← #72216330
Subgroups — #72216330
Order 1024.
459 conjugacy classes of subgroups.
The subgroup lattice has 303 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#12067 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #151969 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#7390 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#7333 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #129164 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #94057 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #94030 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#2177 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#1580 | — | 2 | 1 | no | — |
| 10 | C2×#1419 | C4 | 1 | 1 | yes | characteristic |
| 11 | #13617 | C4 | 1 | 1 | yes | characteristic |
| 12 | C2×#1398 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#1398 | C22 | 1 | 1 | yes | characteristic |
| 14 | #12671 | C22 | 1 | 1 | yes | characteristic |
| 15 | #12665 | — | 2 | 1 | no | — |
| 16 | #12067 | — | 2 | 2 | no | — |
| 17 | #12067 | C4 | 1 | 4 | yes | complemented |
| 18 | C2×#1369 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#1368 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#1333 | — | 2 | 1 | no | — |
| 21 | C2×#1313 | C22 | 1 | 1 | yes | characteristic |
| 22 | #7477 | C4 | 1 | 1 | yes | characteristic |
| 23 | #7474 | C4 | 1 | 1 | yes | characteristic |
| 24 | #7390 | C4 | 1 | 4 | yes | complemented |
| 25 | #7333 | — | 2 | 2 | no | — |
| 26 | #7182 | — | 2 | 1 | no | — |
| 27 | C22×D16 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C2×C4×D8 | — | 2 | 1 | no | — |
| 29 | C2×C8×D4 | — | 4 | 1 | no | — |
| 30 | C2×M64 | — | 2 | 1 | no | — |
| 31 | C2×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C2×M64 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | #2177 | — | 2 | 1 | no | — |
| 34 | #2177 | — | 2 | 1 | no | — |
| 35 | #2177 | C2×C4 | 1 | 2 | yes | — |
| 36 | #2177 | C2×C4 | 1 | 2 | yes | — |
| 37 | C2×#368 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | #2158 | C2×C4 | 1 | 1 | yes | characteristic |
| 39 | C2×#367 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C2×#364 | — | 2 | 1 | no | — |
| 41 | C2×#357 | — | 2 | 1 | no | — |
| 42 | #2108 | — | 4 | 1 | no | — |
| 43 | C2×#346 | D4 | 1 | 1 | yes | characteristic |
| 44 | C2×#346 | C23 | 1 | 1 | yes | characteristic |
| 45 | #1772 | — | 2 | 1 | no | — |
| 46 | #1580 | — | 2 | 4 | no | — |
| 47 | #1580 | — | 4 | 2 | no | — |
| 48 | C2×#334 | — | 2 | 1 | no | — |
| 49 | C2×D4⋊C8 | — | 4 | 1 | no | — |
| 50 | C2×#322 | D4 | 1 | 1 | yes | characteristic |
| 51 | #1419 | C8 | 1 | 4 | yes | complemented |
| 52 | #1415 | C8 | 1 | 4 | yes | complemented |
| 53 | #1398 | — | 2 | 1 | no | — |
| 54 | #1398 | — | 2 | 1 | no | — |
| 55 | #1398 | — | 2 | 1 | no | — |
| 56 | #1398 | C2×C4 | 1 | 2 | yes | — |
| 57 | #1382 | — | 2 | 1 | no | — |
| 58 | #1379 | — | 4 | 1 | no | — |
| 59 | #1369 | — | 2 | 1 | no | — |
| 60 | #1369 | C2×C4 | 1 | 2 | yes | — |
| 61 | #1368 | — | 2 | 1 | no | — |
| 62 | #1368 | C2×C4 | 1 | 2 | yes | — |
| 63 | #1333 | — | 2 | 4 | no | — |
| 64 | #1313 | C2×C4 | 1 | 2 | yes | — |
| 65 | #1313 | D4 | 1 | 2 | yes | — |
| 66 | C22×D8 | — | 2 | 1 | no | — |
| 67 | C23×C8 | — | 8 | 1 | no | — |
| 68 | C2×C4×D4 | — | 4 | 1 | no | — |
| 69 | C2×D16 | — | 2 | 1 | no | — |
| 70 | C2×D16 | — | 2 | 1 | no | — |
| 71 | C2×D16 | — | 2 | 1 | no | — |
| 72 | C2×D16 | C42 | 1 | 2 | yes | — |
| 73 | C2×D16 | C2×C8 | 1 | 2 | yes | — |
| 74 | C2×D16 | C2×C8 | 1 | 2 | yes | — |
| 75 | C2×M32 | — | 4 | 1 | no | — |
| 76 | C22×C16 | — | 2 | 1 | no | — |
| 77 | C22×C16 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C22×C16 | #41 | 1 | 1 | yes | characteristic |
| 79 | C4×D8 | — | 2 | 2 | no | — |
| 80 | C4×D8 | — | 2 | 2 | no | — |
| 81 | C4×D8 | — | 4 | 1 | no | — |
| 82 | C4×D8 | — | 4 | 1 | no | — |
| 83 | C8×D4 | — | 4 | 4 | no | — |
| 84 | C8×D4 | — | 8 | 2 | no | — |
| 85 | C2×#108 | — | 2 | 1 | no | — |
| 86 | #427 | — | 2 | 1 | no | — |
| 87 | C2×#107 | — | 2 | 1 | no | — |
| 88 | C2×C4⋊C8 | — | 4 | 1 | no | — |
| 89 | C2×#102 | — | 4 | 1 | no | — |
| 90 | C2×C22⋊C8 | — | 8 | 1 | no | — |
| 91 | #405 | — | 4 | 1 | no | — |
| 92 | C2×C4×C8 | — | 2 | 1 | no | — |
| 93 | C2×C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | M64 | — | 2 | 1 | no | — |
| 95 | M64 | — | 2 | 1 | no | — |
| 96 | M64 | — | 2 | 1 | no | — |
| 97 | M64 | — | 2 | 2 | no | — |
| 98 | M64 | — | 4 | 1 | no | — |
| 99 | M64 | C2×C8 | 1 | 2 | yes | complemented |
| 100 | C2×C32 | — | 2 | 1 | no | — |
| 101 | C2×C32 | — | 4 | 1 | no | — |
| 102 | C2×C32 | C2×C8 | 1 | 2 | yes | complemented |
| 103 | #368 | — | 2 | 1 | no | — |
| 104 | #368 | C2×C8 | 1 | 2 | yes | complemented |
| 105 | #367 | — | 2 | 1 | no | — |
| 106 | #367 | — | 2 | 1 | no | — |
| 107 | #367 | C42 | 1 | 2 | yes | — |
| 108 | #366 | C2×C8 | 1 | 2 | yes | complemented |
| 109 | #364 | — | 2 | 2 | no | — |
| 110 | #364 | — | 4 | 1 | no | — |
| 111 | #357 | — | 2 | 2 | no | — |
| 112 | #357 | — | 2 | 2 | no | — |
| 113 | #346 | — | 2 | 1 | no | — |
| 114 | #346 | — | 8 | 1 | no | — |
| 115 | #346 | C22⋊C4 | 1 | 2 | yes | — |
| 116 | #346 | C22×C4 | 1 | 1 | yes | characteristic |
| 117 | #346 | C22×C4 | 1 | 1 | yes | characteristic |
| 118 | #346 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | #346 | #41 | 1 | 1 | yes | characteristic |
| 120 | C4×C16 | — | 8 | 1 | no | — |
| 121 | #339 | — | 4 | 1 | no | — |
| 122 | #334 | — | 2 | 2 | no | — |
| 123 | #334 | — | 4 | 1 | no | — |
| 124 | D4⋊C8 | — | 4 | 4 | no | — |
| 125 | #322 | — | 2 | 1 | no | — |
| 126 | #322 | C22⋊C4 | 1 | 2 | yes | — |
| 127 | C22×D4 | — | 4 | 1 | no | — |
| 128 | C23×C4 | — | 8 | 1 | no | — |
| 129 | C2×D8 | — | 2 | 2 | no | — |
| 130 | C2×D8 | — | 2 | 2 | no | — |
| 131 | C2×D8 | — | 2 | 2 | no | — |
| 132 | C2×D8 | — | 4 | 1 | no | — |
| 133 | C2×D8 | — | 4 | 1 | no | — |
| 134 | C2×D8 | — | 4 | 1 | no | — |
| 135 | C2×M16 | — | 4 | 1 | no | — |
| 136 | C22×C8 | — | 4 | 1 | no | — |
| 137 | C22×C8 | — | 8 | 4 | no | — |
| 138 | C22×C8 | — | 8 | 1 | no | — |
| 139 | C22×C8 | — | 16 | 4 | no | — |
| 140 | C22×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 141 | C22×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 142 | C4×D4 | — | 4 | 2 | no | — |
| 143 | C4×D4 | — | 4 | 2 | no | — |
| 144 | C4×D4 | — | 8 | 1 | no | — |
| 145 | C4×D4 | — | 8 | 1 | no | — |
| 146 | #117 | — | 4 | 1 | no | — |
| 147 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 148 | C2×C22⋊C4 | — | 8 | 1 | no | — |
| 149 | C2×C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 150 | D16 | — | 2 | 2 | no | — |
| 151 | D16 | — | 2 | 2 | no | — |
| 152 | D16 | — | 2 | 2 | no | — |
| 153 | D16 | C4×C8 | 1 | 4 | yes | — |
| 154 | M32 | — | 4 | 2 | no | — |
| 155 | M32 | — | 8 | 1 | no | — |
| 156 | C2×C16 | — | 2 | 1 | no | — |
| 157 | C2×C16 | — | 2 | 1 | no | — |
| 158 | C2×C16 | — | 2 | 2 | no | — |
| 159 | C2×C16 | — | 4 | 1 | no | — |
| 160 | C2×C16 | — | 4 | 1 | no | — |
| 161 | C2×C16 | — | 8 | 1 | no | — |
| 162 | C2×C16 | C2×C42 | 1 | 1 | yes | characteristic |
| 163 | C2×C16 | #117 | 1 | 1 | yes | characteristic |
| 164 | C2×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 165 | C2×C16 | C4×D4 | 1 | 1 | yes | characteristic |
| 166 | C2×C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 167 | C2×C16 | C22×C8 | 1 | 1 | yes | characteristic |
| 168 | C2×C16 | #131 | 1 | 1 | yes | characteristic |
| 169 | C2×C16 | #131 | 1 | 1 | yes | characteristic |
| 170 | #108 | — | 2 | 2 | no | — |
| 171 | #108 | — | 4 | 1 | no | — |
| 172 | #107 | — | 2 | 2 | no | — |
| 173 | #107 | — | 2 | 2 | no | — |
| 174 | #107 | — | 4 | 1 | no | — |
| 175 | #106 | — | 4 | 1 | no | — |
| 176 | C4⋊C8 | — | 4 | 2 | no | — |
| 177 | C4⋊C8 | — | 8 | 1 | no | — |
| 178 | #102 | — | 4 | 2 | no | — |
| 179 | #102 | — | 4 | 2 | no | — |
| 180 | C22⋊C8 | — | 8 | 4 | no | — |
| 181 | #97 | — | 8 | 1 | no | — |
| 182 | C4×C8 | — | 2 | 2 | no | — |
| 183 | C4×C8 | — | 4 | 1 | no | — |
| 184 | C4×C8 | — | 8 | 1 | no | — |
| 185 | C4×C8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 186 | C4×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 187 | C4×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 188 | C4×C8 | #137 | 1 | 1 | yes | characteristic |
| 189 | C32 | — | 2 | 2 | no | — |
| 190 | C32 | — | 2 | 2 | no | — |
| 191 | C32 | — | 4 | 2 | no | — |
| 192 | C24 | — | 8 | 1 | no | — |
| 193 | C2×D4 | — | 4 | 2 | no | — |
| 194 | C2×D4 | — | 4 | 2 | no | — |
| 195 | C2×D4 | — | 4 | 2 | no | — |
| 196 | C2×D4 | — | 8 | 1 | no | — |
| 197 | C2×D4 | — | 8 | 1 | no | — |
| 198 | C2×D4 | — | 8 | 1 | no | — |
| 199 | C22×C4 | — | 2 | 1 | no | — |
| 200 | C22×C4 | — | 8 | 2 | no | — |
| 201 | C22×C4 | — | 8 | 2 | no | — |
| 202 | C22×C4 | — | 8 | 1 | no | — |
| 203 | C22×C4 | — | 16 | 2 | no | — |
| 204 | C22×C4 | — | 16 | 2 | no | — |
| 205 | C22×C4 | #442 | 1 | 1 | yes | characteristic |
| 206 | D8 | — | 2 | 4 | no | — |
| 207 | D8 | — | 4 | 2 | no | — |
| 208 | D8 | — | 4 | 2 | no | — |
| 209 | D8 | — | 4 | 2 | no | — |
| 210 | M16 | — | 4 | 2 | no | — |
| 211 | M16 | — | 8 | 1 | no | — |
| 212 | C2×C8 | — | 2 | 1 | no | — |
| 213 | C2×C8 | — | 2 | 1 | no | — |
| 214 | C2×C8 | — | 4 | 2 | no | — |
| 215 | C2×C8 | — | 8 | 2 | no | — |
| 216 | C2×C8 | — | 8 | 1 | no | — |
| 217 | C2×C8 | — | 8 | 2 | no | — |
| 218 | C2×C8 | — | 16 | 8 | no | — |
| 219 | C2×C8 | — | 16 | 2 | no | — |
| 220 | C2×C8 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 221 | C2×C8 | #417 | 1 | 1 | yes | characteristic |
| 222 | C2×C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 223 | C2×C8 | C8×D4 | 1 | 1 | yes | characteristic |
| 224 | C2×C8 | #440 | 1 | 1 | yes | characteristic |
| 225 | C2×C8 | #442 | 1 | 1 | yes | characteristic |
| 226 | C2×C8 | #444 | 1 | 1 | yes | characteristic |
| 227 | C2×C8 | #444 | 1 | 1 | yes | characteristic |
| 228 | C4⋊C4 | — | 4 | 2 | no | — |
| 229 | C4⋊C4 | — | 4 | 2 | no | — |
| 230 | C4⋊C4 | — | 8 | 1 | no | — |
| 231 | C22⋊C4 | — | 8 | 2 | no | — |
| 232 | C22⋊C4 | — | 8 | 2 | no | — |
| 233 | C22⋊C4 | — | 8 | 1 | no | — |
| 234 | C42 | — | 2 | 1 | no | — |
| 235 | C42 | — | 8 | 1 | no | — |
| 236 | C42 | #418 | 1 | 1 | yes | characteristic |
| 237 | C42 | #440 | 1 | 1 | yes | characteristic |
| 238 | C16 | — | 2 | 1 | no | — |
| 239 | C16 | — | 2 | 1 | no | — |
| 240 | C16 | — | 4 | 2 | no | — |
| 241 | C16 | — | 8 | 2 | no | — |
| 242 | C16 | C2×C4×C8 | 1 | 1 | yes | characteristic |
| 243 | C16 | #405 | 1 | 1 | yes | characteristic |
| 244 | C16 | C8×D4 | 1 | 1 | yes | characteristic |
| 245 | C16 | C8×D4 | 1 | 1 | yes | characteristic |
| 246 | C23 | — | 8 | 2 | no | — |
| 247 | C23 | — | 8 | 2 | no | — |
| 248 | C23 | — | 8 | 2 | no | — |
| 249 | C23 | — | 16 | 2 | no | — |
| 250 | C23 | — | 16 | 2 | no | — |
| 251 | C23 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 252 | D4 | — | 4 | 4 | no | — |
| 253 | D4 | — | 8 | 2 | no | — |
| 254 | D4 | — | 8 | 2 | no | — |
| 255 | D4 | — | 8 | 2 | no | — |
| 256 | C2×C4 | — | 2 | 1 | no | — |
| 257 | C2×C4 | — | 2 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 2 | no | — |
| 259 | C2×C4 | — | 8 | 2 | no | — |
| 260 | C2×C4 | — | 8 | 2 | no | — |
| 261 | C2×C4 | — | 16 | 2 | no | — |
| 262 | C2×C4 | — | 16 | 2 | no | — |
| 263 | C2×C4 | — | 16 | 4 | no | — |
| 264 | C2×C4 | — | 16 | 2 | no | — |
| 265 | C2×C4 | #1580 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | #1580 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | #1762 | 1 | 1 | yes | characteristic |
| 268 | C2×C4 | #2180 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | AGL(1,ℤ/16) | 1 | 1 | yes | characteristic |
| 271 | C8 | — | 2 | 1 | no | — |
| 272 | C8 | — | 2 | 1 | no | — |
| 273 | C8 | — | 8 | 2 | no | — |
| 274 | C8 | — | 8 | 2 | no | — |
| 275 | C8 | — | 16 | 2 | no | — |
| 276 | C8 | SD16⋊C8 | 1 | 1 | yes | characteristic |
| 277 | C8 | SD16⋊C8 | 1 | 1 | yes | characteristic |
| 278 | C8 | C8×C22⋊C4 | 1 | 1 | yes | characteristic |
| 279 | C8 | #1765 | 1 | 1 | yes | characteristic |
| 280 | C22 | — | 2 | 2 | no | — |
| 281 | C22 | — | 8 | 4 | no | — |
| 282 | C22 | — | 16 | 2 | no | — |
| 283 | C22 | — | 16 | 2 | no | — |
| 284 | C22 | — | 16 | 4 | no | — |
| 285 | C22 | — | 16 | 4 | no | — |
| 286 | C22 | #12065 | 1 | 1 | yes | characteristic |
| 287 | C22 | #12067 | 1 | 1 | yes | characteristic |
| 288 | C22 | #12647 | 1 | 1 | yes | characteristic |
| 289 | C4 | — | 4 | 1 | no | — |
| 290 | C4 | — | 4 | 1 | no | — |
| 291 | C4 | — | 16 | 2 | no | — |
| 292 | C4 | — | 16 | 2 | no | — |
| 293 | C4 | #8200 | 1 | 1 | yes | characteristic |
| 294 | C4 | #12065 | 1 | 1 | yes | characteristic |
| 295 | C4 | #12067 | 1 | 1 | yes | characteristic |
| 296 | C4 | #12650 | 1 | 1 | yes | characteristic |
| 297 | C2 | — | 2 | 1 | no | — |
| 298 | C2 | — | 2 | 1 | no | — |
| 299 | C2 | — | 16 | 4 | no | — |
| 300 | C2 | #128688 | 1 | 1 | yes | characteristic |
| 301 | C2 | AGL(1,ℤ/32) | 1 | 2 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta