Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1611 conjugacy classes of subgroups.
The subgroup lattice has 338 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #45437 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #45436 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #32680 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #28738 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #28461 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #28260 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #24646 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #24644 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #22806 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #21715 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #21695 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | C2×#492 | C22 | 1 | 2 | yes | — |
| 13 | C2×#492 | C22 | 1 | 4 | yes | — |
| 14 | C2×#460 | C22 | 1 | 4 | yes | — |
| 15 | C2×#461 | C22 | 1 | 4 | yes | — |
| 16 | C2×C8×Q8 | C22 | 1 | 2 | yes | — |
| 17 | C2×#440 | C22 | 1 | 4 | yes | — |
| 18 | C2×C4×SD16 | C22 | 1 | 4 | yes | — |
| 19 | #2769 | — | 2 | 4 | no | — |
| 20 | #2699 | — | 2 | 4 | no | — |
| 21 | #2612 | — | 2 | 4 | no | — |
| 22 | #2568 | C22 | 1 | 4 | yes | — |
| 23 | #2555 | — | 2 | 4 | no | — |
| 24 | #2548 | C22 | 1 | 4 | yes | — |
| 25 | #2546 | C22 | 1 | 4 | yes | — |
| 26 | #2541 | C22 | 1 | 4 | yes | — |
| 27 | #2399 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2357 | C22 | 1 | 4 | yes | — |
| 29 | #2356 | C22 | 1 | 4 | yes | — |
| 30 | #2328 | C22 | 1 | 2 | yes | — |
| 31 | #2327 | C22 | 1 | 2 | yes | — |
| 32 | #2055 | C22 | 1 | 8 | yes | — |
| 33 | #2055 | C22 | 1 | 8 | yes | — |
| 34 | #2014 | C22 | 1 | 8 | yes | — |
| 35 | #2013 | C22 | 1 | 8 | yes | — |
| 36 | #1985 | C22 | 1 | 2 | yes | — |
| 37 | #1982 | C22 | 1 | 4 | yes | — |
| 38 | #1981 | C22 | 1 | 4 | yes | — |
| 39 | #1973 | C22 | 1 | 4 | yes | — |
| 40 | #1968 | C22 | 1 | 8 | yes | — |
| 41 | #1968 | C22 | 1 | 4 | yes | — |
| 42 | #1961 | C22 | 1 | 2 | yes | — |
| 43 | #1960 | C22 | 1 | 2 | yes | — |
| 44 | #1940 | C22 | 1 | 4 | yes | — |
| 45 | #1928 | C22 | 1 | 8 | yes | — |
| 46 | #1925 | C22 | 1 | 8 | yes | — |
| 47 | #1846 | C22 | 1 | 2 | yes | — |
| 48 | #1843 | C22 | 1 | 1 | yes | characteristic |
| 49 | #1805 | C22 | 1 | 4 | yes | — |
| 50 | #1804 | C22 | 1 | 4 | yes | — |
| 51 | C4×#106 | C22 | 1 | 4 | yes | — |
| 52 | C8×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 53 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 54 | C22×SD16 | C23 | 1 | 4 | yes | — |
| 55 | C22×SD16 | C23 | 1 | 4 | yes | — |
| 56 | C22×SD16 | C23 | 1 | 8 | yes | — |
| 57 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 58 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 59 | C2×#127 | C23 | 1 | 2 | yes | — |
| 60 | C2×#127 | C23 | 1 | 4 | yes | — |
| 61 | C2×#124 | — | 2 | 4 | no | — |
| 62 | C2×#121 | — | 2 | 8 | no | — |
| 63 | C2×C4×Q8 | C23 | 1 | 4 | yes | — |
| 64 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 65 | C2×C4×D4 | C23 | 1 | 4 | yes | — |
| 66 | #492 | — | 2 | 8 | no | — |
| 67 | #492 | — | 4 | 8 | no | — |
| 68 | #461 | — | 4 | 8 | no | — |
| 69 | #460 | — | 4 | 8 | no | — |
| 70 | C8×Q8 | — | 4 | 4 | no | — |
| 71 | #440 | — | 4 | 8 | no | — |
| 72 | C4×SD16 | — | 2 | 16 | no | — |
| 73 | C2×#107 | C23 | 1 | 4 | yes | — |
| 74 | C2×#106 | C23 | 1 | 4 | yes | — |
| 75 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 76 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 77 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 78 | C2×#103 | C23 | 1 | 4 | yes | — |
| 79 | C2×#103 | C23 | 1 | 4 | yes | — |
| 80 | C2×#103 | C23 | 1 | 8 | yes | — |
| 81 | C2×#102 | C23 | 1 | 4 | yes | — |
| 82 | C2×#102 | C23 | 1 | 4 | yes | — |
| 83 | C2×#102 | C23 | 1 | 8 | yes | — |
| 84 | C2×#97 | C23 | 1 | 2 | yes | — |
| 85 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 87 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 89 | #400 | — | 2 | 4 | no | — |
| 90 | #400 | — | 2 | 4 | no | — |
| 91 | #398 | — | 2 | 4 | no | — |
| 92 | #398 | — | 2 | 4 | no | — |
| 93 | #398 | D4 | 1 | 4 | yes | — |
| 94 | #395 | D4 | 1 | 4 | yes | — |
| 95 | #394 | — | 2 | 4 | no | — |
| 96 | #394 | — | 2 | 4 | no | — |
| 97 | #392 | — | 2 | 4 | no | — |
| 98 | #392 | — | 2 | 4 | no | — |
| 99 | #391 | — | 2 | 4 | no | — |
| 100 | #391 | C23 | 1 | 4 | yes | — |
| 101 | #391 | C23 | 1 | 8 | yes | — |
| 102 | #390 | — | 2 | 4 | no | — |
| 103 | #390 | C23 | 1 | 4 | yes | — |
| 104 | #390 | C23 | 1 | 8 | yes | — |
| 105 | #389 | — | 2 | 8 | no | — |
| 106 | #388 | — | 2 | 8 | no | — |
| 107 | #387 | — | 2 | 8 | no | — |
| 108 | #387 | — | 2 | 8 | no | — |
| 109 | #386 | — | 2 | 8 | no | — |
| 110 | #383 | — | 2 | 4 | no | — |
| 111 | #382 | — | 2 | 4 | no | — |
| 112 | #382 | C23 | 1 | 2 | yes | — |
| 113 | #382 | C23 | 1 | 2 | yes | — |
| 114 | #381 | — | 2 | 4 | no | — |
| 115 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 116 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 117 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 118 | C4×C22⋊C4 | — | 2 | 4 | no | — |
| 119 | #376 | — | 2 | 4 | no | — |
| 120 | #376 | C23 | 1 | 1 | yes | characteristic |
| 121 | #340 | C23 | 1 | 8 | yes | — |
| 122 | #336 | C23 | 1 | 8 | yes | — |
| 123 | C22×Q8 | C2×D4 | 1 | 2 | yes | — |
| 124 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 125 | C22×D4 | — | 2 | 4 | no | — |
| 126 | C22×D4 | — | 2 | 4 | no | — |
| 127 | C22×D4 | — | 2 | 4 | no | — |
| 128 | C22×D4 | C24 | 1 | 4 | yes | — |
| 129 | C23×C4 | — | 2 | 4 | no | — |
| 130 | C2×SD16 | — | 2 | 8 | no | — |
| 131 | C2×SD16 | — | 2 | 8 | no | — |
| 132 | C2×SD16 | — | 2 | 8 | no | — |
| 133 | C2×SD16 | — | 2 | 8 | no | — |
| 134 | C2×SD16 | — | 4 | 4 | no | — |
| 135 | C2×SD16 | — | 4 | 4 | no | — |
| 136 | C2×SD16 | — | 4 | 8 | no | — |
| 137 | C2×SD16 | — | 4 | 8 | no | — |
| 138 | C2×SD16 | — | 4 | 8 | no | — |
| 139 | C22×C8 | C24 | 1 | 4 | yes | — |
| 140 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 141 | C22×C8 | C24 | 1 | 2 | yes | — |
| 142 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 143 | C4⋊Q8 | — | 2 | 4 | no | — |
| 144 | C4⋊Q8 | — | 2 | 4 | no | — |
| 145 | C4⋊Q8 | — | 4 | 4 | no | — |
| 146 | C4⋊Q8 | — | 4 | 4 | no | — |
| 147 | #127 | — | 2 | 4 | no | — |
| 148 | #127 | — | 2 | 4 | no | — |
| 149 | #127 | — | 4 | 4 | no | — |
| 150 | #127 | — | 4 | 4 | no | — |
| 151 | #124 | — | 8 | 8 | no | — |
| 152 | #121 | — | 8 | 16 | no | — |
| 153 | C4×Q8 | — | 2 | 8 | no | — |
| 154 | C4×Q8 | — | 2 | 8 | no | — |
| 155 | C4×Q8 | — | 4 | 2 | no | — |
| 156 | C4×Q8 | — | 4 | 2 | no | — |
| 157 | C4×D4 | — | 2 | 8 | no | — |
| 158 | C4×D4 | — | 2 | 8 | no | — |
| 159 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 160 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 161 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 162 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 163 | C2×C4⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 164 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 165 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 166 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 167 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 168 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 169 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 170 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 171 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 172 | C2×C42 | — | 2 | 4 | no | — |
| 173 | C2×C42 | — | 2 | 4 | no | — |
| 174 | C2×C42 | — | 2 | 4 | no | — |
| 175 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 176 | C2×C42 | C24 | 1 | 4 | yes | — |
| 177 | C2×C42 | C24 | 1 | 2 | yes | — |
| 178 | #107 | — | 4 | 4 | no | — |
| 179 | #106 | — | 2 | 8 | no | — |
| 180 | C4⋊C8 | — | 2 | 8 | no | — |
| 181 | C4⋊C8 | — | 4 | 2 | no | — |
| 182 | C4⋊C8 | — | 4 | 4 | no | — |
| 183 | #103 | — | 2 | 8 | no | — |
| 184 | #103 | — | 4 | 4 | no | — |
| 185 | #103 | — | 4 | 8 | no | — |
| 186 | #102 | — | 2 | 8 | no | — |
| 187 | #102 | — | 4 | 4 | no | — |
| 188 | #102 | — | 4 | 8 | no | — |
| 189 | #97 | — | 2 | 4 | no | — |
| 190 | #97 | — | 4 | 1 | no | — |
| 191 | C4×C8 | — | 2 | 2 | no | — |
| 192 | C4×C8 | — | 2 | 8 | no | — |
| 193 | C4×C8 | #41 | 1 | 8 | yes | — |
| 194 | #95 | — | 2 | 4 | no | — |
| 195 | #95 | — | 2 | 8 | no | — |
| 196 | #95 | — | 2 | 4 | no | — |
| 197 | #95 | — | 2 | 4 | no | — |
| 198 | #95 | — | 2 | 4 | no | — |
| 199 | #95 | — | 2 | 8 | no | — |
| 200 | #95 | — | 2 | 8 | no | — |
| 201 | #95 | C2×D4 | 1 | 4 | yes | — |
| 202 | #95 | C2×D4 | 1 | 4 | yes | — |
| 203 | #95 | C2×D4 | 1 | 4 | yes | — |
| 204 | C24 | — | 2 | 4 | no | — |
| 205 | C2×Q8 | — | 2 | 8 | no | — |
| 206 | C2×Q8 | — | 2 | 4 | no | — |
| 207 | C2×Q8 | — | 2 | 4 | no | — |
| 208 | C2×Q8 | — | 2 | 4 | no | — |
| 209 | C2×Q8 | — | 2 | 4 | no | — |
| 210 | C2×Q8 | — | 2 | 4 | no | — |
| 211 | C2×Q8 | — | 2 | 4 | no | — |
| 212 | C2×Q8 | — | 4 | 2 | no | — |
| 213 | C2×D4 | — | 2 | 4 | no | — |
| 214 | C2×D4 | — | 2 | 4 | no | — |
| 215 | C2×D4 | — | 2 | 4 | no | — |
| 216 | C2×D4 | — | 2 | 4 | no | — |
| 217 | C2×D4 | — | 2 | 4 | no | — |
| 218 | C2×D4 | — | 2 | 4 | no | — |
| 219 | C2×D4 | — | 4 | 8 | no | — |
| 220 | C2×D4 | — | 4 | 8 | no | — |
| 221 | C2×D4 | — | 8 | 8 | no | — |
| 222 | C2×D4 | — | 8 | 8 | no | — |
| 223 | C2×D4 | — | 8 | 4 | no | — |
| 224 | C2×D4 | — | 8 | 4 | no | — |
| 225 | C2×D4 | — | 8 | 4 | no | — |
| 226 | C22×C4 | — | 2 | 4 | no | — |
| 227 | C22×C4 | — | 2 | 4 | no | — |
| 228 | C22×C4 | — | 2 | 4 | no | — |
| 229 | C22×C4 | — | 4 | 8 | no | — |
| 230 | C22×C4 | — | 8 | 8 | no | — |
| 231 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 232 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 233 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 234 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 235 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 236 | SD16 | — | 4 | 16 | no | — |
| 237 | SD16 | — | 4 | 16 | no | — |
| 238 | SD16 | — | 8 | 16 | no | — |
| 239 | C2×C8 | — | 2 | 2 | no | — |
| 240 | C2×C8 | — | 2 | 2 | no | — |
| 241 | C2×C8 | — | 2 | 1 | no | — |
| 242 | C2×C8 | — | 2 | 4 | no | — |
| 243 | C2×C8 | — | 2 | 2 | no | — |
| 244 | C2×C8 | — | 2 | 8 | no | — |
| 245 | C2×C8 | C2×#41 | 1 | 4 | yes | — |
| 246 | C2×C8 | C2×#41 | 1 | 4 | yes | — |
| 247 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 248 | C4⋊C4 | — | 2 | 16 | no | — |
| 249 | C4⋊C4 | — | 2 | 4 | no | — |
| 250 | C4⋊C4 | — | 2 | 4 | no | — |
| 251 | C4⋊C4 | — | 2 | 4 | no | — |
| 252 | C4⋊C4 | — | 2 | 4 | no | — |
| 253 | C4⋊C4 | — | 4 | 4 | no | — |
| 254 | C4⋊C4 | — | 4 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 8 | no | — |
| 256 | C4⋊C4 | — | 4 | 8 | no | — |
| 257 | C4⋊C4 | — | 8 | 4 | no | — |
| 258 | C4⋊C4 | — | 8 | 4 | no | — |
| 259 | C22⋊C4 | — | 4 | 8 | no | — |
| 260 | C22⋊C4 | — | 4 | 16 | no | — |
| 261 | C22⋊C4 | — | 8 | 8 | no | — |
| 262 | C22⋊C4 | — | 8 | 8 | no | — |
| 263 | C42 | — | 2 | 1 | no | — |
| 264 | C42 | — | 2 | 4 | no | — |
| 265 | C42 | — | 2 | 1 | no | — |
| 266 | C42 | — | 2 | 4 | no | — |
| 267 | C42 | — | 4 | 8 | no | — |
| 268 | C42 | — | 4 | 8 | no | — |
| 269 | C42 | — | 8 | 4 | no | — |
| 270 | C42 | C2×#41 | 1 | 4 | yes | — |
| 271 | C42 | H(5,𝔽2) | 1 | 4 | yes | — |
| 272 | C23 | — | 4 | 4 | no | — |
| 273 | C23 | — | 4 | 4 | no | — |
| 274 | C23 | — | 4 | 4 | no | — |
| 275 | C23 | — | 4 | 8 | no | — |
| 276 | C23 | — | 4 | 8 | no | — |
| 277 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 278 | Q8 | — | 4 | 8 | no | — |
| 279 | Q8 | — | 4 | 4 | no | — |
| 280 | Q8 | — | 4 | 4 | no | — |
| 281 | Q8 | — | 4 | 4 | no | — |
| 282 | Q8 | — | 4 | 4 | no | — |
| 283 | D4 | — | 4 | 4 | no | — |
| 284 | D4 | — | 4 | 4 | no | — |
| 285 | D4 | — | 4 | 4 | no | — |
| 286 | D4 | — | 4 | 4 | no | — |
| 287 | D4 | — | 16 | 8 | no | — |
| 288 | D4 | — | 16 | 8 | no | — |
| 289 | D4 | — | 16 | 8 | no | — |
| 290 | C2×C4 | — | 2 | 4 | no | — |
| 291 | C2×C4 | — | 2 | 4 | no | — |
| 292 | C2×C4 | — | 2 | 8 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 2 | 4 | no | — |
| 295 | C2×C4 | — | 2 | 4 | no | — |
| 296 | C2×C4 | — | 2 | 8 | no | — |
| 297 | C2×C4 | — | 4 | 4 | no | — |
| 298 | C2×C4 | — | 4 | 4 | no | — |
| 299 | C2×C4 | — | 4 | 4 | no | — |
| 300 | C2×C4 | — | 4 | 4 | no | — |
| 301 | C2×C4 | — | 4 | 4 | no | — |
| 302 | C2×C4 | — | 4 | 8 | no | — |
| 303 | C2×C4 | — | 4 | 8 | no | — |
| 304 | C2×C4 | — | 16 | 8 | no | — |
| 305 | C2×C4 | #550 | 1 | 8 | yes | — |
| 306 | C2×C4 | #577 | 1 | 8 | yes | — |
| 307 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 308 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | C2×H(5,𝔽2) | 1 | 2 | yes | — |
| 310 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 311 | C8 | — | 2 | 4 | no | — |
| 312 | C8 | — | 4 | 1 | no | — |
| 313 | C8 | — | 4 | 4 | no | — |
| 314 | C8 | #555 | 1 | 4 | yes | — |
| 315 | C22 | — | 8 | 8 | no | — |
| 316 | C22 | — | 8 | 4 | no | — |
| 317 | C22 | — | 8 | 4 | no | — |
| 318 | C22 | — | 8 | 8 | no | — |
| 319 | C22 | — | 8 | 4 | no | — |
| 320 | C22 | C2×#550 | 1 | 2 | yes | — |
| 321 | C22 | #3496 | 1 | 1 | yes | characteristic |
| 322 | C22 | C2×#577 | 1 | 4 | yes | — |
| 323 | C4 | — | 2 | 8 | no | — |
| 324 | C4 | — | 2 | 8 | no | — |
| 325 | C4 | — | 4 | 4 | no | — |
| 326 | C4 | — | 4 | 2 | no | — |
| 327 | C4 | — | 8 | 4 | no | — |
| 328 | C4 | — | 8 | 4 | no | — |
| 329 | C4 | — | 8 | 4 | no | — |
| 330 | C4 | C2×#555 | 1 | 1 | yes | characteristic |
| 331 | C4 | #3506 | 1 | 2 | yes | — |
| 332 | C4 | #3549 | 1 | 1 | yes | characteristic |
| 333 | C2 | — | 16 | 4 | no | — |
| 334 | C2 | #59822 | 1 | 1 | yes | characteristic |
| 335 | C2 | #61410 | 1 | 4 | yes | — |
| 336 | C2 | #62268 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45437 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #45436 | C2 | semidirect |
| 3. #32680 | C2 | semidirect |
| 4. #28738 | C2 | semidirect |
| 5. #28461 | C2 | semidirect |
| 6. #28260 | C2 | semidirect |
| 7. #24646 | C2 | semidirect |
| 8. #24644 | C2 | semidirect |
| 9. #22806 | C2 | semidirect |
| 10. #21715 | C2 | semidirect |
| 11. #21695 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #45437 | 2 |
| #45436 | 2 |
| #32680 | 2 |
| #28738 | 2 |
| #28461 | 2 |
| #28260 | 2 |
| #24646 | 2 |
| #24644 | 2 |
| #22806 | 2 |
| #21715 | 2 |
| #21695 | 2 |