Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
519 conjugacy classes of subgroups.
The subgroup lattice has 337 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #28731 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22747 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #19176 | C2 | 1 | 2 | yes | maximal |
| 4 | C2×#1696 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #17936 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #10605 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10551 | C2 | 1 | 2 | yes | maximal |
| 8 | #9778 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #9609 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #9250 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #8686 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 13 | #3141 | C22 | 1 | 2 | yes | — |
| 14 | #3103 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#477 | C22 | 1 | 2 | yes | — |
| 16 | Q8×M16 | C22 | 1 | 2 | yes | — |
| 17 | C2×#446 | C22 | 1 | 1 | yes | characteristic |
| 18 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 19 | #2078 | — | 2 | 1 | no | — |
| 20 | #2078 | — | 2 | 1 | no | — |
| 21 | #1997 | C22 | 1 | 2 | yes | — |
| 22 | #1950 | — | 2 | 2 | no | — |
| 23 | #1950 | C22 | 1 | 2 | yes | — |
| 24 | #1838 | — | 2 | 1 | no | — |
| 25 | #1837 | C22 | 1 | 1 | yes | characteristic |
| 26 | #1816 | — | 2 | 2 | no | — |
| 27 | #1778 | — | 2 | 1 | no | — |
| 28 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1704 | — | 2 | 2 | no | — |
| 30 | #1698 | — | 2 | 2 | no | — |
| 31 | #1696 | — | 2 | 2 | no | — |
| 32 | #1696 | — | 2 | 2 | no | — |
| 33 | #1684 | — | 2 | 2 | no | — |
| 34 | #1678 | — | 2 | 2 | no | — |
| 35 | #1592 | — | 2 | 2 | no | — |
| 36 | #1585 | — | 2 | 2 | no | — |
| 37 | #1571 | C22 | 1 | 1 | yes | characteristic |
| 38 | #1565 | C22 | 1 | 2 | yes | — |
| 39 | C2×#334 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1549 | C22 | 1 | 2 | yes | — |
| 41 | #1548 | C22 | 1 | 2 | yes | — |
| 42 | #1547 | C22 | 1 | 2 | yes | — |
| 43 | C2×#333 | C22 | 1 | 2 | yes | — |
| 44 | #1487 | C22 | 1 | 1 | yes | characteristic |
| 45 | #1480 | C22 | 1 | 1 | yes | characteristic |
| 46 | #1452 | C22 | 1 | 1 | yes | characteristic |
| 47 | #1292 | C22 | 1 | 2 | yes | — |
| 48 | #1283 | C22 | 1 | 2 | yes | — |
| 49 | C2×#131 | — | 2 | 1 | no | — |
| 50 | #532 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 52 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 53 | #500 | — | 2 | 1 | no | — |
| 54 | #500 | — | 2 | 1 | no | — |
| 55 | #500 | — | 2 | 1 | no | — |
| 56 | #500 | — | 2 | 1 | no | — |
| 57 | #489 | — | 4 | 2 | no | — |
| 58 | #487 | — | 2 | 2 | no | — |
| 59 | #486 | — | 2 | 2 | no | — |
| 60 | #481 | — | 2 | 1 | no | — |
| 61 | #477 | — | 2 | 2 | no | — |
| 62 | #477 | — | 2 | 2 | no | — |
| 63 | #477 | — | 2 | 2 | no | — |
| 64 | #477 | — | 2 | 2 | no | — |
| 65 | #475 | D4 | 1 | 2 | yes | — |
| 66 | #460 | D4 | 1 | 2 | yes | — |
| 67 | #448 | — | 2 | 1 | no | — |
| 68 | #446 | — | 2 | 2 | no | — |
| 69 | #446 | — | 2 | 1 | no | — |
| 70 | #446 | — | 2 | 1 | no | — |
| 71 | #446 | — | 4 | 2 | no | — |
| 72 | #446 | D4 | 1 | 2 | yes | — |
| 73 | #446 | D4 | 1 | 2 | yes | — |
| 74 | C8×Q8 | — | 2 | 2 | no | — |
| 75 | C4×SD16 | — | 2 | 2 | no | — |
| 76 | #436 | — | 2 | 1 | no | — |
| 77 | #436 | — | 2 | 1 | no | — |
| 78 | C8×D4 | — | 2 | 1 | no | — |
| 79 | C8×D4 | — | 2 | 1 | no | — |
| 80 | #431 | C23 | 1 | 1 | yes | characteristic |
| 81 | #431 | C23 | 1 | 1 | yes | characteristic |
| 82 | C2×#108 | — | 2 | 2 | no | — |
| 83 | C2×#108 | D4 | 1 | 2 | yes | — |
| 84 | C2×#107 | — | 2 | 1 | no | — |
| 85 | C2×#107 | — | 2 | 1 | no | — |
| 86 | C2×#107 | — | 2 | 2 | no | — |
| 87 | #423 | — | 2 | 2 | no | — |
| 88 | #423 | C23 | 1 | 2 | yes | — |
| 89 | #423 | C23 | 1 | 1 | yes | characteristic |
| 90 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 91 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 92 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 93 | #417 | — | 2 | 2 | no | — |
| 94 | #417 | — | 2 | 2 | no | — |
| 95 | C2×#103 | — | 2 | 2 | no | — |
| 96 | #408 | — | 2 | 1 | no | — |
| 97 | C4×M16 | — | 2 | 2 | no | — |
| 98 | C4×M16 | C23 | 1 | 2 | yes | — |
| 99 | C2×#97 | — | 2 | 1 | no | — |
| 100 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 101 | #343 | — | 2 | 2 | no | — |
| 102 | #343 | — | 2 | 2 | no | — |
| 103 | #341 | — | 2 | 2 | no | — |
| 104 | #341 | — | 2 | 1 | no | — |
| 105 | #341 | — | 2 | 1 | no | — |
| 106 | #341 | D4 | 1 | 1 | yes | characteristic |
| 107 | #341 | D4 | 1 | 1 | yes | characteristic |
| 108 | #335 | — | 2 | 2 | no | — |
| 109 | #335 | — | 2 | 2 | no | — |
| 110 | #334 | — | 2 | 2 | no | — |
| 111 | #334 | — | 2 | 1 | no | — |
| 112 | #334 | — | 2 | 1 | no | — |
| 113 | #333 | — | 2 | 4 | no | — |
| 114 | #333 | — | 2 | 2 | no | — |
| 115 | #333 | — | 2 | 2 | no | — |
| 116 | #332 | — | 2 | 2 | no | — |
| 117 | #332 | — | 4 | 2 | no | — |
| 118 | #331 | — | 2 | 2 | no | — |
| 119 | Q8⋊C8 | — | 2 | 2 | no | — |
| 120 | D4⋊C8 | — | 2 | 2 | no | — |
| 121 | #322 | — | 2 | 1 | no | — |
| 122 | #322 | — | 2 | 1 | no | — |
| 123 | C2×#41 | — | 2 | 1 | no | — |
| 124 | C22×Q8 | — | 2 | 2 | no | — |
| 125 | C22×Q8 | #41 | 1 | 1 | yes | characteristic |
| 126 | C2×SD16 | — | 2 | 2 | no | — |
| 127 | #131 | — | 4 | 4 | no | — |
| 128 | C2×M16 | — | 2 | 1 | no | — |
| 129 | C2×M16 | — | 2 | 2 | no | — |
| 130 | C2×M16 | — | 4 | 1 | no | — |
| 131 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 132 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 133 | C22×C8 | — | 2 | 1 | no | — |
| 134 | C22×C8 | — | 2 | 2 | no | — |
| 135 | C22×C8 | — | 4 | 1 | no | — |
| 136 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 137 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 138 | C4⋊Q8 | — | 2 | 2 | no | — |
| 139 | C4⋊Q8 | — | 2 | 2 | no | — |
| 140 | C4⋊Q8 | — | 2 | 2 | no | — |
| 141 | C4⋊Q8 | — | 2 | 2 | no | — |
| 142 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 143 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 144 | #124 | — | 2 | 1 | no | — |
| 145 | #121 | — | 2 | 1 | no | — |
| 146 | #121 | — | 2 | 1 | no | — |
| 147 | C4×Q8 | — | 2 | 1 | no | — |
| 148 | C4×Q8 | — | 2 | 1 | no | — |
| 149 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 150 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 151 | C4×D4 | — | 2 | 1 | no | — |
| 152 | C4×D4 | — | 2 | 1 | no | — |
| 153 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 154 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 155 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 156 | C2×C4⋊C4 | #41 | 1 | 1 | yes | characteristic |
| 157 | C2×C42 | — | 2 | 1 | no | — |
| 158 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 159 | #108 | — | 4 | 2 | no | — |
| 160 | #108 | — | 4 | 4 | no | — |
| 161 | #107 | — | 4 | 1 | no | — |
| 162 | #107 | — | 4 | 1 | no | — |
| 163 | #107 | — | 4 | 1 | no | — |
| 164 | #107 | — | 4 | 1 | no | — |
| 165 | #107 | — | 4 | 2 | no | — |
| 166 | #107 | — | 4 | 2 | no | — |
| 167 | #106 | — | 2 | 2 | no | — |
| 168 | C4⋊C8 | — | 2 | 2 | no | — |
| 169 | C4⋊C8 | — | 2 | 2 | no | — |
| 170 | C4⋊C8 | — | 2 | 1 | no | — |
| 171 | C4⋊C8 | — | 2 | 2 | no | — |
| 172 | C4⋊C8 | — | 2 | 2 | no | — |
| 173 | C4⋊C8 | — | 2 | 2 | no | — |
| 174 | C4⋊C8 | — | 2 | 1 | no | — |
| 175 | C4⋊C8 | — | 2 | 1 | no | — |
| 176 | C4⋊C8 | — | 2 | 1 | no | — |
| 177 | C4⋊C8 | — | 2 | 2 | no | — |
| 178 | C4⋊C8 | — | 4 | 2 | no | — |
| 179 | C4⋊C8 | — | 4 | 2 | no | — |
| 180 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 181 | C4⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 182 | #103 | — | 2 | 2 | no | — |
| 183 | #103 | — | 4 | 2 | no | — |
| 184 | #103 | — | 4 | 2 | no | — |
| 185 | #103 | — | 4 | 2 | no | — |
| 186 | #103 | — | 4 | 2 | no | — |
| 187 | #102 | — | 2 | 2 | no | — |
| 188 | #102 | — | 4 | 2 | no | — |
| 189 | #102 | — | 4 | 2 | no | — |
| 190 | C22⋊C8 | — | 2 | 1 | no | — |
| 191 | C22⋊C8 | — | 2 | 1 | no | — |
| 192 | C22⋊C8 | — | 4 | 1 | no | — |
| 193 | C22⋊C8 | — | 4 | 1 | no | — |
| 194 | #97 | — | 2 | 2 | no | — |
| 195 | #97 | — | 2 | 2 | no | — |
| 196 | #97 | — | 2 | 2 | no | — |
| 197 | #97 | — | 4 | 2 | no | — |
| 198 | C4×C8 | — | 2 | 1 | no | — |
| 199 | C4×C8 | — | 2 | 2 | no | — |
| 200 | C4×C8 | — | 2 | 1 | no | — |
| 201 | C4×C8 | — | 4 | 2 | no | — |
| 202 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 203 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 204 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 205 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 206 | #41 | — | 4 | 2 | no | — |
| 207 | #41 | — | 4 | 2 | no | — |
| 208 | C2×Q8 | — | 2 | 1 | no | — |
| 209 | C2×Q8 | — | 2 | 1 | no | — |
| 210 | C2×Q8 | — | 4 | 2 | no | — |
| 211 | C2×Q8 | — | 4 | 2 | no | — |
| 212 | C2×Q8 | — | 4 | 2 | no | — |
| 213 | C2×Q8 | — | 16 | 2 | no | — |
| 214 | C2×Q8 | #121 | 1 | 2 | yes | — |
| 215 | C2×D4 | — | 2 | 1 | no | — |
| 216 | C2×D4 | — | 2 | 1 | no | — |
| 217 | C2×D4 | — | 2 | 1 | no | — |
| 218 | C2×D4 | — | 4 | 1 | no | — |
| 219 | C22×C4 | — | 2 | 1 | no | — |
| 220 | C22×C4 | — | 2 | 1 | no | — |
| 221 | C22×C4 | — | 4 | 1 | no | — |
| 222 | C22×C4 | — | 4 | 2 | no | — |
| 223 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 224 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 225 | C22×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 226 | SD16 | — | 8 | 2 | no | — |
| 227 | M16 | — | 2 | 2 | no | — |
| 228 | M16 | — | 2 | 2 | no | — |
| 229 | M16 | — | 2 | 2 | no | — |
| 230 | M16 | — | 2 | 2 | no | — |
| 231 | M16 | — | 4 | 2 | no | — |
| 232 | M16 | — | 4 | 2 | no | — |
| 233 | M16 | — | 4 | 2 | no | — |
| 234 | M16 | — | 8 | 2 | no | — |
| 235 | C2×C8 | — | 2 | 2 | no | — |
| 236 | C2×C8 | — | 2 | 2 | no | — |
| 237 | C2×C8 | — | 2 | 2 | no | — |
| 238 | C2×C8 | — | 2 | 2 | no | — |
| 239 | C2×C8 | — | 2 | 2 | no | — |
| 240 | C2×C8 | — | 2 | 2 | no | — |
| 241 | C2×C8 | — | 4 | 2 | no | — |
| 242 | C2×C8 | — | 4 | 1 | no | — |
| 243 | C2×C8 | — | 4 | 2 | no | — |
| 244 | C2×C8 | — | 4 | 1 | no | — |
| 245 | C2×C8 | — | 4 | 2 | no | — |
| 246 | C2×C8 | — | 8 | 2 | no | — |
| 247 | C2×C8 | — | 8 | 2 | no | — |
| 248 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 252 | C4⋊C4 | — | 2 | 2 | no | — |
| 253 | C4⋊C4 | — | 2 | 2 | no | — |
| 254 | C4⋊C4 | — | 2 | 1 | no | — |
| 255 | C4⋊C4 | — | 2 | 1 | no | — |
| 256 | C4⋊C4 | — | 2 | 1 | no | — |
| 257 | C4⋊C4 | — | 4 | 2 | no | — |
| 258 | C4⋊C4 | — | 4 | 2 | no | — |
| 259 | C4⋊C4 | — | 4 | 2 | no | — |
| 260 | C4⋊C4 | — | 4 | 2 | no | — |
| 261 | C4⋊C4 | #121 | 1 | 2 | yes | — |
| 262 | C22⋊C4 | — | 4 | 1 | no | — |
| 263 | C22⋊C4 | — | 4 | 1 | no | — |
| 264 | C42 | — | 2 | 2 | no | — |
| 265 | C42 | — | 2 | 2 | no | — |
| 266 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 267 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 268 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 269 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 270 | C23 | — | 4 | 1 | no | — |
| 271 | C23 | #548 | 1 | 1 | yes | characteristic |
| 272 | Q8 | — | 2 | 2 | no | — |
| 273 | Q8 | — | 2 | 2 | no | — |
| 274 | Q8 | — | 4 | 2 | no | — |
| 275 | Q8 | — | 8 | 1 | no | — |
| 276 | Q8 | — | 16 | 2 | no | — |
| 277 | Q8 | — | 16 | 2 | no | — |
| 278 | D4 | — | 4 | 2 | no | — |
| 279 | D4 | — | 4 | 2 | no | — |
| 280 | D4 | — | 8 | 1 | no | — |
| 281 | D4 | — | 8 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 2 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 2 | 2 | no | — |
| 285 | C2×C4 | — | 4 | 1 | no | — |
| 286 | C2×C4 | — | 4 | 1 | no | — |
| 287 | C2×C4 | — | 8 | 1 | no | — |
| 288 | C2×C4 | — | 8 | 1 | no | — |
| 289 | C2×C4 | — | 8 | 1 | no | — |
| 290 | C2×C4 | — | 8 | 1 | no | — |
| 291 | C2×C4 | — | 8 | 2 | no | — |
| 292 | C2×C4 | — | 16 | 2 | no | — |
| 293 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #537 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #547 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #553 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | #575 | 1 | 2 | yes | — |
| 300 | C2×C4 | #578 | 1 | 2 | yes | — |
| 301 | C8 | — | 2 | 1 | no | — |
| 302 | C8 | — | 2 | 1 | no | — |
| 303 | C8 | — | 2 | 1 | no | — |
| 304 | C8 | — | 2 | 1 | no | — |
| 305 | C8 | — | 4 | 2 | no | — |
| 306 | C8 | — | 4 | 2 | no | — |
| 307 | C8 | — | 4 | 2 | no | — |
| 308 | C8 | — | 4 | 2 | no | — |
| 309 | C8 | — | 8 | 1 | no | — |
| 310 | C8 | — | 8 | 2 | no | — |
| 311 | C8 | — | 8 | 1 | no | — |
| 312 | C8 | — | 8 | 2 | no | — |
| 313 | C22 | — | 2 | 1 | no | — |
| 314 | C22 | — | 2 | 1 | no | — |
| 315 | C22 | — | 8 | 1 | no | — |
| 316 | C22 | — | 8 | 1 | no | — |
| 317 | C22 | — | 8 | 1 | no | — |
| 318 | C22 | #2416 | 1 | 1 | yes | characteristic |
| 319 | C22 | #3377 | 1 | 2 | yes | — |
| 320 | C4 | — | 2 | 1 | no | — |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 2 | 1 | no | — |
| 323 | C4 | — | 4 | 2 | no | — |
| 324 | C4 | — | 4 | 2 | no | — |
| 325 | C4 | — | 8 | 1 | no | — |
| 326 | C4 | — | 16 | 2 | no | — |
| 327 | C4 | #3040 | 1 | 1 | yes | characteristic |
| 328 | C4 | #3044 | 1 | 1 | yes | characteristic |
| 329 | C4 | #3069 | 1 | 2 | yes | — |
| 330 | C4 | #3269 | 1 | 2 | yes | — |
| 331 | C2 | — | 2 | 2 | no | — |
| 332 | C2 | — | 16 | 1 | no | — |
| 333 | C2 | #24743 | 1 | 1 | yes | characteristic |
| 334 | C2 | #24807 | 1 | 1 | yes | characteristic |
| 335 | C2 | #24809 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #532 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #22747 | C2 | semidirect |
| 4. C2×#1696 | C2 | semidirect |
| 6. #10605 | C2 | semidirect |
| 8. #9778 | C2 | semidirect |
| 9. #9609 | C2 | semidirect |
| 10. #9250 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28731 | 2 |
| #22747 | 2 |
| #19176 | 2 |
| C2×#1696 | 2 |
| #17936 | 2 |
| #10605 | 2 |
| #10551 | 2 |
| #9778 | 2 |
| #9609 | 2 |
| #9250 | 2 |
| #8686 | 2 |