Finite Group DatabasebetaFinite Group DatabasebetaOrder 1280.
362 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 61 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 | #10611624 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #10601720 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10600099 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10598891 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C5×#1336 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #10598061 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #10597934 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #66104 | — | 2 | 1 | no | — |
| 9 | #66085 | — | 2 | 1 | no | — |
| 10 | C5×#443 | C22 | 1 | 1 | yes | characteristic |
| 11 | #65484 | — | 2 | 1 | no | — |
| 12 | #65430 | — | 2 | 1 | no | — |
| 13 | #65198 | — | 2 | 1 | no | — |
| 14 | #65187 | — | 2 | 1 | no | — |
| 15 | #65115 | — | 2 | 1 | no | — |
| 16 | #65111 | — | 2 | 1 | no | — |
| 17 | #64989 | C22 | 1 | 1 | yes | characteristic |
| 18 | #64988 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | #64983 | — | 2 | 1 | no | — |
| 20 | #64982 | C22 | 1 | 1 | yes | characteristic |
| 21 | #64971 | — | 2 | 1 | no | — |
| 22 | C5×#364 | C22 | 1 | 1 | yes | characteristic, complemented |
| 23 | C5×#322 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #64737 | — | 2 | 1 | no | — |
| 25 | #64716 | — | 2 | 1 | no | — |
| 26 | #64683 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | #12315 | — | 5 | 1 | no | maximal, Sylow |
| 28 | #4150 | — | 2 | 1 | no | — |
| 29 | #4149 | — | 2 | 1 | no | — |
| 30 | C5×#135 | D4 | 1 | 1 | yes | characteristic |
| 31 | C5×#131 | — | 2 | 1 | no | — |
| 32 | #4097 | — | 2 | 1 | no | — |
| 33 | #4095 | — | 2 | 1 | no | — |
| 34 | #4064 | — | 4 | 1 | no | — |
| 35 | #4062 | — | 4 | 1 | no | — |
| 36 | D5×SD16 | — | 4 | 1 | no | — |
| 37 | #4057 | — | 4 | 1 | no | — |
| 38 | #4055 | — | 2 | 1 | no | — |
| 39 | #4055 | — | 4 | 1 | no | — |
| 40 | #4054 | — | 2 | 1 | no | — |
| 41 | #4054 | — | 4 | 1 | no | — |
| 42 | C2×Dic20 | — | 2 | 1 | no | — |
| 43 | C2×Dic20 | — | 2 | 1 | no | — |
| 44 | #4050 | — | 4 | 1 | no | — |
| 45 | #4050 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×#677 | — | 2 | 1 | no | — |
| 47 | C2×#677 | — | 2 | 1 | no | — |
| 48 | C2×#677 | — | 4 | 1 | no | — |
| 49 | #4047 | — | 2 | 1 | no | — |
| 50 | #4020 | D4 | 1 | 1 | yes | characteristic |
| 51 | #4015 | — | 2 | 1 | no | — |
| 52 | #4015 | D4 | 1 | 1 | yes | characteristic |
| 53 | C5×M32 | — | 2 | 1 | no | — |
| 54 | C5×#108 | D4 | 1 | 1 | yes | characteristic |
| 55 | C5×#104 | — | 2 | 1 | no | — |
| 56 | C4×C5×C8 | — | 2 | 1 | no | — |
| 57 | C4×C5×C8 | C23 | 1 | 1 | yes | characteristic |
| 58 | #3961 | — | 4 | 1 | no | — |
| 59 | #3959 | — | 4 | 1 | no | — |
| 60 | #3956 | — | 2 | 1 | no | — |
| 61 | #3955 | — | 2 | 1 | no | — |
| 62 | #3953 | — | 4 | 2 | no | — |
| 63 | #3951 | D4 | 1 | 1 | yes | characteristic |
| 64 | #3950 | — | 4 | 2 | no | — |
| 65 | #3944 | — | 2 | 1 | no | — |
| 66 | #3937 | — | 2 | 1 | no | — |
| 67 | Dic40 | — | 4 | 1 | no | — |
| 68 | #3932 | — | 4 | 1 | no | — |
| 69 | #3291 | — | 5 | 1 | no | — |
| 70 | #2214 | — | 5 | 1 | no | — |
| 71 | #2214 | — | 5 | 1 | no | — |
| 72 | #1722 | — | 5 | 1 | no | — |
| 73 | #1359 | — | 5 | 1 | no | — |
| 74 | #1336 | — | 5 | 1 | no | — |
| 75 | #1336 | D5 | 1 | 1 | yes | characteristic, complemented |
| 76 | C5×#41 | — | 2 | 1 | no | — |
| 77 | #713 | — | 4 | 1 | no | — |
| 78 | Q8×D5 | — | 4 | 1 | no | — |
| 79 | #711 | — | 4 | 1 | no | — |
| 80 | #711 | — | 8 | 1 | no | — |
| 81 | D4×D5 | — | 4 | 1 | no | — |
| 82 | D4×D5 | — | 8 | 1 | no | — |
| 83 | #709 | — | 2 | 1 | no | — |
| 84 | #709 | — | 4 | 1 | no | — |
| 85 | #709 | — | 4 | 1 | no | — |
| 86 | C2×D20 | — | 2 | 1 | no | — |
| 87 | C2×D20 | — | 4 | 1 | no | — |
| 88 | C2×D20 | — | 4 | 1 | no | — |
| 89 | C2×Dic10 | — | 2 | 1 | no | — |
| 90 | C2×Dic10 | — | 4 | 1 | no | — |
| 91 | C2×Dic10 | — | 4 | 1 | no | — |
| 92 | C5×Q16 | D8 | 1 | 1 | yes | characteristic |
| 93 | C5×SD16 | — | 2 | 1 | no | — |
| 94 | C5×D8 | D8 | 1 | 1 | yes | characteristic |
| 95 | C5×M16 | — | 2 | 1 | no | — |
| 96 | C5×M16 | — | 4 | 1 | no | — |
| 97 | C2×C5×C8 | — | 2 | 2 | no | — |
| 98 | C2×C5×C8 | — | 4 | 1 | no | — |
| 99 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C2×C5×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C42×C5 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | #689 | — | 4 | 1 | no | — |
| 103 | #688 | — | 4 | 1 | no | — |
| 104 | #687 | — | 4 | 1 | no | — |
| 105 | #686 | — | 4 | 1 | no | — |
| 106 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 107 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 108 | C4⋊Dic5 | — | 4 | 1 | no | — |
| 109 | #681 | — | 2 | 1 | no | — |
| 110 | Dic20 | — | 4 | 1 | no | — |
| 111 | Dic20 | — | 4 | 1 | no | — |
| 112 | Dic20 | — | 4 | 1 | no | — |
| 113 | Dic20 | — | 4 | 1 | no | — |
| 114 | Dic20 | — | 4 | 1 | no | — |
| 115 | Dic20 | D8 | 1 | 1 | yes | characteristic |
| 116 | D40 | — | 4 | 1 | no | — |
| 117 | D40 | D8 | 1 | 1 | yes | characteristic |
| 118 | #677 | — | 2 | 1 | no | — |
| 119 | #677 | — | 4 | 1 | no | — |
| 120 | #677 | — | 4 | 1 | no | — |
| 121 | #677 | — | 4 | 1 | no | — |
| 122 | #677 | — | 4 | 1 | no | — |
| 123 | #677 | — | 4 | 1 | no | — |
| 124 | #677 | — | 4 | 1 | no | — |
| 125 | #676 | — | 4 | 1 | no | — |
| 126 | C8×D5 | — | 4 | 1 | no | — |
| 127 | C5×C16 | — | 4 | 1 | no | — |
| 128 | #672 | — | 4 | 1 | no | — |
| 129 | #577 | — | 10 | 1 | no | — |
| 130 | #577 | — | 10 | 1 | no | — |
| 131 | #510 | — | 10 | 1 | no | — |
| 132 | #510 | — | 10 | 1 | no | — |
| 133 | #500 | — | 10 | 1 | no | — |
| 134 | #494 | — | 5 | 1 | no | — |
| 135 | #492 | — | 5 | 1 | no | — |
| 136 | #486 | — | 10 | 1 | no | — |
| 137 | #471 | — | 10 | 1 | no | — |
| 138 | #471 | — | 10 | 1 | no | — |
| 139 | #456 | — | 10 | 1 | no | — |
| 140 | #453 | — | 10 | 1 | no | — |
| 141 | #443 | — | 5 | 1 | no | — |
| 142 | #443 | C2×D5 | 1 | 1 | yes | characteristic |
| 143 | #364 | — | 5 | 1 | no | — |
| 144 | #364 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 145 | #362 | — | 10 | 1 | no | — |
| 146 | #362 | — | 10 | 1 | no | — |
| 147 | #322 | C2×D5 | 1 | 1 | yes | characteristic, complemented |
| 148 | C22×D5 | — | 4 | 1 | no | — |
| 149 | C22×D5 | — | 8 | 1 | no | — |
| 150 | C5×Q8 | — | 2 | 1 | no | — |
| 151 | C5×D4 | — | 2 | 1 | no | — |
| 152 | C5×D4 | — | 4 | 1 | no | — |
| 153 | C2×C4×C5 | — | 2 | 1 | no | — |
| 154 | C2×C4×C5 | — | 4 | 1 | no | — |
| 155 | C2×C4×C5 | #120 | 1 | 1 | yes | characteristic |
| 156 | C22⋊D5 | — | 4 | 1 | no | — |
| 157 | C22⋊D5 | — | 8 | 1 | no | — |
| 158 | C22⋊D5 | — | 8 | 1 | no | — |
| 159 | C2×Dic5 | — | 4 | 1 | no | — |
| 160 | C2×Dic5 | — | 8 | 1 | no | — |
| 161 | C2×Dic5 | — | 8 | 1 | no | — |
| 162 | D20 | — | 2 | 1 | no | — |
| 163 | D20 | — | 4 | 1 | no | — |
| 164 | D20 | — | 4 | 1 | no | — |
| 165 | D20 | — | 8 | 2 | no | — |
| 166 | D20 | — | 8 | 1 | no | — |
| 167 | C4×D5 | — | 4 | 1 | no | — |
| 168 | C4×D5 | — | 8 | 1 | no | — |
| 169 | C4×D5 | — | 8 | 1 | no | — |
| 170 | Dic10 | — | 2 | 1 | no | — |
| 171 | Dic10 | — | 4 | 1 | no | — |
| 172 | Dic10 | — | 4 | 1 | no | — |
| 173 | Dic10 | — | 8 | 1 | no | — |
| 174 | Dic10 | — | 8 | 1 | no | — |
| 175 | Dic10 | — | 8 | 1 | no | — |
| 176 | C5×C8 | — | 4 | 2 | no | — |
| 177 | C5×C8 | — | 4 | 1 | no | — |
| 178 | C5×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 179 | C5×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 180 | C5×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 181 | C5×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 182 | #168 | — | 4 | 1 | no | — |
| 183 | 2−1+4 | — | 10 | 1 | no | — |
| 184 | H(5,𝔽2) | — | 10 | 1 | no | — |
| 185 | #137 | — | 10 | 1 | no | — |
| 186 | #137 | — | 10 | 1 | no | — |
| 187 | #137 | — | 20 | 1 | no | — |
| 188 | #137 | — | 20 | 1 | no | — |
| 189 | AGL(1,ℤ/8) | — | 10 | 1 | no | — |
| 190 | AGL(1,ℤ/8) | — | 10 | 1 | no | — |
| 191 | AGL(1,ℤ/8) | — | 20 | 1 | no | — |
| 192 | AGL(1,ℤ/8) | — | 20 | 1 | no | — |
| 193 | #135 | — | 5 | 1 | no | — |
| 194 | #135 | — | 20 | 1 | no | — |
| 195 | #135 | — | 20 | 1 | no | — |
| 196 | #135 | D20 | 1 | 1 | yes | characteristic |
| 197 | C2×Q16 | — | 10 | 1 | no | — |
| 198 | C2×Q16 | — | 10 | 1 | no | — |
| 199 | C2×SD16 | — | 10 | 1 | no | — |
| 200 | C2×SD16 | — | 10 | 1 | no | — |
| 201 | C2×SD16 | — | 20 | 1 | no | — |
| 202 | C2×SD16 | — | 20 | 1 | no | — |
| 203 | #131 | — | 2 | 1 | no | — |
| 204 | #131 | — | 10 | 1 | no | — |
| 205 | C4⋊Q8 | — | 5 | 1 | no | — |
| 206 | C4⋊Q8 | — | 10 | 1 | no | — |
| 207 | #127 | — | 5 | 1 | no | — |
| 208 | Q32 | — | 20 | 1 | no | — |
| 209 | Q32 | — | 20 | 1 | no | — |
| 210 | SD32 | — | 20 | 1 | no | — |
| 211 | SD32 | — | 20 | 1 | no | — |
| 212 | M32 | — | 2 | 1 | no | — |
| 213 | M32 | — | 10 | 1 | no | — |
| 214 | #108 | — | 5 | 1 | no | — |
| 215 | #108 | D20 | 1 | 1 | yes | characteristic |
| 216 | #107 | — | 20 | 2 | no | — |
| 217 | #104 | — | 2 | 1 | no | — |
| 218 | #104 | — | 10 | 1 | no | — |
| 219 | #102 | — | 20 | 2 | no | — |
| 220 | #101 | — | 10 | 1 | no | — |
| 221 | #100 | — | 10 | 1 | no | — |
| 222 | C4×C8 | — | 2 | 1 | no | — |
| 223 | C4×C8 | C22×D5 | 1 | 1 | yes | characteristic |
| 224 | C22×C5 | — | 4 | 1 | no | — |
| 225 | C22×C5 | #453 | 1 | 1 | yes | characteristic |
| 226 | C2×D5 | — | 4 | 1 | no | — |
| 227 | C2×D5 | — | 8 | 1 | no | — |
| 228 | C2×D5 | — | 16 | 1 | no | — |
| 229 | C2×D5 | — | 32 | 1 | no | — |
| 230 | C4×C5 | — | 2 | 2 | no | — |
| 231 | C4×C5 | — | 4 | 1 | no | — |
| 232 | C4×C5 | #447 | 1 | 1 | yes | characteristic |
| 233 | C4×C5 | #447 | 1 | 1 | yes | characteristic |
| 234 | Dic5 | — | 4 | 1 | no | — |
| 235 | Dic5 | — | 8 | 1 | no | — |
| 236 | Dic5 | — | 16 | 1 | no | — |
| 237 | #41 | — | 2 | 1 | no | — |
| 238 | #41 | — | 10 | 1 | no | — |
| 239 | #41 | — | 20 | 1 | no | — |
| 240 | #41 | — | 20 | 1 | no | — |
| 241 | #41 | — | 20 | 1 | no | — |
| 242 | #41 | — | 20 | 1 | no | — |
| 243 | #41 | — | 40 | 1 | no | — |
| 244 | C2×Q8 | — | 10 | 1 | no | — |
| 245 | C2×Q8 | — | 20 | 1 | no | — |
| 246 | C2×Q8 | — | 20 | 1 | no | — |
| 247 | C2×Q8 | — | 20 | 1 | no | — |
| 248 | C2×D4 | — | 10 | 1 | no | — |
| 249 | C2×D4 | — | 20 | 1 | no | — |
| 250 | C2×D4 | — | 20 | 1 | no | — |
| 251 | C2×D4 | — | 20 | 1 | no | — |
| 252 | C2×D4 | — | 40 | 1 | no | — |
| 253 | Q16 | — | 5 | 1 | no | — |
| 254 | Q16 | — | 20 | 1 | no | — |
| 255 | Q16 | — | 20 | 1 | no | — |
| 256 | Q16 | — | 20 | 1 | no | — |
| 257 | Q16 | — | 20 | 1 | no | — |
| 258 | Q16 | — | 20 | 1 | no | — |
| 259 | Q16 | — | 20 | 1 | no | — |
| 260 | Q16 | D40 | 1 | 1 | yes | characteristic |
| 261 | SD16 | — | 2 | 1 | no | — |
| 262 | SD16 | — | 10 | 1 | no | — |
| 263 | SD16 | — | 20 | 1 | no | — |
| 264 | SD16 | — | 20 | 1 | no | — |
| 265 | SD16 | — | 20 | 1 | no | — |
| 266 | SD16 | — | 20 | 1 | no | — |
| 267 | SD16 | — | 20 | 1 | no | — |
| 268 | SD16 | — | 20 | 1 | no | — |
| 269 | SD16 | — | 20 | 1 | no | — |
| 270 | SD16 | — | 20 | 1 | no | — |
| 271 | D8 | — | 5 | 1 | no | — |
| 272 | D8 | — | 20 | 1 | no | — |
| 273 | D8 | — | 20 | 1 | no | — |
| 274 | D8 | D40 | 1 | 1 | yes | characteristic |
| 275 | M16 | — | 2 | 1 | no | — |
| 276 | M16 | — | 4 | 1 | no | — |
| 277 | M16 | — | 10 | 1 | no | — |
| 278 | M16 | — | 20 | 1 | no | — |
| 279 | C2×C8 | — | 2 | 2 | no | — |
| 280 | C2×C8 | — | 4 | 1 | no | — |
| 281 | C2×C8 | — | 20 | 1 | no | — |
| 282 | C2×C8 | C2×D20 | 1 | 1 | yes | characteristic |
| 283 | C2×C8 | D4×D5 | 1 | 1 | yes | characteristic |
| 284 | C4⋊C4 | — | 20 | 1 | no | — |
| 285 | C4⋊C4 | — | 20 | 1 | no | — |
| 286 | C4⋊C4 | — | 20 | 1 | no | — |
| 287 | C42 | D4×D5 | 1 | 1 | yes | characteristic |
| 288 | C16 | — | 4 | 1 | no | — |
| 289 | C16 | — | 20 | 1 | no | — |
| 290 | C2×C5 | — | 2 | 1 | no | — |
| 291 | C2×C5 | — | 8 | 1 | no | — |
| 292 | C2×C5 | #1619 | 1 | 1 | yes | characteristic |
| 293 | D5 | — | 8 | 1 | no | — |
| 294 | D5 | — | 16 | 1 | no | — |
| 295 | C23 | — | 20 | 1 | no | — |
| 296 | C23 | — | 40 | 1 | no | — |
| 297 | Q8 | — | 2 | 1 | no | — |
| 298 | Q8 | — | 10 | 1 | no | — |
| 299 | Q8 | — | 20 | 1 | no | — |
| 300 | Q8 | — | 20 | 1 | no | — |
| 301 | Q8 | — | 40 | 1 | no | — |
| 302 | Q8 | — | 40 | 1 | no | — |
| 303 | Q8 | — | 40 | 1 | no | — |
| 304 | D4 | — | 2 | 1 | no | — |
| 305 | D4 | — | 4 | 1 | no | — |
| 306 | D4 | — | 10 | 1 | no | — |
| 307 | D4 | — | 20 | 1 | no | — |
| 308 | D4 | — | 20 | 1 | no | — |
| 309 | D4 | — | 20 | 1 | no | — |
| 310 | D4 | — | 40 | 2 | no | — |
| 311 | D4 | — | 40 | 1 | no | — |
| 312 | D4 | — | 40 | 1 | no | — |
| 313 | D4 | — | 40 | 1 | no | — |
| 314 | C2×C4 | — | 2 | 1 | no | — |
| 315 | C2×C4 | — | 4 | 1 | no | — |
| 316 | C2×C4 | — | 20 | 1 | no | — |
| 317 | C2×C4 | — | 20 | 1 | no | — |
| 318 | C2×C4 | — | 40 | 1 | no | — |
| 319 | C2×C4 | — | 40 | 1 | no | — |
| 320 | C2×C4 | — | 40 | 1 | no | — |
| 321 | C2×C4 | — | 40 | 1 | no | — |
| 322 | C2×C4 | #4028 | 1 | 1 | yes | characteristic |
| 323 | C8 | — | 4 | 2 | no | — |
| 324 | C8 | — | 4 | 1 | no | — |
| 325 | C8 | — | 20 | 1 | no | — |
| 326 | C8 | C2×D40 | 1 | 1 | yes | characteristic |
| 327 | C8 | #4054 | 1 | 1 | yes | characteristic |
| 328 | C8 | D5×D8 | 1 | 1 | yes | characteristic |
| 329 | C8 | #4060 | 1 | 1 | yes | characteristic |
| 330 | C5 | #12315 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 331 | C22 | — | 4 | 1 | no | — |
| 332 | C22 | — | 20 | 1 | no | — |
| 333 | C22 | — | 40 | 1 | no | — |
| 334 | C22 | — | 80 | 1 | no | — |
| 335 | C22 | — | 160 | 1 | no | — |
| 336 | C22 | #65111 | 1 | 1 | yes | characteristic |
| 337 | C4 | — | 2 | 2 | no | — |
| 338 | C4 | — | 4 | 1 | no | — |
| 339 | C4 | — | 20 | 1 | no | — |
| 340 | C4 | — | 40 | 1 | no | — |
| 341 | C4 | — | 80 | 1 | no | — |
| 342 | C4 | #65021 | 1 | 1 | yes | characteristic |
| 343 | C4 | #65062 | 1 | 1 | yes | characteristic |
| 344 | C2 | — | 2 | 1 | no | — |
| 345 | C2 | — | 8 | 1 | no | — |
| 346 | C2 | — | 40 | 1 | no | — |
| 347 | C2 | — | 80 | 1 | no | — |
| 348 | C2 | #10598987 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #10601720 | C2 | semidirect |
| 3. #10600099 | C2 | semidirect |
| 4. #10598891 | C2 | semidirect |
| 5. C5×#1336 | C2 | semidirect |
| 6. #10598061 | C2 | semidirect |
| 7. #10597934 | C2 | semidirect |
| 18. #64988 | C22 | semidirect |
| 22. C5×#364 | C22 | semidirect |
| 23. C5×#322 | C22 | semidirect |
| 26. #64683 | C22 | semidirect |
| 75. #1336 | D5 | semidirect |
| 144. #364 | C2×D5 | semidirect |
| 147. #322 | C2×D5 | semidirect |
| 330. C5 | #12315 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10611624 | 2 |
| #10601720 | 2 |
| #10600099 | 2 |
| #10598891 | 2 |
| C5×#1336 | 2 |
| #10598061 | 2 |
| #10597934 | 2 |
| #12315 | 5 |