Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
723 conjugacy classes of subgroups.
The subgroup lattice has 342 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #33196 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #33033 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #30262 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #20572 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#1680 | C2 | 1 | 2 | yes | maximal |
| 6 | C2×#1670 | C2 | 1 | 2 | yes | maximal |
| 7 | #9748 | C2 | 1 | 2 | yes | maximal |
| 8 | #9742 | C2 | 1 | 2 | yes | maximal |
| 9 | #9708 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #9705 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #8812 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×#494 | C22 | 1 | 2 | yes | — |
| 13 | C2×#487 | C22 | 1 | 2 | yes | — |
| 14 | C2×#475 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#475 | C22 | 1 | 2 | yes | — |
| 16 | C2×#462 | C22 | 1 | 1 | yes | characteristic |
| 17 | C2×C8×Q8 | C22 | 1 | 2 | yes | — |
| 18 | #2868 | C22 | 1 | 1 | yes | characteristic |
| 19 | #2862 | C22 | 1 | 1 | yes | characteristic |
| 20 | C42×Q8 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2016 | — | 2 | 1 | no | — |
| 22 | #2016 | — | 2 | 1 | no | — |
| 23 | #2016 | — | 2 | 2 | no | — |
| 24 | #1988 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1988 | C22 | 1 | 2 | yes | — |
| 26 | #1981 | C22 | 1 | 1 | yes | characteristic |
| 27 | #1948 | — | 2 | 2 | no | — |
| 28 | #1927 | — | 2 | 2 | no | — |
| 29 | #1878 | — | 2 | 2 | no | — |
| 30 | #1842 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1831 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1803 | — | 2 | 2 | no | — |
| 33 | #1770 | — | 2 | 2 | no | — |
| 34 | #1770 | C22 | 1 | 2 | yes | — |
| 35 | #1767 | — | 2 | 1 | no | — |
| 36 | #1767 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1744 | — | 2 | 1 | no | — |
| 38 | #1744 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1680 | — | 2 | 8 | no | — |
| 40 | #1670 | — | 2 | 8 | no | — |
| 41 | C2×#335 | C22 | 1 | 2 | yes | — |
| 42 | C2×#334 | C22 | 1 | 2 | yes | — |
| 43 | C2×#332 | C22 | 1 | 2 | yes | — |
| 44 | C2×#332 | C22 | 1 | 2 | yes | — |
| 45 | C2×#332 | C22 | 1 | 2 | yes | — |
| 46 | C2×#332 | C22 | 1 | 2 | yes | — |
| 47 | C22×Q16 | — | 2 | 1 | no | — |
| 48 | C22×Q16 | — | 2 | 2 | no | — |
| 49 | C22×Q16 | — | 2 | 2 | no | — |
| 50 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 51 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 52 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 53 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 54 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 55 | C2×C4×Q8 | D4 | 1 | 2 | yes | — |
| 56 | C2×C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 57 | #494 | — | 2 | 4 | no | — |
| 58 | #494 | — | 2 | 4 | no | — |
| 59 | #487 | — | 2 | 4 | no | — |
| 60 | #487 | — | 2 | 4 | no | — |
| 61 | #475 | — | 2 | 4 | no | — |
| 62 | #475 | — | 2 | 4 | no | — |
| 63 | #475 | — | 2 | 4 | no | — |
| 64 | #462 | — | 2 | 4 | no | — |
| 65 | C8×Q8 | — | 2 | 4 | no | — |
| 66 | C8×Q8 | D4 | 1 | 8 | yes | — |
| 67 | C2×#106 | — | 2 | 1 | no | — |
| 68 | C2×#106 | — | 2 | 2 | no | — |
| 69 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 70 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 71 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 72 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 73 | C2×C4⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 74 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 75 | C2×#103 | — | 2 | 2 | no | — |
| 76 | C2×#103 | — | 2 | 1 | no | — |
| 77 | C2×#103 | — | 2 | 2 | no | — |
| 78 | C2×#103 | — | 2 | 2 | no | — |
| 79 | C2×#103 | — | 2 | 2 | no | — |
| 80 | C2×#103 | — | 2 | 1 | no | — |
| 81 | C2×#97 | — | 2 | 2 | no | — |
| 82 | C2×#97 | — | 2 | 1 | no | — |
| 83 | C2×C4×C8 | — | 2 | 2 | no | — |
| 84 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 85 | #391 | — | 2 | 2 | no | — |
| 86 | #391 | — | 2 | 1 | no | — |
| 87 | #391 | — | 2 | 1 | no | — |
| 88 | #384 | — | 2 | 2 | no | — |
| 89 | #384 | — | 2 | 2 | no | — |
| 90 | #382 | — | 2 | 2 | no | — |
| 91 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 92 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 93 | C4×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 94 | C4×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 95 | C43 | — | 2 | 1 | no | — |
| 96 | C43 | C23 | 1 | 1 | yes | characteristic |
| 97 | #340 | — | 2 | 4 | no | — |
| 98 | #340 | — | 2 | 2 | no | — |
| 99 | #336 | — | 2 | 2 | no | — |
| 100 | #336 | — | 2 | 2 | no | — |
| 101 | #336 | — | 2 | 1 | no | — |
| 102 | #336 | D4 | 1 | 1 | yes | characteristic |
| 103 | #336 | D4 | 1 | 1 | yes | characteristic |
| 104 | #336 | D4 | 1 | 2 | yes | — |
| 105 | #335 | — | 2 | 4 | no | — |
| 106 | #334 | — | 2 | 4 | no | — |
| 107 | #332 | — | 2 | 4 | no | — |
| 108 | #332 | — | 2 | 4 | no | — |
| 109 | #332 | — | 2 | 4 | no | — |
| 110 | #332 | — | 2 | 4 | no | — |
| 111 | C22×Q8 | — | 2 | 2 | no | — |
| 112 | C22×Q8 | — | 2 | 1 | no | — |
| 113 | C22×Q8 | — | 2 | 1 | no | — |
| 114 | C22×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 115 | C2×Q16 | — | 4 | 2 | no | — |
| 116 | C2×Q16 | — | 4 | 4 | no | — |
| 117 | C2×Q16 | — | 4 | 4 | no | — |
| 118 | C2×Q16 | — | 4 | 4 | no | — |
| 119 | C2×Q16 | — | 8 | 2 | no | — |
| 120 | C2×Q16 | — | 8 | 2 | no | — |
| 121 | C2×Q16 | — | 8 | 2 | no | — |
| 122 | C2×Q16 | — | 8 | 2 | no | — |
| 123 | C22×C8 | — | 2 | 2 | no | — |
| 124 | C22×C8 | — | 2 | 1 | no | — |
| 125 | C22×C8 | — | 2 | 1 | no | — |
| 126 | C22×C8 | — | 2 | 2 | no | — |
| 127 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 128 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 129 | C4⋊Q8 | — | 2 | 2 | no | — |
| 130 | C4⋊Q8 | — | 2 | 2 | no | — |
| 131 | C4⋊Q8 | — | 2 | 2 | no | — |
| 132 | C4⋊Q8 | — | 2 | 2 | no | — |
| 133 | C4⋊Q8 | — | 2 | 2 | no | — |
| 134 | C4⋊Q8 | — | 2 | 2 | no | — |
| 135 | C4⋊Q8 | — | 2 | 2 | no | — |
| 136 | C4⋊Q8 | — | 2 | 2 | no | — |
| 137 | C4⋊Q8 | — | 8 | 1 | no | — |
| 138 | C4⋊Q8 | — | 8 | 2 | no | — |
| 139 | C4⋊Q8 | — | 8 | 1 | no | — |
| 140 | C4×Q8 | — | 2 | 2 | no | — |
| 141 | C4×Q8 | — | 2 | 4 | no | — |
| 142 | C4×Q8 | — | 4 | 2 | no | — |
| 143 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 144 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 145 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 146 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 147 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 148 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 149 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 150 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 151 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 152 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 153 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 154 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 155 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 156 | C2×C42 | — | 2 | 1 | no | — |
| 157 | C2×C42 | — | 2 | 2 | no | — |
| 158 | C2×C42 | — | 2 | 2 | no | — |
| 159 | C2×C42 | — | 2 | 1 | no | — |
| 160 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 161 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 162 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 163 | C2×C42 | #41 | 1 | 2 | yes | — |
| 164 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 165 | #106 | — | 4 | 2 | no | — |
| 166 | #106 | — | 4 | 4 | no | — |
| 167 | C4⋊C8 | — | 2 | 8 | no | — |
| 168 | C4⋊C8 | — | 2 | 2 | no | — |
| 169 | C4⋊C8 | — | 2 | 2 | no | — |
| 170 | C4⋊C8 | — | 2 | 2 | no | — |
| 171 | C4⋊C8 | — | 2 | 2 | no | — |
| 172 | C4⋊C8 | — | 2 | 2 | no | — |
| 173 | C4⋊C8 | — | 4 | 4 | no | — |
| 174 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 175 | #103 | — | 4 | 4 | no | — |
| 176 | #103 | — | 4 | 4 | no | — |
| 177 | #103 | — | 4 | 2 | no | — |
| 178 | #103 | — | 4 | 4 | no | — |
| 179 | #103 | — | 4 | 4 | no | — |
| 180 | #103 | — | 4 | 2 | no | — |
| 181 | #97 | — | 8 | 1 | no | — |
| 182 | #97 | — | 8 | 2 | no | — |
| 183 | C4×C8 | — | 2 | 8 | no | — |
| 184 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 185 | C4×C8 | #41 | 1 | 4 | yes | — |
| 186 | #95 | — | 2 | 2 | no | — |
| 187 | #95 | — | 2 | 1 | no | — |
| 188 | #95 | — | 4 | 2 | no | — |
| 189 | #95 | — | 4 | 2 | no | — |
| 190 | C2×Q8 | — | 2 | 1 | no | — |
| 191 | C2×Q8 | — | 2 | 2 | no | — |
| 192 | C2×Q8 | — | 4 | 2 | no | — |
| 193 | C2×Q8 | — | 4 | 2 | no | — |
| 194 | C2×Q8 | — | 4 | 2 | no | — |
| 195 | C2×Q8 | — | 4 | 1 | no | — |
| 196 | C2×Q8 | — | 4 | 1 | no | — |
| 197 | C2×Q8 | — | 4 | 2 | no | — |
| 198 | C2×Q8 | — | 4 | 2 | no | — |
| 199 | C2×Q8 | — | 4 | 2 | no | — |
| 200 | C2×Q8 | — | 4 | 1 | no | — |
| 201 | C2×Q8 | — | 4 | 2 | no | — |
| 202 | C2×Q8 | — | 4 | 2 | no | — |
| 203 | C2×Q8 | — | 4 | 2 | no | — |
| 204 | C2×Q8 | — | 4 | 2 | no | — |
| 205 | C2×Q8 | — | 4 | 1 | no | — |
| 206 | C2×Q8 | #127 | 1 | 2 | yes | — |
| 207 | C2×Q8 | #137 | 1 | 4 | yes | — |
| 208 | C22×C4 | — | 2 | 1 | no | — |
| 209 | C22×C4 | — | 2 | 1 | no | — |
| 210 | C22×C4 | — | 2 | 2 | no | — |
| 211 | C22×C4 | — | 4 | 1 | no | — |
| 212 | C22×C4 | — | 4 | 1 | no | — |
| 213 | C22×C4 | — | 4 | 2 | no | — |
| 214 | C22×C4 | #121 | 1 | 2 | yes | — |
| 215 | C22×C4 | #121 | 1 | 2 | yes | — |
| 216 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 217 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 218 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 219 | Q16 | — | 8 | 8 | no | — |
| 220 | Q16 | — | 8 | 8 | no | — |
| 221 | Q16 | — | 8 | 4 | no | — |
| 222 | C2×C8 | — | 2 | 2 | no | — |
| 223 | C2×C8 | — | 2 | 2 | no | — |
| 224 | C2×C8 | — | 2 | 2 | no | — |
| 225 | C2×C8 | — | 2 | 2 | no | — |
| 226 | C2×C8 | — | 2 | 4 | no | — |
| 227 | C2×C8 | — | 4 | 2 | no | — |
| 228 | C2×C8 | — | 4 | 2 | no | — |
| 229 | C2×C8 | — | 4 | 2 | no | — |
| 230 | C2×C8 | — | 4 | 1 | no | — |
| 231 | C2×C8 | — | 4 | 1 | no | — |
| 232 | C2×C8 | — | 4 | 2 | no | — |
| 233 | C2×C8 | — | 4 | 2 | no | — |
| 234 | C2×C8 | — | 4 | 4 | no | — |
| 235 | C2×C8 | #121 | 1 | 4 | yes | — |
| 236 | C2×C8 | #121 | 1 | 4 | yes | — |
| 237 | C4⋊C4 | — | 2 | 4 | no | — |
| 238 | C4⋊C4 | — | 2 | 1 | no | — |
| 239 | C4⋊C4 | — | 2 | 2 | no | — |
| 240 | C4⋊C4 | — | 2 | 2 | no | — |
| 241 | C4⋊C4 | — | 4 | 4 | no | — |
| 242 | C4⋊C4 | — | 4 | 2 | no | — |
| 243 | C4⋊C4 | — | 4 | 2 | no | — |
| 244 | C4⋊C4 | — | 4 | 1 | no | — |
| 245 | C4⋊C4 | — | 4 | 2 | no | — |
| 246 | C4⋊C4 | — | 4 | 2 | no | — |
| 247 | C4⋊C4 | — | 4 | 2 | no | — |
| 248 | C4⋊C4 | — | 4 | 2 | no | — |
| 249 | C4⋊C4 | — | 4 | 2 | no | — |
| 250 | C4⋊C4 | — | 4 | 1 | no | — |
| 251 | C4⋊C4 | — | 8 | 4 | no | — |
| 252 | C4⋊C4 | — | 8 | 4 | no | — |
| 253 | C4⋊C4 | #127 | 1 | 2 | yes | — |
| 254 | C42 | — | 2 | 4 | no | — |
| 255 | C42 | — | 2 | 2 | no | — |
| 256 | C42 | — | 2 | 2 | no | — |
| 257 | C42 | — | 2 | 1 | no | — |
| 258 | C42 | — | 2 | 2 | no | — |
| 259 | C42 | — | 2 | 1 | no | — |
| 260 | C42 | — | 2 | 4 | no | — |
| 261 | C42 | — | 4 | 4 | no | — |
| 262 | C42 | — | 4 | 1 | no | — |
| 263 | C42 | — | 4 | 1 | no | — |
| 264 | C42 | — | 4 | 4 | no | — |
| 265 | C42 | C22×D4 | 1 | 2 | yes | — |
| 266 | C42 | C2×#41 | 1 | 2 | yes | — |
| 267 | C23 | C2×#121 | 1 | 1 | yes | characteristic |
| 268 | Q8 | — | 2 | 2 | no | — |
| 269 | Q8 | — | 2 | 4 | no | — |
| 270 | Q8 | — | 8 | 1 | no | — |
| 271 | Q8 | — | 8 | 1 | no | — |
| 272 | Q8 | — | 8 | 2 | no | — |
| 273 | Q8 | — | 8 | 2 | no | — |
| 274 | Q8 | — | 8 | 2 | no | — |
| 275 | Q8 | — | 8 | 2 | no | — |
| 276 | Q8 | — | 8 | 2 | no | — |
| 277 | Q8 | — | 8 | 2 | no | — |
| 278 | Q8 | — | 8 | 2 | no | — |
| 279 | Q8 | #497 | 1 | 4 | yes | — |
| 280 | C2×C4 | — | 2 | 2 | no | — |
| 281 | C2×C4 | — | 2 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 2 | no | — |
| 283 | C2×C4 | — | 2 | 4 | no | — |
| 284 | C2×C4 | — | 2 | 2 | no | — |
| 285 | C2×C4 | — | 2 | 2 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 2 | 2 | no | — |
| 288 | C2×C4 | — | 2 | 2 | no | — |
| 289 | C2×C4 | — | 4 | 2 | no | — |
| 290 | C2×C4 | — | 4 | 2 | no | — |
| 291 | C2×C4 | — | 4 | 2 | no | — |
| 292 | C2×C4 | — | 4 | 2 | no | — |
| 293 | C2×C4 | — | 8 | 1 | no | — |
| 294 | C2×C4 | — | 8 | 2 | no | — |
| 295 | C2×C4 | — | 8 | 1 | no | — |
| 296 | C2×C4 | — | 8 | 2 | no | — |
| 297 | C2×C4 | — | 8 | 1 | no | — |
| 298 | C2×C4 | — | 8 | 2 | no | — |
| 299 | C2×C4 | — | 8 | 1 | no | — |
| 300 | C2×C4 | — | 8 | 2 | no | — |
| 301 | C2×C4 | #473 | 1 | 4 | yes | — |
| 302 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 303 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | #532 | 1 | 2 | yes | — |
| 305 | C2×C4 | #532 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | C2×#137 | 1 | 2 | yes | — |
| 307 | C2×C4 | #575 | 1 | 2 | yes | — |
| 308 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | #577 | 1 | 2 | yes | — |
| 311 | C8 | — | 2 | 4 | no | — |
| 312 | C8 | — | 2 | 4 | no | — |
| 313 | C8 | — | 4 | 4 | no | — |
| 314 | C8 | — | 8 | 1 | no | — |
| 315 | C8 | — | 8 | 1 | no | — |
| 316 | C8 | — | 8 | 2 | no | — |
| 317 | C22 | #2856 | 1 | 1 | yes | characteristic |
| 318 | C22 | #3043 | 1 | 1 | yes | characteristic |
| 319 | C22 | #3045 | 1 | 1 | yes | characteristic |
| 320 | C22 | #3056 | 1 | 2 | yes | — |
| 321 | C22 | C2×#473 | 1 | 2 | yes | — |
| 322 | C4 | — | 2 | 4 | no | — |
| 323 | C4 | — | 2 | 4 | no | — |
| 324 | C4 | — | 2 | 2 | no | — |
| 325 | C4 | — | 4 | 2 | no | — |
| 326 | C4 | — | 4 | 2 | no | — |
| 327 | C4 | — | 8 | 2 | no | — |
| 328 | C4 | — | 16 | 2 | no | — |
| 329 | C4 | — | 16 | 1 | no | — |
| 330 | C4 | — | 16 | 1 | no | — |
| 331 | C4 | #3043 | 1 | 2 | yes | — |
| 332 | C4 | #3045 | 1 | 2 | yes | — |
| 333 | C4 | C2×#497 | 1 | 1 | yes | characteristic |
| 334 | C4 | #3150 | 1 | 1 | yes | characteristic |
| 335 | C4 | #3181 | 1 | 2 | yes | — |
| 336 | C2 | #19471 | 1 | 2 | yes | — |
| 337 | C2 | #19497 | 1 | 2 | yes | — |
| 338 | C2 | #33167 | 1 | 1 | yes | characteristic |
| 339 | C2 | #33184 | 1 | 1 | yes | characteristic |
| 340 | C2 | #33191 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ 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 |
|---|---|
| #33196 | 2 |
| #33033 | 2 |
| #30262 | 2 |
| #20572 | 2 |
| C2×#1680 | 2 |
| C2×#1670 | 2 |
| #9748 | 2 |
| #9742 | 2 |
| #9708 | 2 |
| #9705 | 2 |
| #8812 | 2 |