Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1668 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60688 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #60208 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22281 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22257 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22235 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #22206 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #21416 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #21139 | C2 | 1 | 8 | yes | maximal, complemented |
| 9 | #20830 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #20367 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #3444 | C22 | 1 | 2 | yes | — |
| 12 | #3181 | C4 | 1 | 8 | yes | complemented |
| 13 | #3180 | C4 | 1 | 8 | yes | complemented |
| 14 | #3162 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3150 | C22 | 1 | 1 | yes | characteristic |
| 16 | #3135 | C22 | 1 | 2 | yes | — |
| 17 | #3050 | C22 | 1 | 4 | yes | — |
| 18 | #2971 | — | 2 | 4 | no | — |
| 19 | #2970 | — | 2 | 4 | no | — |
| 20 | #2969 | — | 2 | 4 | no | — |
| 21 | #2969 | — | 2 | 4 | no | — |
| 22 | #2960 | C22 | 1 | 4 | yes | — |
| 23 | #2943 | C22 | 1 | 4 | yes | — |
| 24 | #2931 | C22 | 1 | 4 | yes | — |
| 25 | #2930 | C22 | 1 | 8 | yes | — |
| 26 | #2923 | C22 | 1 | 4 | yes | — |
| 27 | #2920 | C22 | 1 | 2 | yes | — |
| 28 | #2875 | C22 | 1 | 4 | yes | — |
| 29 | C2×C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 30 | C22×#97 | C22 | 1 | 1 | yes | characteristic |
| 31 | C4×#117 | C22 | 1 | 1 | yes | characteristic |
| 32 | #1995 | C22 | 1 | 4 | yes | — |
| 33 | #1963 | C22 | 1 | 4 | yes | — |
| 34 | #1929 | C22 | 1 | 16 | yes | — |
| 35 | #1922 | C22 | 1 | 16 | yes | — |
| 36 | #1897 | C22 | 1 | 8 | yes | — |
| 37 | #1876 | C22 | 1 | 16 | yes | — |
| 38 | #1866 | C22 | 1 | 16 | yes | — |
| 39 | #1849 | C22 | 1 | 2 | yes | — |
| 40 | #1838 | C22 | 1 | 2 | yes | — |
| 41 | #1810 | C22 | 1 | 4 | yes | — |
| 42 | #1778 | C22 | 1 | 4 | yes | — |
| 43 | C4×#104 | C22 | 1 | 8 | yes | — |
| 44 | #1746 | C22 | 1 | 2 | yes | — |
| 45 | #1746 | C22 | 1 | 2 | yes | — |
| 46 | #1745 | C22 | 1 | 4 | yes | — |
| 47 | #1729 | C22 | 1 | 4 | yes | — |
| 48 | C2×#135 | C23 | 1 | 4 | yes | — |
| 49 | C2×#131 | C23 | 1 | 8 | yes | — |
| 50 | C22×M16 | C23 | 1 | 2 | yes | — |
| 51 | C23×C8 | C23 | 1 | 2 | yes | — |
| 52 | #533 | C2×C4 | 1 | 4 | yes | — |
| 53 | #533 | C23 | 1 | 2 | yes | — |
| 54 | #532 | C2×C4 | 1 | 4 | yes | — |
| 55 | #532 | C23 | 1 | 2 | yes | — |
| 56 | #529 | — | 2 | 4 | no | — |
| 57 | #529 | — | 2 | 4 | no | — |
| 58 | #529 | C2×C4 | 1 | 8 | yes | — |
| 59 | #525 | — | 2 | 4 | no | — |
| 60 | #525 | C23 | 1 | 4 | yes | — |
| 61 | C4×#41 | — | 2 | 4 | no | — |
| 62 | C4×#41 | C23 | 1 | 4 | yes | — |
| 63 | C2×#117 | C23 | 1 | 2 | yes | — |
| 64 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 65 | #501 | C2×C4 | 1 | 8 | yes | — |
| 66 | #497 | C2×C4 | 1 | 4 | yes | — |
| 67 | #496 | C2×C4 | 1 | 4 | yes | — |
| 68 | #491 | C2×C4 | 1 | 4 | yes | — |
| 69 | #490 | C2×C4 | 1 | 4 | yes | — |
| 70 | #489 | C2×C4 | 1 | 8 | yes | — |
| 71 | #467 | C2×C4 | 1 | 8 | yes | — |
| 72 | #466 | C2×C4 | 1 | 8 | yes | — |
| 73 | #465 | C2×C4 | 1 | 16 | yes | — |
| 74 | #446 | — | 2 | 4 | no | — |
| 75 | #446 | — | 2 | 4 | no | — |
| 76 | C8×Q8 | — | 2 | 4 | no | — |
| 77 | C8×Q8 | — | 2 | 4 | no | — |
| 78 | #442 | C2×C4 | 1 | 8 | yes | — |
| 79 | #441 | C2×C4 | 1 | 8 | yes | — |
| 80 | #440 | C2×C4 | 1 | 16 | yes | — |
| 81 | #436 | — | 2 | 8 | no | — |
| 82 | #436 | — | 2 | 8 | no | — |
| 83 | #436 | — | 2 | 4 | no | — |
| 84 | #436 | — | 2 | 4 | no | — |
| 85 | #435 | — | 2 | 8 | no | — |
| 86 | #435 | — | 2 | 8 | no | — |
| 87 | C8×D4 | — | 2 | 4 | no | — |
| 88 | C8×D4 | — | 2 | 4 | no | — |
| 89 | #433 | — | 2 | 4 | no | — |
| 90 | #433 | — | 2 | 4 | no | — |
| 91 | #432 | — | 2 | 4 | no | — |
| 92 | #432 | — | 2 | 8 | no | — |
| 93 | #431 | — | 2 | 4 | no | — |
| 94 | C2×#108 | C23 | 1 | 8 | yes | — |
| 95 | #427 | C23 | 1 | 4 | yes | — |
| 96 | #424 | C23 | 1 | 8 | yes | — |
| 97 | #423 | C23 | 1 | 4 | yes | — |
| 98 | C2×#104 | C23 | 1 | 16 | yes | — |
| 99 | #416 | C23 | 1 | 8 | yes | — |
| 100 | #408 | C23 | 1 | 8 | yes | — |
| 101 | #407 | C23 | 1 | 4 | yes | — |
| 102 | #405 | — | 2 | 8 | no | — |
| 103 | #405 | C23 | 1 | 8 | yes | — |
| 104 | C4×M16 | — | 2 | 4 | no | — |
| 105 | C4×M16 | D4 | 1 | 4 | yes | — |
| 106 | C4×M16 | D4 | 1 | 4 | yes | — |
| 107 | C4×M16 | C23 | 1 | 4 | yes | — |
| 108 | C2×#97 | — | 2 | 4 | no | — |
| 109 | C2×#97 | C2×C4 | 1 | 8 | yes | — |
| 110 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 111 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 112 | C2×#97 | C23 | 1 | 2 | yes | — |
| 113 | C2×#97 | C23 | 1 | 2 | yes | — |
| 114 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 115 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 116 | C4×C22⋊C4 | — | 2 | 2 | no | — |
| 117 | #376 | C23 | 1 | 2 | yes | — |
| 118 | C43 | C23 | 1 | 2 | yes | — |
| 119 | #341 | C23 | 1 | 8 | yes | — |
| 120 | #339 | C23 | 1 | 16 | yes | — |
| 121 | #336 | C23 | 1 | 8 | yes | — |
| 122 | #336 | C23 | 1 | 8 | yes | — |
| 123 | C2×#41 | — | 2 | 4 | no | — |
| 124 | C2×#41 | C24 | 1 | 4 | yes | — |
| 125 | C23×C4 | #41 | 1 | 2 | yes | — |
| 126 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 127 | #135 | — | 4 | 8 | no | — |
| 128 | C2×Q16 | C22×C4 | 1 | 4 | yes | — |
| 129 | C2×SD16 | C22×C4 | 1 | 8 | yes | — |
| 130 | C2×D8 | C22×C4 | 1 | 4 | yes | — |
| 131 | #131 | — | 4 | 16 | no | — |
| 132 | C2×M16 | — | 2 | 8 | no | — |
| 133 | C2×M16 | — | 4 | 4 | no | — |
| 134 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 135 | C2×M16 | C2×D4 | 1 | 4 | yes | — |
| 136 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 137 | C2×M16 | C24 | 1 | 8 | yes | — |
| 138 | C22×C8 | — | 2 | 8 | no | — |
| 139 | C22×C8 | — | 4 | 4 | no | — |
| 140 | C22×C8 | C22×C4 | 1 | 8 | yes | — |
| 141 | C22×C8 | C24 | 1 | 2 | yes | — |
| 142 | C22×C8 | C24 | 1 | 2 | yes | — |
| 143 | C22×C8 | C24 | 1 | 4 | yes | — |
| 144 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 145 | C4⋊Q8 | C22×C4 | 1 | 4 | yes | — |
| 146 | #127 | C22×C4 | 1 | 2 | yes | — |
| 147 | #126 | — | 2 | 8 | no | — |
| 148 | #125 | — | 2 | 4 | no | — |
| 149 | #125 | C22×C4 | 1 | 4 | yes | — |
| 150 | #124 | — | 2 | 4 | no | — |
| 151 | #124 | C22×C4 | 1 | 4 | yes | — |
| 152 | #123 | — | 2 | 8 | no | — |
| 153 | #123 | — | 2 | 4 | no | — |
| 154 | C22⋊Q8 | — | 2 | 4 | no | — |
| 155 | C22⋊Q8 | — | 2 | 4 | no | — |
| 156 | C22⋊Q8 | C22×C4 | 1 | 8 | yes | — |
| 157 | #121 | — | 2 | 4 | no | — |
| 158 | #121 | — | 2 | 4 | no | — |
| 159 | #121 | C22×C4 | 1 | 8 | yes | — |
| 160 | #120 | — | 2 | 4 | no | — |
| 161 | C4×Q8 | — | 2 | 4 | no | — |
| 162 | C4×Q8 | — | 2 | 4 | no | — |
| 163 | C4×Q8 | C22×C4 | 1 | 8 | yes | — |
| 164 | C4×D4 | — | 2 | 8 | no | — |
| 165 | C4×D4 | — | 2 | 8 | no | — |
| 166 | C4×D4 | — | 2 | 8 | no | — |
| 167 | C4×D4 | — | 2 | 4 | no | — |
| 168 | C4×D4 | — | 2 | 4 | no | — |
| 169 | C4×D4 | C22×C4 | 1 | 8 | yes | — |
| 170 | #117 | — | 2 | 4 | no | — |
| 171 | #117 | — | 2 | 8 | no | — |
| 172 | #117 | — | 2 | 4 | no | — |
| 173 | #117 | — | 4 | 4 | no | — |
| 174 | #117 | C24 | 1 | 4 | yes | — |
| 175 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 176 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 177 | C2×C42 | — | 2 | 4 | no | — |
| 178 | C2×C42 | — | 2 | 4 | no | — |
| 179 | C2×C42 | C22×C4 | 1 | 4 | yes | — |
| 180 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 181 | C2×C42 | #41 | 1 | 2 | yes | — |
| 182 | C2×C42 | C24 | 1 | 4 | yes | — |
| 183 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 184 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 185 | #108 | — | 4 | 8 | no | — |
| 186 | #107 | C22×C4 | 1 | 8 | yes | — |
| 187 | #106 | C22×C4 | 1 | 8 | yes | — |
| 188 | C4⋊C8 | — | 2 | 8 | no | — |
| 189 | C4⋊C8 | — | 2 | 8 | no | — |
| 190 | C4⋊C8 | — | 2 | 4 | no | — |
| 191 | C4⋊C8 | — | 2 | 4 | no | — |
| 192 | #104 | — | 4 | 16 | no | — |
| 193 | #103 | C22×C4 | 1 | 16 | yes | — |
| 194 | #102 | C22×C4 | 1 | 16 | yes | — |
| 195 | C22⋊C8 | — | 2 | 8 | no | — |
| 196 | C22⋊C8 | — | 2 | 8 | no | — |
| 197 | C22⋊C8 | — | 2 | 8 | no | — |
| 198 | #97 | — | 2 | 8 | no | — |
| 199 | #97 | — | 2 | 4 | no | — |
| 200 | #97 | C22×C4 | 1 | 8 | yes | — |
| 201 | #97 | C22×C4 | 1 | 4 | yes | — |
| 202 | #97 | C22×C4 | 1 | 4 | yes | — |
| 203 | #97 | C2×D4 | 1 | 2 | yes | — |
| 204 | #97 | C2×D4 | 1 | 2 | yes | — |
| 205 | #97 | C2×D4 | 1 | 4 | yes | — |
| 206 | C4×C8 | — | 2 | 8 | no | — |
| 207 | C4×C8 | — | 2 | 4 | no | — |
| 208 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 209 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 210 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 211 | #95 | — | 2 | 4 | no | — |
| 212 | C24 | C2×#41 | 1 | 1 | yes | characteristic |
| 213 | #41 | — | 2 | 16 | no | — |
| 214 | #41 | — | 8 | 8 | no | — |
| 215 | C2×Q8 | — | 2 | 4 | no | — |
| 216 | C2×Q8 | C23×C4 | 1 | 4 | yes | — |
| 217 | C2×D4 | — | 2 | 4 | no | — |
| 218 | C2×D4 | — | 2 | 8 | no | — |
| 219 | C2×D4 | — | 2 | 4 | no | — |
| 220 | C2×D4 | C23×C4 | 1 | 4 | yes | — |
| 221 | C22×C4 | — | 2 | 4 | no | — |
| 222 | C22×C4 | — | 2 | 4 | no | — |
| 223 | C22×C4 | — | 2 | 8 | no | — |
| 224 | C22×C4 | — | 2 | 4 | no | — |
| 225 | C22×C4 | — | 8 | 2 | no | — |
| 226 | C22×C4 | C4×D4 | 1 | 4 | yes | — |
| 227 | C22×C4 | C4×D4 | 1 | 4 | yes | — |
| 228 | C22×C4 | C23×C4 | 1 | 2 | yes | — |
| 229 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 230 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 231 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 232 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 233 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 234 | Q16 | — | 4 | 4 | no | — |
| 235 | SD16 | — | 4 | 8 | no | — |
| 236 | D8 | — | 4 | 4 | no | — |
| 237 | M16 | — | 2 | 16 | no | — |
| 238 | M16 | — | 4 | 2 | no | — |
| 239 | M16 | — | 4 | 4 | no | — |
| 240 | M16 | — | 4 | 2 | no | — |
| 241 | M16 | — | 4 | 16 | no | — |
| 242 | C2×C8 | — | 2 | 4 | no | — |
| 243 | C2×C8 | — | 2 | 4 | no | — |
| 244 | C2×C8 | — | 2 | 8 | no | — |
| 245 | C2×C8 | — | 2 | 8 | no | — |
| 246 | C2×C8 | — | 4 | 8 | no | — |
| 247 | C2×C8 | — | 4 | 16 | no | — |
| 248 | C2×C8 | C23×C4 | 1 | 4 | yes | — |
| 249 | C2×C8 | C23×C4 | 1 | 4 | yes | — |
| 250 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 251 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 252 | C4⋊C4 | — | 2 | 4 | no | — |
| 253 | C4⋊C4 | — | 2 | 4 | no | — |
| 254 | C4⋊C4 | — | 2 | 8 | no | — |
| 255 | C4⋊C4 | — | 2 | 8 | no | — |
| 256 | C4⋊C4 | — | 2 | 8 | no | — |
| 257 | C4⋊C4 | — | 4 | 4 | no | — |
| 258 | C4⋊C4 | C23×C4 | 1 | 8 | yes | — |
| 259 | C22⋊C4 | — | 2 | 8 | no | — |
| 260 | C22⋊C4 | — | 2 | 4 | no | — |
| 261 | C22⋊C4 | — | 2 | 8 | no | — |
| 262 | C22⋊C4 | — | 2 | 8 | no | — |
| 263 | C22⋊C4 | — | 2 | 8 | no | — |
| 264 | C22⋊C4 | — | 4 | 4 | no | — |
| 265 | C42 | — | 2 | 8 | no | — |
| 266 | C42 | — | 2 | 4 | no | — |
| 267 | C42 | — | 2 | 8 | no | — |
| 268 | C42 | — | 8 | 4 | no | — |
| 269 | C42 | C4×D4 | 1 | 4 | yes | — |
| 270 | C42 | C4×D4 | 1 | 4 | yes | — |
| 271 | C42 | C23×C4 | 1 | 2 | yes | — |
| 272 | C42 | C23×C4 | 1 | 2 | yes | — |
| 273 | C42 | C22×D4 | 1 | 2 | yes | — |
| 274 | C42 | C22×D4 | 1 | 2 | yes | — |
| 275 | C23 | — | 2 | 2 | no | — |
| 276 | C23 | — | 2 | 4 | no | — |
| 277 | C23 | — | 4 | 2 | no | — |
| 278 | C23 | C2×C4×D4 | 1 | 2 | yes | — |
| 279 | C23 | C22×#41 | 1 | 1 | yes | characteristic |
| 280 | Q8 | — | 2 | 8 | no | — |
| 281 | Q8 | — | 4 | 8 | no | — |
| 282 | D4 | — | 2 | 8 | no | — |
| 283 | D4 | — | 4 | 8 | no | — |
| 284 | D4 | — | 4 | 8 | no | — |
| 285 | D4 | — | 8 | 8 | no | — |
| 286 | C2×C4 | — | 2 | 2 | no | — |
| 287 | C2×C4 | — | 2 | 4 | no | — |
| 288 | C2×C4 | — | 2 | 2 | no | — |
| 289 | C2×C4 | — | 2 | 2 | no | — |
| 290 | C2×C4 | — | 2 | 4 | no | — |
| 291 | C2×C4 | — | 2 | 8 | no | — |
| 292 | C2×C4 | — | 4 | 2 | no | — |
| 293 | C2×C4 | — | 4 | 4 | no | — |
| 294 | C2×C4 | — | 4 | 4 | no | — |
| 295 | C2×C4 | — | 4 | 2 | no | — |
| 296 | C2×C4 | — | 4 | 8 | no | — |
| 297 | C2×C4 | — | 4 | 8 | no | — |
| 298 | C2×C4 | — | 4 | 4 | no | — |
| 299 | C2×C4 | — | 8 | 8 | no | — |
| 300 | C2×C4 | C2×C4×D4 | 1 | 4 | yes | — |
| 301 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 302 | C2×C4 | C2×C4×D4 | 1 | 4 | yes | — |
| 303 | C2×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #585 | 1 | 4 | yes | — |
| 306 | C8 | — | 2 | 4 | no | — |
| 307 | C8 | — | 2 | 4 | no | — |
| 308 | C8 | — | 2 | 4 | no | — |
| 309 | C8 | — | 2 | 4 | no | — |
| 310 | C8 | — | 4 | 8 | no | — |
| 311 | C22 | — | 2 | 2 | no | — |
| 312 | C22 | — | 2 | 2 | no | — |
| 313 | C22 | — | 2 | 2 | no | — |
| 314 | C22 | — | 2 | 2 | no | — |
| 315 | C22 | — | 4 | 4 | no | — |
| 316 | C22 | — | 4 | 4 | no | — |
| 317 | C22 | — | 4 | 8 | no | — |
| 318 | C22 | C22×C4×D4 | 1 | 1 | yes | characteristic |
| 319 | C22 | #3455 | 1 | 2 | yes | — |
| 320 | C4 | — | 2 | 2 | no | — |
| 321 | C4 | — | 2 | 4 | no | — |
| 322 | C4 | — | 2 | 4 | no | — |
| 323 | C4 | — | 4 | 4 | no | — |
| 324 | C4 | — | 4 | 8 | no | — |
| 325 | C4 | — | 8 | 4 | no | — |
| 326 | C4 | #3428 | 1 | 2 | yes | — |
| 327 | C4 | #3449 | 1 | 2 | yes | — |
| 328 | C2 | — | 2 | 2 | no | — |
| 329 | C2 | — | 4 | 2 | no | — |
| 330 | C2 | — | 8 | 4 | no | — |
| 331 | C2 | #34070 | 1 | 1 | yes | characteristic |
| 332 | C2 | #60182 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60688 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60208 | C2 | semidirect |
| 3. #22281 | C2 | semidirect |
| 4. #22257 | C2 | semidirect |
| 5. #22235 | C2 | semidirect |
| 6. #22206 | C2 | semidirect |
| 7. #21416 | C2 | semidirect |
| 8. #21139 | C2 | semidirect |
| 9. #20830 | C2 | semidirect |
| 10. #20367 | C2 | semidirect |
| 12. #3181 | C4 | semidirect |
| 13. #3180 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60688 | 2 |
| #60208 | 2 |
| #22281 | 2 |
| #22257 | 2 |
| #22235 | 2 |
| #22206 | 2 |
| #21416 | 2 |
| #21139 | 2 |
| #20830 | 2 |
| #20367 | 2 |