Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
624 conjugacy classes of subgroups.
The subgroup lattice has 350 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #28762 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #28699 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19681 | C2 | 1 | 2 | yes | maximal |
| 4 | C2×#1707 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #19209 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #19204 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #17958 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #10173 | C2 | 1 | 2 | yes | maximal |
| 9 | #10167 | C2 | 1 | 2 | yes | maximal |
| 10 | #9809 | C2 | 1 | 2 | yes | maximal |
| 11 | #8293 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #3230 | C22 | 1 | 2 | yes | — |
| 13 | #3229 | C22 | 1 | 2 | yes | — |
| 14 | #3170 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3165 | C22 | 1 | 2 | yes | — |
| 16 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×#498 | C22 | 1 | 2 | yes | — |
| 18 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 19 | #3139 | C22 | 1 | 2 | yes | — |
| 20 | C2×#487 | C22 | 1 | 2 | yes | — |
| 21 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 22 | #2960 | C22 | 1 | 1 | yes | characteristic |
| 23 | #2089 | — | 2 | 2 | no | — |
| 24 | #2047 | — | 2 | 2 | no | — |
| 25 | #1997 | C22 | 1 | 2 | yes | — |
| 26 | #1979 | — | 2 | 2 | no | — |
| 27 | #1966 | C22 | 1 | 2 | yes | — |
| 28 | #1952 | — | 2 | 2 | no | — |
| 29 | C4×#108 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1723 | — | 2 | 2 | no | — |
| 31 | #1718 | — | 2 | 2 | no | — |
| 32 | #1718 | — | 2 | 2 | no | — |
| 33 | #1718 | — | 2 | 2 | no | — |
| 34 | #1707 | — | 2 | 2 | no | — |
| 35 | #1707 | — | 2 | 2 | no | — |
| 36 | #1696 | — | 2 | 2 | no | — |
| 37 | #1690 | — | 2 | 2 | no | — |
| 38 | #1676 | — | 2 | 2 | no | — |
| 39 | #1670 | — | 2 | 2 | no | — |
| 40 | #1591 | — | 2 | 2 | no | — |
| 41 | C8×Q16 | — | 2 | 2 | no | — |
| 42 | #1567 | C22 | 1 | 1 | yes | characteristic |
| 43 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1548 | C22 | 1 | 2 | yes | — |
| 45 | #1546 | C22 | 1 | 2 | yes | — |
| 46 | #1493 | C22 | 1 | 1 | yes | characteristic |
| 47 | #1493 | C22 | 1 | 1 | yes | characteristic |
| 48 | #1451 | C22 | 1 | 2 | yes | — |
| 49 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 50 | #1278 | C22 | 1 | 2 | yes | — |
| 51 | C22×Q16 | — | 2 | 1 | no | — |
| 52 | C22×Q16 | — | 2 | 1 | no | — |
| 53 | #533 | C23 | 1 | 1 | yes | characteristic |
| 54 | #533 | C23 | 1 | 1 | yes | characteristic |
| 55 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 56 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 57 | #501 | — | 4 | 2 | no | — |
| 58 | #500 | — | 2 | 2 | no | — |
| 59 | #500 | — | 2 | 2 | no | — |
| 60 | #500 | — | 2 | 1 | no | — |
| 61 | #500 | — | 2 | 1 | no | — |
| 62 | #500 | — | 2 | 1 | no | — |
| 63 | #500 | — | 2 | 1 | no | — |
| 64 | #500 | D4 | 1 | 1 | yes | characteristic |
| 65 | #498 | — | 2 | 2 | no | — |
| 66 | #498 | — | 2 | 2 | no | — |
| 67 | #498 | — | 2 | 4 | no | — |
| 68 | #498 | — | 4 | 2 | no | — |
| 69 | #494 | — | 2 | 1 | no | — |
| 70 | #494 | — | 2 | 1 | no | — |
| 71 | #494 | — | 2 | 1 | no | — |
| 72 | #494 | — | 2 | 1 | no | — |
| 73 | #494 | D4 | 1 | 1 | yes | characteristic |
| 74 | #491 | — | 4 | 2 | no | — |
| 75 | #487 | — | 2 | 2 | no | — |
| 76 | #487 | — | 2 | 2 | no | — |
| 77 | #487 | — | 2 | 2 | no | — |
| 78 | #487 | — | 2 | 2 | no | — |
| 79 | #487 | — | 2 | 4 | no | — |
| 80 | #487 | — | 2 | 2 | no | — |
| 81 | #487 | — | 2 | 2 | no | — |
| 82 | #487 | D4 | 1 | 1 | yes | characteristic |
| 83 | #487 | D4 | 1 | 1 | yes | characteristic |
| 84 | #484 | — | 4 | 2 | no | — |
| 85 | #483 | — | 4 | 2 | no | — |
| 86 | #477 | — | 2 | 4 | no | — |
| 87 | #475 | — | 2 | 4 | no | — |
| 88 | #467 | — | 2 | 2 | no | — |
| 89 | #446 | — | 2 | 1 | no | — |
| 90 | #446 | — | 2 | 1 | no | — |
| 91 | #446 | — | 2 | 1 | no | — |
| 92 | #446 | — | 2 | 1 | no | — |
| 93 | C8×Q8 | — | 2 | 1 | no | — |
| 94 | C8×Q8 | — | 2 | 1 | no | — |
| 95 | C8×Q8 | — | 2 | 1 | no | — |
| 96 | C8×Q8 | — | 2 | 1 | no | — |
| 97 | C4×Q16 | — | 2 | 2 | no | — |
| 98 | #431 | C23 | 1 | 2 | yes | — |
| 99 | C2×#108 | D4 | 1 | 2 | yes | — |
| 100 | #428 | — | 2 | 4 | no | — |
| 101 | C2×#107 | — | 2 | 1 | no | — |
| 102 | C2×#107 | — | 2 | 1 | no | — |
| 103 | C2×#106 | — | 4 | 2 | no | — |
| 104 | #424 | — | 2 | 1 | no | — |
| 105 | #424 | — | 2 | 1 | no | — |
| 106 | #423 | C23 | 1 | 1 | yes | characteristic |
| 107 | #423 | C23 | 1 | 1 | yes | characteristic |
| 108 | #419 | — | 2 | 4 | no | — |
| 109 | C2×#103 | — | 2 | 4 | no | — |
| 110 | C2×#101 | — | 2 | 4 | no | — |
| 111 | #405 | — | 2 | 1 | no | — |
| 112 | #405 | — | 2 | 1 | no | — |
| 113 | C4×M16 | C23 | 1 | 2 | yes | — |
| 114 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 117 | #343 | — | 2 | 4 | no | — |
| 118 | #341 | D4 | 1 | 2 | yes | — |
| 119 | #335 | — | 2 | 2 | no | — |
| 120 | #335 | D4 | 1 | 2 | yes | — |
| 121 | #334 | D4 | 1 | 2 | yes | — |
| 122 | #333 | — | 2 | 4 | no | — |
| 123 | #332 | — | 2 | 4 | no | — |
| 124 | Q8⋊C8 | — | 2 | 2 | no | — |
| 125 | Q8⋊C8 | — | 2 | 2 | no | — |
| 126 | #322 | — | 2 | 2 | no | — |
| 127 | #322 | — | 2 | 2 | no | — |
| 128 | C82 | D4 | 1 | 2 | yes | — |
| 129 | C82 | D4 | 1 | 2 | yes | — |
| 130 | C22×Q8 | — | 2 | 2 | no | — |
| 131 | C22×Q8 | — | 2 | 2 | no | — |
| 132 | C2×Q16 | — | 2 | 2 | no | — |
| 133 | C2×Q16 | — | 4 | 1 | no | — |
| 134 | C2×Q16 | — | 4 | 1 | no | — |
| 135 | C2×Q16 | — | 4 | 1 | no | — |
| 136 | C2×Q16 | — | 4 | 1 | no | — |
| 137 | C2×Q16 | — | 4 | 4 | no | — |
| 138 | C2×Q16 | — | 8 | 2 | no | — |
| 139 | C2×M16 | — | 2 | 1 | no | — |
| 140 | C2×M16 | — | 2 | 1 | no | — |
| 141 | C2×M16 | C2×D4 | 1 | 4 | yes | — |
| 142 | C22×C8 | — | 2 | 1 | no | — |
| 143 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 144 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 145 | C22×C8 | #41 | 1 | 2 | yes | — |
| 146 | C4⋊Q8 | — | 2 | 2 | no | — |
| 147 | C4⋊Q8 | — | 2 | 2 | no | — |
| 148 | C4⋊Q8 | — | 2 | 2 | no | — |
| 149 | C4⋊Q8 | — | 2 | 2 | no | — |
| 150 | C4⋊Q8 | — | 2 | 2 | no | — |
| 151 | C4⋊Q8 | — | 2 | 2 | no | — |
| 152 | C4⋊Q8 | — | 2 | 2 | no | — |
| 153 | C4⋊Q8 | — | 2 | 2 | no | — |
| 154 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 155 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 156 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 157 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 158 | #125 | — | 2 | 1 | no | — |
| 159 | #125 | — | 2 | 1 | no | — |
| 160 | C22⋊Q8 | — | 2 | 1 | no | — |
| 161 | C22⋊Q8 | — | 2 | 1 | no | — |
| 162 | C22⋊Q8 | — | 2 | 1 | no | — |
| 163 | C22⋊Q8 | — | 2 | 1 | no | — |
| 164 | C4×Q8 | — | 2 | 1 | no | — |
| 165 | C4×Q8 | — | 2 | 1 | no | — |
| 166 | C4×Q8 | — | 2 | 1 | no | — |
| 167 | C4×Q8 | — | 2 | 1 | no | — |
| 168 | #117 | — | 2 | 1 | no | — |
| 169 | #117 | — | 2 | 1 | no | — |
| 170 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 171 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 172 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 173 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 174 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 175 | #108 | — | 2 | 4 | no | — |
| 176 | #107 | — | 2 | 2 | no | — |
| 177 | #107 | — | 4 | 4 | no | — |
| 178 | #107 | — | 4 | 4 | no | — |
| 179 | #107 | — | 4 | 1 | no | — |
| 180 | #107 | — | 4 | 1 | no | — |
| 181 | #107 | — | 4 | 1 | no | — |
| 182 | #107 | — | 4 | 1 | no | — |
| 183 | #106 | — | 4 | 4 | no | — |
| 184 | #106 | — | 4 | 4 | no | — |
| 185 | #106 | — | 4 | 8 | no | — |
| 186 | C4⋊C8 | — | 2 | 1 | no | — |
| 187 | C4⋊C8 | — | 2 | 1 | no | — |
| 188 | C4⋊C8 | — | 2 | 1 | no | — |
| 189 | C4⋊C8 | — | 2 | 1 | no | — |
| 190 | C4⋊C8 | — | 4 | 1 | no | — |
| 191 | C4⋊C8 | — | 4 | 1 | no | — |
| 192 | C4⋊C8 | — | 4 | 1 | no | — |
| 193 | C4⋊C8 | — | 4 | 1 | no | — |
| 194 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 195 | #103 | — | 2 | 2 | no | — |
| 196 | #103 | — | 2 | 2 | no | — |
| 197 | #103 | — | 4 | 4 | no | — |
| 198 | #103 | — | 4 | 4 | no | — |
| 199 | #103 | — | 4 | 4 | no | — |
| 200 | #103 | — | 4 | 4 | no | — |
| 201 | #103 | — | 4 | 4 | no | — |
| 202 | #103 | — | 4 | 4 | no | — |
| 203 | #101 | — | 8 | 4 | no | — |
| 204 | C22⋊C8 | — | 2 | 2 | no | — |
| 205 | #97 | — | 2 | 2 | no | — |
| 206 | #97 | — | 4 | 1 | no | — |
| 207 | #97 | — | 4 | 1 | no | — |
| 208 | C4×C8 | — | 2 | 2 | no | — |
| 209 | C4×C8 | — | 2 | 2 | no | — |
| 210 | C4×C8 | — | 2 | 1 | no | — |
| 211 | C4×C8 | — | 4 | 1 | no | — |
| 212 | C4×C8 | — | 4 | 1 | no | — |
| 213 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 214 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 215 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 216 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 217 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 218 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 219 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 220 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 221 | C2×Q8 | — | 2 | 1 | no | — |
| 222 | C2×Q8 | — | 2 | 1 | no | — |
| 223 | C2×Q8 | — | 2 | 1 | no | — |
| 224 | C2×Q8 | — | 2 | 1 | no | — |
| 225 | C2×Q8 | — | 4 | 2 | no | — |
| 226 | C2×Q8 | — | 4 | 2 | no | — |
| 227 | C2×Q8 | — | 4 | 2 | no | — |
| 228 | C2×Q8 | — | 4 | 2 | no | — |
| 229 | C2×Q8 | — | 4 | 4 | no | — |
| 230 | C2×Q8 | — | 4 | 4 | no | — |
| 231 | C2×Q8 | — | 8 | 4 | no | — |
| 232 | C22×C4 | — | 4 | 2 | no | — |
| 233 | C22×C4 | — | 4 | 2 | no | — |
| 234 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 235 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 236 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 237 | Q16 | — | 4 | 4 | no | — |
| 238 | Q16 | — | 8 | 2 | no | — |
| 239 | Q16 | — | 8 | 2 | no | — |
| 240 | Q16 | — | 8 | 2 | no | — |
| 241 | Q16 | — | 8 | 2 | no | — |
| 242 | M16 | — | 2 | 4 | no | — |
| 243 | M16 | — | 2 | 4 | no | — |
| 244 | M16 | — | 4 | 2 | no | — |
| 245 | M16 | — | 4 | 2 | no | — |
| 246 | C2×C8 | — | 2 | 4 | no | — |
| 247 | C2×C8 | — | 2 | 2 | no | — |
| 248 | C2×C8 | — | 2 | 2 | no | — |
| 249 | C2×C8 | — | 2 | 4 | no | — |
| 250 | C2×C8 | — | 2 | 4 | no | — |
| 251 | C2×C8 | — | 4 | 2 | no | — |
| 252 | C2×C8 | — | 4 | 1 | no | — |
| 253 | C2×C8 | — | 4 | 1 | no | — |
| 254 | C2×C8 | #121 | 1 | 2 | yes | — |
| 255 | C2×C8 | #121 | 1 | 2 | yes | — |
| 256 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 257 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 258 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | #137 | 1 | 4 | yes | — |
| 261 | C4⋊C4 | — | 2 | 1 | no | — |
| 262 | C4⋊C4 | — | 2 | 1 | no | — |
| 263 | C4⋊C4 | — | 2 | 1 | no | — |
| 264 | C4⋊C4 | — | 2 | 1 | no | — |
| 265 | C4⋊C4 | — | 2 | 1 | no | — |
| 266 | C4⋊C4 | — | 2 | 1 | no | — |
| 267 | C4⋊C4 | — | 2 | 1 | no | — |
| 268 | C4⋊C4 | — | 2 | 1 | no | — |
| 269 | C4⋊C4 | — | 4 | 1 | no | — |
| 270 | C4⋊C4 | — | 4 | 1 | no | — |
| 271 | C4⋊C4 | — | 4 | 1 | no | — |
| 272 | C4⋊C4 | — | 4 | 1 | no | — |
| 273 | C4⋊C4 | — | 4 | 2 | no | — |
| 274 | C4⋊C4 | — | 4 | 2 | no | — |
| 275 | C4⋊C4 | — | 4 | 2 | no | — |
| 276 | C4⋊C4 | — | 4 | 2 | no | — |
| 277 | C4⋊C4 | — | 4 | 2 | no | — |
| 278 | C4⋊C4 | — | 4 | 2 | no | — |
| 279 | C4⋊C4 | — | 4 | 2 | no | — |
| 280 | C4⋊C4 | — | 4 | 2 | no | — |
| 281 | C22⋊C4 | — | 4 | 1 | no | — |
| 282 | C22⋊C4 | — | 4 | 1 | no | — |
| 283 | C42 | — | 4 | 1 | no | — |
| 284 | C42 | — | 4 | 1 | no | — |
| 285 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 286 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 287 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 288 | C42 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 289 | C23 | #550 | 1 | 1 | yes | characteristic |
| 290 | Q8 | — | 4 | 2 | no | — |
| 291 | Q8 | — | 4 | 2 | no | — |
| 292 | Q8 | — | 4 | 2 | no | — |
| 293 | Q8 | — | 4 | 2 | no | — |
| 294 | Q8 | — | 8 | 2 | no | — |
| 295 | Q8 | — | 8 | 2 | no | — |
| 296 | Q8 | — | 8 | 4 | no | — |
| 297 | Q8 | — | 8 | 4 | no | — |
| 298 | Q8 | — | 16 | 2 | no | — |
| 299 | C2×C4 | — | 2 | 2 | no | — |
| 300 | C2×C4 | — | 2 | 2 | no | — |
| 301 | C2×C4 | — | 4 | 1 | no | — |
| 302 | C2×C4 | — | 4 | 1 | no | — |
| 303 | C2×C4 | — | 4 | 1 | no | — |
| 304 | C2×C4 | — | 4 | 1 | no | — |
| 305 | C2×C4 | — | 8 | 4 | no | — |
| 306 | C2×C4 | — | 8 | 2 | no | — |
| 307 | C2×C4 | — | 8 | 2 | no | — |
| 308 | C2×C4 | — | 16 | 2 | no | — |
| 309 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | #535 | 1 | 1 | yes | characteristic |
| 312 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 313 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 314 | C2×C4 | #550 | 1 | 1 | yes | characteristic |
| 315 | C2×C4 | C2×#137 | 1 | 2 | yes | — |
| 316 | C2×C4 | #577 | 1 | 2 | yes | — |
| 317 | C8 | — | 2 | 4 | no | — |
| 318 | C8 | — | 2 | 1 | no | — |
| 319 | C8 | — | 2 | 1 | no | — |
| 320 | C8 | — | 2 | 2 | no | — |
| 321 | C8 | — | 2 | 2 | no | — |
| 322 | C8 | — | 4 | 4 | no | — |
| 323 | C8 | — | 4 | 4 | no | — |
| 324 | C8 | — | 8 | 1 | no | — |
| 325 | C8 | — | 8 | 1 | no | — |
| 326 | C8 | #497 | 1 | 2 | yes | — |
| 327 | C8 | #497 | 1 | 2 | yes | — |
| 328 | C22 | — | 2 | 1 | no | — |
| 329 | C22 | — | 2 | 1 | no | — |
| 330 | C22 | #2541 | 1 | 1 | yes | characteristic |
| 331 | C22 | #3400 | 1 | 2 | yes | — |
| 332 | C4 | — | 2 | 1 | no | — |
| 333 | C4 | — | 2 | 1 | no | — |
| 334 | C4 | — | 8 | 1 | no | — |
| 335 | C4 | — | 8 | 1 | no | — |
| 336 | C4 | — | 8 | 1 | no | — |
| 337 | C4 | — | 8 | 1 | no | — |
| 338 | C4 | — | 16 | 2 | no | — |
| 339 | C4 | — | 16 | 2 | no | — |
| 340 | C4 | #3067 | 1 | 2 | yes | — |
| 341 | C4 | C2×#497 | 1 | 1 | yes | characteristic |
| 342 | C4 | #3154 | 1 | 1 | yes | characteristic |
| 343 | C4 | #3278 | 1 | 2 | yes | — |
| 344 | C4 | #3293 | 1 | 2 | yes | — |
| 345 | C2 | — | 2 | 2 | no | — |
| 346 | C2 | #28473 | 1 | 1 | yes | characteristic |
| 347 | C2 | #28496 | 1 | 1 | yes | characteristic |
| 348 | C2 | #28505 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2541 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #550 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #28762 | 2 |
| #28699 | 2 |
| #19681 | 2 |
| C2×#1707 | 2 |
| #19209 | 2 |
| #19204 | 2 |
| #17958 | 2 |
| #10173 | 2 |
| #10167 | 2 |
| #9809 | 2 |
| #8293 | 2 |