← #213305
Subgroups — #213305
Order 512.
650 conjugacy classes of subgroups.
The subgroup lattice has 298 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22234 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #19613 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #17976 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #17962 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #17961 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #9858 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #9853 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #9852 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #9843 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #8825 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #7843 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 13 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3167 | C22 | 1 | 2 | yes | — |
| 15 | #3131 | C22 | 1 | 2 | yes | — |
| 16 | #2960 | C22 | 1 | 2 | yes | — |
| 17 | #2931 | C22 | 1 | 2 | yes | — |
| 18 | #1994 | C22 | 1 | 2 | yes | — |
| 19 | #1962 | C22 | 1 | 2 | yes | — |
| 20 | #1929 | — | 2 | 4 | no | — |
| 21 | #1866 | — | 2 | 4 | no | — |
| 22 | #1848 | C22 | 1 | 2 | yes | — |
| 23 | #1744 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1744 | C22 | 1 | 1 | yes | characteristic |
| 25 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1720 | C4 | 1 | 2 | yes | complemented |
| 27 | #1719 | C4 | 1 | 2 | yes | complemented |
| 28 | #1718 | C4 | 1 | 2 | yes | complemented |
| 29 | #1717 | C4 | 1 | 2 | yes | complemented |
| 30 | #1594 | — | 2 | 2 | no | — |
| 31 | #1593 | — | 2 | 2 | no | — |
| 32 | #1583 | — | 2 | 4 | no | — |
| 33 | #1582 | — | 2 | 4 | no | — |
| 34 | #1580 | — | 2 | 2 | no | — |
| 35 | #1579 | — | 2 | 2 | no | — |
| 36 | #1566 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1566 | C22 | 1 | 1 | yes | characteristic |
| 38 | #1566 | C22 | 1 | 2 | yes | — |
| 39 | #1493 | C22 | 1 | 2 | yes | — |
| 40 | #1493 | C22 | 1 | 2 | yes | — |
| 41 | #1492 | C22 | 1 | 2 | yes | — |
| 42 | #1492 | C22 | 1 | 2 | yes | — |
| 43 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 44 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 45 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 46 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 47 | C2×#131 | — | 2 | 2 | no | — |
| 48 | #533 | C23 | 1 | 2 | yes | — |
| 49 | #532 | C23 | 1 | 2 | yes | — |
| 50 | #500 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | #500 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | #498 | C2×C4 | 1 | 2 | yes | — |
| 53 | #494 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | #493 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #492 | C2×C4 | 1 | 2 | yes | — |
| 56 | #488 | — | 2 | 4 | no | — |
| 57 | #487 | C2×C4 | 1 | 2 | yes | — |
| 58 | #487 | C2×C4 | 1 | 2 | yes | — |
| 59 | #487 | C2×C4 | 1 | 2 | yes | — |
| 60 | #487 | C2×C4 | 1 | 2 | yes | — |
| 61 | #486 | C2×C4 | 1 | 2 | yes | — |
| 62 | #486 | C2×C4 | 1 | 2 | yes | — |
| 63 | #486 | C2×C4 | 1 | 2 | yes | — |
| 64 | #486 | C2×C4 | 1 | 2 | yes | — |
| 65 | #467 | — | 2 | 2 | no | — |
| 66 | #466 | — | 2 | 2 | no | — |
| 67 | #465 | — | 2 | 4 | no | — |
| 68 | #446 | — | 2 | 2 | no | — |
| 69 | #446 | — | 2 | 2 | no | — |
| 70 | C8×Q8 | — | 2 | 2 | no | — |
| 71 | C8×Q8 | — | 2 | 2 | no | — |
| 72 | C4×Q16 | — | 2 | 2 | no | — |
| 73 | C4×SD16 | — | 2 | 4 | no | — |
| 74 | C4×D8 | — | 2 | 2 | no | — |
| 75 | #436 | — | 2 | 2 | no | — |
| 76 | #436 | — | 2 | 2 | no | — |
| 77 | C8×D4 | — | 2 | 2 | no | — |
| 78 | C8×D4 | — | 2 | 2 | no | — |
| 79 | C2×#108 | — | 2 | 4 | no | — |
| 80 | #424 | — | 2 | 2 | no | — |
| 81 | #423 | C23 | 1 | 2 | yes | — |
| 82 | #423 | C23 | 1 | 2 | yes | — |
| 83 | C2×#104 | — | 2 | 4 | no | — |
| 84 | #416 | — | 2 | 4 | no | — |
| 85 | #408 | — | 2 | 2 | no | — |
| 86 | #405 | — | 2 | 2 | no | — |
| 87 | C2×C4×C8 | — | 2 | 2 | no | — |
| 88 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 94 | C43 | C23 | 1 | 1 | yes | characteristic |
| 95 | #339 | — | 2 | 4 | no | — |
| 96 | #336 | — | 2 | 2 | no | — |
| 97 | #336 | D4 | 1 | 4 | yes | — |
| 98 | #334 | — | 2 | 2 | no | — |
| 99 | #334 | — | 2 | 2 | no | — |
| 100 | #334 | — | 2 | 4 | no | — |
| 101 | Q8⋊C8 | — | 2 | 4 | no | — |
| 102 | Q8⋊C8 | — | 2 | 4 | no | — |
| 103 | D4⋊C8 | — | 2 | 4 | no | — |
| 104 | D4⋊C8 | — | 2 | 4 | no | — |
| 105 | #322 | — | 2 | 2 | no | — |
| 106 | #322 | C2×C4 | 1 | 2 | yes | — |
| 107 | #322 | C2×C4 | 1 | 2 | yes | — |
| 108 | #322 | D4 | 1 | 2 | yes | — |
| 109 | #322 | D4 | 1 | 2 | yes | — |
| 110 | #322 | D4 | 1 | 4 | yes | — |
| 111 | C2×#41 | — | 2 | 2 | no | — |
| 112 | C2×Q16 | — | 2 | 2 | no | — |
| 113 | C2×SD16 | — | 2 | 4 | no | — |
| 114 | C2×D8 | — | 2 | 2 | no | — |
| 115 | #131 | — | 8 | 4 | no | — |
| 116 | C2×M16 | — | 2 | 2 | no | — |
| 117 | C2×M16 | — | 2 | 2 | no | — |
| 118 | C2×M16 | — | 4 | 2 | no | — |
| 119 | C22×C8 | — | 2 | 1 | no | — |
| 120 | C22×C8 | — | 2 | 1 | no | — |
| 121 | C22×C8 | — | 4 | 2 | no | — |
| 122 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 123 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 124 | C22×C8 | #41 | 1 | 2 | yes | — |
| 125 | C22×C8 | #41 | 1 | 2 | yes | — |
| 126 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 127 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 128 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 129 | #127 | C22×C4 | 1 | 2 | yes | — |
| 130 | #125 | — | 2 | 2 | no | — |
| 131 | #124 | — | 2 | 2 | no | — |
| 132 | C22⋊Q8 | — | 2 | 2 | no | — |
| 133 | C22⋊Q8 | — | 2 | 2 | no | — |
| 134 | #121 | — | 2 | 2 | no | — |
| 135 | #121 | — | 2 | 2 | no | — |
| 136 | C4×Q8 | — | 2 | 2 | no | — |
| 137 | C4×Q8 | — | 2 | 2 | no | — |
| 138 | C4×D4 | — | 2 | 2 | no | — |
| 139 | C4×D4 | — | 2 | 2 | no | — |
| 140 | #117 | — | 2 | 2 | no | — |
| 141 | C2×C42 | — | 2 | 1 | no | — |
| 142 | C2×C42 | — | 2 | 1 | no | — |
| 143 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 144 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 145 | #108 | — | 8 | 4 | no | — |
| 146 | #107 | — | 2 | 2 | no | — |
| 147 | #107 | — | 2 | 2 | no | — |
| 148 | #106 | — | 2 | 4 | no | — |
| 149 | C4⋊C8 | — | 2 | 2 | no | — |
| 150 | C4⋊C8 | — | 2 | 2 | no | — |
| 151 | C4⋊C8 | — | 2 | 2 | no | — |
| 152 | C4⋊C8 | — | 2 | 2 | no | — |
| 153 | C4⋊C8 | — | 4 | 2 | no | — |
| 154 | C4⋊C8 | — | 4 | 2 | no | — |
| 155 | #104 | — | 8 | 4 | no | — |
| 156 | #103 | — | 2 | 4 | no | — |
| 157 | #103 | — | 2 | 4 | no | — |
| 158 | #103 | — | 2 | 4 | no | — |
| 159 | #103 | — | 2 | 4 | no | — |
| 160 | #102 | — | 2 | 4 | no | — |
| 161 | #102 | — | 2 | 4 | no | — |
| 162 | #102 | — | 2 | 4 | no | — |
| 163 | #102 | — | 2 | 4 | no | — |
| 164 | C22⋊C8 | — | 4 | 2 | no | — |
| 165 | C22⋊C8 | — | 4 | 2 | no | — |
| 166 | #97 | — | 4 | 2 | no | — |
| 167 | C4×C8 | — | 2 | 1 | no | — |
| 168 | C4×C8 | — | 2 | 1 | no | — |
| 169 | C4×C8 | — | 4 | 4 | no | — |
| 170 | C4×C8 | — | 4 | 2 | no | — |
| 171 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 172 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 173 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 174 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 175 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 176 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 177 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 178 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 179 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 180 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 181 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 182 | C4×C8 | #41 | 1 | 2 | yes | — |
| 183 | #41 | — | 4 | 4 | no | — |
| 184 | #41 | — | 4 | 4 | no | — |
| 185 | C2×Q8 | — | 2 | 2 | no | — |
| 186 | C2×Q8 | — | 2 | 2 | no | — |
| 187 | C2×Q8 | — | 2 | 2 | no | — |
| 188 | C2×D4 | — | 2 | 2 | no | — |
| 189 | C2×D4 | — | 2 | 2 | no | — |
| 190 | C2×D4 | — | 2 | 2 | no | — |
| 191 | C2×D4 | — | 4 | 2 | no | — |
| 192 | C22×C4 | — | 2 | 2 | no | — |
| 193 | C22×C4 | — | 4 | 2 | no | — |
| 194 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 195 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 196 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 197 | Q16 | — | 4 | 4 | no | — |
| 198 | SD16 | — | 8 | 4 | no | — |
| 199 | D8 | — | 4 | 4 | no | — |
| 200 | M16 | — | 4 | 4 | no | — |
| 201 | M16 | — | 4 | 4 | no | — |
| 202 | M16 | — | 8 | 4 | no | — |
| 203 | C2×C8 | — | 2 | 4 | no | — |
| 204 | C2×C8 | — | 2 | 4 | no | — |
| 205 | C2×C8 | — | 2 | 2 | no | — |
| 206 | C2×C8 | — | 2 | 2 | no | — |
| 207 | C2×C8 | — | 4 | 2 | no | — |
| 208 | C2×C8 | — | 4 | 2 | no | — |
| 209 | C2×C8 | — | 4 | 2 | no | — |
| 210 | C2×C8 | — | 4 | 2 | no | — |
| 211 | C2×C8 | — | 4 | 2 | no | — |
| 212 | C2×C8 | — | 4 | 2 | no | — |
| 213 | C2×C8 | — | 8 | 4 | no | — |
| 214 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 215 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 216 | C2×C8 | C4×D4 | 1 | 2 | yes | — |
| 217 | C2×C8 | C4×D4 | 1 | 4 | yes | — |
| 218 | C2×C8 | #127 | 1 | 2 | yes | — |
| 219 | C2×C8 | #127 | 1 | 2 | yes | — |
| 220 | C4⋊C4 | — | 2 | 2 | no | — |
| 221 | C4⋊C4 | — | 2 | 2 | no | — |
| 222 | C4⋊C4 | — | 2 | 2 | no | — |
| 223 | C4⋊C4 | — | 2 | 2 | no | — |
| 224 | C4⋊C4 | — | 2 | 2 | no | — |
| 225 | C4⋊C4 | — | 2 | 2 | no | — |
| 226 | C4⋊C4 | — | 4 | 2 | no | — |
| 227 | C4⋊C4 | — | 4 | 2 | no | — |
| 228 | C22⋊C4 | — | 4 | 2 | no | — |
| 229 | C22⋊C4 | — | 4 | 2 | no | — |
| 230 | C22⋊C4 | — | 4 | 2 | no | — |
| 231 | C42 | — | 2 | 4 | no | — |
| 232 | C42 | — | 2 | 4 | no | — |
| 233 | C42 | — | 4 | 2 | no | — |
| 234 | C42 | — | 4 | 2 | no | — |
| 235 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 236 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 237 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 238 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 239 | C23 | — | 4 | 2 | no | — |
| 240 | C23 | #532 | 1 | 1 | yes | characteristic |
| 241 | Q8 | — | 4 | 4 | no | — |
| 242 | Q8 | — | 4 | 4 | no | — |
| 243 | Q8 | — | 4 | 4 | no | — |
| 244 | D4 | — | 4 | 4 | no | — |
| 245 | D4 | — | 4 | 4 | no | — |
| 246 | D4 | — | 4 | 4 | no | — |
| 247 | D4 | — | 8 | 4 | no | — |
| 248 | C2×C4 | — | 2 | 2 | no | — |
| 249 | C2×C4 | — | 2 | 2 | no | — |
| 250 | C2×C4 | — | 2 | 4 | no | — |
| 251 | C2×C4 | — | 4 | 2 | no | — |
| 252 | C2×C4 | — | 4 | 2 | no | — |
| 253 | C2×C4 | — | 4 | 2 | no | — |
| 254 | C2×C4 | — | 4 | 2 | no | — |
| 255 | C2×C4 | — | 4 | 2 | no | — |
| 256 | C2×C4 | — | 8 | 2 | no | — |
| 257 | C2×C4 | — | 8 | 2 | no | — |
| 258 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 260 | C2×C4 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 261 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 263 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 264 | C2×C4 | #577 | 1 | 4 | yes | — |
| 265 | C8 | — | 2 | 2 | no | — |
| 266 | C8 | — | 2 | 2 | no | — |
| 267 | C8 | — | 2 | 4 | no | — |
| 268 | C8 | — | 2 | 2 | no | — |
| 269 | C8 | — | 2 | 2 | no | — |
| 270 | C8 | — | 2 | 4 | no | — |
| 271 | C8 | — | 4 | 2 | no | — |
| 272 | C8 | — | 4 | 2 | no | — |
| 273 | C8 | — | 8 | 2 | no | — |
| 274 | C8 | — | 8 | 2 | no | — |
| 275 | C22 | — | 2 | 1 | no | — |
| 276 | C22 | — | 2 | 1 | no | — |
| 277 | C22 | — | 8 | 2 | no | — |
| 278 | C22 | — | 8 | 2 | no | — |
| 279 | C22 | — | 8 | 2 | no | — |
| 280 | C22 | C4×#127 | 1 | 1 | yes | characteristic |
| 281 | C22 | #3173 | 1 | 2 | yes | — |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 2 | 1 | no | — |
| 284 | C4 | — | 4 | 4 | no | — |
| 285 | C4 | — | 8 | 2 | no | — |
| 286 | C4 | — | 8 | 2 | no | — |
| 287 | C4 | — | 8 | 2 | no | — |
| 288 | C4 | #2952 | 1 | 2 | yes | — |
| 289 | C4 | #2955 | 1 | 2 | yes | — |
| 290 | C4 | #2956 | 1 | 2 | yes | — |
| 291 | C4 | #3154 | 1 | 2 | yes | — |
| 292 | C2 | — | 2 | 2 | no | — |
| 293 | C2 | — | 16 | 2 | no | — |
| 294 | C2 | #21694 | 1 | 1 | yes | characteristic |
| 295 | C2 | #21709 | 1 | 1 | yes | characteristic |
| 296 | C2 | #21710 | 1 | 1 | yes | characteristic |
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