Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1099 conjugacy classes of subgroups.
The subgroup lattice has 316 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #53147 | C2 | 1 | 2 | yes | maximal |
| 2 | #33285 | C2 | 1 | 2 | yes | maximal |
| 3 | #32857 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #32816 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #32255 | C2 | 1 | 4 | yes | maximal |
| 6 | #31828 | C2 | 1 | 4 | yes | maximal |
| 7 | #31664 | C2 | 1 | 4 | yes | maximal |
| 8 | #29082 | C2 | 1 | 4 | yes | maximal |
| 9 | #28904 | C2 | 1 | 4 | yes | maximal |
| 10 | #27831 | C2 | 1 | 4 | yes | maximal |
| 11 | #21677 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #2839 | — | 2 | 4 | no | — |
| 13 | #2839 | — | 2 | 4 | no | — |
| 14 | #2794 | — | 2 | 4 | no | — |
| 15 | #2791 | — | 2 | 4 | no | — |
| 16 | #2647 | C22 | 1 | 2 | yes | — |
| 17 | #2642 | C22 | 1 | 2 | yes | — |
| 18 | #2565 | C22 | 1 | 4 | yes | — |
| 19 | #2552 | C22 | 1 | 4 | yes | — |
| 20 | #2511 | C22 | 1 | 4 | yes | — |
| 21 | #2511 | C22 | 1 | 4 | yes | — |
| 22 | #2453 | C22 | 1 | 4 | yes | — |
| 23 | #2446 | C22 | 1 | 4 | yes | — |
| 24 | #2402 | C22 | 1 | 1 | yes | characteristic |
| 25 | #2101 | C22 | 1 | 4 | yes | — |
| 26 | #2100 | C22 | 1 | 4 | yes | — |
| 27 | #2088 | C22 | 1 | 4 | yes | — |
| 28 | #2084 | C22 | 1 | 4 | yes | — |
| 29 | #2072 | C22 | 1 | 4 | yes | — |
| 30 | #2066 | C22 | 1 | 4 | yes | — |
| 31 | #2056 | C22 | 1 | 4 | yes | — |
| 32 | #2056 | C22 | 1 | 4 | yes | — |
| 33 | #2026 | C22 | 1 | 4 | yes | — |
| 34 | #2026 | C22 | 1 | 4 | yes | — |
| 35 | #2026 | C22 | 1 | 4 | yes | — |
| 36 | #2016 | C22 | 1 | 4 | yes | — |
| 37 | #1992 | C22 | 1 | 2 | yes | — |
| 38 | #1992 | C22 | 1 | 4 | yes | — |
| 39 | #1991 | C22 | 1 | 2 | yes | — |
| 40 | #1989 | C22 | 1 | 4 | yes | — |
| 41 | #1971 | C22 | 1 | 4 | yes | — |
| 42 | #1960 | C22 | 1 | 2 | yes | — |
| 43 | #1956 | C22 | 1 | 2 | yes | — |
| 44 | #1940 | C22 | 1 | 2 | yes | — |
| 45 | #1935 | C22 | 1 | 4 | yes | — |
| 46 | #1935 | C22 | 1 | 4 | yes | — |
| 47 | #1935 | C22 | 1 | 4 | yes | — |
| 48 | #1933 | C22 | 1 | 4 | yes | — |
| 49 | #1933 | C22 | 1 | 4 | yes | — |
| 50 | #1927 | C22 | 1 | 4 | yes | — |
| 51 | #1920 | C22 | 1 | 4 | yes | — |
| 52 | #1917 | C22 | 1 | 4 | yes | — |
| 53 | #1878 | C22 | 1 | 4 | yes | — |
| 54 | #1878 | C22 | 1 | 4 | yes | — |
| 55 | #1832 | C22 | 1 | 1 | yes | characteristic |
| 56 | #1803 | C22 | 1 | 4 | yes | — |
| 57 | #1771 | C22 | 1 | 4 | yes | — |
| 58 | #1770 | C22 | 1 | 2 | yes | — |
| 59 | #1763 | C22 | 1 | 4 | yes | — |
| 60 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 61 | C22×Q16 | C23 | 1 | 4 | yes | — |
| 62 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 63 | C2×#125 | C23 | 1 | 4 | yes | — |
| 64 | C2×#125 | C23 | 1 | 2 | yes | — |
| 65 | C2×C4×Q8 | C23 | 1 | 4 | yes | — |
| 66 | C2×#106 | C23 | 1 | 4 | yes | — |
| 67 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 68 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 69 | C2×#103 | C23 | 1 | 4 | yes | — |
| 70 | C2×#103 | C23 | 1 | 4 | yes | — |
| 71 | C2×#103 | C23 | 1 | 4 | yes | — |
| 72 | C2×#103 | C23 | 1 | 4 | yes | — |
| 73 | C2×#103 | C23 | 1 | 4 | yes | — |
| 74 | C2×#103 | C23 | 1 | 4 | yes | — |
| 75 | C2×#97 | C23 | 1 | 2 | yes | — |
| 76 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 77 | C2×#97 | C23 | 1 | 4 | yes | — |
| 78 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 80 | #400 | — | 2 | 4 | no | — |
| 81 | #400 | — | 2 | 4 | no | — |
| 82 | #400 | — | 2 | 4 | no | — |
| 83 | #400 | — | 2 | 4 | no | — |
| 84 | #400 | — | 2 | 4 | no | — |
| 85 | #400 | — | 2 | 4 | no | — |
| 86 | #400 | — | 2 | 4 | no | — |
| 87 | #400 | — | 2 | 4 | no | — |
| 88 | #400 | — | 2 | 4 | no | — |
| 89 | #400 | D4 | 1 | 4 | yes | — |
| 90 | #398 | — | 2 | 4 | no | — |
| 91 | #398 | — | 2 | 4 | no | — |
| 92 | #398 | — | 2 | 4 | no | — |
| 93 | #398 | — | 2 | 4 | no | — |
| 94 | #398 | — | 2 | 4 | no | — |
| 95 | #398 | — | 2 | 4 | no | — |
| 96 | #398 | — | 2 | 4 | no | — |
| 97 | #398 | D4 | 1 | 4 | yes | — |
| 98 | #391 | C23 | 1 | 4 | yes | — |
| 99 | #391 | C23 | 1 | 4 | yes | — |
| 100 | #391 | C23 | 1 | 4 | yes | — |
| 101 | #391 | C23 | 1 | 4 | yes | — |
| 102 | #391 | C23 | 1 | 4 | yes | — |
| 103 | #389 | — | 2 | 4 | no | — |
| 104 | #389 | — | 2 | 4 | no | — |
| 105 | #389 | C23 | 1 | 4 | yes | — |
| 106 | #389 | C23 | 1 | 4 | yes | — |
| 107 | #389 | C23 | 1 | 4 | yes | — |
| 108 | #387 | — | 2 | 4 | no | — |
| 109 | #387 | — | 2 | 4 | no | — |
| 110 | #387 | — | 2 | 4 | no | — |
| 111 | #387 | — | 2 | 4 | no | — |
| 112 | #387 | — | 2 | 4 | no | — |
| 113 | #387 | — | 2 | 4 | no | — |
| 114 | #387 | — | 2 | 4 | no | — |
| 115 | #387 | — | 2 | 4 | no | — |
| 116 | #387 | — | 2 | 4 | no | — |
| 117 | #384 | — | 2 | 4 | no | — |
| 118 | #384 | C23 | 1 | 4 | yes | — |
| 119 | #384 | C23 | 1 | 2 | yes | — |
| 120 | #382 | C23 | 1 | 2 | yes | — |
| 121 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 122 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 123 | C4×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 124 | #376 | C23 | 1 | 1 | yes | characteristic |
| 125 | #340 | C23 | 1 | 4 | yes | — |
| 126 | #340 | C23 | 1 | 4 | yes | — |
| 127 | #340 | C23 | 1 | 4 | yes | — |
| 128 | #340 | C23 | 1 | 4 | yes | — |
| 129 | #340 | C23 | 1 | 4 | yes | — |
| 130 | #340 | C23 | 1 | 4 | yes | — |
| 131 | #340 | C23 | 1 | 4 | yes | — |
| 132 | #340 | C23 | 1 | 4 | yes | — |
| 133 | #336 | C23 | 1 | 4 | yes | — |
| 134 | #336 | C23 | 1 | 4 | yes | — |
| 135 | #336 | C23 | 1 | 4 | yes | — |
| 136 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 137 | C2×Q16 | — | 4 | 4 | no | — |
| 138 | C2×Q16 | — | 4 | 4 | no | — |
| 139 | C2×Q16 | — | 4 | 4 | no | — |
| 140 | C22×C8 | C24 | 1 | 4 | yes | — |
| 141 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 142 | C22×C8 | C24 | 1 | 2 | yes | — |
| 143 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 144 | C4⋊Q8 | — | 4 | 2 | no | — |
| 145 | C4⋊Q8 | — | 4 | 2 | no | — |
| 146 | #125 | — | 4 | 2 | no | — |
| 147 | #125 | — | 4 | 4 | no | — |
| 148 | #125 | — | 4 | 2 | no | — |
| 149 | #125 | — | 4 | 4 | no | — |
| 150 | C4×Q8 | — | 4 | 4 | no | — |
| 151 | C4×Q8 | — | 4 | 4 | no | — |
| 152 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 153 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 154 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 155 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 156 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 157 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 158 | C2×C4⋊C4 | — | 2 | 4 | 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 | 4 | yes | — |
| 164 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 165 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 166 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 167 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 168 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 169 | C2×C42 | — | 2 | 4 | no | — |
| 170 | C2×C42 | — | 2 | 4 | no | — |
| 171 | C2×C42 | C24 | 1 | 4 | yes | — |
| 172 | C2×C42 | C24 | 1 | 2 | yes | — |
| 173 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 174 | #106 | — | 4 | 4 | no | — |
| 175 | C4⋊C8 | — | 4 | 2 | no | — |
| 176 | C4⋊C8 | — | 4 | 4 | no | — |
| 177 | #103 | — | 2 | 8 | no | — |
| 178 | #103 | — | 4 | 4 | no | — |
| 179 | #103 | — | 4 | 4 | no | — |
| 180 | #103 | — | 4 | 4 | no | — |
| 181 | #103 | — | 4 | 4 | no | — |
| 182 | #103 | — | 4 | 4 | no | — |
| 183 | #97 | — | 4 | 1 | no | — |
| 184 | #97 | — | 4 | 4 | no | — |
| 185 | #97 | — | 4 | 2 | no | — |
| 186 | C4×C8 | — | 2 | 2 | no | — |
| 187 | C4×C8 | — | 4 | 2 | no | — |
| 188 | #95 | — | 2 | 4 | no | — |
| 189 | #95 | — | 2 | 4 | no | — |
| 190 | #95 | — | 2 | 4 | no | — |
| 191 | #95 | — | 2 | 4 | no | — |
| 192 | #95 | — | 2 | 4 | no | — |
| 193 | #95 | — | 2 | 4 | no | — |
| 194 | #95 | — | 2 | 4 | no | — |
| 195 | #95 | — | 2 | 4 | no | — |
| 196 | #95 | — | 2 | 4 | no | — |
| 197 | #95 | — | 2 | 4 | no | — |
| 198 | #95 | — | 2 | 4 | no | — |
| 199 | #95 | — | 2 | 4 | no | — |
| 200 | #95 | — | 2 | 4 | no | — |
| 201 | #95 | — | 2 | 4 | no | — |
| 202 | #95 | — | 2 | 4 | no | — |
| 203 | #95 | C2×D4 | 1 | 4 | yes | — |
| 204 | #95 | C2×D4 | 1 | 4 | yes | — |
| 205 | #95 | C2×D4 | 1 | 4 | yes | — |
| 206 | #95 | C2×D4 | 1 | 2 | yes | — |
| 207 | #95 | C2×D4 | 1 | 2 | yes | — |
| 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 | — | 2 | 4 | no | — |
| 213 | C2×Q8 | — | 2 | 4 | no | — |
| 214 | C22×C4 | — | 2 | 4 | no | — |
| 215 | C22×C4 | — | 2 | 4 | no | — |
| 216 | C22×C4 | — | 2 | 4 | no | — |
| 217 | C22×C4 | — | 2 | 4 | no | — |
| 218 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 219 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 220 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 221 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 222 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 223 | Q16 | — | 8 | 8 | no | — |
| 224 | C2×C8 | — | 2 | 2 | no | — |
| 225 | C2×C8 | — | 2 | 1 | no | — |
| 226 | C2×C8 | — | 2 | 2 | no | — |
| 227 | C2×C8 | — | 2 | 2 | no | — |
| 228 | C2×C8 | — | 2 | 2 | no | — |
| 229 | C2×C8 | — | 2 | 4 | no | — |
| 230 | C2×C8 | — | 2 | 2 | no | — |
| 231 | C2×C8 | — | 2 | 4 | no | — |
| 232 | C2×C8 | — | 2 | 4 | no | — |
| 233 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 234 | C4⋊C4 | — | 2 | 4 | no | — |
| 235 | C4⋊C4 | — | 2 | 4 | no | — |
| 236 | C4⋊C4 | — | 2 | 4 | no | — |
| 237 | C4⋊C4 | — | 2 | 4 | no | — |
| 238 | C4⋊C4 | — | 2 | 4 | no | — |
| 239 | C4⋊C4 | — | 2 | 4 | no | — |
| 240 | C4⋊C4 | — | 4 | 4 | no | — |
| 241 | C4⋊C4 | — | 4 | 4 | no | — |
| 242 | C4⋊C4 | — | 4 | 4 | no | — |
| 243 | C4⋊C4 | — | 4 | 8 | no | — |
| 244 | C4⋊C4 | — | 8 | 4 | no | — |
| 245 | C4⋊C4 | — | 8 | 4 | no | — |
| 246 | C4⋊C4 | — | 8 | 4 | no | — |
| 247 | C4⋊C4 | — | 8 | 4 | no | — |
| 248 | C4⋊C4 | — | 8 | 4 | no | — |
| 249 | C4⋊C4 | — | 8 | 4 | no | — |
| 250 | C4⋊C4 | — | 8 | 4 | no | — |
| 251 | C4⋊C4 | — | 8 | 4 | no | — |
| 252 | C4⋊C4 | — | 8 | 4 | no | — |
| 253 | C4⋊C4 | — | 8 | 4 | no | — |
| 254 | C42 | — | 2 | 1 | no | — |
| 255 | C42 | — | 2 | 2 | no | — |
| 256 | C42 | — | 2 | 1 | no | — |
| 257 | C42 | — | 2 | 4 | no | — |
| 258 | C42 | — | 2 | 2 | no | — |
| 259 | C42 | — | 2 | 4 | no | — |
| 260 | C42 | — | 8 | 4 | no | — |
| 261 | C42 | — | 8 | 4 | no | — |
| 262 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 263 | Q8 | — | 4 | 4 | no | — |
| 264 | Q8 | — | 4 | 4 | no | — |
| 265 | Q8 | — | 4 | 4 | no | — |
| 266 | Q8 | — | 4 | 4 | no | — |
| 267 | C2×C4 | — | 2 | 4 | no | — |
| 268 | C2×C4 | — | 2 | 4 | no | — |
| 269 | C2×C4 | — | 2 | 4 | no | — |
| 270 | C2×C4 | — | 2 | 4 | no | — |
| 271 | C2×C4 | — | 2 | 4 | no | — |
| 272 | C2×C4 | — | 2 | 4 | no | — |
| 273 | C2×C4 | — | 2 | 4 | no | — |
| 274 | C2×C4 | — | 2 | 2 | no | — |
| 275 | C2×C4 | — | 2 | 4 | no | — |
| 276 | C2×C4 | — | 2 | 4 | no | — |
| 277 | C2×C4 | — | 2 | 4 | no | — |
| 278 | C2×C4 | — | 4 | 4 | no | — |
| 279 | C2×C4 | — | 4 | 4 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 4 | 4 | no | — |
| 282 | C2×C4 | — | 4 | 4 | no | — |
| 283 | C2×C4 | — | 4 | 4 | no | — |
| 284 | C2×C4 | — | 4 | 4 | no | — |
| 285 | C2×C4 | — | 4 | 4 | no | — |
| 286 | C2×C4 | — | 4 | 4 | no | — |
| 287 | C2×C4 | — | 4 | 4 | no | — |
| 288 | C2×C4 | — | 4 | 4 | no | — |
| 289 | C2×C4 | — | 4 | 4 | no | — |
| 290 | C2×C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | #585 | 1 | 2 | yes | — |
| 292 | C2×C4 | #585 | 1 | 2 | yes | — |
| 293 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 294 | C8 | — | 2 | 2 | no | — |
| 295 | C8 | — | 4 | 1 | no | — |
| 296 | C8 | — | 4 | 2 | no | — |
| 297 | C8 | — | 4 | 4 | no | — |
| 298 | C22 | #3497 | 1 | 1 | yes | characteristic |
| 299 | C22 | #3500 | 1 | 2 | yes | — |
| 300 | C22 | #3586 | 1 | 4 | yes | — |
| 301 | C4 | — | 4 | 4 | no | — |
| 302 | C4 | — | 4 | 4 | no | — |
| 303 | C4 | — | 4 | 4 | no | — |
| 304 | C4 | — | 4 | 2 | no | — |
| 305 | C4 | — | 8 | 4 | no | — |
| 306 | C4 | — | 8 | 4 | no | — |
| 307 | C4 | — | 8 | 4 | no | — |
| 308 | C4 | — | 8 | 4 | no | — |
| 309 | C4 | #3531 | 1 | 1 | yes | characteristic |
| 310 | C4 | #3553 | 1 | 1 | yes | characteristic |
| 311 | C4 | #3568 | 1 | 2 | yes | — |
| 312 | C2 | #46239 | 1 | 1 | yes | characteristic |
| 313 | C2 | #62388 | 1 | 2 | yes | — |
| 314 | C2 | #62572 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #53147 | 2 |
| #33285 | 2 |
| #32857 | 2 |
| #32816 | 2 |
| #32255 | 2 |
| #31828 | 2 |
| #31664 | 2 |
| #29082 | 2 |
| #28904 | 2 |
| #27831 | 2 |
| #21677 | 2 |