Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1434 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #35345 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #35338 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33288 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #33279 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #30207 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #29363 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #29117 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #29115 | C2 | 1 | 4 | yes | maximal |
| 9 | #21837 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #21806 | C2 | 1 | 2 | yes | maximal |
| 11 | #20462 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#533 | C22 | 1 | 1 | yes | characteristic |
| 13 | C22×#124 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#431 | C22 | 1 | 2 | yes | — |
| 15 | #2864 | C22 | 1 | 2 | yes | complemented |
| 16 | #2861 | C22 | 1 | 2 | yes | — |
| 17 | #2861 | C22 | 1 | 2 | yes | — |
| 18 | #2859 | C22 | 1 | 2 | yes | complemented |
| 19 | #2640 | C22 | 1 | 4 | yes | — |
| 20 | #2629 | C22 | 1 | 4 | yes | — |
| 21 | #2608 | — | 2 | 4 | no | — |
| 22 | #2606 | — | 2 | 4 | no | — |
| 23 | #2592 | — | 2 | 4 | no | — |
| 24 | #2561 | — | 2 | 4 | no | — |
| 25 | #2330 | C22 | 1 | 4 | yes | complemented |
| 26 | #2310 | C22 | 1 | 2 | yes | — |
| 27 | #2309 | C22 | 1 | 2 | yes | — |
| 28 | #2283 | C22 | 1 | 2 | yes | — |
| 29 | C4×#117 | C22 | 1 | 1 | yes | characteristic |
| 30 | #2072 | C22 | 1 | 8 | yes | complemented |
| 31 | #2071 | C22 | 1 | 8 | yes | — |
| 32 | #2026 | C22 | 1 | 8 | yes | complemented |
| 33 | #2025 | C22 | 1 | 8 | yes | — |
| 34 | #1991 | C22 | 1 | 4 | yes | complemented |
| 35 | #1991 | C22 | 1 | 4 | yes | complemented |
| 36 | #1988 | C22 | 1 | 4 | yes | complemented |
| 37 | #1987 | C22 | 1 | 4 | yes | complemented |
| 38 | #1983 | C22 | 1 | 8 | yes | — |
| 39 | #1958 | C22 | 1 | 4 | yes | — |
| 40 | #1958 | C22 | 1 | 4 | yes | — |
| 41 | #1936 | C22 | 1 | 8 | yes | — |
| 42 | #1935 | C22 | 1 | 8 | yes | complemented |
| 43 | #1934 | C22 | 1 | 8 | yes | — |
| 44 | #1933 | C22 | 1 | 8 | yes | complemented |
| 45 | #1825 | C22 | 1 | 8 | yes | — |
| 46 | #1787 | C22 | 1 | 4 | yes | — |
| 47 | #1786 | C22 | 1 | 4 | yes | — |
| 48 | #1768 | C22 | 1 | 4 | yes | complemented |
| 49 | #1750 | C22 | 1 | 4 | yes | — |
| 50 | #1744 | C22 | 1 | 2 | yes | complemented |
| 51 | C4×#97 | C22 | 1 | 1 | yes | characteristic, complemented |
| 52 | C42×C8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 53 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 54 | C23×D4 | C23 | 1 | 1 | yes | characteristic |
| 55 | #533 | — | 2 | 8 | no | — |
| 56 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 57 | C2×#125 | C23 | 1 | 2 | yes | — |
| 58 | C2×#125 | C23 | 1 | 4 | yes | — |
| 59 | C2×#124 | — | 2 | 2 | no | — |
| 60 | C2×#124 | — | 2 | 8 | no | — |
| 61 | C2×#124 | C23 | 1 | 2 | yes | — |
| 62 | C2×#124 | C23 | 1 | 2 | yes | — |
| 63 | C2×C22⋊Q8 | — | 2 | 2 | no | — |
| 64 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 65 | C2×#117 | C23 | 1 | 2 | yes | — |
| 66 | C2×#117 | C23 | 1 | 2 | yes | — |
| 67 | C22×C22⋊C4 | — | 2 | 2 | no | — |
| 68 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 69 | #431 | — | 2 | 8 | no | — |
| 70 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 71 | C2×#103 | C23 | 1 | 8 | yes | — |
| 72 | C2×#103 | C23 | 1 | 8 | yes | — |
| 73 | C2×#102 | C23 | 1 | 8 | yes | — |
| 74 | C2×#102 | C23 | 1 | 8 | yes | — |
| 75 | C2×C22⋊C8 | C23 | 1 | 4 | yes | — |
| 76 | C2×#97 | C23 | 1 | 4 | yes | — |
| 77 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 78 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 79 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 80 | #398 | — | 2 | 8 | no | — |
| 81 | #397 | — | 2 | 8 | no | — |
| 82 | #396 | — | 2 | 8 | no | — |
| 83 | #393 | — | 2 | 8 | no | — |
| 84 | #391 | — | 2 | 4 | no | — |
| 85 | #391 | C23 | 1 | 4 | yes | — |
| 86 | #391 | C23 | 1 | 4 | yes | — |
| 87 | #390 | — | 2 | 4 | no | — |
| 88 | #390 | C23 | 1 | 4 | yes | — |
| 89 | #390 | C23 | 1 | 4 | yes | — |
| 90 | #389 | — | 2 | 8 | no | — |
| 91 | #389 | — | 2 | 8 | no | — |
| 92 | #389 | C23 | 1 | 8 | yes | — |
| 93 | #388 | — | 2 | 8 | no | — |
| 94 | #388 | — | 2 | 8 | no | — |
| 95 | #387 | — | 2 | 8 | no | — |
| 96 | #387 | — | 2 | 8 | no | — |
| 97 | #384 | C23 | 1 | 4 | yes | — |
| 98 | #383 | — | 2 | 2 | no | — |
| 99 | #382 | C23 | 1 | 4 | yes | — |
| 100 | #381 | — | 2 | 2 | no | — |
| 101 | #381 | — | 2 | 4 | no | — |
| 102 | #380 | C23 | 1 | 4 | yes | — |
| 103 | #379 | — | 2 | 4 | no | — |
| 104 | C4×C4⋊C4 | D4 | 1 | 4 | yes | — |
| 105 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 106 | C4×C22⋊C4 | — | 2 | 2 | no | — |
| 107 | C4×C22⋊C4 | D4 | 1 | 4 | yes | — |
| 108 | #376 | — | 2 | 2 | no | — |
| 109 | #376 | C23 | 1 | 2 | yes | — |
| 110 | C43 | C23 | 1 | 1 | yes | characteristic |
| 111 | C43 | C23 | 1 | 1 | yes | characteristic |
| 112 | #340 | C23 | 1 | 8 | yes | — |
| 113 | #340 | C23 | 1 | 8 | yes | — |
| 114 | #340 | C23 | 1 | 8 | yes | — |
| 115 | #340 | C23 | 1 | 8 | yes | — |
| 116 | #336 | C23 | 1 | 8 | yes | — |
| 117 | C25 | — | 2 | 1 | no | — |
| 118 | C22×Q8 | — | 2 | 1 | no | — |
| 119 | C22×Q8 | — | 2 | 2 | no | — |
| 120 | C22×Q8 | — | 2 | 4 | no | — |
| 121 | C22×Q8 | — | 4 | 4 | no | — |
| 122 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 123 | C22×D4 | — | 2 | 1 | no | — |
| 124 | C22×D4 | — | 2 | 4 | no | — |
| 125 | C22×D4 | — | 4 | 4 | no | — |
| 126 | C22×D4 | C24 | 1 | 2 | yes | — |
| 127 | C23×C4 | — | 2 | 1 | no | — |
| 128 | C23×C4 | C2×D4 | 1 | 2 | yes | — |
| 129 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 130 | C22×C8 | C24 | 1 | 4 | yes | — |
| 131 | C22×C8 | C24 | 1 | 4 | yes | — |
| 132 | C4⋊Q8 | — | 2 | 8 | no | — |
| 133 | #125 | — | 2 | 8 | no | — |
| 134 | #125 | — | 4 | 4 | no | — |
| 135 | #125 | — | 4 | 4 | no | — |
| 136 | #124 | — | 2 | 8 | no | — |
| 137 | #124 | — | 2 | 8 | no | — |
| 138 | #124 | — | 4 | 8 | no | — |
| 139 | C22⋊Q8 | — | 4 | 8 | no | — |
| 140 | C4×Q8 | — | 4 | 8 | no | — |
| 141 | #117 | — | 2 | 8 | no | — |
| 142 | #117 | — | 4 | 4 | no | — |
| 143 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 144 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 145 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 146 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 147 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 148 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 149 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 150 | C2×C4⋊C4 | C24 | 1 | 8 | yes | — |
| 151 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 152 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 153 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 154 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 155 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 156 | C2×C22⋊C4 | — | 2 | 8 | no | — |
| 157 | C2×C22⋊C4 | — | 4 | 8 | no | — |
| 158 | C2×C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 159 | C2×C42 | — | 2 | 2 | no | — |
| 160 | C2×C42 | — | 2 | 4 | no | — |
| 161 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 162 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 163 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 164 | C2×C42 | #41 | 1 | 8 | yes | — |
| 165 | C2×C42 | C24 | 1 | 2 | yes | — |
| 166 | C2×C42 | C24 | 1 | 2 | yes | — |
| 167 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 168 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 169 | C4⋊C8 | — | 2 | 8 | no | — |
| 170 | #103 | — | 4 | 8 | no | — |
| 171 | #103 | — | 4 | 8 | no | — |
| 172 | #102 | — | 4 | 8 | no | — |
| 173 | #102 | — | 4 | 8 | no | — |
| 174 | C22⋊C8 | — | 2 | 8 | no | — |
| 175 | #97 | — | 4 | 4 | no | — |
| 176 | C4×C8 | — | 2 | 4 | no | — |
| 177 | C4×C8 | — | 2 | 4 | no | — |
| 178 | C4×C8 | — | 4 | 4 | no | — |
| 179 | #95 | — | 2 | 4 | no | — |
| 180 | #95 | — | 2 | 4 | no | — |
| 181 | #95 | — | 2 | 4 | no | — |
| 182 | #95 | — | 2 | 4 | no | — |
| 183 | #95 | — | 2 | 8 | no | — |
| 184 | #95 | — | 2 | 8 | no | — |
| 185 | #95 | — | 2 | 8 | no | — |
| 186 | #95 | C2×D4 | 1 | 8 | yes | — |
| 187 | C24 | — | 2 | 2 | no | — |
| 188 | C24 | — | 4 | 4 | no | — |
| 189 | C24 | — | 4 | 2 | no | — |
| 190 | C24 | — | 8 | 4 | no | — |
| 191 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 192 | C2×Q8 | — | 2 | 4 | no | — |
| 193 | C2×Q8 | — | 2 | 8 | no | — |
| 194 | C2×Q8 | — | 4 | 4 | no | — |
| 195 | C2×Q8 | — | 4 | 4 | no | — |
| 196 | C2×Q8 | — | 4 | 2 | no | — |
| 197 | C2×Q8 | — | 8 | 4 | no | — |
| 198 | C2×Q8 | — | 8 | 8 | no | — |
| 199 | C2×D4 | — | 2 | 4 | no | — |
| 200 | C2×D4 | — | 2 | 8 | no | — |
| 201 | C2×D4 | — | 4 | 4 | no | — |
| 202 | C2×D4 | — | 4 | 2 | no | — |
| 203 | C2×D4 | — | 8 | 8 | no | — |
| 204 | C22×C4 | — | 2 | 4 | no | — |
| 205 | C22×C4 | — | 2 | 2 | no | — |
| 206 | C22×C4 | — | 2 | 8 | no | — |
| 207 | C22×C4 | — | 2 | 4 | no | — |
| 208 | C22×C4 | — | 4 | 4 | no | — |
| 209 | C22×C4 | — | 4 | 4 | no | — |
| 210 | C22×C4 | — | 4 | 2 | no | — |
| 211 | C22×C4 | #124 | 1 | 8 | yes | — |
| 212 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 213 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 214 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 215 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 216 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 217 | C22×C4 | C2×#41 | 1 | 4 | yes | — |
| 218 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 219 | C2×C8 | — | 2 | 4 | no | — |
| 220 | C2×C8 | — | 2 | 4 | no | — |
| 221 | C2×C8 | — | 2 | 8 | no | — |
| 222 | C2×C8 | — | 2 | 8 | no | — |
| 223 | C4⋊C4 | — | 2 | 8 | no | — |
| 224 | C4⋊C4 | — | 2 | 8 | no | — |
| 225 | C4⋊C4 | — | 2 | 8 | no | — |
| 226 | C4⋊C4 | — | 4 | 4 | no | — |
| 227 | C4⋊C4 | — | 4 | 16 | no | — |
| 228 | C4⋊C4 | — | 4 | 8 | no | — |
| 229 | C4⋊C4 | — | 4 | 4 | no | — |
| 230 | C4⋊C4 | — | 8 | 8 | no | — |
| 231 | C4⋊C4 | — | 8 | 8 | no | — |
| 232 | C22⋊C4 | — | 4 | 4 | no | — |
| 233 | C22⋊C4 | — | 4 | 4 | no | — |
| 234 | C22⋊C4 | — | 4 | 8 | no | — |
| 235 | C22⋊C4 | — | 4 | 8 | no | — |
| 236 | C22⋊C4 | — | 8 | 8 | no | — |
| 237 | C22⋊C4 | — | 8 | 8 | no | — |
| 238 | C22⋊C4 | — | 8 | 4 | no | — |
| 239 | C42 | — | 2 | 4 | no | — |
| 240 | C42 | — | 2 | 4 | no | — |
| 241 | C42 | — | 4 | 4 | no | — |
| 242 | C42 | — | 4 | 4 | no | — |
| 243 | C42 | — | 4 | 4 | no | — |
| 244 | C42 | — | 8 | 4 | no | — |
| 245 | C42 | #124 | 1 | 8 | yes | — |
| 246 | C42 | C2×#41 | 1 | 4 | yes | — |
| 247 | C42 | C2×#41 | 1 | 4 | yes | — |
| 248 | C23 | — | 2 | 4 | no | — |
| 249 | C23 | — | 2 | 2 | no | — |
| 250 | C23 | — | 4 | 8 | no | — |
| 251 | C23 | — | 4 | 2 | no | — |
| 252 | C23 | — | 4 | 4 | no | — |
| 253 | C23 | — | 8 | 4 | no | — |
| 254 | C23 | — | 8 | 8 | no | — |
| 255 | C23 | — | 16 | 8 | no | — |
| 256 | C23 | C2×#124 | 1 | 2 | yes | — |
| 257 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 258 | Q8 | — | 4 | 8 | no | — |
| 259 | Q8 | — | 8 | 4 | no | — |
| 260 | Q8 | — | 8 | 8 | no | — |
| 261 | D4 | — | 4 | 8 | no | — |
| 262 | D4 | — | 8 | 4 | no | — |
| 263 | C2×C4 | — | 2 | 4 | no | — |
| 264 | C2×C4 | — | 2 | 4 | no | — |
| 265 | C2×C4 | — | 2 | 2 | no | — |
| 266 | C2×C4 | — | 2 | 8 | no | — |
| 267 | C2×C4 | — | 2 | 8 | no | — |
| 268 | C2×C4 | — | 2 | 8 | no | — |
| 269 | C2×C4 | — | 2 | 4 | no | — |
| 270 | C2×C4 | — | 2 | 4 | no | — |
| 271 | C2×C4 | — | 4 | 8 | no | — |
| 272 | C2×C4 | — | 4 | 8 | no | — |
| 273 | C2×C4 | — | 4 | 8 | no | — |
| 274 | C2×C4 | — | 4 | 8 | no | — |
| 275 | C2×C4 | — | 4 | 4 | no | — |
| 276 | C2×C4 | — | 4 | 2 | no | — |
| 277 | C2×C4 | — | 4 | 8 | no | — |
| 278 | C2×C4 | — | 4 | 4 | no | — |
| 279 | C2×C4 | — | 8 | 4 | no | — |
| 280 | C2×C4 | C2×#124 | 1 | 2 | yes | — |
| 281 | C2×C4 | C2×#124 | 1 | 4 | yes | — |
| 282 | C2×C4 | C2×#124 | 1 | 4 | yes | — |
| 283 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 284 | C2×C4 | #585 | 1 | 4 | yes | — |
| 285 | C8 | — | 4 | 4 | no | — |
| 286 | C8 | — | 4 | 4 | no | — |
| 287 | C22 | — | 2 | 2 | no | — |
| 288 | C22 | — | 2 | 4 | no | — |
| 289 | C22 | — | 4 | 4 | no | — |
| 290 | C22 | — | 8 | 2 | no | — |
| 291 | C22 | — | 8 | 8 | no | — |
| 292 | C22 | — | 8 | 4 | no | — |
| 293 | C22 | — | 16 | 8 | no | — |
| 294 | C22 | C22×#124 | 1 | 1 | yes | characteristic |
| 295 | C22 | #3449 | 1 | 2 | yes | — |
| 296 | C22 | #3584 | 1 | 4 | yes | — |
| 297 | C4 | — | 2 | 2 | no | — |
| 298 | C4 | — | 2 | 4 | no | — |
| 299 | C4 | — | 2 | 4 | no | — |
| 300 | C4 | — | 4 | 4 | no | — |
| 301 | C4 | — | 4 | 4 | no | — |
| 302 | C4 | — | 8 | 2 | no | — |
| 303 | C4 | — | 8 | 8 | no | — |
| 304 | C4 | — | 8 | 4 | no | — |
| 305 | C4 | #3510 | 1 | 4 | yes | — |
| 306 | C2 | — | 4 | 2 | no | — |
| 307 | C2 | — | 16 | 2 | no | — |
| 308 | C2 | #35338 | 1 | 1 | yes | characteristic |
| 309 | C2 | #60612 | 1 | 2 | yes | — |
| 310 | C2 | #61010 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #35345 | C2 | semidirect |
| 3. #33288 | C2 | semidirect |
| 4. #33279 | C2 | semidirect |
| 5. #30207 | C2 | semidirect |
| 6. #29363 | C2 | semidirect |
| 7. #29117 | C2 | semidirect |
| 9. #21837 | C2 | semidirect |
| 11. #20462 | C2 | semidirect |
| 15. #2864 | C22 | semidirect |
| 18. #2859 | C22 | semidirect |
| 25. #2330 | C22 | semidirect |
| 30. #2072 | C22 | semidirect |
| 32. #2026 | C22 | semidirect |
| 34. #1991 | C22 | semidirect |
| 35. #1991 | C22 | semidirect |
| 36. #1988 | C22 | semidirect |
| 37. #1987 | C22 | semidirect |
| 42. #1935 | C22 | semidirect |
| 44. #1933 | C22 | semidirect |
| 48. #1768 | C22 | semidirect |
| 50. #1744 | C22 | semidirect |
| 51. C4×#97 | C22 | semidirect |
| 52. C42×C8 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #35345 | 2 |
| #35338 | 2 |
| #33288 | 2 |
| #33279 | 2 |
| #30207 | 2 |
| #29363 | 2 |
| #29117 | 2 |
| #29115 | 2 |
| #21837 | 2 |
| #21806 | 2 |
| #20462 | 2 |