← #491046
Subgroups — #491046
Order 512.
456 conjugacy classes of subgroups.
The subgroup lattice has 313 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #33812 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33793 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19633 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #18066 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #12499 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12487 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12447 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #12420 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12015 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #11949 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C2×#1365 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3209 | — | 2 | 1 | no | — |
| 13 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 14 | #3169 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3165 | — | 2 | 1 | no | — |
| 16 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2935 | — | 2 | 1 | no | — |
| 18 | #2254 | — | 2 | 1 | no | — |
| 19 | #2240 | — | 2 | 1 | no | — |
| 20 | #2233 | — | 2 | 2 | no | — |
| 21 | #2226 | C22 | 1 | 2 | yes | — |
| 22 | #2226 | C22 | 1 | 2 | yes | — |
| 23 | #2226 | C22 | 1 | 2 | yes | — |
| 24 | #2225 | — | 2 | 1 | no | — |
| 25 | #2217 | — | 2 | 2 | no | — |
| 26 | #2208 | C22 | 1 | 2 | yes | — |
| 27 | #2207 | — | 2 | 1 | no | — |
| 28 | #2162 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1723 | — | 2 | 2 | no | — |
| 31 | #1713 | C22 | 1 | 2 | yes | — |
| 32 | #1707 | C22 | 1 | 2 | yes | — |
| 33 | #1589 | C22 | 1 | 2 | yes | — |
| 34 | #1581 | — | 2 | 2 | no | — |
| 35 | C8×D8 | C22 | 1 | 2 | yes | — |
| 36 | C2×#335 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1480 | — | 2 | 1 | no | — |
| 38 | C8×M16 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1370 | C22 | 1 | 2 | yes | — |
| 40 | #1367 | C22 | 1 | 2 | yes | — |
| 41 | #1365 | C22 | 1 | 2 | yes | — |
| 42 | #1365 | C22 | 1 | 1 | yes | characteristic |
| 43 | #1365 | C22 | 1 | 2 | yes | — |
| 44 | #1365 | C22 | 1 | 1 | yes | characteristic |
| 45 | #1362 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1362 | C22 | 1 | 1 | yes | characteristic |
| 47 | C2×#131 | — | 4 | 1 | no | — |
| 48 | #533 | — | 2 | 1 | no | — |
| 49 | #532 | — | 2 | 1 | no | — |
| 50 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 51 | C2×SD32 | — | 2 | 2 | no | — |
| 52 | C22×C16 | — | 2 | 1 | no | — |
| 53 | #501 | — | 4 | 2 | no | — |
| 54 | #500 | — | 2 | 1 | no | — |
| 55 | #500 | — | 2 | 1 | no | — |
| 56 | #500 | — | 2 | 2 | no | — |
| 57 | #500 | — | 2 | 1 | no | — |
| 58 | #500 | D4 | 1 | 1 | yes | characteristic |
| 59 | #500 | C23 | 1 | 2 | yes | — |
| 60 | #500 | C23 | 1 | 1 | yes | characteristic |
| 61 | #498 | — | 2 | 2 | no | — |
| 62 | #494 | — | 2 | 2 | no | — |
| 63 | #494 | C23 | 1 | 1 | yes | characteristic |
| 64 | #493 | D4 | 1 | 1 | yes | characteristic |
| 65 | #487 | — | 2 | 1 | no | — |
| 66 | #487 | — | 2 | 2 | no | — |
| 67 | #486 | — | 2 | 1 | no | — |
| 68 | #467 | — | 2 | 1 | no | — |
| 69 | #467 | — | 2 | 1 | no | — |
| 70 | #466 | — | 2 | 1 | no | — |
| 71 | #466 | D4 | 1 | 2 | yes | — |
| 72 | #462 | — | 4 | 2 | no | — |
| 73 | #451 | — | 4 | 2 | no | — |
| 74 | C4×Q16 | — | 2 | 1 | no | — |
| 75 | C4×Q16 | — | 2 | 1 | no | — |
| 76 | C4×D8 | — | 2 | 1 | no | — |
| 77 | C4×D8 | C23 | 1 | 2 | yes | — |
| 78 | #436 | — | 2 | 2 | no | — |
| 79 | #435 | — | 4 | 2 | no | — |
| 80 | C8×D4 | — | 2 | 2 | no | — |
| 81 | #431 | D4 | 1 | 1 | yes | characteristic |
| 82 | #428 | — | 4 | 1 | no | — |
| 83 | #428 | — | 4 | 1 | no | — |
| 84 | C2×#107 | — | 2 | 1 | no | — |
| 85 | C2×#107 | — | 2 | 1 | no | — |
| 86 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 87 | #408 | — | 4 | 1 | no | — |
| 88 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 90 | #367 | — | 2 | 2 | no | — |
| 91 | #367 | — | 2 | 2 | no | — |
| 92 | #367 | — | 2 | 2 | no | — |
| 93 | C4⋊C16 | — | 2 | 1 | no | — |
| 94 | C4⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 95 | C4⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 96 | C4⋊C16 | C23 | 1 | 2 | yes | — |
| 97 | #358 | — | 2 | 2 | no | — |
| 98 | #357 | — | 2 | 2 | no | — |
| 99 | #357 | — | 2 | 2 | no | — |
| 100 | #357 | — | 2 | 2 | no | — |
| 101 | C22⋊C16 | — | 2 | 1 | no | — |
| 102 | C4×C16 | — | 2 | 1 | no | — |
| 103 | #335 | — | 2 | 1 | no | — |
| 104 | #335 | C23 | 1 | 1 | yes | characteristic |
| 105 | #335 | C23 | 1 | 1 | yes | characteristic |
| 106 | D4⋊C8 | — | 2 | 2 | no | — |
| 107 | D4⋊C8 | — | 2 | 2 | no | — |
| 108 | #322 | D4 | 1 | 2 | yes | — |
| 109 | C82 | C23 | 1 | 2 | yes | — |
| 110 | C2×#41 | — | 4 | 1 | no | — |
| 111 | C22×Q8 | — | 4 | 1 | no | — |
| 112 | C2×Q16 | — | 2 | 1 | no | — |
| 113 | C2×Q16 | — | 2 | 1 | no | — |
| 114 | C2×Q16 | — | 4 | 4 | no | — |
| 115 | C2×D8 | — | 2 | 1 | no | — |
| 116 | C2×D8 | C2×D4 | 1 | 2 | yes | — |
| 117 | #131 | — | 8 | 4 | no | — |
| 118 | C2×M16 | — | 4 | 2 | no | — |
| 119 | C2×M16 | D8 | 1 | 1 | yes | characteristic |
| 120 | C2×M16 | D8 | 1 | 1 | yes | characteristic |
| 121 | C22×C8 | — | 4 | 1 | no | — |
| 122 | C22×C8 | — | 4 | 2 | no | — |
| 123 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 125 | C4⋊Q8 | — | 2 | 1 | no | — |
| 126 | C4⋊Q8 | — | 2 | 2 | no | — |
| 127 | C4⋊Q8 | — | 2 | 1 | no | — |
| 128 | C4⋊Q8 | — | 2 | 1 | no | — |
| 129 | C4⋊Q8 | — | 4 | 1 | no | — |
| 130 | C4⋊Q8 | — | 4 | 1 | no | — |
| 131 | C4⋊Q8 | — | 4 | 1 | no | — |
| 132 | #127 | — | 2 | 1 | no | — |
| 133 | #125 | — | 4 | 1 | no | — |
| 134 | #124 | — | 4 | 1 | no | — |
| 135 | C22⋊Q8 | — | 4 | 1 | no | — |
| 136 | C22⋊Q8 | — | 4 | 1 | no | — |
| 137 | #121 | — | 2 | 2 | no | — |
| 138 | #121 | — | 4 | 1 | no | — |
| 139 | C4×Q8 | — | 4 | 1 | no | — |
| 140 | C4×Q8 | — | 4 | 1 | no | — |
| 141 | C4×D4 | — | 2 | 2 | no | — |
| 142 | C4×D4 | — | 4 | 1 | no | — |
| 143 | #117 | — | 4 | 1 | no | — |
| 144 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 145 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 146 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 147 | SD32 | — | 8 | 2 | no | — |
| 148 | C2×C16 | — | 2 | 1 | no | — |
| 149 | C2×C16 | — | 2 | 1 | no | — |
| 150 | C2×C16 | — | 2 | 2 | no | — |
| 151 | C2×C16 | — | 8 | 1 | no | — |
| 152 | #107 | — | 2 | 1 | no | — |
| 153 | #107 | — | 2 | 2 | no | — |
| 154 | #107 | — | 2 | 1 | no | — |
| 155 | #107 | — | 2 | 2 | no | — |
| 156 | #107 | — | 2 | 1 | no | — |
| 157 | #107 | — | 4 | 1 | no | — |
| 158 | #107 | — | 4 | 1 | no | — |
| 159 | #107 | — | 4 | 2 | no | — |
| 160 | #107 | — | 4 | 2 | no | — |
| 161 | #107 | C2×D4 | 1 | 2 | yes | — |
| 162 | #106 | — | 4 | 2 | no | — |
| 163 | #106 | — | 4 | 2 | no | — |
| 164 | C4⋊C8 | — | 2 | 1 | no | — |
| 165 | C4⋊C8 | — | 2 | 1 | no | — |
| 166 | C4⋊C8 | — | 2 | 1 | no | — |
| 167 | C4⋊C8 | — | 4 | 1 | no | — |
| 168 | #103 | — | 4 | 4 | no | — |
| 169 | #103 | — | 4 | 1 | no | — |
| 170 | #103 | — | 4 | 1 | no | — |
| 171 | #102 | — | 2 | 2 | no | — |
| 172 | #102 | — | 2 | 2 | no | — |
| 173 | C22⋊C8 | — | 2 | 1 | no | — |
| 174 | C22⋊C8 | — | 4 | 2 | no | — |
| 175 | C22⋊C8 | — | 4 | 2 | no | — |
| 176 | #97 | D8 | 1 | 2 | yes | — |
| 177 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 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 | #41 | 1 | 1 | yes | characteristic |
| 182 | C4×C8 | C24 | 1 | 1 | yes | characteristic |
| 183 | #41 | — | 8 | 2 | no | — |
| 184 | #41 | — | 8 | 2 | no | — |
| 185 | C2×Q8 | — | 4 | 1 | no | — |
| 186 | C2×Q8 | — | 4 | 1 | no | — |
| 187 | C2×Q8 | — | 4 | 2 | no | — |
| 188 | C2×Q8 | — | 4 | 1 | no | — |
| 189 | C2×Q8 | — | 8 | 1 | no | — |
| 190 | C2×Q8 | — | 16 | 2 | no | — |
| 191 | C2×D4 | — | 2 | 2 | no | — |
| 192 | C2×D4 | — | 4 | 2 | no | — |
| 193 | C2×D4 | — | 4 | 1 | no | — |
| 194 | C2×D4 | — | 4 | 1 | no | — |
| 195 | C22×C4 | — | 4 | 2 | no | — |
| 196 | C22×C4 | — | 8 | 1 | no | — |
| 197 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 198 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 199 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 200 | Q16 | — | 4 | 1 | no | — |
| 201 | Q16 | — | 4 | 1 | no | — |
| 202 | Q16 | — | 8 | 1 | no | — |
| 203 | Q16 | — | 16 | 4 | no | — |
| 204 | D8 | — | 2 | 2 | no | — |
| 205 | D8 | — | 2 | 2 | no | — |
| 206 | D8 | — | 8 | 1 | no | — |
| 207 | M16 | — | 2 | 2 | no | — |
| 208 | M16 | — | 8 | 4 | no | — |
| 209 | M16 | SD32 | 1 | 4 | yes | complemented |
| 210 | C2×C8 | — | 2 | 1 | no | — |
| 211 | C2×C8 | — | 2 | 1 | no | — |
| 212 | C2×C8 | — | 2 | 2 | no | — |
| 213 | C2×C8 | — | 4 | 1 | no | — |
| 214 | C2×C8 | — | 4 | 1 | no | — |
| 215 | C2×C8 | — | 4 | 1 | no | — |
| 216 | C2×C8 | — | 8 | 2 | no | — |
| 217 | C2×C8 | — | 8 | 4 | no | — |
| 218 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 219 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 220 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 221 | C2×C8 | C2×D8 | 1 | 2 | yes | — |
| 222 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 223 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 224 | C4⋊C4 | — | 2 | 2 | no | — |
| 225 | C4⋊C4 | — | 4 | 1 | no | — |
| 226 | C4⋊C4 | — | 4 | 1 | no | — |
| 227 | C4⋊C4 | — | 4 | 2 | no | — |
| 228 | C4⋊C4 | — | 4 | 1 | no | — |
| 229 | C4⋊C4 | — | 4 | 2 | no | — |
| 230 | C4⋊C4 | — | 4 | 1 | no | — |
| 231 | C4⋊C4 | — | 4 | 1 | no | — |
| 232 | C4⋊C4 | — | 8 | 1 | no | — |
| 233 | C4⋊C4 | — | 8 | 1 | no | — |
| 234 | C4⋊C4 | — | 8 | 1 | no | — |
| 235 | C4⋊C4 | — | 8 | 1 | no | — |
| 236 | C22⋊C4 | — | 4 | 2 | no | — |
| 237 | C22⋊C4 | — | 4 | 2 | no | — |
| 238 | C22⋊C4 | — | 8 | 1 | no | — |
| 239 | C42 | — | 8 | 1 | no | — |
| 240 | C42 | #121 | 1 | 1 | yes | characteristic |
| 241 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 242 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 243 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 244 | C16 | — | 4 | 1 | no | — |
| 245 | C16 | — | 4 | 2 | no | — |
| 246 | C16 | — | 4 | 1 | no | — |
| 247 | C23 | — | 4 | 2 | no | — |
| 248 | C23 | #459 | 1 | 1 | yes | characteristic |
| 249 | Q8 | — | 8 | 2 | no | — |
| 250 | Q8 | — | 8 | 4 | no | — |
| 251 | Q8 | — | 8 | 2 | no | — |
| 252 | Q8 | — | 8 | 1 | no | — |
| 253 | Q8 | — | 8 | 1 | no | — |
| 254 | Q8 | — | 16 | 2 | no | — |
| 255 | D4 | — | 4 | 2 | no | — |
| 256 | D4 | — | 4 | 2 | no | — |
| 257 | D4 | — | 8 | 4 | no | — |
| 258 | D4 | — | 8 | 2 | no | — |
| 259 | D4 | — | 16 | 1 | no | — |
| 260 | C2×C4 | — | 2 | 1 | no | — |
| 261 | C2×C4 | — | 2 | 1 | no | — |
| 262 | C2×C4 | — | 4 | 1 | no | — |
| 263 | C2×C4 | — | 4 | 2 | no | — |
| 264 | C2×C4 | — | 8 | 2 | no | — |
| 265 | C2×C4 | — | 8 | 2 | no | — |
| 266 | C2×C4 | — | 8 | 1 | no | — |
| 267 | C2×C4 | — | 8 | 1 | no | — |
| 268 | C2×C4 | — | 8 | 2 | no | — |
| 269 | C2×C4 | — | 16 | 2 | no | — |
| 270 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | #459 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C2×SD32 | 1 | 2 | yes | — |
| 274 | C2×C4 | #510 | 1 | 2 | yes | — |
| 275 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 278 | C8 | — | 2 | 2 | no | — |
| 279 | C8 | — | 2 | 1 | no | — |
| 280 | C8 | — | 2 | 1 | no | — |
| 281 | C8 | — | 2 | 1 | no | — |
| 282 | C8 | — | 4 | 2 | no | — |
| 283 | C8 | — | 8 | 1 | no | — |
| 284 | C8 | — | 8 | 1 | no | — |
| 285 | C8 | C2×SD32 | 1 | 4 | yes | — |
| 286 | C8 | #577 | 1 | 1 | yes | characteristic |
| 287 | C8 | #578 | 1 | 1 | yes | characteristic |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 2 | 1 | no | — |
| 290 | C22 | — | 8 | 2 | no | — |
| 291 | C22 | — | 8 | 2 | no | — |
| 292 | C22 | — | 8 | 2 | no | — |
| 293 | C22 | #2208 | 1 | 2 | yes | — |
| 294 | C22 | C2×#459 | 1 | 1 | yes | characteristic |
| 295 | C4 | — | 2 | 1 | no | — |
| 296 | C4 | — | 2 | 1 | no | — |
| 297 | C4 | — | 2 | 1 | no | — |
| 298 | C4 | — | 8 | 2 | no | — |
| 299 | C4 | — | 16 | 1 | no | — |
| 300 | C4 | — | 16 | 2 | no | — |
| 301 | C4 | — | 16 | 1 | no | — |
| 302 | C4 | #2208 | 1 | 2 | yes | — |
| 303 | C4 | #3063 | 1 | 1 | yes | characteristic |
| 304 | C4 | #3069 | 1 | 1 | yes | characteristic |
| 305 | C4 | C22×SD32 | 1 | 1 | yes | characteristic |
| 306 | C4 | C2×#510 | 1 | 1 | yes | characteristic |
| 307 | C2 | — | 2 | 2 | no | — |
| 308 | C2 | — | 16 | 2 | no | — |
| 309 | C2 | #19231 | 1 | 1 | yes | characteristic |
| 310 | C2 | C2×#2208 | 1 | 1 | yes | characteristic |
| 311 | C2 | #33595 | 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