Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1724 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#3185 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #34221 | C2 | 1 | 2 | yes | maximal |
| 3 | #22277 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22253 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22229 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #22189 | C2 | 1 | 4 | yes | maximal |
| 7 | #22188 | C2 | 1 | 4 | yes | maximal |
| 8 | #21404 | C2 | 1 | 2 | yes | maximal |
| 9 | #21127 | C2 | 1 | 8 | yes | maximal, complemented |
| 10 | #20825 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #20365 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×#533 | C22 | 1 | 2 | yes | — |
| 13 | #3185 | C4 | 1 | 16 | yes | complemented |
| 14 | C2×#498 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#487 | C22 | 1 | 2 | yes | — |
| 17 | C2×#470 | C22 | 1 | 4 | yes | — |
| 18 | C2×#441 | C22 | 1 | 4 | yes | — |
| 19 | C2×C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2371 | — | 2 | 4 | no | — |
| 22 | #2368 | — | 2 | 4 | no | — |
| 23 | #2356 | — | 2 | 4 | no | — |
| 24 | #2343 | — | 2 | 4 | no | — |
| 25 | #2331 | C22 | 1 | 4 | yes | — |
| 26 | #2327 | C22 | 1 | 4 | yes | — |
| 27 | #2320 | C22 | 1 | 4 | yes | — |
| 28 | C4×C22⋊Q8 | C22 | 1 | 8 | yes | — |
| 29 | #2283 | C22 | 1 | 2 | yes | — |
| 30 | #2276 | C22 | 1 | 4 | yes | — |
| 31 | C2×#376 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1993 | C22 | 1 | 4 | yes | — |
| 33 | #1960 | C22 | 1 | 4 | yes | — |
| 34 | #1928 | C22 | 1 | 8 | yes | — |
| 35 | #1927 | C22 | 1 | 8 | yes | — |
| 36 | #1920 | C22 | 1 | 8 | yes | — |
| 37 | #1919 | C22 | 1 | 8 | yes | — |
| 38 | #1894 | C22 | 1 | 4 | yes | — |
| 39 | #1894 | C22 | 1 | 4 | yes | — |
| 40 | #1878 | C22 | 1 | 8 | yes | — |
| 41 | #1878 | C22 | 1 | 8 | yes | — |
| 42 | #1865 | C22 | 1 | 8 | yes | — |
| 43 | #1863 | C22 | 1 | 8 | yes | — |
| 44 | #1846 | C22 | 1 | 2 | yes | — |
| 45 | #1835 | C22 | 1 | 2 | yes | — |
| 46 | #1809 | C22 | 1 | 4 | yes | — |
| 47 | #1776 | C22 | 1 | 4 | yes | — |
| 48 | C4×#103 | C22 | 1 | 8 | yes | — |
| 49 | #1746 | C22 | 1 | 2 | yes | — |
| 50 | #1746 | C22 | 1 | 4 | yes | — |
| 51 | #1746 | C22 | 1 | 2 | yes | — |
| 52 | #1728 | C22 | 1 | 2 | yes | — |
| 53 | #1728 | C22 | 1 | 2 | yes | — |
| 54 | C22×Q16 | C23 | 1 | 4 | yes | — |
| 55 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 56 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 57 | C22×M16 | C23 | 1 | 2 | yes | — |
| 58 | #533 | — | 2 | 8 | no | — |
| 59 | #533 | C2×C4 | 1 | 16 | yes | — |
| 60 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 61 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 62 | C2×#125 | — | 2 | 2 | no | — |
| 63 | C2×C22⋊Q8 | — | 2 | 2 | no | — |
| 64 | C2×C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 65 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 66 | C2×C4×Q8 | C23 | 1 | 4 | yes | — |
| 67 | C2×C4×Q8 | C23 | 1 | 8 | yes | — |
| 68 | C2×#117 | — | 2 | 2 | no | — |
| 69 | C22×C42 | C23 | 1 | 2 | yes | — |
| 70 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 71 | #498 | C2×C4 | 1 | 8 | yes | complemented |
| 72 | #494 | C2×C4 | 1 | 8 | yes | complemented |
| 73 | #487 | C2×C4 | 1 | 16 | yes | complemented |
| 74 | #470 | C2×C4 | 1 | 32 | yes | — |
| 75 | #441 | C2×C4 | 1 | 32 | yes | complemented |
| 76 | C2×#107 | C23 | 1 | 4 | yes | — |
| 77 | C2×#106 | C23 | 1 | 4 | yes | — |
| 78 | C2×#106 | C23 | 1 | 4 | yes | — |
| 79 | C2×#103 | C23 | 1 | 8 | yes | — |
| 80 | C2×#103 | C23 | 1 | 8 | yes | — |
| 81 | C2×#103 | C23 | 1 | 8 | yes | — |
| 82 | C4×M16 | — | 2 | 4 | no | — |
| 83 | C4×M16 | C2×C4 | 1 | 8 | yes | — |
| 84 | C2×#97 | C23 | 1 | 2 | yes | — |
| 85 | C2×#97 | C23 | 1 | 2 | yes | — |
| 86 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 87 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | #391 | — | 2 | 8 | no | — |
| 90 | #391 | C23 | 1 | 8 | yes | — |
| 91 | #389 | C23 | 1 | 8 | yes | — |
| 92 | #387 | — | 2 | 8 | no | — |
| 93 | #387 | — | 2 | 8 | no | — |
| 94 | #387 | — | 2 | 8 | no | — |
| 95 | #387 | — | 2 | 8 | no | — |
| 96 | #387 | — | 2 | 8 | no | — |
| 97 | #385 | — | 2 | 8 | no | — |
| 98 | #383 | — | 2 | 2 | no | — |
| 99 | #383 | — | 2 | 4 | no | — |
| 100 | #382 | — | 2 | 4 | no | — |
| 101 | #382 | C23 | 1 | 4 | yes | — |
| 102 | #381 | — | 2 | 2 | no | — |
| 103 | #380 | C23 | 1 | 4 | yes | — |
| 104 | C4×C4⋊C4 | — | 2 | 8 | no | — |
| 105 | C4×C4⋊C4 | — | 2 | 4 | no | — |
| 106 | C4×C4⋊C4 | — | 2 | 8 | no | — |
| 107 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 108 | C4×C4⋊C4 | C23 | 1 | 8 | yes | — |
| 109 | C4×C22⋊C4 | — | 2 | 8 | no | — |
| 110 | C4×C22⋊C4 | — | 2 | 2 | no | — |
| 111 | #376 | — | 2 | 2 | no | — |
| 112 | #376 | — | 2 | 4 | no | — |
| 113 | #376 | D4 | 1 | 4 | yes | — |
| 114 | #376 | C23 | 1 | 2 | yes | — |
| 115 | #376 | C23 | 1 | 2 | yes | — |
| 116 | C2×#95 | D4 | 1 | 4 | yes | — |
| 117 | #340 | C23 | 1 | 8 | yes | — |
| 118 | #340 | C23 | 1 | 8 | yes | — |
| 119 | #340 | C23 | 1 | 8 | yes | — |
| 120 | #336 | C23 | 1 | 4 | yes | — |
| 121 | #336 | C23 | 1 | 4 | yes | — |
| 122 | #336 | C23 | 1 | 4 | yes | — |
| 123 | #336 | C23 | 1 | 4 | yes | — |
| 124 | C22×Q8 | — | 2 | 2 | no | — |
| 125 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 126 | C23×C4 | C2×D4 | 1 | 4 | yes | — |
| 127 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 128 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 129 | C2×Q16 | — | 4 | 4 | no | — |
| 130 | C2×Q16 | — | 4 | 4 | no | — |
| 131 | C2×Q16 | C22×C4 | 1 | 16 | yes | — |
| 132 | C2×M16 | — | 2 | 2 | no | — |
| 133 | C2×M16 | — | 2 | 2 | no | — |
| 134 | C2×M16 | — | 4 | 2 | no | — |
| 135 | C2×M16 | — | 4 | 2 | no | — |
| 136 | C2×M16 | — | 4 | 2 | no | — |
| 137 | C2×M16 | — | 4 | 2 | no | — |
| 138 | C2×M16 | C22×C4 | 1 | 8 | yes | — |
| 139 | C22×C8 | C24 | 1 | 2 | yes | — |
| 140 | C22×C8 | C24 | 1 | 2 | yes | — |
| 141 | C22×C8 | C24 | 1 | 2 | yes | — |
| 142 | C22×C8 | C24 | 1 | 2 | yes | — |
| 143 | C4⋊Q8 | — | 2 | 4 | no | — |
| 144 | C4⋊Q8 | — | 2 | 4 | no | — |
| 145 | C4⋊Q8 | C22×C4 | 1 | 8 | yes | — |
| 146 | C4⋊Q8 | C22×C4 | 1 | 8 | yes | — |
| 147 | #125 | — | 2 | 16 | no | — |
| 148 | C22⋊Q8 | — | 2 | 16 | no | — |
| 149 | C22⋊Q8 | — | 2 | 8 | no | — |
| 150 | C22⋊Q8 | C22×C4 | 1 | 16 | yes | — |
| 151 | C4×Q8 | — | 2 | 16 | no | — |
| 152 | C4×Q8 | — | 2 | 8 | no | — |
| 153 | C4×Q8 | — | 4 | 8 | no | — |
| 154 | C4×Q8 | — | 4 | 8 | no | — |
| 155 | C4×Q8 | C22×C4 | 1 | 16 | yes | — |
| 156 | #117 | — | 2 | 16 | no | — |
| 157 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 158 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 159 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 160 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 161 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 162 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 163 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 164 | C2×C4⋊C4 | C24 | 1 | 8 | yes | — |
| 165 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 166 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 167 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 168 | C2×C42 | — | 2 | 4 | no | — |
| 169 | C2×C42 | — | 2 | 8 | no | — |
| 170 | C2×C42 | — | 2 | 4 | no | — |
| 171 | C2×C42 | — | 2 | 8 | no | — |
| 172 | C2×C42 | — | 4 | 2 | no | — |
| 173 | C2×C42 | — | 4 | 2 | no | — |
| 174 | C2×C42 | C22×C4 | 1 | 4 | yes | — |
| 175 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 176 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 177 | C2×C42 | #41 | 1 | 4 | yes | — |
| 178 | C2×C42 | C24 | 1 | 4 | yes | — |
| 179 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 180 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 181 | #107 | — | 4 | 4 | no | — |
| 182 | #106 | — | 4 | 4 | no | — |
| 183 | #106 | C22×C4 | 1 | 16 | yes | — |
| 184 | #103 | — | 4 | 8 | no | — |
| 185 | #103 | — | 4 | 8 | no | — |
| 186 | #103 | C22×C4 | 1 | 32 | yes | — |
| 187 | #97 | — | 2 | 4 | no | — |
| 188 | #97 | C22×C4 | 1 | 8 | yes | — |
| 189 | C4×C8 | — | 2 | 4 | no | — |
| 190 | C4×C8 | C22×C4 | 1 | 4 | yes | — |
| 191 | C4×C8 | C22×C4 | 1 | 4 | yes | — |
| 192 | #95 | — | 2 | 4 | no | — |
| 193 | #95 | — | 2 | 4 | no | — |
| 194 | #95 | — | 2 | 8 | no | — |
| 195 | #95 | — | 2 | 4 | no | — |
| 196 | #95 | — | 2 | 8 | no | — |
| 197 | #95 | — | 2 | 4 | no | — |
| 198 | #95 | — | 2 | 8 | no | — |
| 199 | #95 | — | 2 | 8 | no | — |
| 200 | #95 | C2×D4 | 1 | 4 | yes | — |
| 201 | #95 | C2×D4 | 1 | 4 | yes | — |
| 202 | #95 | C2×D4 | 1 | 8 | yes | — |
| 203 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 204 | C2×Q8 | — | 2 | 8 | no | — |
| 205 | C2×Q8 | — | 2 | 4 | no | — |
| 206 | C2×Q8 | — | 2 | 8 | no | — |
| 207 | C2×Q8 | — | 2 | 8 | no | — |
| 208 | C2×Q8 | — | 8 | 4 | no | — |
| 209 | C2×Q8 | C23×C4 | 1 | 8 | yes | — |
| 210 | C22×C4 | — | 2 | 2 | no | — |
| 211 | C22×C4 | — | 2 | 2 | no | — |
| 212 | C22×C4 | — | 2 | 4 | no | — |
| 213 | C22×C4 | — | 2 | 4 | no | — |
| 214 | C22×C4 | — | 2 | 8 | no | — |
| 215 | C22×C4 | — | 4 | 4 | no | — |
| 216 | C22×C4 | — | 4 | 4 | no | — |
| 217 | C22×C4 | — | 4 | 8 | no | — |
| 218 | C22×C4 | C4×D4 | 1 | 8 | yes | — |
| 219 | C22×C4 | C23×C4 | 1 | 2 | yes | — |
| 220 | C22×C4 | C22×D4 | 1 | 8 | yes | — |
| 221 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 222 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 223 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 224 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 225 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 226 | Q16 | — | 4 | 16 | no | — |
| 227 | M16 | — | 4 | 8 | no | — |
| 228 | M16 | — | 4 | 4 | no | — |
| 229 | M16 | — | 4 | 4 | no | — |
| 230 | C2×C8 | — | 2 | 2 | no | — |
| 231 | C2×C8 | — | 2 | 2 | no | — |
| 232 | C2×C8 | — | 2 | 2 | no | — |
| 233 | C2×C8 | — | 2 | 2 | no | — |
| 234 | C2×C8 | — | 2 | 2 | no | — |
| 235 | C2×C8 | — | 2 | 2 | no | — |
| 236 | C2×C8 | — | 2 | 4 | no | — |
| 237 | C2×C8 | — | 2 | 4 | no | — |
| 238 | C2×C8 | C23×C4 | 1 | 4 | yes | — |
| 239 | C2×C8 | C23×C4 | 1 | 4 | yes | — |
| 240 | C4⋊C4 | — | 2 | 8 | no | — |
| 241 | C4⋊C4 | — | 2 | 8 | no | — |
| 242 | C4⋊C4 | — | 2 | 8 | no | — |
| 243 | C4⋊C4 | — | 2 | 16 | no | — |
| 244 | C4⋊C4 | — | 2 | 16 | no | — |
| 245 | C4⋊C4 | — | 2 | 4 | no | — |
| 246 | C4⋊C4 | — | 2 | 8 | no | — |
| 247 | C4⋊C4 | — | 2 | 8 | no | — |
| 248 | C4⋊C4 | — | 4 | 16 | no | — |
| 249 | C4⋊C4 | — | 8 | 8 | no | — |
| 250 | C4⋊C4 | C23×C4 | 1 | 8 | yes | — |
| 251 | C22⋊C4 | — | 2 | 16 | no | — |
| 252 | C22⋊C4 | — | 4 | 8 | no | — |
| 253 | C42 | — | 2 | 16 | no | — |
| 254 | C42 | — | 2 | 4 | no | — |
| 255 | C42 | — | 2 | 4 | no | — |
| 256 | C42 | — | 4 | 4 | no | — |
| 257 | C42 | — | 4 | 16 | no | — |
| 258 | C42 | — | 8 | 4 | no | — |
| 259 | C42 | — | 8 | 8 | no | — |
| 260 | C42 | C4×D4 | 1 | 8 | yes | — |
| 261 | C42 | C23×C4 | 1 | 2 | yes | — |
| 262 | C42 | C23×C4 | 1 | 2 | yes | — |
| 263 | C42 | C2×#41 | 1 | 2 | yes | — |
| 264 | C42 | C2×#41 | 1 | 2 | yes | — |
| 265 | C23 | — | 2 | 4 | no | — |
| 266 | C23 | — | 2 | 2 | no | — |
| 267 | C23 | C2×C4×D4 | 1 | 2 | yes | — |
| 268 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 269 | Q8 | — | 2 | 16 | no | — |
| 270 | Q8 | — | 4 | 4 | no | — |
| 271 | Q8 | — | 4 | 4 | no | — |
| 272 | Q8 | — | 8 | 8 | no | — |
| 273 | C2×C4 | — | 2 | 4 | no | — |
| 274 | C2×C4 | — | 2 | 4 | no | — |
| 275 | C2×C4 | — | 2 | 2 | no | — |
| 276 | C2×C4 | — | 2 | 4 | no | — |
| 277 | C2×C4 | — | 2 | 2 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 2 | 8 | no | — |
| 280 | C2×C4 | — | 2 | 8 | no | — |
| 281 | C2×C4 | — | 2 | 8 | no | — |
| 282 | C2×C4 | — | 2 | 8 | no | — |
| 283 | C2×C4 | — | 2 | 8 | no | — |
| 284 | C2×C4 | — | 4 | 8 | no | — |
| 285 | C2×C4 | — | 4 | 4 | no | — |
| 286 | C2×C4 | — | 4 | 4 | no | — |
| 287 | C2×C4 | — | 4 | 8 | no | — |
| 288 | C2×C4 | — | 4 | 8 | no | — |
| 289 | C2×C4 | — | 4 | 8 | no | — |
| 290 | C2×C4 | — | 4 | 8 | no | — |
| 291 | C2×C4 | — | 8 | 8 | no | — |
| 292 | C2×C4 | C2×C4×D4 | 1 | 4 | yes | — |
| 293 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 294 | C2×C4 | C2×C4×D4 | 1 | 4 | yes | — |
| 295 | C2×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C22×#41 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #585 | 1 | 2 | yes | — |
| 298 | C2×C4 | #585 | 1 | 2 | yes | — |
| 299 | C8 | — | 2 | 4 | no | — |
| 300 | C8 | — | 2 | 4 | no | — |
| 301 | C8 | — | 4 | 2 | no | — |
| 302 | C8 | — | 4 | 2 | no | — |
| 303 | C22 | — | 2 | 4 | no | — |
| 304 | C22 | — | 2 | 2 | no | — |
| 305 | C22 | — | 4 | 4 | no | — |
| 306 | C22 | C22×C4×D4 | 1 | 1 | yes | characteristic |
| 307 | C22 | #3449 | 1 | 2 | yes | — |
| 308 | C22 | #3586 | 1 | 4 | yes | — |
| 309 | C4 | — | 2 | 4 | no | — |
| 310 | C4 | — | 2 | 4 | no | — |
| 311 | C4 | — | 2 | 2 | no | — |
| 312 | C4 | — | 4 | 8 | no | — |
| 313 | C4 | — | 4 | 8 | no | — |
| 314 | C4 | — | 4 | 8 | no | — |
| 315 | C4 | — | 8 | 8 | no | — |
| 316 | C4 | #3428 | 1 | 2 | yes | — |
| 317 | C4 | #3455 | 1 | 1 | yes | characteristic |
| 318 | C4 | #3455 | 1 | 1 | yes | characteristic |
| 319 | C2 | — | 4 | 2 | no | — |
| 320 | C2 | #34070 | 1 | 1 | yes | characteristic |
| 321 | C2 | #60169 | 1 | 2 | yes | — |
| 322 | C2 | #61005 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #22277 | C2 | semidirect |
| 4. #22253 | C2 | semidirect |
| 5. #22229 | C2 | semidirect |
| 9. #21127 | C2 | semidirect |
| 13. #3185 | C4 | semidirect |
| 71. #498 | C2×C4 | semidirect |
| 72. #494 | C2×C4 | semidirect |
| 73. #487 | C2×C4 | semidirect |
| 75. #441 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#3185 | 2 |
| #34221 | 2 |
| #22277 | 2 |
| #22253 | 2 |
| #22229 | 2 |
| #22189 | 2 |
| #22188 | 2 |
| #21404 | 2 |
| #21127 | 2 |
| #20825 | 2 |
| #20365 | 2 |