Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
324 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 56 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 | #10604320 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10604290 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10598936 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C22×#700 | C4 | 1 | 1 | yes | characteristic |
| 5 | C2×#4039 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×#4003 | — | 2 | 1 | no | — |
| 7 | C2×#4003 | C4 | 1 | 1 | yes | characteristic |
| 8 | #65748 | — | 2 | 1 | no | — |
| 9 | D5×C22⋊C8 | — | 2 | 1 | no | — |
| 10 | #64918 | — | 2 | 1 | no | — |
| 11 | #64915 | — | 2 | 1 | no | — |
| 12 | #64915 | C4 | 1 | 2 | yes | — |
| 13 | C5×#324 | C4 | 1 | 1 | yes | characteristic |
| 14 | #64748 | C4 | 1 | 1 | yes | characteristic |
| 15 | #64688 | — | 2 | 1 | no | — |
| 16 | #7285 | — | 5 | 1 | no | maximal, Sylow |
| 17 | C22×D20 | Q8 | 1 | 1 | yes | characteristic |
| 18 | C22×C4×D5 | C2×C4 | 1 | 1 | yes | characteristic |
| 19 | C22×C4×D5 | D4 | 1 | 1 | yes | characteristic |
| 20 | C22×#170 | — | 2 | 1 | no | — |
| 21 | C22×#170 | — | 2 | 1 | no | — |
| 22 | C2×#700 | — | 4 | 1 | no | — |
| 23 | C2×#700 | — | 4 | 2 | no | — |
| 24 | C2×C5×C4⋊C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C2×#685 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×#685 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×Dic5 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×C8×D5 | — | 4 | 1 | no | — |
| 29 | #4039 | — | 4 | 1 | no | — |
| 30 | #4039 | — | 4 | 1 | no | — |
| 31 | C22⋊#170 | — | 2 | 2 | no | — |
| 32 | #4003 | — | 4 | 2 | no | — |
| 33 | #4003 | — | 4 | 1 | no | — |
| 34 | C4⋊#170 | — | 4 | 1 | no | — |
| 35 | C4×#170 | — | 4 | 1 | no | — |
| 36 | C5×C22⋊C8 | — | 2 | 1 | no | — |
| 37 | C22⋊#168 | — | 2 | 1 | no | — |
| 38 | #3952 | — | 4 | 1 | no | — |
| 39 | #1870 | — | 5 | 1 | no | — |
| 40 | #1460 | — | 5 | 1 | no | — |
| 41 | #1459 | — | 5 | 1 | no | — |
| 42 | C23×D5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 43 | C23×D5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 44 | C22×C4×C5 | C42 | 1 | 1 | yes | characteristic |
| 45 | C22×C4×C5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 46 | C22×C4×C5 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 47 | C22×Dic5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 48 | C22×Dic5 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 49 | C2×D20 | — | 2 | 2 | no | — |
| 50 | C2×D20 | — | 2 | 1 | no | — |
| 51 | C2×D20 | — | 2 | 1 | no | — |
| 52 | C2×D20 | — | 4 | 1 | no | — |
| 53 | C2×C4×D5 | — | 2 | 1 | no | — |
| 54 | C2×C4×D5 | — | 2 | 1 | no | — |
| 55 | C2×C4×D5 | — | 2 | 1 | no | — |
| 56 | C2×C4×D5 | — | 2 | 1 | no | — |
| 57 | C2×C4×D5 | — | 4 | 1 | no | — |
| 58 | C2×C4×D5 | — | 4 | 1 | no | — |
| 59 | C2×C4×D5 | — | 4 | 1 | no | — |
| 60 | C2×C4×D5 | — | 4 | 1 | no | — |
| 61 | C2×#170 | — | 2 | 2 | no | — |
| 62 | C2×#170 | — | 4 | 1 | no | — |
| 63 | C2×#170 | — | 4 | 1 | no | — |
| 64 | C2×#170 | — | 4 | 1 | no | — |
| 65 | C2×#170 | — | 4 | 1 | no | — |
| 66 | C2×#170 | — | 4 | 2 | no | — |
| 67 | C2×#170 | — | 4 | 1 | no | — |
| 68 | #700 | — | 4 | 2 | no | — |
| 69 | #700 | — | 4 | 2 | no | — |
| 70 | C2×C5×C8 | — | 4 | 1 | no | — |
| 71 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 72 | #685 | — | 4 | 1 | no | — |
| 73 | #685 | — | 4 | 1 | no | — |
| 74 | C4×Dic5 | — | 2 | 1 | no | — |
| 75 | C4×Dic5 | — | 2 | 1 | no | — |
| 76 | C2×#168 | — | 4 | 1 | no | — |
| 77 | C8×D5 | — | 8 | 1 | no | — |
| 78 | C8×D5 | — | 8 | 1 | no | — |
| 79 | C22×M16 | — | 5 | 1 | no | — |
| 80 | C2×C4×D4 | — | 5 | 1 | no | — |
| 81 | #431 | — | 10 | 1 | no | — |
| 82 | C2×C22⋊C8 | — | 5 | 1 | no | — |
| 83 | C2×C22⋊C8 | — | 10 | 1 | no | — |
| 84 | C2×C22⋊C8 | — | 10 | 1 | no | — |
| 85 | #336 | — | 10 | 1 | no | — |
| 86 | #324 | — | 5 | 1 | no | — |
| 87 | #324 | AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 88 | #323 | — | 5 | 2 | no | — |
| 89 | #323 | — | 10 | 1 | no | — |
| 90 | #323 | — | 10 | 1 | no | — |
| 91 | C23×C5 | #95 | 1 | 1 | yes | characteristic |
| 92 | C22×D5 | — | 2 | 1 | no | — |
| 93 | C22×D5 | — | 2 | 1 | no | — |
| 94 | C22×D5 | — | 2 | 1 | no | — |
| 95 | C22×D5 | — | 2 | 1 | no | — |
| 96 | C22×D5 | — | 2 | 1 | no | — |
| 97 | C22×D5 | — | 4 | 1 | no | — |
| 98 | C22×D5 | — | 4 | 1 | no | — |
| 99 | C22×D5 | — | 4 | 1 | no | — |
| 100 | C22×D5 | — | 4 | 1 | no | — |
| 101 | C22×D5 | #108 | 1 | 2 | yes | — |
| 102 | C2×C4×C5 | — | 2 | 1 | no | — |
| 103 | C2×C4×C5 | — | 2 | 1 | no | — |
| 104 | C2×C4×C5 | — | 2 | 1 | no | — |
| 105 | C2×C4×C5 | — | 2 | 1 | no | — |
| 106 | C2×C4×C5 | — | 4 | 1 | no | — |
| 107 | C2×C4×C5 | — | 4 | 1 | no | — |
| 108 | C2×C4×C5 | #108 | 1 | 1 | yes | characteristic |
| 109 | C2×C4×C5 | #108 | 1 | 1 | yes | characteristic |
| 110 | C2×Dic5 | — | 2 | 1 | no | — |
| 111 | C2×Dic5 | — | 2 | 1 | no | — |
| 112 | C2×Dic5 | — | 2 | 1 | no | — |
| 113 | C2×Dic5 | — | 4 | 1 | no | — |
| 114 | C2×Dic5 | #99 | 1 | 1 | yes | characteristic |
| 115 | C2×Dic5 | #99 | 1 | 1 | yes | characteristic |
| 116 | D20 | — | 4 | 2 | no | — |
| 117 | D20 | — | 4 | 2 | no | — |
| 118 | C4×D5 | — | 4 | 2 | no | — |
| 119 | C4×D5 | — | 4 | 1 | no | — |
| 120 | C4×D5 | — | 4 | 1 | no | — |
| 121 | C4×D5 | — | 4 | 1 | no | — |
| 122 | C4×D5 | — | 4 | 1 | no | — |
| 123 | C4×D5 | — | 8 | 1 | no | — |
| 124 | #170 | — | 4 | 2 | no | — |
| 125 | #170 | — | 8 | 1 | no | — |
| 126 | #170 | — | 8 | 2 | no | — |
| 127 | C5×C8 | — | 8 | 1 | no | — |
| 128 | #168 | — | 8 | 1 | no | — |
| 129 | C22×D4 | — | 5 | 1 | no | — |
| 130 | C23×C4 | — | 5 | 1 | no | — |
| 131 | C23×C4 | — | 5 | 1 | no | — |
| 132 | C2×M16 | — | 20 | 2 | no | — |
| 133 | C2×M16 | — | 20 | 1 | no | — |
| 134 | C22×C8 | — | 10 | 1 | no | — |
| 135 | C22×C8 | — | 10 | 1 | no | — |
| 136 | C22×C8 | — | 20 | 1 | no | — |
| 137 | C4×D4 | — | 20 | 1 | no | — |
| 138 | C4×D4 | — | 20 | 1 | no | — |
| 139 | C2×C4⋊C4 | C2×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 140 | C2×C22⋊C4 | — | 5 | 1 | no | — |
| 141 | C2×C22⋊C4 | — | 5 | 1 | no | — |
| 142 | C2×C42 | — | 5 | 1 | no | — |
| 143 | C4⋊C8 | — | 20 | 1 | no | — |
| 144 | C22⋊C8 | — | 2 | 1 | no | — |
| 145 | C22⋊C8 | — | 10 | 2 | no | — |
| 146 | C22⋊C8 | — | 10 | 1 | no | — |
| 147 | C22⋊C8 | — | 20 | 1 | no | — |
| 148 | C22⋊C8 | — | 20 | 2 | no | — |
| 149 | C22⋊C8 | — | 20 | 1 | no | — |
| 150 | C4×C8 | — | 20 | 1 | no | — |
| 151 | C22×C5 | — | 2 | 1 | no | — |
| 152 | C22×C5 | — | 2 | 1 | no | — |
| 153 | C22×C5 | #341 | 1 | 1 | yes | characteristic |
| 154 | C22×C5 | #341 | 1 | 1 | yes | characteristic |
| 155 | C22×C5 | #342 | 1 | 1 | yes | characteristic |
| 156 | C2×D5 | — | 2 | 2 | no | — |
| 157 | C2×D5 | — | 4 | 1 | no | — |
| 158 | C2×D5 | — | 4 | 1 | no | — |
| 159 | C2×D5 | — | 4 | 2 | no | — |
| 160 | C2×D5 | — | 4 | 1 | no | — |
| 161 | C2×D5 | — | 4 | 1 | no | — |
| 162 | C2×D5 | — | 4 | 1 | no | — |
| 163 | C2×D5 | — | 8 | 1 | no | — |
| 164 | C2×D5 | — | 8 | 1 | no | — |
| 165 | C2×D5 | — | 8 | 1 | no | — |
| 166 | C4×C5 | — | 2 | 1 | no | — |
| 167 | C4×C5 | — | 2 | 1 | no | — |
| 168 | C4×C5 | — | 4 | 1 | no | — |
| 169 | C4×C5 | — | 4 | 1 | no | — |
| 170 | Dic5 | — | 2 | 1 | no | — |
| 171 | Dic5 | — | 4 | 1 | no | — |
| 172 | Dic5 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 173 | Dic5 | #352 | 1 | 1 | yes | characteristic |
| 174 | C24 | — | 5 | 1 | no | — |
| 175 | C24 | — | 5 | 1 | no | — |
| 176 | C2×D4 | — | 10 | 2 | no | — |
| 177 | C2×D4 | — | 10 | 1 | no | — |
| 178 | C2×D4 | — | 10 | 1 | no | — |
| 179 | C2×D4 | — | 20 | 1 | no | — |
| 180 | C22×C4 | — | 5 | 1 | no | — |
| 181 | C22×C4 | — | 5 | 1 | no | — |
| 182 | C22×C4 | — | 10 | 1 | no | — |
| 183 | C22×C4 | — | 10 | 1 | no | — |
| 184 | C22×C4 | — | 10 | 1 | no | — |
| 185 | C22×C4 | — | 10 | 1 | no | — |
| 186 | C22×C4 | — | 20 | 1 | no | — |
| 187 | C22×C4 | — | 20 | 1 | no | — |
| 188 | C22×C4 | — | 20 | 1 | no | — |
| 189 | C22×C4 | — | 20 | 1 | no | — |
| 190 | C22×C4 | C4×AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 191 | C22×C4 | C4⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 192 | C22×C4 | C22⋊AGL(1,𝔽5) | 1 | 1 | yes | characteristic |
| 193 | M16 | — | 20 | 2 | no | — |
| 194 | M16 | — | 20 | 2 | no | — |
| 195 | C2×C8 | — | 4 | 1 | no | — |
| 196 | C2×C8 | — | 10 | 2 | no | — |
| 197 | C2×C8 | — | 20 | 1 | no | — |
| 198 | C2×C8 | — | 20 | 1 | no | — |
| 199 | C2×C8 | — | 20 | 1 | no | — |
| 200 | C2×C8 | — | 20 | 1 | no | — |
| 201 | C2×C8 | — | 20 | 2 | no | — |
| 202 | C2×C8 | — | 20 | 1 | no | — |
| 203 | C2×C8 | — | 20 | 1 | no | — |
| 204 | C2×C8 | — | 40 | 1 | no | — |
| 205 | C2×C8 | — | 40 | 1 | no | — |
| 206 | C4⋊C4 | — | 4 | 1 | no | — |
| 207 | C22⋊C4 | — | 20 | 1 | no | — |
| 208 | C22⋊C4 | — | 20 | 1 | no | — |
| 209 | C42 | — | 10 | 1 | no | — |
| 210 | C42 | — | 10 | 1 | no | — |
| 211 | C2×C5 | — | 2 | 1 | no | — |
| 212 | C2×C5 | — | 2 | 1 | no | — |
| 213 | C2×C5 | #1298 | 1 | 1 | yes | characteristic |
| 214 | C2×C5 | #1390 | 1 | 1 | yes | characteristic |
| 215 | C2×C5 | #1395 | 1 | 1 | yes | characteristic |
| 216 | D5 | — | 4 | 2 | no | — |
| 217 | D5 | — | 8 | 1 | no | — |
| 218 | C23 | — | 5 | 2 | no | — |
| 219 | C23 | — | 10 | 1 | no | — |
| 220 | C23 | — | 10 | 1 | no | — |
| 221 | C23 | — | 10 | 1 | no | — |
| 222 | C23 | — | 10 | 1 | no | — |
| 223 | C23 | — | 10 | 1 | no | — |
| 224 | C23 | — | 20 | 1 | no | — |
| 225 | C23 | — | 20 | 1 | no | — |
| 226 | C23 | — | 20 | 1 | no | — |
| 227 | C23 | — | 20 | 1 | no | — |
| 228 | C23 | (C2×AGL(1,𝔽5))⋊C4 | 1 | 1 | yes | characteristic |
| 229 | D4 | — | 20 | 2 | no | — |
| 230 | D4 | — | 20 | 2 | no | — |
| 231 | C2×C4 | — | 2 | 1 | no | — |
| 232 | C2×C4 | — | 2 | 1 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 2 | 1 | no | — |
| 235 | C2×C4 | — | 4 | 1 | no | — |
| 236 | C2×C4 | — | 4 | 1 | no | — |
| 237 | C2×C4 | — | 5 | 1 | no | — |
| 238 | C2×C4 | — | 5 | 1 | no | — |
| 239 | C2×C4 | — | 10 | 1 | no | — |
| 240 | C2×C4 | — | 10 | 1 | no | — |
| 241 | C2×C4 | — | 10 | 1 | no | — |
| 242 | C2×C4 | — | 20 | 1 | no | — |
| 243 | C2×C4 | — | 20 | 1 | no | — |
| 244 | C2×C4 | — | 20 | 1 | no | — |
| 245 | C2×C4 | — | 20 | 1 | no | — |
| 246 | C2×C4 | — | 20 | 2 | no | — |
| 247 | C2×C4 | — | 20 | 1 | no | — |
| 248 | C2×C4 | — | 40 | 1 | no | — |
| 249 | C2×C4 | #3995 | 1 | 1 | yes | characteristic |
| 250 | C2×C4 | #3996 | 1 | 1 | yes | characteristic |
| 251 | C8 | — | 8 | 1 | no | — |
| 252 | C8 | — | 20 | 2 | no | — |
| 253 | C8 | — | 40 | 1 | no | — |
| 254 | C8 | — | 40 | 1 | no | — |
| 255 | C8 | — | 40 | 2 | no | — |
| 256 | C5 | #7285 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 257 | C22 | — | 2 | 1 | no | — |
| 258 | C22 | — | 2 | 1 | no | — |
| 259 | C22 | — | 10 | 2 | no | — |
| 260 | C22 | — | 20 | 2 | no | — |
| 261 | C22 | — | 20 | 1 | no | — |
| 262 | C22 | — | 20 | 1 | no | — |
| 263 | C22 | — | 20 | 1 | no | — |
| 264 | C22 | — | 20 | 1 | no | — |
| 265 | C22 | — | 20 | 1 | no | — |
| 266 | C22 | — | 40 | 1 | no | — |
| 267 | C22 | — | 40 | 1 | no | — |
| 268 | C22 | — | 40 | 1 | no | — |
| 269 | C22 | #64866 | 1 | 1 | yes | characteristic |
| 270 | C22 | #64896 | 1 | 1 | yes | characteristic |
| 271 | C22 | #64901 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 2 | 1 | no | — |
| 273 | C4 | — | 2 | 1 | no | — |
| 274 | C4 | — | 4 | 1 | no | — |
| 275 | C4 | — | 4 | 1 | no | — |
| 276 | C4 | — | 5 | 1 | no | — |
| 277 | C4 | — | 5 | 1 | no | — |
| 278 | C4 | — | 10 | 1 | no | — |
| 279 | C4 | — | 20 | 1 | no | — |
| 280 | C2 | — | 2 | 1 | no | — |
| 281 | C2 | — | 2 | 1 | no | — |
| 282 | C2 | — | 20 | 2 | no | — |
| 283 | C2 | — | 40 | 1 | no | — |
| 284 | C2 | #10598561 | 1 | 1 | yes | characteristic |
| 285 | C2 | #10598567 | 1 | 1 | yes | characteristic |
| 286 | C2 | #10598578 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 256. C5 | #7285 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10604320 | 2 |
| #10604290 | 2 |
| #10598936 | 2 |
| #7285 | 5 |