Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
368 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 368 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 | #10611461 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10601709 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10599877 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10599334 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C5×#1334 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10598006 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10597958 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#4064 | — | 2 | 1 | no | — |
| 9 | #65910 | — | 2 | 1 | no | — |
| 10 | C4×C5×Q16 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#3961 | — | 2 | 1 | no | — |
| 12 | #65317 | — | 2 | 1 | no | — |
| 13 | C2×Dic40 | — | 2 | 1 | no | — |
| 14 | C2×#3932 | — | 2 | 1 | no | — |
| 15 | #65175 | — | 2 | 1 | no | — |
| 16 | #65135 | — | 2 | 1 | no | — |
| 17 | #65100 | — | 2 | 1 | no | — |
| 18 | #65097 | — | 2 | 1 | no | — |
| 19 | #64988 | C22 | 1 | 1 | yes | characteristic |
| 20 | #64985 | C22 | 1 | 1 | yes | characteristic |
| 21 | #64977 | C22 | 1 | 1 | yes | characteristic |
| 22 | C5×#346 | C22 | 1 | 1 | yes | characteristic |
| 23 | C5×#334 | C22 | 1 | 1 | yes | characteristic |
| 24 | #64712 | — | 2 | 1 | no | — |
| 25 | #64706 | — | 2 | 1 | no | — |
| 26 | C4⋊#672 | C22 | 1 | 1 | yes | characteristic |
| 27 | #12327 | — | 5 | 1 | no | maximal, Sylow |
| 28 | C2×#713 | — | 2 | 1 | no | — |
| 29 | C2×Q8×D5 | — | 2 | 1 | no | — |
| 30 | C2×C5×Q16 | D4 | 1 | 1 | yes | characteristic |
| 31 | C4×C5×Q8 | — | 2 | 1 | no | — |
| 32 | C2×#689 | — | 2 | 1 | no | — |
| 33 | C2×#688 | — | 2 | 1 | no | — |
| 34 | #4064 | — | 4 | 2 | no | — |
| 35 | #4064 | — | 4 | 2 | no | — |
| 36 | C2×Dic20 | — | 2 | 1 | no | — |
| 37 | C2×Dic20 | — | 2 | 1 | no | — |
| 38 | C2×Dic20 | D4 | 1 | 1 | yes | characteristic |
| 39 | C2×D40 | D4 | 1 | 1 | yes | characteristic |
| 40 | C2×#677 | — | 2 | 1 | no | — |
| 41 | C2×#676 | — | 2 | 1 | no | — |
| 42 | #4043 | — | 2 | 1 | no | — |
| 43 | #4043 | — | 4 | 1 | no | — |
| 44 | #4041 | — | 2 | 1 | no | — |
| 45 | #4041 | — | 4 | 1 | no | — |
| 46 | #4021 | — | 2 | 1 | no | — |
| 47 | #4021 | — | 4 | 1 | no | — |
| 48 | C4×D20 | — | 4 | 1 | no | — |
| 49 | C4×D20 | D4 | 1 | 1 | yes | characteristic |
| 50 | #4015 | — | 2 | 1 | no | — |
| 51 | C2×C5×C16 | — | 2 | 1 | no | — |
| 52 | C5×#107 | D4 | 1 | 1 | yes | characteristic |
| 53 | C5×C4⋊C8 | — | 2 | 1 | no | — |
| 54 | C5×#103 | — | 2 | 1 | no | — |
| 55 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 56 | #3961 | — | 4 | 2 | no | — |
| 57 | #3953 | — | 4 | 1 | no | — |
| 58 | #3952 | — | 2 | 1 | no | — |
| 59 | #3950 | — | 4 | 1 | no | — |
| 60 | C2×#672 | — | 2 | 1 | no | — |
| 61 | #3942 | — | 2 | 1 | no | — |
| 62 | #3941 | — | 2 | 1 | no | — |
| 63 | C4⋊#168 | D4 | 1 | 1 | yes | characteristic |
| 64 | Dic40 | — | 4 | 2 | no | — |
| 65 | #3932 | — | 4 | 2 | no | — |
| 66 | #3277 | — | 5 | 1 | no | — |
| 67 | #2251 | — | 5 | 1 | no | — |
| 68 | #2209 | — | 5 | 1 | no | — |
| 69 | #1689 | — | 5 | 1 | no | — |
| 70 | #1360 | — | 5 | 1 | no | — |
| 71 | #1334 | D5 | 1 | 1 | yes | characteristic, complemented |
| 72 | #1330 | — | 5 | 1 | no | — |
| 73 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 74 | #713 | — | 4 | 2 | no | — |
| 75 | #713 | — | 8 | 1 | no | — |
| 76 | Q8×D5 | — | 4 | 2 | no | — |
| 77 | Q8×D5 | — | 8 | 1 | no | — |
| 78 | C2×D20 | — | 2 | 1 | no | — |
| 79 | C2×D20 | — | 4 | 1 | no | — |
| 80 | C2×D20 | — | 4 | 1 | no | — |
| 81 | C2×D20 | D8 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×D5 | — | 2 | 1 | no | — |
| 83 | C2×C4×D5 | — | 4 | 1 | no | — |
| 84 | C2×C4×D5 | — | 4 | 1 | no | — |
| 85 | C2×Dic10 | — | 2 | 1 | no | — |
| 86 | C2×Dic10 | — | 4 | 1 | no | — |
| 87 | C2×Dic10 | — | 4 | 1 | no | — |
| 88 | C5×Q16 | — | 2 | 1 | no | — |
| 89 | C5×Q16 | D8 | 1 | 2 | yes | — |
| 90 | C2×C5×C8 | — | 4 | 1 | no | — |
| 91 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C5×C4⋊C4 | — | 2 | 1 | no | — |
| 94 | C5×C4⋊C4 | — | 4 | 1 | no | — |
| 95 | C42×C5 | — | 4 | 1 | no | — |
| 96 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | #689 | — | 4 | 2 | no | — |
| 98 | #688 | — | 4 | 2 | no | — |
| 99 | #685 | — | 2 | 1 | no | — |
| 100 | #685 | — | 4 | 1 | no | — |
| 101 | #685 | — | 4 | 1 | no | — |
| 102 | #685 | — | 4 | 1 | no | — |
| 103 | #685 | — | 4 | 1 | no | — |
| 104 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 105 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 106 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 107 | C4⋊Dic5 | D8 | 1 | 1 | yes | characteristic |
| 108 | C2×#168 | — | 2 | 1 | no | — |
| 109 | Dic20 | — | 4 | 1 | no | — |
| 110 | Dic20 | — | 4 | 1 | no | — |
| 111 | Dic20 | — | 4 | 1 | no | — |
| 112 | Dic20 | — | 4 | 1 | no | — |
| 113 | Dic20 | — | 4 | 1 | no | — |
| 114 | D40 | — | 4 | 1 | no | — |
| 115 | #677 | — | 4 | 1 | no | — |
| 116 | #677 | — | 4 | 1 | no | — |
| 117 | #676 | — | 4 | 1 | no | — |
| 118 | #676 | — | 4 | 1 | no | — |
| 119 | C5×C16 | — | 2 | 2 | no | — |
| 120 | #672 | — | 4 | 1 | no | — |
| 121 | C2×#137 | — | 10 | 1 | no | — |
| 122 | #548 | — | 10 | 1 | no | — |
| 123 | C2×Q32 | — | 10 | 1 | no | — |
| 124 | C2×Q32 | — | 10 | 1 | no | — |
| 125 | C2×SD32 | — | 10 | 1 | no | — |
| 126 | #495 | — | 5 | 1 | no | — |
| 127 | #494 | — | 5 | 1 | no | — |
| 128 | #474 | — | 10 | 1 | no | — |
| 129 | #468 | — | 10 | 1 | no | — |
| 130 | #464 | — | 10 | 1 | no | — |
| 131 | #451 | — | 10 | 1 | no | — |
| 132 | #449 | — | 10 | 1 | no | — |
| 133 | C4×Q16 | C2×D5 | 1 | 1 | yes | characteristic |
| 134 | #436 | — | 5 | 1 | no | — |
| 135 | C4⋊C16 | — | 5 | 1 | no | — |
| 136 | #358 | — | 10 | 1 | no | — |
| 137 | #346 | C2×D5 | 1 | 1 | yes | characteristic |
| 138 | #334 | C2×D5 | 1 | 1 | yes | characteristic |
| 139 | #332 | — | 10 | 1 | no | — |
| 140 | C22×D5 | — | 2 | 1 | no | — |
| 141 | C22×D5 | — | 4 | 1 | no | — |
| 142 | C5×Q8 | — | 2 | 2 | no | — |
| 143 | C5×Q8 | — | 4 | 1 | no | — |
| 144 | C2×C4×C5 | — | 4 | 1 | no | — |
| 145 | C2×C4×C5 | — | 4 | 1 | no | — |
| 146 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 147 | C2×C4×C5 | C2×D8 | 1 | 1 | yes | characteristic |
| 148 | C2×C4×C5 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 149 | C2×Dic5 | — | 2 | 1 | no | — |
| 150 | C2×Dic5 | — | 4 | 1 | no | — |
| 151 | C2×Dic5 | — | 8 | 1 | no | — |
| 152 | D20 | — | 4 | 1 | no | — |
| 153 | D20 | — | 4 | 1 | no | — |
| 154 | D20 | — | 4 | 1 | no | — |
| 155 | D20 | — | 8 | 2 | no | — |
| 156 | D20 | — | 8 | 2 | no | — |
| 157 | C4×D5 | — | 4 | 1 | no | — |
| 158 | C4×D5 | — | 4 | 1 | no | — |
| 159 | C4×D5 | — | 8 | 2 | no | — |
| 160 | C4×D5 | — | 8 | 2 | no | — |
| 161 | Dic10 | — | 4 | 1 | no | — |
| 162 | Dic10 | — | 4 | 1 | no | — |
| 163 | Dic10 | — | 8 | 2 | no | — |
| 164 | Dic10 | — | 8 | 1 | no | — |
| 165 | Dic10 | — | 8 | 1 | no | — |
| 166 | C5×C8 | — | 2 | 1 | no | — |
| 167 | C5×C8 | — | 8 | 1 | no | — |
| 168 | C5×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 169 | C5×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 170 | #168 | — | 4 | 1 | no | — |
| 171 | C2×#41 | — | 10 | 1 | no | — |
| 172 | C22×Q8 | — | 10 | 1 | no | — |
| 173 | #137 | — | 20 | 2 | no | — |
| 174 | #137 | — | 20 | 2 | no | — |
| 175 | C2×Q16 | — | 5 | 1 | no | — |
| 176 | C2×Q16 | — | 10 | 1 | no | — |
| 177 | C2×Q16 | — | 10 | 1 | no | — |
| 178 | C2×Q16 | — | 10 | 1 | no | — |
| 179 | C2×Q16 | D20 | 1 | 1 | yes | characteristic |
| 180 | C2×SD16 | — | 10 | 1 | no | — |
| 181 | C2×SD16 | — | 10 | 1 | no | — |
| 182 | C2×D8 | — | 5 | 1 | no | — |
| 183 | C2×M16 | — | 10 | 1 | no | — |
| 184 | C4⋊Q8 | — | 10 | 1 | no | — |
| 185 | #124 | — | 10 | 1 | no | — |
| 186 | #124 | — | 20 | 1 | no | — |
| 187 | C22⋊Q8 | — | 10 | 1 | no | — |
| 188 | C22⋊Q8 | — | 20 | 1 | no | — |
| 189 | #121 | — | 10 | 1 | no | — |
| 190 | #121 | — | 20 | 1 | no | — |
| 191 | C4×Q8 | — | 2 | 1 | no | — |
| 192 | C4×D4 | — | 5 | 1 | no | — |
| 193 | C4×D4 | — | 20 | 1 | no | — |
| 194 | Q32 | — | 20 | 2 | no | — |
| 195 | Q32 | — | 20 | 2 | no | — |
| 196 | SD32 | — | 20 | 2 | no | — |
| 197 | C2×C16 | — | 2 | 1 | no | — |
| 198 | C2×C16 | — | 10 | 1 | no | — |
| 199 | #107 | — | 20 | 1 | no | — |
| 200 | #107 | D20 | 1 | 1 | yes | characteristic |
| 201 | C4⋊C8 | — | 2 | 1 | no | — |
| 202 | C4⋊C8 | — | 5 | 1 | no | — |
| 203 | #103 | — | 2 | 1 | no | — |
| 204 | #103 | — | 10 | 1 | no | — |
| 205 | #102 | — | 10 | 1 | no | — |
| 206 | #102 | — | 20 | 1 | no | — |
| 207 | C22⋊C8 | — | 10 | 1 | no | — |
| 208 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 209 | C22×C5 | #447 | 1 | 1 | yes | characteristic |
| 210 | C2×D5 | — | 4 | 1 | no | — |
| 211 | C2×D5 | — | 4 | 1 | no | — |
| 212 | C2×D5 | — | 4 | 1 | no | — |
| 213 | C2×D5 | — | 8 | 1 | no | — |
| 214 | C2×D5 | — | 8 | 1 | no | — |
| 215 | C2×D5 | — | 8 | 1 | no | — |
| 216 | C4×C5 | — | 2 | 1 | no | — |
| 217 | C4×C5 | — | 2 | 1 | no | — |
| 218 | C4×C5 | — | 4 | 2 | no | — |
| 219 | C4×C5 | — | 8 | 1 | no | — |
| 220 | C4×C5 | #447 | 1 | 1 | yes | characteristic |
| 221 | C4×C5 | #453 | 1 | 1 | yes | characteristic |
| 222 | Dic5 | — | 4 | 1 | no | — |
| 223 | Dic5 | — | 8 | 1 | no | — |
| 224 | Dic5 | — | 16 | 1 | no | — |
| 225 | #41 | — | 20 | 2 | no | — |
| 226 | #41 | — | 40 | 1 | no | — |
| 227 | C2×Q8 | — | 2 | 1 | no | — |
| 228 | C2×Q8 | — | 10 | 1 | no | — |
| 229 | C2×Q8 | — | 20 | 1 | no | — |
| 230 | C2×Q8 | — | 20 | 2 | no | — |
| 231 | C2×Q8 | — | 20 | 1 | no | — |
| 232 | C2×Q8 | — | 40 | 1 | no | — |
| 233 | C2×D4 | — | 5 | 1 | no | — |
| 234 | C2×D4 | — | 10 | 1 | no | — |
| 235 | C2×D4 | — | 20 | 1 | no | — |
| 236 | C2×D4 | — | 20 | 1 | no | — |
| 237 | C22×C4 | — | 10 | 1 | no | — |
| 238 | C22×C4 | — | 20 | 1 | no | — |
| 239 | C22×C4 | — | 20 | 1 | no | — |
| 240 | Q16 | — | 2 | 1 | no | — |
| 241 | Q16 | — | 20 | 1 | no | — |
| 242 | Q16 | — | 20 | 1 | no | — |
| 243 | Q16 | — | 20 | 1 | no | — |
| 244 | Q16 | — | 20 | 2 | no | — |
| 245 | Q16 | — | 20 | 1 | no | — |
| 246 | Q16 | — | 20 | 1 | no | — |
| 247 | Q16 | D40 | 1 | 2 | yes | — |
| 248 | SD16 | — | 20 | 2 | no | — |
| 249 | SD16 | — | 20 | 1 | no | — |
| 250 | SD16 | — | 20 | 1 | no | — |
| 251 | D8 | — | 20 | 1 | no | — |
| 252 | M16 | — | 20 | 1 | no | — |
| 253 | M16 | — | 20 | 1 | no | — |
| 254 | C2×C8 | — | 4 | 1 | no | — |
| 255 | C2×C8 | — | 10 | 1 | no | — |
| 256 | C2×C8 | C2×D20 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 258 | C4⋊C4 | — | 2 | 1 | no | — |
| 259 | C4⋊C4 | — | 4 | 1 | no | — |
| 260 | C4⋊C4 | — | 5 | 1 | no | — |
| 261 | C4⋊C4 | — | 20 | 1 | no | — |
| 262 | C4⋊C4 | — | 20 | 1 | no | — |
| 263 | C4⋊C4 | — | 20 | 1 | no | — |
| 264 | C22⋊C4 | — | 10 | 1 | no | — |
| 265 | C22⋊C4 | — | 20 | 1 | no | — |
| 266 | C22⋊C4 | — | 20 | 1 | no | — |
| 267 | C22⋊C4 | — | 20 | 1 | no | — |
| 268 | C22⋊C4 | — | 20 | 1 | no | — |
| 269 | C42 | — | 4 | 1 | no | — |
| 270 | C42 | D4×D5 | 1 | 1 | yes | characteristic |
| 271 | C16 | — | 2 | 2 | no | — |
| 272 | C16 | — | 20 | 1 | no | — |
| 273 | C2×C5 | #1619 | 1 | 1 | yes | characteristic |
| 274 | C2×C5 | #2191 | 1 | 1 | yes | characteristic |
| 275 | C2×C5 | #2195 | 1 | 1 | yes | characteristic |
| 276 | D5 | — | 8 | 1 | no | — |
| 277 | D5 | — | 16 | 1 | no | — |
| 278 | C23 | — | 10 | 1 | no | — |
| 279 | C23 | — | 20 | 1 | no | — |
| 280 | Q8 | — | 2 | 2 | no | — |
| 281 | Q8 | — | 4 | 1 | no | — |
| 282 | Q8 | — | 20 | 1 | no | — |
| 283 | Q8 | — | 20 | 1 | no | — |
| 284 | Q8 | — | 40 | 2 | no | — |
| 285 | Q8 | — | 40 | 1 | no | — |
| 286 | Q8 | — | 40 | 1 | no | — |
| 287 | D4 | — | 20 | 1 | no | — |
| 288 | D4 | — | 20 | 1 | no | — |
| 289 | D4 | — | 20 | 1 | no | — |
| 290 | D4 | — | 40 | 2 | no | — |
| 291 | D4 | — | 40 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 1 | no | — |
| 293 | C2×C4 | — | 4 | 1 | no | — |
| 294 | C2×C4 | — | 10 | 1 | no | — |
| 295 | C2×C4 | — | 20 | 1 | no | — |
| 296 | C2×C4 | — | 20 | 1 | no | — |
| 297 | C2×C4 | — | 20 | 1 | no | — |
| 298 | C2×C4 | — | 40 | 2 | no | — |
| 299 | C2×C4 | — | 40 | 2 | no | — |
| 300 | C2×C4 | — | 40 | 1 | no | — |
| 301 | C2×C4 | #4028 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | D5×D8 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | #4060 | 1 | 1 | yes | characteristic |
| 304 | C8 | — | 2 | 1 | no | — |
| 305 | C8 | — | 8 | 1 | no | — |
| 306 | C8 | — | 20 | 1 | no | — |
| 307 | C8 | C2×D40 | 1 | 1 | yes | characteristic |
| 308 | C8 | #4054 | 1 | 1 | yes | characteristic |
| 309 | C5 | #12327 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 310 | C22 | — | 20 | 1 | no | — |
| 311 | C22 | — | 20 | 1 | no | — |
| 312 | C22 | — | 20 | 1 | no | — |
| 313 | C22 | — | 40 | 1 | no | — |
| 314 | C22 | — | 40 | 1 | no | — |
| 315 | C22 | — | 40 | 1 | no | — |
| 316 | C22 | #65062 | 1 | 1 | yes | characteristic |
| 317 | C4 | — | 2 | 1 | no | — |
| 318 | C4 | — | 2 | 1 | no | — |
| 319 | C4 | — | 4 | 2 | no | — |
| 320 | C4 | — | 8 | 1 | no | — |
| 321 | C4 | — | 20 | 1 | no | — |
| 322 | C4 | — | 40 | 1 | no | — |
| 323 | C4 | — | 80 | 1 | no | — |
| 324 | C4 | #65021 | 1 | 1 | yes | characteristic |
| 325 | C4 | #65111 | 1 | 1 | yes | characteristic |
| 326 | C2 | — | 40 | 1 | no | — |
| 327 | C2 | — | 80 | 1 | no | — |
| 328 | C2 | #10598987 | 1 | 1 | yes | characteristic |
| 329 | C2 | #10600046 | 1 | 1 | yes | characteristic |
| 330 | C2 | #10600062 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10601709 | C2 | semidirect |
| 3. #10599877 | C2 | semidirect |
| 4. #10599334 | C2 | semidirect |
| 5. C5×#1334 | C2 | semidirect |
| 6. #10598006 | C2 | semidirect |
| 7. #10597958 | C2 | semidirect |
| 71. #1334 | D5 | semidirect |
| 309. C5 | #12327 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10611461 | 2 |
| #10601709 | 2 |
| #10599877 | 2 |
| #10599334 | 2 |
| C5×#1334 | 2 |
| #10598006 | 2 |
| #10597958 | 2 |
| #12327 | 5 |