Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
1630 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 195 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 | C2×C5×D42 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10614890 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10614046 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10602701 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #10602553 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10602392 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C23×C5×D4 | — | 2 | 2 | no | — |
| 8 | C5×D42 | — | 2 | 2 | no | — |
| 9 | C5×D42 | — | 2 | 4 | no | — |
| 10 | C5×D42 | C22 | 1 | 4 | yes | — |
| 11 | C2×C5×#127 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C5×#121 | — | 2 | 2 | no | — |
| 13 | C2×C5×#120 | — | 2 | 2 | no | — |
| 14 | C2×C4×C5×D4 | C22 | 1 | 2 | yes | — |
| 15 | #65804 | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | #65507 | — | 2 | 2 | no | — |
| 17 | #65504 | — | 2 | 2 | no | — |
| 18 | #65360 | — | 2 | 2 | no | — |
| 19 | #65359 | C22 | 1 | 4 | yes | complemented |
| 20 | #65308 | C22 | 1 | 4 | yes | complemented |
| 21 | #65307 | — | 2 | 4 | no | — |
| 22 | #65296 | C22 | 1 | 4 | yes | complemented |
| 23 | #65259 | — | 2 | 2 | no | — |
| 24 | #65215 | C22 | 1 | 2 | yes | complemented |
| 25 | C4×#681 | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | #64764 | C22 | 1 | 4 | yes | complemented |
| 27 | D4⋊#168 | C22 | 1 | 8 | yes | complemented |
| 28 | #18529 | — | 5 | 1 | no | maximal, Sylow |
| 29 | C25×C5 | — | 2 | 2 | no | — |
| 30 | C22×C5×D4 | — | 2 | 4 | no | — |
| 31 | C22×C5×D4 | — | 2 | 8 | no | — |
| 32 | C22×C5×D4 | — | 2 | 8 | no | — |
| 33 | C22×C5×D4 | — | 2 | 2 | no | — |
| 34 | C22×C5×D4 | — | 4 | 16 | no | — |
| 35 | C22×C5×D4 | D4 | 1 | 2 | yes | — |
| 36 | C22×C5×D4 | D4 | 1 | 2 | yes | — |
| 37 | C23×C4×C5 | — | 2 | 2 | no | — |
| 38 | C5×#127 | — | 2 | 2 | no | — |
| 39 | C5×#127 | D4 | 1 | 2 | yes | — |
| 40 | C5×#127 | C23 | 1 | 2 | yes | — |
| 41 | C5×#121 | — | 2 | 8 | no | — |
| 42 | C5×#121 | — | 2 | 8 | no | — |
| 43 | C5×#120 | — | 2 | 8 | no | — |
| 44 | C5×#120 | — | 2 | 8 | no | — |
| 45 | C4×C5×D4 | — | 2 | 2 | no | — |
| 46 | C4×C5×D4 | — | 2 | 2 | no | — |
| 47 | C4×C5×D4 | D4 | 1 | 4 | yes | — |
| 48 | C4×C5×D4 | C23 | 1 | 4 | yes | — |
| 49 | C2×C5×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 50 | C2×C5×C22⋊C4 | — | 2 | 2 | no | — |
| 51 | C2×C5×C22⋊C4 | — | 2 | 2 | no | — |
| 52 | C2×C42×C5 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×#687 | — | 2 | 4 | no | — |
| 54 | #4072 | — | 2 | 1 | no | — |
| 55 | #4070 | — | 2 | 2 | no | — |
| 56 | C2×#681 | — | 2 | 1 | no | — |
| 57 | #4016 | — | 2 | 1 | no | — |
| 58 | #4015 | C23 | 1 | 2 | yes | complemented |
| 59 | C4×Dic10 | — | 2 | 2 | no | — |
| 60 | #3964 | — | 2 | 4 | no | — |
| 61 | #3964 | — | 2 | 8 | no | — |
| 62 | C22⋊#168 | — | 2 | 2 | no | — |
| 63 | #3942 | — | 2 | 4 | no | — |
| 64 | #3939 | — | 2 | 4 | no | — |
| 65 | C4⋊#168 | C23 | 1 | 4 | yes | complemented |
| 66 | #3935 | — | 2 | 2 | no | — |
| 67 | #3935 | — | 2 | 1 | no | — |
| 68 | C4×#168 | C23 | 1 | 2 | yes | complemented |
| 69 | C2×D42 | D5 | 1 | 1 | yes | characteristic |
| 70 | #3138 | — | 5 | 1 | no | — |
| 71 | #3110 | — | 5 | 2 | no | — |
| 72 | #1639 | — | 5 | 8 | no | — |
| 73 | #1555 | — | 5 | 1 | no | — |
| 74 | #1490 | — | 5 | 2 | no | — |
| 75 | C24×C5 | — | 2 | 2 | no | — |
| 76 | C24×C5 | — | 2 | 8 | no | — |
| 77 | C24×C5 | — | 2 | 2 | no | — |
| 78 | C24×C5 | — | 4 | 16 | no | — |
| 79 | C24×C5 | — | 8 | 4 | no | — |
| 80 | C2×C5×D4 | — | 2 | 8 | no | — |
| 81 | C2×C5×D4 | — | 2 | 16 | no | — |
| 82 | C2×C5×D4 | — | 2 | 2 | no | — |
| 83 | C2×C5×D4 | — | 2 | 4 | no | — |
| 84 | C2×C5×D4 | — | 2 | 2 | no | — |
| 85 | C2×C5×D4 | — | 2 | 4 | no | — |
| 86 | C2×C5×D4 | — | 4 | 8 | no | — |
| 87 | C2×C5×D4 | — | 4 | 4 | no | — |
| 88 | C2×C5×D4 | — | 4 | 32 | no | — |
| 89 | C2×C5×D4 | — | 4 | 16 | no | — |
| 90 | C2×C5×D4 | — | 4 | 16 | no | — |
| 91 | C2×C5×D4 | — | 4 | 16 | no | — |
| 92 | C2×C5×D4 | C2×D4 | 1 | 4 | yes | — |
| 93 | C2×C5×D4 | C2×D4 | 1 | 4 | yes | — |
| 94 | C22×C4×C5 | — | 2 | 2 | no | — |
| 95 | C22×C4×C5 | — | 2 | 4 | no | — |
| 96 | C22×C4×C5 | — | 2 | 4 | no | — |
| 97 | C22×C4×C5 | — | 4 | 8 | no | — |
| 98 | C22×C4×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C22×C4×C5 | C2×D4 | 1 | 2 | yes | — |
| 100 | C2×Dic10 | — | 2 | 2 | no | — |
| 101 | C5×C4⋊C4 | — | 2 | 2 | no | — |
| 102 | C5×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 103 | C5×C22⋊C4 | — | 2 | 8 | no | — |
| 104 | C5×C22⋊C4 | — | 2 | 4 | no | — |
| 105 | C5×C22⋊C4 | — | 2 | 4 | no | — |
| 106 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C42×C5 | C2×D4 | 1 | 2 | yes | — |
| 108 | C42×C5 | C24 | 1 | 1 | yes | characteristic |
| 109 | C22⋊Dic5 | — | 4 | 1 | no | — |
| 110 | #687 | — | 4 | 8 | no | — |
| 111 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 112 | C4⋊Dic5 | — | 2 | 2 | no | — |
| 113 | #683 | — | 4 | 2 | no | — |
| 114 | C4×Dic5 | — | 4 | 1 | no | — |
| 115 | #681 | — | 4 | 2 | no | — |
| 116 | C2×#168 | — | 2 | 2 | no | — |
| 117 | C2×#168 | — | 2 | 4 | no | — |
| 118 | C23×D4 | — | 2 | 2 | no | — |
| 119 | D42 | — | 2 | 2 | no | — |
| 120 | D42 | — | 2 | 4 | no | — |
| 121 | D42 | C2×D5 | 1 | 4 | yes | — |
| 122 | #533 | — | 5 | 1 | no | — |
| 123 | C2×#127 | C2×D5 | 1 | 1 | yes | characteristic |
| 124 | C2×#121 | — | 2 | 2 | no | — |
| 125 | C2×#120 | — | 2 | 2 | no | — |
| 126 | C2×C4×D4 | C2×D5 | 1 | 2 | yes | — |
| 127 | #490 | — | 10 | 2 | no | — |
| 128 | #486 | — | 5 | 4 | no | — |
| 129 | #484 | — | 10 | 2 | no | — |
| 130 | #474 | — | 5 | 4 | no | — |
| 131 | #461 | — | 5 | 4 | no | — |
| 132 | #450 | — | 10 | 2 | no | — |
| 133 | #440 | — | 10 | 4 | no | — |
| 134 | #432 | — | 5 | 2 | no | — |
| 135 | #418 | — | 10 | 2 | no | — |
| 136 | C4×M16 | — | 5 | 1 | no | — |
| 137 | #331 | — | 5 | 4 | no | — |
| 138 | D4⋊C8 | — | 5 | 8 | no | — |
| 139 | C23×C5 | — | 2 | 8 | no | — |
| 140 | C23×C5 | — | 2 | 4 | no | — |
| 141 | C23×C5 | — | 2 | 4 | no | — |
| 142 | C23×C5 | — | 4 | 4 | no | — |
| 143 | C23×C5 | — | 4 | 8 | no | — |
| 144 | C23×C5 | — | 4 | 32 | no | — |
| 145 | C23×C5 | — | 4 | 4 | no | — |
| 146 | C23×C5 | — | 4 | 4 | no | — |
| 147 | C23×C5 | — | 8 | 8 | no | — |
| 148 | C23×C5 | — | 8 | 32 | no | — |
| 149 | C23×C5 | #120 | 1 | 1 | yes | characteristic |
| 150 | C5×D4 | — | 2 | 8 | no | — |
| 151 | C5×D4 | — | 2 | 4 | no | — |
| 152 | C5×D4 | — | 4 | 4 | no | — |
| 153 | C5×D4 | — | 4 | 8 | no | — |
| 154 | C5×D4 | — | 4 | 32 | no | — |
| 155 | C5×D4 | — | 4 | 8 | no | — |
| 156 | C5×D4 | — | 8 | 2 | no | — |
| 157 | C5×D4 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 158 | C2×C4×C5 | — | 2 | 2 | no | — |
| 159 | C2×C4×C5 | — | 2 | 4 | no | — |
| 160 | C2×C4×C5 | — | 2 | 4 | no | — |
| 161 | C2×C4×C5 | — | 4 | 8 | no | — |
| 162 | C2×C4×C5 | — | 4 | 4 | no | — |
| 163 | C2×C4×C5 | — | 4 | 8 | no | — |
| 164 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 165 | C2×C4×C5 | #120 | 1 | 2 | yes | — |
| 166 | C2×C4×C5 | C22×D4 | 1 | 1 | yes | characteristic |
| 167 | C2×C4×C5 | C22×D4 | 1 | 2 | yes | — |
| 168 | C2×Dic5 | — | 4 | 2 | no | — |
| 169 | Dic10 | — | 4 | 4 | no | — |
| 170 | #168 | — | 4 | 2 | no | — |
| 171 | #168 | — | 4 | 4 | no | — |
| 172 | C25 | — | 2 | 2 | no | — |
| 173 | C22×D4 | — | 2 | 4 | no | — |
| 174 | C22×D4 | — | 2 | 8 | no | — |
| 175 | C22×D4 | — | 2 | 8 | no | — |
| 176 | C22×D4 | — | 2 | 2 | no | — |
| 177 | C22×D4 | — | 4 | 16 | no | — |
| 178 | C22×D4 | C22⋊D5 | 1 | 2 | yes | — |
| 179 | C22×D4 | C22⋊D5 | 1 | 2 | yes | — |
| 180 | C23×C4 | — | 2 | 2 | no | — |
| 181 | C2×SD16 | — | 10 | 4 | no | — |
| 182 | C2×M16 | — | 10 | 1 | no | — |
| 183 | C4⋊Q8 | — | 5 | 2 | no | — |
| 184 | #127 | — | 2 | 2 | no | — |
| 185 | #127 | C22⋊D5 | 1 | 2 | yes | — |
| 186 | #127 | C22×D5 | 1 | 2 | yes | — |
| 187 | #125 | — | 10 | 1 | no | — |
| 188 | C22⋊Q8 | — | 10 | 2 | no | — |
| 189 | #121 | — | 2 | 8 | no | — |
| 190 | #121 | — | 2 | 8 | no | — |
| 191 | #120 | — | 2 | 8 | no | — |
| 192 | #120 | — | 2 | 8 | no | — |
| 193 | C4×Q8 | — | 10 | 2 | no | — |
| 194 | C4×D4 | — | 2 | 2 | no | — |
| 195 | C4×D4 | — | 2 | 2 | no | — |
| 196 | C4×D4 | C22⋊D5 | 1 | 4 | yes | — |
| 197 | C4×D4 | C22×D5 | 1 | 4 | yes | — |
| 198 | #117 | — | 10 | 1 | no | — |
| 199 | C2×C4⋊C4 | C22⋊D5 | 1 | 2 | yes | — |
| 200 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 201 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 202 | C2×C42 | C22×D5 | 1 | 1 | yes | characteristic |
| 203 | #107 | — | 10 | 4 | no | — |
| 204 | C4⋊C8 | — | 5 | 4 | no | — |
| 205 | #103 | — | 10 | 4 | no | — |
| 206 | #102 | — | 10 | 8 | no | — |
| 207 | #102 | — | 10 | 4 | no | — |
| 208 | C22⋊C8 | — | 10 | 2 | no | — |
| 209 | #97 | — | 10 | 1 | no | — |
| 210 | #97 | — | 10 | 2 | no | — |
| 211 | C4×C8 | — | 5 | 2 | no | — |
| 212 | C22×C5 | — | 2 | 2 | no | — |
| 213 | C22×C5 | — | 2 | 8 | no | — |
| 214 | C22×C5 | — | 4 | 16 | no | — |
| 215 | C22×C5 | — | 4 | 4 | no | — |
| 216 | C22×C5 | — | 4 | 4 | no | — |
| 217 | C22×C5 | — | 4 | 8 | no | — |
| 218 | C22×C5 | — | 4 | 4 | no | — |
| 219 | C22×C5 | — | 8 | 8 | no | — |
| 220 | C22×C5 | — | 8 | 32 | no | — |
| 221 | C22×C5 | #455 | 1 | 2 | yes | — |
| 222 | C22×C5 | C2×#120 | 1 | 1 | yes | characteristic |
| 223 | C4×C5 | — | 2 | 2 | no | — |
| 224 | C4×C5 | — | 2 | 1 | no | — |
| 225 | C4×C5 | — | 2 | 1 | no | — |
| 226 | C4×C5 | — | 4 | 4 | no | — |
| 227 | C4×C5 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 228 | Dic5 | — | 8 | 2 | no | — |
| 229 | C24 | — | 2 | 2 | no | — |
| 230 | C24 | — | 2 | 8 | no | — |
| 231 | C24 | — | 2 | 2 | no | — |
| 232 | C24 | — | 4 | 16 | no | — |
| 233 | C24 | — | 8 | 4 | no | — |
| 234 | C2×Q8 | — | 10 | 2 | no | — |
| 235 | C2×D4 | — | 2 | 8 | no | — |
| 236 | C2×D4 | — | 2 | 16 | no | — |
| 237 | C2×D4 | — | 2 | 2 | no | — |
| 238 | C2×D4 | — | 2 | 4 | no | — |
| 239 | C2×D4 | — | 2 | 2 | no | — |
| 240 | C2×D4 | — | 2 | 4 | no | — |
| 241 | C2×D4 | — | 4 | 8 | no | — |
| 242 | C2×D4 | — | 4 | 4 | no | — |
| 243 | C2×D4 | — | 4 | 32 | no | — |
| 244 | C2×D4 | — | 4 | 16 | no | — |
| 245 | C2×D4 | — | 4 | 16 | no | — |
| 246 | C2×D4 | — | 4 | 16 | no | — |
| 247 | C2×D4 | C2×C22⋊D5 | 1 | 4 | yes | — |
| 248 | C2×D4 | C2×C22⋊D5 | 1 | 4 | yes | — |
| 249 | C22×C4 | — | 2 | 2 | no | — |
| 250 | C22×C4 | — | 2 | 4 | no | — |
| 251 | C22×C4 | — | 2 | 4 | no | — |
| 252 | C22×C4 | — | 4 | 8 | no | — |
| 253 | C22×C4 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 254 | C22×C4 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 255 | SD16 | — | 20 | 8 | no | — |
| 256 | M16 | — | 20 | 2 | no | — |
| 257 | C2×C8 | — | 10 | 2 | no | — |
| 258 | C2×C8 | — | 10 | 4 | no | — |
| 259 | C4⋊C4 | — | 2 | 2 | no | — |
| 260 | C4⋊C4 | — | 10 | 2 | no | — |
| 261 | C4⋊C4 | — | 10 | 2 | no | — |
| 262 | C4⋊C4 | — | 20 | 2 | no | — |
| 263 | C4⋊C4 | C2×C22⋊D5 | 1 | 4 | yes | — |
| 264 | C22⋊C4 | — | 2 | 8 | no | — |
| 265 | C22⋊C4 | — | 2 | 4 | no | — |
| 266 | C22⋊C4 | — | 2 | 4 | no | — |
| 267 | C22⋊C4 | — | 20 | 1 | no | — |
| 268 | C42 | — | 20 | 1 | no | — |
| 269 | C42 | C2×C22⋊D5 | 1 | 1 | yes | characteristic |
| 270 | C42 | C2×C22⋊D5 | 1 | 2 | yes | — |
| 271 | C42 | C23×D5 | 1 | 1 | yes | characteristic |
| 272 | C2×C5 | — | 2 | 2 | no | — |
| 273 | C2×C5 | — | 4 | 8 | no | — |
| 274 | C2×C5 | — | 8 | 4 | no | — |
| 275 | C2×C5 | #3003 | 1 | 2 | yes | — |
| 276 | C2×C5 | C2×#455 | 1 | 1 | yes | characteristic |
| 277 | C23 | — | 2 | 8 | no | — |
| 278 | C23 | — | 2 | 4 | no | — |
| 279 | C23 | — | 2 | 4 | no | — |
| 280 | C23 | — | 4 | 4 | no | — |
| 281 | C23 | — | 4 | 8 | no | — |
| 282 | C23 | — | 4 | 32 | no | — |
| 283 | C23 | — | 4 | 4 | no | — |
| 284 | C23 | — | 4 | 4 | no | — |
| 285 | C23 | — | 8 | 8 | no | — |
| 286 | C23 | — | 8 | 32 | no | — |
| 287 | C23 | #4099 | 1 | 1 | yes | characteristic |
| 288 | Q8 | — | 20 | 4 | no | — |
| 289 | D4 | — | 2 | 8 | no | — |
| 290 | D4 | — | 2 | 4 | no | — |
| 291 | D4 | — | 4 | 4 | no | — |
| 292 | D4 | — | 4 | 8 | no | — |
| 293 | D4 | — | 4 | 32 | no | — |
| 294 | D4 | — | 4 | 8 | no | — |
| 295 | D4 | — | 8 | 2 | no | — |
| 296 | D4 | #4078 | 1 | 8 | yes | — |
| 297 | C2×C4 | — | 2 | 2 | no | — |
| 298 | C2×C4 | — | 2 | 4 | no | — |
| 299 | C2×C4 | — | 2 | 4 | no | — |
| 300 | C2×C4 | — | 4 | 8 | no | — |
| 301 | C2×C4 | — | 4 | 4 | no | — |
| 302 | C2×C4 | — | 4 | 8 | no | — |
| 303 | C2×C4 | — | 20 | 2 | no | — |
| 304 | C2×C4 | #4099 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #4099 | 1 | 2 | yes | — |
| 306 | C2×C4 | C22×C22⋊D5 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | C22×C22⋊D5 | 1 | 2 | yes | — |
| 308 | C8 | — | 20 | 2 | no | — |
| 309 | C8 | — | 20 | 4 | no | — |
| 310 | C5 | #18529 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 311 | C22 | — | 2 | 2 | no | — |
| 312 | C22 | — | 2 | 8 | no | — |
| 313 | C22 | — | 4 | 16 | no | — |
| 314 | C22 | — | 4 | 4 | no | — |
| 315 | C22 | — | 4 | 4 | no | — |
| 316 | C22 | — | 4 | 4 | no | — |
| 317 | C22 | — | 4 | 8 | no | — |
| 318 | C22 | — | 8 | 8 | no | — |
| 319 | C22 | — | 8 | 32 | no | — |
| 320 | C22 | #65531 | 1 | 2 | yes | — |
| 321 | C22 | C2×#4099 | 1 | 1 | yes | characteristic |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 2 | 1 | no | — |
| 324 | C4 | — | 2 | 2 | no | — |
| 325 | C4 | — | 4 | 4 | no | — |
| 326 | C4 | — | 40 | 2 | no | — |
| 327 | C4 | C2×#4078 | 1 | 4 | yes | — |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 4 | 8 | no | — |
| 330 | C2 | — | 8 | 4 | no | — |
| 331 | C2 | #10615769 | 1 | 2 | yes | — |
| 332 | C2 | C2×#65531 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10614890 | C2 | semidirect |
| 3. #10614046 | C2 | semidirect |
| 4. #10602701 | C2 | semidirect |
| 5. #10602553 | C2 | semidirect |
| 6. #10602392 | C2 | semidirect |
| 15. #65804 | C22 | semidirect |
| 19. #65359 | C22 | semidirect |
| 20. #65308 | C22 | semidirect |
| 22. #65296 | C22 | semidirect |
| 24. #65215 | C22 | semidirect |
| 25. C4×#681 | C22 | semidirect |
| 26. #64764 | C22 | semidirect |
| 27. D4⋊#168 | C22 | semidirect |
| 58. #4015 | C23 | semidirect |
| 65. C4⋊#168 | C23 | semidirect |
| 68. C4×#168 | C23 | semidirect |
| 310. C5 | #18529 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×D42 | 2 |
| #10614890 | 2 |
| #10614046 | 2 |
| #10602701 | 2 |
| #10602553 | 2 |
| #10602392 | 2 |
| #18529 | 5 |