |
| 293 | C2 | — | 40 | 2 | no | — |
| 294 | C2 | #10602560 | 1 | 1 | yes | characteristic |
| 295 | C2 | #10602911 | 1 | 1 | yes | characteristic |
| 296 | C2 | #10602914 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×#3150 | C2 | semidirect |
| 2. #10615029 | C2 | semidirect |
| 3. #10602593 | C2 | semidirect |
| 4. #10600630 | C2 | semidirect |
| 5. #10598151 | C2 | semidirect |
| 54. #3150 | D5 | semidirect |
| 83. C2×C5×C8 | D8 | semidirect |
| 223. C2×C8 | #686 | semidirect |
| 271. C5 | #10032 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×#3150 | 2 |
| #10615029 | 2 |
| #10602593 | 2 |
| #10600630 | 2 |
| #10598151 | 2 |
| #10032 | 5 |