Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
623 conjugacy classes of subgroups.
The subgroup lattice has 337 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #21329 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11387 | C2 | 1 | 2 | yes | maximal |
| 3 | #11310 | C2 | 1 | 2 | yes | maximal |
| 4 | #11309 | C2 | 1 | 2 | yes | maximal |
| 5 | #11278 | C2 | 1 | 2 | yes | maximal |
| 6 | #11277 | C2 | 1 | 2 | yes | maximal |
| 7 | #11244 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #11243 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #10824 | C2 | 1 | 2 | yes | maximal |
| 10 | C2×#533 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#430 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2911 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#423 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×#117 | C22 | 1 | 1 | yes | characteristic |
| 15 | #1995 | C4 | 1 | 4 | yes | complemented |
| 16 | #1994 | C4 | 1 | 4 | yes | complemented |
| 17 | C2×#344 | C22 | 1 | 2 | yes | — |
| 18 | C2×#344 | C22 | 1 | 2 | yes | — |
| 19 | #1742 | C22 | 1 | 2 | yes | — |
| 20 | #1742 | C22 | 1 | 2 | yes | — |
| 21 | #1740 | C22 | 1 | 2 | yes | — |
| 22 | #1740 | C22 | 1 | 2 | yes | — |
| 23 | #1733 | C22 | 1 | 2 | yes | — |
| 24 | #1733 | C22 | 1 | 2 | yes | — |
| 25 | #1733 | C22 | 1 | 2 | yes | — |
| 26 | C2×#339 | C22 | 1 | 2 | yes | — |
| 27 | C2×#339 | C22 | 1 | 2 | yes | — |
| 28 | C2×#338 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×H(3,ℤ/4) | C22 | 1 | 2 | yes | — |
| 30 | C2×H(3,ℤ/4) | C22 | 1 | 2 | yes | — |
| 31 | C2×H(3,ℤ/4) | C22 | 1 | 1 | yes | characteristic |
| 32 | C2×H(3,ℤ/4) | C22 | 1 | 2 | yes | — |
| 33 | C22×M16 | C23 | 1 | 2 | yes | — |
| 34 | C22×M16 | C23 | 1 | 2 | yes | — |
| 35 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 36 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 37 | #533 | — | 2 | 1 | no | — |
| 38 | #533 | — | 2 | 1 | no | — |
| 39 | #533 | — | 4 | 1 | no | — |
| 40 | #533 | — | 4 | 1 | no | — |
| 41 | #533 | C2×C4 | 1 | 2 | yes | — |
| 42 | #533 | C2×C4 | 1 | 2 | yes | — |
| 43 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 44 | C2×#125 | — | 2 | 1 | no | — |
| 45 | C2×C22⋊Q8 | — | 4 | 1 | no | — |
| 46 | C2×C4×Q8 | — | 4 | 1 | no | — |
| 47 | C2×#117 | C23 | 1 | 2 | yes | — |
| 48 | C2×#117 | C23 | 1 | 2 | yes | — |
| 49 | C2×#117 | C23 | 1 | 2 | yes | — |
| 50 | C2×#117 | C23 | 1 | 2 | yes | — |
| 51 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 52 | #430 | — | 4 | 2 | no | — |
| 53 | C2×#108 | — | 4 | 1 | no | — |
| 54 | #428 | — | 4 | 2 | no | — |
| 55 | #427 | — | 4 | 1 | no | — |
| 56 | #423 | — | 4 | 1 | no | — |
| 57 | #423 | C2×C4 | 1 | 2 | yes | — |
| 58 | #423 | C2×C4 | 1 | 2 | yes | — |
| 59 | C2×C4⋊C8 | — | 4 | 1 | no | — |
| 60 | C4×C4⋊C4 | — | 4 | 1 | no | — |
| 61 | C4×C22⋊C4 | — | 4 | 1 | no | — |
| 62 | #376 | C2×C4 | 1 | 2 | yes | — |
| 63 | C43 | C2×C4 | 1 | 2 | yes | — |
| 64 | #344 | — | 4 | 2 | no | — |
| 65 | #344 | — | 4 | 2 | no | — |
| 66 | #343 | — | 4 | 2 | no | — |
| 67 | #343 | — | 4 | 2 | no | — |
| 68 | #342 | — | 4 | 2 | no | — |
| 69 | #342 | — | 4 | 2 | no | — |
| 70 | #339 | — | 2 | 2 | no | — |
| 71 | #339 | — | 2 | 2 | no | — |
| 72 | #339 | — | 2 | 2 | no | — |
| 73 | #339 | — | 2 | 2 | no | — |
| 74 | #339 | — | 4 | 2 | no | — |
| 75 | #339 | C2×C4 | 1 | 8 | yes | — |
| 76 | #339 | C2×C4 | 1 | 8 | yes | — |
| 77 | #338 | — | 4 | 1 | no | — |
| 78 | H(3,ℤ/4) | — | 4 | 2 | no | — |
| 79 | H(3,ℤ/4) | — | 4 | 2 | no | — |
| 80 | H(3,ℤ/4) | — | 4 | 2 | no | — |
| 81 | H(3,ℤ/4) | — | 4 | 1 | no | — |
| 82 | C22×Q8 | — | 4 | 1 | no | — |
| 83 | C23×C4 | — | 2 | 1 | no | — |
| 84 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 85 | C2×M16 | — | 2 | 1 | no | — |
| 86 | C2×M16 | — | 2 | 1 | no | — |
| 87 | C2×M16 | — | 2 | 2 | no | — |
| 88 | C2×M16 | — | 2 | 2 | no | — |
| 89 | C2×M16 | — | 2 | 2 | no | — |
| 90 | C2×M16 | — | 2 | 2 | no | — |
| 91 | C2×M16 | — | 2 | 2 | no | — |
| 92 | C2×M16 | — | 2 | 2 | no | — |
| 93 | C2×M16 | — | 2 | 2 | no | — |
| 94 | C2×M16 | — | 2 | 2 | no | — |
| 95 | C2×M16 | — | 2 | 2 | no | — |
| 96 | C2×M16 | — | 2 | 2 | no | — |
| 97 | C2×M16 | — | 2 | 2 | no | — |
| 98 | C2×M16 | — | 2 | 2 | no | — |
| 99 | C2×M16 | — | 2 | 2 | no | — |
| 100 | C2×M16 | — | 2 | 2 | no | — |
| 101 | C2×M16 | — | 2 | 2 | no | — |
| 102 | C2×M16 | — | 2 | 2 | no | — |
| 103 | C2×M16 | C22×C4 | 1 | 2 | yes | — |
| 104 | C2×M16 | C22×C4 | 1 | 2 | yes | — |
| 105 | C22×C8 | — | 2 | 1 | no | — |
| 106 | C22×C8 | — | 2 | 1 | no | — |
| 107 | C22×C8 | — | 2 | 2 | no | — |
| 108 | C22×C8 | — | 2 | 2 | no | — |
| 109 | C4⋊Q8 | — | 2 | 2 | no | — |
| 110 | C4⋊Q8 | — | 2 | 2 | no | — |
| 111 | C4⋊Q8 | — | 4 | 1 | no | — |
| 112 | C4⋊Q8 | — | 4 | 1 | no | — |
| 113 | #125 | — | 2 | 2 | no | — |
| 114 | #125 | — | 2 | 2 | no | — |
| 115 | #125 | — | 4 | 2 | no | — |
| 116 | C22⋊Q8 | — | 4 | 2 | no | — |
| 117 | C22⋊Q8 | — | 4 | 2 | no | — |
| 118 | C22⋊Q8 | — | 4 | 2 | no | — |
| 119 | C22⋊Q8 | — | 4 | 2 | no | — |
| 120 | C4×Q8 | — | 4 | 2 | no | — |
| 121 | C4×Q8 | — | 4 | 2 | no | — |
| 122 | C4×Q8 | — | 4 | 2 | no | — |
| 123 | C4×Q8 | — | 4 | 2 | no | — |
| 124 | #117 | — | 2 | 2 | no | — |
| 125 | #117 | — | 2 | 2 | no | — |
| 126 | #117 | — | 2 | 2 | no | — |
| 127 | #117 | — | 2 | 2 | no | — |
| 128 | #117 | — | 2 | 2 | no | — |
| 129 | #117 | — | 2 | 2 | no | — |
| 130 | #117 | — | 2 | 2 | no | — |
| 131 | #117 | — | 2 | 2 | no | — |
| 132 | #117 | — | 2 | 2 | no | — |
| 133 | #117 | — | 2 | 2 | no | — |
| 134 | #117 | — | 2 | 2 | no | — |
| 135 | #117 | — | 2 | 2 | no | — |
| 136 | #117 | — | 2 | 2 | no | — |
| 137 | #117 | — | 2 | 2 | no | — |
| 138 | #117 | — | 2 | 2 | no | — |
| 139 | #117 | C22×C4 | 1 | 4 | yes | — |
| 140 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 141 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 142 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 143 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 144 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 145 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 146 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 147 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 148 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 149 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 150 | C2×C42 | — | 2 | 1 | no | — |
| 151 | C2×C42 | — | 2 | 1 | no | — |
| 152 | C2×C42 | — | 2 | 1 | no | — |
| 153 | C2×C42 | — | 2 | 2 | no | — |
| 154 | C2×C42 | — | 2 | 2 | no | — |
| 155 | C2×C42 | — | 2 | 2 | no | — |
| 156 | C2×C42 | — | 4 | 2 | no | — |
| 157 | C2×C42 | C22×C4 | 1 | 4 | yes | — |
| 158 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 159 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 160 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 161 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 162 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 163 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 164 | #108 | — | 4 | 4 | no | — |
| 165 | #107 | — | 4 | 2 | no | — |
| 166 | #106 | — | 4 | 2 | no | — |
| 167 | C4⋊C8 | — | 4 | 2 | no | — |
| 168 | C4⋊C8 | — | 4 | 2 | no | — |
| 169 | #95 | — | 4 | 2 | no | — |
| 170 | C24 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 171 | C2×Q8 | — | 4 | 2 | no | — |
| 172 | C2×Q8 | — | 4 | 2 | no | — |
| 173 | C2×Q8 | — | 8 | 4 | no | — |
| 174 | C22×C4 | — | 2 | 1 | no | — |
| 175 | C22×C4 | — | 2 | 1 | no | — |
| 176 | C22×C4 | — | 2 | 1 | no | — |
| 177 | C22×C4 | — | 2 | 1 | no | — |
| 178 | C22×C4 | — | 2 | 1 | no | — |
| 179 | C22×C4 | — | 2 | 1 | no | — |
| 180 | C22×C4 | — | 2 | 4 | no | — |
| 181 | C22×C4 | — | 4 | 2 | no | — |
| 182 | C22×C4 | — | 4 | 2 | no | — |
| 183 | C22×C4 | — | 4 | 2 | no | — |
| 184 | C22×C4 | — | 8 | 2 | no | — |
| 185 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 186 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 187 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 188 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 189 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 190 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 191 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 192 | C22×C4 | 2−1+4 | 1 | 2 | yes | — |
| 193 | C22×C4 | 2−1+4 | 1 | 2 | yes | — |
| 194 | C22×C4 | 2−1+4 | 1 | 2 | yes | — |
| 195 | M16 | — | 2 | 4 | no | — |
| 196 | M16 | — | 2 | 4 | no | — |
| 197 | M16 | — | 4 | 2 | no | — |
| 198 | M16 | — | 4 | 2 | no | — |
| 199 | M16 | — | 4 | 2 | no | — |
| 200 | M16 | — | 4 | 2 | no | — |
| 201 | M16 | — | 4 | 2 | no | — |
| 202 | M16 | — | 4 | 2 | no | — |
| 203 | M16 | — | 4 | 2 | no | — |
| 204 | M16 | — | 4 | 2 | no | — |
| 205 | M16 | — | 4 | 2 | no | — |
| 206 | M16 | — | 4 | 2 | no | — |
| 207 | C2×C8 | — | 2 | 2 | no | — |
| 208 | C2×C8 | — | 2 | 2 | no | — |
| 209 | C2×C8 | — | 4 | 1 | no | — |
| 210 | C2×C8 | — | 4 | 1 | no | — |
| 211 | C2×C8 | — | 4 | 1 | no | — |
| 212 | C2×C8 | — | 4 | 1 | no | — |
| 213 | C2×C8 | — | 4 | 2 | no | — |
| 214 | C2×C8 | — | 4 | 2 | no | — |
| 215 | C2×C8 | — | 4 | 2 | no | — |
| 216 | C2×C8 | — | 4 | 2 | no | — |
| 217 | C2×C8 | — | 4 | 2 | no | — |
| 218 | C2×C8 | — | 4 | 2 | no | — |
| 219 | C4⋊C4 | — | 2 | 4 | no | — |
| 220 | C4⋊C4 | — | 4 | 4 | no | — |
| 221 | C4⋊C4 | — | 4 | 4 | no | — |
| 222 | C4⋊C4 | — | 4 | 2 | no | — |
| 223 | C4⋊C4 | — | 4 | 2 | no | — |
| 224 | C4⋊C4 | — | 4 | 2 | no | — |
| 225 | C4⋊C4 | — | 4 | 2 | no | — |
| 226 | C4⋊C4 | — | 4 | 2 | no | — |
| 227 | C4⋊C4 | — | 4 | 2 | no | — |
| 228 | C4⋊C4 | — | 4 | 2 | no | — |
| 229 | C4⋊C4 | — | 4 | 2 | no | — |
| 230 | C4⋊C4 | — | 4 | 2 | no | — |
| 231 | C22⋊C4 | — | 2 | 4 | no | — |
| 232 | C22⋊C4 | — | 4 | 2 | no | — |
| 233 | C22⋊C4 | — | 4 | 2 | no | — |
| 234 | C22⋊C4 | — | 4 | 2 | no | — |
| 235 | C22⋊C4 | — | 4 | 2 | no | — |
| 236 | C22⋊C4 | — | 4 | 2 | no | — |
| 237 | C22⋊C4 | — | 4 | 2 | no | — |
| 238 | C22⋊C4 | — | 4 | 2 | no | — |
| 239 | C42 | — | 2 | 1 | no | — |
| 240 | C42 | — | 2 | 1 | no | — |
| 241 | C42 | — | 2 | 1 | no | — |
| 242 | C42 | — | 2 | 1 | no | — |
| 243 | C42 | — | 2 | 1 | no | — |
| 244 | C42 | — | 2 | 1 | no | — |
| 245 | C42 | — | 2 | 4 | no | — |
| 246 | C42 | — | 2 | 4 | no | — |
| 247 | C42 | — | 2 | 4 | no | — |
| 248 | C42 | — | 4 | 2 | no | — |
| 249 | C42 | — | 4 | 2 | no | — |
| 250 | C42 | — | 4 | 2 | no | — |
| 251 | C42 | — | 4 | 2 | no | — |
| 252 | C42 | — | 4 | 2 | no | — |
| 253 | C42 | — | 8 | 4 | no | — |
| 254 | C42 | #117 | 1 | 1 | yes | characteristic |
| 255 | C42 | #117 | 1 | 1 | yes | characteristic |
| 256 | C42 | #117 | 1 | 1 | yes | characteristic |
| 257 | C42 | #117 | 1 | 1 | yes | characteristic |
| 258 | C23 | — | 8 | 1 | no | — |
| 259 | C23 | #518 | 1 | 1 | yes | characteristic |
| 260 | C23 | #538 | 1 | 1 | yes | characteristic |
| 261 | C23 | #538 | 1 | 1 | yes | characteristic |
| 262 | C23 | #562 | 1 | 2 | yes | — |
| 263 | C23 | #563 | 1 | 2 | yes | — |
| 264 | Q8 | — | 8 | 4 | no | — |
| 265 | Q8 | — | 8 | 4 | no | — |
| 266 | C2×C4 | — | 2 | 1 | no | — |
| 267 | C2×C4 | — | 2 | 1 | no | — |
| 268 | C2×C4 | — | 2 | 1 | no | — |
| 269 | C2×C4 | — | 2 | 1 | no | — |
| 270 | C2×C4 | — | 4 | 2 | no | — |
| 271 | C2×C4 | — | 4 | 2 | no | — |
| 272 | C2×C4 | — | 4 | 2 | no | — |
| 273 | C2×C4 | — | 4 | 4 | no | — |
| 274 | C2×C4 | — | 4 | 4 | no | — |
| 275 | C2×C4 | — | 8 | 1 | no | — |
| 276 | C2×C4 | — | 8 | 1 | no | — |
| 277 | C2×C4 | — | 8 | 1 | no | — |
| 278 | C2×C4 | — | 8 | 4 | no | — |
| 279 | C2×C4 | — | 8 | 4 | no | — |
| 280 | C2×C4 | — | 8 | 2 | no | — |
| 281 | C2×C4 | — | 8 | 2 | no | — |
| 282 | C2×C4 | — | 16 | 2 | no | — |
| 283 | C2×C4 | — | 16 | 2 | no | — |
| 284 | C2×C4 | C2×#117 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 286 | C2×C4 | #519 | 1 | 2 | yes | — |
| 287 | C2×C4 | #520 | 1 | 2 | yes | — |
| 288 | C2×C4 | #538 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | #538 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #541 | 1 | 2 | yes | — |
| 291 | C2×C4 | #541 | 1 | 2 | yes | — |
| 292 | C2×C4 | #542 | 1 | 2 | yes | — |
| 293 | C2×C4 | #542 | 1 | 2 | yes | — |
| 294 | C2×C4 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #562 | 1 | 2 | yes | — |
| 296 | C2×C4 | #562 | 1 | 2 | yes | — |
| 297 | C2×C4 | #563 | 1 | 2 | yes | — |
| 298 | C2×C4 | #563 | 1 | 2 | yes | — |
| 299 | C2×C4 | #563 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | #563 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | #563 | 1 | 1 | yes | characteristic |
| 302 | C8 | — | 4 | 2 | no | — |
| 303 | C8 | — | 4 | 2 | no | — |
| 304 | C8 | — | 8 | 2 | no | — |
| 305 | C8 | — | 8 | 2 | no | — |
| 306 | C22 | — | 4 | 1 | no | — |
| 307 | C22 | — | 4 | 1 | no | — |
| 308 | C22 | — | 4 | 1 | no | — |
| 309 | C22 | — | 4 | 2 | no | — |
| 310 | C22 | — | 4 | 2 | no | — |
| 311 | C22 | #2312 | 1 | 1 | yes | characteristic |
| 312 | C22 | #2378 | 1 | 2 | yes | — |
| 313 | C22 | #2831 | 1 | 2 | yes | — |
| 314 | C22 | #2832 | 1 | 1 | yes | characteristic |
| 315 | C22 | #2833 | 1 | 1 | yes | characteristic |
| 316 | C4 | — | 2 | 2 | no | — |
| 317 | C4 | — | 4 | 1 | no | — |
| 318 | C4 | — | 4 | 1 | no | — |
| 319 | C4 | — | 4 | 1 | no | — |
| 320 | C4 | — | 8 | 4 | no | — |
| 321 | C4 | — | 8 | 4 | no | — |
| 322 | C4 | — | 16 | 2 | no | — |
| 323 | C4 | — | 16 | 2 | no | — |
| 324 | C4 | #2320 | 1 | 2 | yes | — |
| 325 | C4 | #2379 | 1 | 1 | yes | characteristic |
| 326 | C4 | #2381 | 1 | 1 | yes | characteristic |
| 327 | C4 | #2831 | 1 | 2 | yes | — |
| 328 | C4 | #2832 | 1 | 1 | yes | characteristic |
| 329 | C4 | #2833 | 1 | 1 | yes | characteristic |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | — | 2 | 2 | no | — |
| 332 | C2 | — | 2 | 1 | no | — |
| 333 | C2 | — | 2 | 2 | no | — |
| 334 | C2 | — | 2 | 1 | no | — |
| 335 | C2 | #14668 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14668 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 15. #1995 | C4 | semidirect |
| 16. #1994 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21329 | 2 |
| #11387 | 2 |
| #11310 | 2 |
| #11309 | 2 |
| #11278 | 2 |
| #11277 | 2 |
| #11244 | 2 |
| #11243 | 2 |
| #10824 | 2 |