Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
724 conjugacy classes of subgroups.
The subgroup lattice has 347 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #33255 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #21706 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #19662 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #19654 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #19199 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #19197 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #10288 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #10286 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #9772 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #9767 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #8268 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3227 | C22 | 1 | 2 | yes | — |
| 13 | #3226 | C22 | 1 | 2 | yes | — |
| 14 | #3226 | C22 | 1 | 2 | yes | — |
| 15 | #3206 | C22 | 1 | 2 | yes | — |
| 16 | C2×#498 | C22 | 1 | 1 | yes | characteristic |
| 17 | #3147 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#487 | C22 | 1 | 2 | yes | — |
| 20 | C2×#444 | — | 2 | 2 | no | — |
| 21 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 22 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 23 | #2862 | C22 | 1 | 2 | yes | — |
| 24 | #1975 | — | 2 | 2 | no | — |
| 25 | #1971 | — | 2 | 2 | no | — |
| 26 | #1971 | — | 2 | 2 | no | — |
| 27 | #1962 | C22 | 1 | 1 | yes | characteristic |
| 28 | #1962 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 30 | C4×#97 | C22 | 1 | 1 | yes | characteristic |
| 31 | #1722 | — | 2 | 1 | no | — |
| 32 | #1722 | — | 2 | 1 | no | — |
| 33 | #1716 | — | 2 | 1 | no | — |
| 34 | #1715 | — | 2 | 1 | no | — |
| 35 | #1713 | — | 2 | 1 | no | — |
| 36 | #1709 | — | 2 | 1 | no | — |
| 37 | #1708 | — | 2 | 1 | no | — |
| 38 | #1705 | — | 2 | 1 | no | — |
| 39 | #1703 | — | 2 | 4 | no | — |
| 40 | #1699 | — | 2 | 4 | no | — |
| 41 | #1683 | — | 2 | 4 | no | — |
| 42 | #1679 | — | 2 | 4 | no | — |
| 43 | #1564 | C22 | 1 | 2 | yes | — |
| 44 | #1549 | C22 | 1 | 2 | yes | — |
| 45 | #1548 | C22 | 1 | 2 | yes | — |
| 46 | #1546 | C22 | 1 | 2 | yes | — |
| 47 | #1546 | C22 | 1 | 2 | yes | — |
| 48 | #1450 | C22 | 1 | 1 | yes | characteristic |
| 49 | #1450 | C22 | 1 | 1 | yes | characteristic |
| 50 | #1276 | C22 | 1 | 2 | yes | — |
| 51 | C2×#135 | — | 2 | 2 | no | — |
| 52 | C22×Q16 | — | 2 | 2 | no | — |
| 53 | C2×#131 | — | 2 | 1 | no | — |
| 54 | C2×#131 | — | 2 | 1 | no | — |
| 55 | #532 | C23 | 1 | 1 | yes | characteristic |
| 56 | #532 | C23 | 1 | 1 | yes | characteristic |
| 57 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 58 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 59 | #500 | — | 2 | 2 | no | — |
| 60 | #500 | — | 2 | 2 | no | — |
| 61 | #498 | — | 2 | 2 | no | — |
| 62 | #498 | — | 2 | 1 | no | — |
| 63 | #498 | — | 2 | 1 | no | — |
| 64 | #498 | — | 2 | 2 | no | — |
| 65 | #495 | — | 2 | 2 | no | — |
| 66 | #494 | — | 2 | 1 | no | — |
| 67 | #494 | — | 2 | 1 | no | — |
| 68 | #494 | — | 2 | 2 | no | — |
| 69 | #494 | D4 | 1 | 1 | yes | characteristic |
| 70 | #493 | D4 | 1 | 1 | yes | characteristic |
| 71 | #492 | — | 2 | 2 | no | — |
| 72 | #492 | D4 | 1 | 1 | yes | characteristic |
| 73 | #492 | D4 | 1 | 1 | yes | characteristic |
| 74 | #487 | — | 2 | 2 | no | — |
| 75 | #487 | — | 2 | 2 | no | — |
| 76 | #487 | — | 2 | 2 | no | — |
| 77 | #487 | — | 2 | 2 | no | — |
| 78 | #487 | — | 2 | 2 | no | — |
| 79 | #486 | — | 2 | 2 | no | — |
| 80 | #486 | — | 2 | 2 | no | — |
| 81 | #486 | — | 2 | 2 | no | — |
| 82 | #481 | — | 4 | 2 | no | — |
| 83 | #480 | — | 4 | 2 | no | — |
| 84 | #480 | — | 4 | 2 | no | — |
| 85 | #476 | — | 2 | 4 | no | — |
| 86 | #474 | — | 2 | 4 | no | — |
| 87 | #474 | — | 2 | 4 | no | — |
| 88 | #461 | — | 2 | 4 | no | — |
| 89 | #448 | — | 4 | 2 | no | — |
| 90 | #444 | — | 4 | 8 | no | — |
| 91 | #436 | — | 2 | 2 | no | — |
| 92 | #436 | — | 2 | 2 | no | — |
| 93 | C8×D4 | — | 2 | 2 | no | — |
| 94 | C8×D4 | — | 2 | 2 | no | — |
| 95 | #431 | C23 | 1 | 2 | yes | — |
| 96 | #431 | C23 | 1 | 2 | yes | — |
| 97 | C2×#108 | — | 2 | 2 | no | — |
| 98 | C2×#106 | — | 2 | 2 | no | — |
| 99 | #423 | C23 | 1 | 1 | yes | characteristic |
| 100 | #423 | C23 | 1 | 1 | yes | characteristic |
| 101 | C2×#104 | — | 2 | 2 | no | — |
| 102 | C2×#104 | — | 2 | 2 | no | — |
| 103 | C2×#103 | — | 2 | 2 | no | — |
| 104 | C2×#103 | — | 2 | 2 | no | — |
| 105 | #408 | — | 2 | 1 | no | — |
| 106 | #408 | — | 2 | 1 | no | — |
| 107 | C2×#97 | D4 | 1 | 2 | yes | — |
| 108 | C2×#97 | D4 | 1 | 2 | yes | — |
| 109 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 110 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 111 | #391 | — | 2 | 4 | no | — |
| 112 | #391 | — | 4 | 2 | no | — |
| 113 | #382 | — | 4 | 2 | no | — |
| 114 | C43 | C23 | 1 | 1 | yes | characteristic |
| 115 | #335 | D4 | 1 | 4 | yes | — |
| 116 | #334 | D4 | 1 | 4 | yes | — |
| 117 | #333 | — | 2 | 4 | no | — |
| 118 | #332 | — | 2 | 4 | no | — |
| 119 | #332 | — | 2 | 4 | no | — |
| 120 | #332 | — | 2 | 4 | no | — |
| 121 | #322 | — | 2 | 1 | no | — |
| 122 | #322 | — | 2 | 1 | no | — |
| 123 | C82 | — | 2 | 1 | no | — |
| 124 | C82 | — | 2 | 1 | no | — |
| 125 | C2×#41 | — | 2 | 1 | no | — |
| 126 | C2×#41 | — | 2 | 1 | no | — |
| 127 | C22×Q8 | — | 2 | 2 | no | — |
| 128 | C22×Q8 | — | 2 | 2 | no | — |
| 129 | #135 | — | 4 | 4 | no | — |
| 130 | #135 | — | 4 | 4 | no | — |
| 131 | C2×Q16 | — | 2 | 2 | no | — |
| 132 | C2×Q16 | — | 4 | 2 | no | — |
| 133 | C2×Q16 | — | 4 | 2 | no | — |
| 134 | C2×Q16 | — | 4 | 4 | no | — |
| 135 | C2×Q16 | — | 4 | 4 | no | — |
| 136 | C2×SD16 | — | 2 | 2 | no | — |
| 137 | C2×SD16 | — | 2 | 2 | no | — |
| 138 | C2×SD16 | — | 4 | 4 | no | — |
| 139 | C2×D8 | — | 2 | 2 | no | — |
| 140 | #131 | — | 4 | 4 | no | — |
| 141 | #131 | — | 4 | 4 | no | — |
| 142 | C2×M16 | — | 2 | 1 | no | — |
| 143 | C2×M16 | — | 2 | 1 | no | — |
| 144 | C2×M16 | — | 4 | 1 | no | — |
| 145 | C2×M16 | — | 4 | 1 | no | — |
| 146 | C22×C8 | — | 4 | 1 | no | — |
| 147 | C22×C8 | — | 4 | 1 | no | — |
| 148 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 149 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 150 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 151 | C4⋊Q8 | — | 2 | 2 | no | — |
| 152 | C4⋊Q8 | — | 2 | 2 | no | — |
| 153 | C4⋊Q8 | — | 2 | 2 | no | — |
| 154 | C4⋊Q8 | — | 2 | 2 | no | — |
| 155 | C4⋊Q8 | — | 2 | 2 | no | — |
| 156 | C4⋊Q8 | — | 2 | 2 | no | — |
| 157 | C4⋊Q8 | — | 2 | 2 | no | — |
| 158 | C4⋊Q8 | — | 2 | 2 | no | — |
| 159 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 160 | C4⋊Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 161 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 162 | #127 | C2×D4 | 1 | 1 | yes | characteristic |
| 163 | #124 | — | 2 | 1 | no | — |
| 164 | #124 | — | 2 | 1 | no | — |
| 165 | #121 | — | 2 | 2 | no | — |
| 166 | #121 | — | 2 | 2 | no | — |
| 167 | C4×D4 | — | 2 | 2 | no | — |
| 168 | C4×D4 | — | 2 | 2 | no | — |
| 169 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 170 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 171 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 172 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 173 | C2×C42 | — | 2 | 2 | no | — |
| 174 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 175 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 176 | #108 | — | 4 | 4 | no | — |
| 177 | #107 | — | 4 | 4 | no | — |
| 178 | #107 | — | 4 | 4 | no | — |
| 179 | #106 | — | 4 | 4 | no | — |
| 180 | #106 | — | 4 | 2 | no | — |
| 181 | #106 | — | 4 | 2 | no | — |
| 182 | C4⋊C8 | — | 2 | 2 | no | — |
| 183 | C4⋊C8 | — | 2 | 2 | no | — |
| 184 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 185 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 186 | #104 | — | 4 | 4 | no | — |
| 187 | #104 | — | 4 | 4 | no | — |
| 188 | #103 | — | 4 | 4 | no | — |
| 189 | #103 | — | 4 | 2 | no | — |
| 190 | #103 | — | 4 | 2 | no | — |
| 191 | #103 | — | 4 | 2 | no | — |
| 192 | #103 | — | 4 | 2 | no | — |
| 193 | #102 | — | 4 | 4 | no | — |
| 194 | #102 | — | 4 | 4 | no | — |
| 195 | #102 | — | 4 | 4 | no | — |
| 196 | C22⋊C8 | — | 2 | 2 | no | — |
| 197 | C22⋊C8 | — | 2 | 2 | no | — |
| 198 | C22⋊C8 | — | 4 | 2 | no | — |
| 199 | C22⋊C8 | — | 4 | 2 | no | — |
| 200 | #97 | — | 2 | 4 | no | — |
| 201 | #97 | — | 2 | 4 | no | — |
| 202 | C4×C8 | — | 2 | 2 | no | — |
| 203 | C4×C8 | — | 2 | 2 | no | — |
| 204 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 205 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 206 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 207 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 208 | C4×C8 | #41 | 1 | 2 | yes | — |
| 209 | #95 | — | 4 | 8 | no | — |
| 210 | #41 | — | 4 | 2 | no | — |
| 211 | #41 | — | 4 | 2 | no | — |
| 212 | #41 | — | 4 | 2 | no | — |
| 213 | #41 | — | 4 | 2 | no | — |
| 214 | C2×Q8 | — | 2 | 1 | no | — |
| 215 | C2×Q8 | — | 2 | 1 | no | — |
| 216 | C2×Q8 | — | 4 | 2 | no | — |
| 217 | C2×Q8 | — | 4 | 2 | no | — |
| 218 | C2×Q8 | — | 4 | 2 | no | — |
| 219 | C2×Q8 | — | 4 | 2 | no | — |
| 220 | C2×Q8 | — | 4 | 4 | no | — |
| 221 | C2×Q8 | — | 4 | 4 | no | — |
| 222 | C2×Q8 | — | 8 | 4 | no | — |
| 223 | C2×D4 | — | 2 | 1 | no | — |
| 224 | C2×D4 | — | 2 | 1 | no | — |
| 225 | C2×D4 | — | 2 | 2 | no | — |
| 226 | C2×D4 | — | 2 | 2 | no | — |
| 227 | C2×D4 | — | 4 | 1 | no | — |
| 228 | C2×D4 | — | 4 | 1 | no | — |
| 229 | C22×C4 | — | 2 | 2 | no | — |
| 230 | C22×C4 | — | 4 | 1 | no | — |
| 231 | C22×C4 | — | 4 | 1 | no | — |
| 232 | C22×C4 | — | 4 | 2 | no | — |
| 233 | C22×C4 | — | 4 | 2 | no | — |
| 234 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 235 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 236 | Q16 | — | 4 | 4 | no | — |
| 237 | Q16 | — | 8 | 4 | no | — |
| 238 | Q16 | — | 8 | 4 | no | — |
| 239 | SD16 | — | 4 | 4 | no | — |
| 240 | SD16 | — | 4 | 4 | no | — |
| 241 | SD16 | — | 16 | 4 | no | — |
| 242 | D8 | — | 4 | 4 | no | — |
| 243 | M16 | — | 4 | 2 | no | — |
| 244 | M16 | — | 4 | 2 | no | — |
| 245 | M16 | — | 8 | 2 | no | — |
| 246 | M16 | — | 8 | 2 | no | — |
| 247 | C2×C8 | — | 2 | 2 | no | — |
| 248 | C2×C8 | — | 2 | 2 | no | — |
| 249 | C2×C8 | — | 2 | 4 | no | — |
| 250 | C2×C8 | — | 2 | 4 | no | — |
| 251 | C2×C8 | — | 4 | 1 | no | — |
| 252 | C2×C8 | — | 4 | 1 | no | — |
| 253 | C2×C8 | — | 8 | 2 | no | — |
| 254 | C2×C8 | — | 8 | 2 | no | — |
| 255 | C2×C8 | #121 | 1 | 2 | yes | — |
| 256 | C2×C8 | #121 | 1 | 2 | yes | — |
| 257 | C2×C8 | #127 | 1 | 2 | yes | — |
| 258 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | #135 | 1 | 4 | yes | — |
| 261 | C2×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 262 | C4⋊C4 | — | 2 | 2 | no | — |
| 263 | C4⋊C4 | — | 2 | 2 | no | — |
| 264 | C4⋊C4 | — | 4 | 2 | no | — |
| 265 | C4⋊C4 | — | 4 | 2 | no | — |
| 266 | C4⋊C4 | — | 4 | 2 | no | — |
| 267 | C4⋊C4 | — | 4 | 2 | no | — |
| 268 | C4⋊C4 | — | 4 | 2 | no | — |
| 269 | C4⋊C4 | — | 4 | 2 | no | — |
| 270 | C4⋊C4 | — | 4 | 2 | no | — |
| 271 | C4⋊C4 | — | 4 | 2 | no | — |
| 272 | C22⋊C4 | — | 4 | 2 | no | — |
| 273 | C22⋊C4 | — | 4 | 2 | no | — |
| 274 | C42 | — | 2 | 2 | no | — |
| 275 | C42 | — | 2 | 2 | no | — |
| 276 | C42 | — | 4 | 4 | no | — |
| 277 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 278 | C42 | C22×D4 | 1 | 2 | yes | — |
| 279 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 280 | C23 | — | 4 | 1 | no | — |
| 281 | C23 | — | 4 | 1 | no | — |
| 282 | C23 | #535 | 1 | 1 | yes | characteristic |
| 283 | Q8 | — | 4 | 2 | no | — |
| 284 | Q8 | — | 4 | 2 | no | — |
| 285 | Q8 | — | 8 | 2 | no | — |
| 286 | Q8 | — | 8 | 2 | no | — |
| 287 | Q8 | — | 8 | 4 | no | — |
| 288 | Q8 | — | 8 | 4 | no | — |
| 289 | Q8 | — | 16 | 2 | no | — |
| 290 | D4 | — | 4 | 2 | no | — |
| 291 | D4 | — | 4 | 2 | no | — |
| 292 | D4 | — | 4 | 4 | no | — |
| 293 | D4 | — | 4 | 4 | no | — |
| 294 | D4 | — | 8 | 2 | no | — |
| 295 | D4 | — | 8 | 2 | no | — |
| 296 | C2×C4 | — | 2 | 4 | no | — |
| 297 | C2×C4 | — | 4 | 4 | no | — |
| 298 | C2×C4 | — | 4 | 2 | no | — |
| 299 | C2×C4 | — | 4 | 1 | no | — |
| 300 | C2×C4 | — | 4 | 1 | no | — |
| 301 | C2×C4 | — | 8 | 1 | no | — |
| 302 | C2×C4 | — | 8 | 1 | no | — |
| 303 | C2×C4 | — | 8 | 1 | no | — |
| 304 | C2×C4 | — | 8 | 1 | no | — |
| 305 | C2×C4 | — | 8 | 4 | no | — |
| 306 | C2×C4 | — | 8 | 4 | no | — |
| 307 | C2×C4 | — | 8 | 2 | no | — |
| 308 | C2×C4 | — | 8 | 2 | no | — |
| 309 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 310 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | D42 | 1 | 2 | yes | — |
| 312 | C2×C4 | #550 | 1 | 2 | yes | — |
| 313 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 314 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 315 | C8 | — | 2 | 2 | no | — |
| 316 | C8 | — | 2 | 1 | no | — |
| 317 | C8 | — | 2 | 1 | no | — |
| 318 | C8 | — | 4 | 4 | no | — |
| 319 | C8 | — | 4 | 4 | no | — |
| 320 | C8 | — | 8 | 1 | no | — |
| 321 | C8 | — | 8 | 1 | no | — |
| 322 | C8 | #463 | 1 | 4 | yes | — |
| 323 | C8 | #463 | 1 | 4 | yes | — |
| 324 | C22 | — | 2 | 2 | no | — |
| 325 | C22 | — | 8 | 1 | no | — |
| 326 | C22 | — | 8 | 1 | no | — |
| 327 | C22 | — | 8 | 2 | no | — |
| 328 | C22 | — | 8 | 2 | no | — |
| 329 | C22 | #2541 | 1 | 1 | yes | characteristic |
| 330 | C22 | #3207 | 1 | 2 | yes | — |
| 331 | C4 | — | 2 | 2 | no | — |
| 332 | C4 | — | 8 | 2 | no | — |
| 333 | C4 | — | 8 | 1 | no | — |
| 334 | C4 | — | 8 | 1 | no | — |
| 335 | C4 | — | 16 | 2 | no | — |
| 336 | C4 | — | 16 | 2 | no | — |
| 337 | C4 | C2×#463 | 1 | 2 | yes | — |
| 338 | C4 | #3147 | 1 | 1 | yes | characteristic |
| 339 | C4 | #3151 | 1 | 1 | yes | characteristic |
| 340 | C4 | #3289 | 1 | 4 | yes | — |
| 341 | C2 | — | 2 | 2 | no | — |
| 342 | C2 | — | 16 | 1 | no | — |
| 343 | C2 | — | 16 | 1 | no | — |
| 344 | C2 | #28443 | 1 | 1 | yes | characteristic |
| 345 | C2 | #28506 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2541 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33255 | C2 | semidirect |
| 5. #19199 | C2 | semidirect |
| 6. #19197 | C2 | semidirect |
| 7. #10288 | C2 | semidirect |
| 8. #10286 | C2 | semidirect |
| 9. #9772 | C2 | semidirect |
| 10. #9767 | C2 | semidirect |
| 11. #8268 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33255 | 2 |
| #21706 | 2 |
| #19662 | 2 |
| #19654 | 2 |
| #19199 | 2 |
| #19197 | 2 |
| #10288 | 2 |
| #10286 | 2 |
| #9772 | 2 |
| #9767 | 2 |
| #8268 | 2 |