Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1214 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #62360 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#3380 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #33952 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #33939 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #33802 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #33796 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #33670 | C2 | 1 | 4 | yes | maximal |
| 8 | #33661 | C2 | 1 | 2 | yes | maximal |
| 9 | #33497 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3502 | — | 2 | 1 | no | — |
| 11 | C2×#554 | — | 2 | 1 | no | — |
| 12 | #3395 | C22 | 1 | 4 | yes | complemented |
| 13 | Q8×Q16 | C22 | 1 | 4 | yes | complemented |
| 14 | #3380 | — | 2 | 4 | no | — |
| 15 | #3380 | C22 | 1 | 8 | yes | — |
| 16 | #3363 | — | 2 | 4 | no | — |
| 17 | #3249 | — | 2 | 2 | no | — |
| 18 | #3249 | — | 2 | 1 | no | — |
| 19 | #3170 | C22 | 1 | 2 | yes | — |
| 20 | #3170 | C22 | 1 | 4 | yes | — |
| 21 | C2×#500 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#499 | C22 | 1 | 2 | yes | — |
| 23 | C2×#478 | — | 2 | 2 | no | — |
| 24 | C2×#476 | — | 2 | 1 | no | — |
| 25 | C2×C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×C4×D8 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×D8 | C22 | 1 | 2 | yes | — |
| 28 | #2255 | C22 | 1 | 8 | yes | complemented |
| 29 | #2252 | C22 | 1 | 4 | yes | complemented |
| 30 | #2245 | C22 | 1 | 8 | yes | complemented |
| 31 | #2241 | C22 | 1 | 4 | yes | complemented |
| 32 | #2236 | C22 | 1 | 8 | yes | — |
| 33 | #2235 | C22 | 1 | 4 | yes | — |
| 34 | #2229 | C22 | 1 | 16 | yes | complemented |
| 35 | #2227 | C22 | 1 | 8 | yes | complemented |
| 36 | #2226 | C22 | 1 | 4 | yes | — |
| 37 | #2224 | C22 | 1 | 8 | yes | — |
| 38 | #2208 | C22 | 1 | 4 | yes | — |
| 39 | #2208 | C22 | 1 | 8 | yes | — |
| 40 | #2187 | C22 | 1 | 4 | yes | — |
| 41 | #2183 | C22 | 1 | 8 | yes | complemented |
| 42 | #2175 | C22 | 1 | 16 | yes | — |
| 43 | C4×SD32 | C22 | 1 | 8 | yes | — |
| 44 | #2163 | C22 | 1 | 4 | yes | — |
| 45 | #2162 | C22 | 1 | 2 | yes | — |
| 46 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 47 | Q82 | — | 2 | 4 | no | — |
| 48 | #557 | — | 2 | 4 | no | — |
| 49 | #557 | — | 4 | 4 | no | — |
| 50 | #554 | — | 2 | 8 | no | — |
| 51 | #554 | — | 4 | 4 | no | — |
| 52 | #544 | — | 4 | 1 | no | — |
| 53 | #544 | — | 4 | 2 | no | — |
| 54 | #533 | — | 2 | 4 | no | — |
| 55 | #533 | — | 2 | 2 | no | — |
| 56 | C2×C4⋊Q8 | — | 2 | 1 | no | — |
| 57 | C2×#125 | — | 2 | 2 | no | — |
| 58 | C2×C22⋊Q8 | — | 4 | 1 | no | — |
| 59 | C2×C22⋊Q8 | — | 4 | 2 | no | — |
| 60 | #519 | — | 4 | 1 | no | — |
| 61 | C2×C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 62 | C2×C4×D4 | — | 2 | 2 | no | — |
| 63 | C2×C4×D4 | — | 2 | 1 | no | — |
| 64 | C22×C4⋊C4 | — | 4 | 1 | no | — |
| 65 | C2×SD32 | C23 | 1 | 8 | yes | — |
| 66 | #500 | — | 2 | 2 | no | — |
| 67 | #500 | C23 | 1 | 4 | yes | — |
| 68 | #500 | C23 | 1 | 4 | yes | — |
| 69 | #500 | C23 | 1 | 2 | yes | — |
| 70 | #499 | — | 2 | 4 | no | — |
| 71 | #499 | C23 | 1 | 8 | yes | — |
| 72 | #494 | C23 | 1 | 2 | yes | — |
| 73 | #494 | C23 | 1 | 4 | yes | — |
| 74 | #487 | — | 2 | 4 | no | — |
| 75 | #487 | — | 2 | 2 | no | — |
| 76 | #478 | — | 2 | 16 | no | — |
| 77 | #477 | — | 2 | 4 | no | — |
| 78 | #477 | — | 2 | 8 | no | — |
| 79 | #476 | — | 2 | 8 | no | — |
| 80 | #467 | — | 2 | 4 | no | — |
| 81 | #467 | — | 2 | 2 | no | — |
| 82 | #467 | C23 | 1 | 4 | yes | — |
| 83 | #462 | — | 2 | 4 | no | — |
| 84 | #462 | — | 2 | 8 | no | — |
| 85 | C8×Q8 | D4 | 1 | 4 | yes | — |
| 86 | C8×Q8 | C23 | 1 | 4 | yes | — |
| 87 | C4×Q16 | — | 2 | 4 | no | — |
| 88 | C4×Q16 | — | 2 | 2 | no | — |
| 89 | C4×Q16 | C23 | 1 | 4 | yes | — |
| 90 | C4×Q16 | C23 | 1 | 8 | yes | — |
| 91 | C4×D8 | — | 2 | 4 | no | — |
| 92 | C4×D8 | — | 2 | 2 | no | — |
| 93 | C4×D8 | C23 | 1 | 4 | yes | — |
| 94 | C4×D8 | C23 | 1 | 8 | yes | — |
| 95 | C2×#107 | — | 2 | 1 | no | — |
| 96 | C2×#107 | C23 | 1 | 1 | yes | characteristic |
| 97 | C2×#107 | C23 | 1 | 2 | yes | — |
| 98 | C2×#106 | — | 2 | 2 | no | — |
| 99 | C2×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 100 | C2×C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 101 | C2×#102 | — | 2 | 1 | no | — |
| 102 | C2×#102 | — | 2 | 2 | no | — |
| 103 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 104 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 105 | #367 | C23 | 1 | 8 | yes | — |
| 106 | #366 | C23 | 1 | 16 | yes | — |
| 107 | C4⋊C16 | C23 | 1 | 4 | yes | — |
| 108 | C4⋊C16 | C23 | 1 | 8 | yes | — |
| 109 | #358 | C23 | 1 | 8 | yes | — |
| 110 | #358 | C23 | 1 | 16 | yes | — |
| 111 | #357 | C23 | 1 | 8 | yes | — |
| 112 | C22⋊C16 | C23 | 1 | 4 | yes | — |
| 113 | #346 | C23 | 1 | 8 | yes | — |
| 114 | C4×C16 | C23 | 1 | 4 | yes | — |
| 115 | C22×Q8 | — | 4 | 1 | no | — |
| 116 | C22×Q8 | D8 | 1 | 1 | yes | characteristic |
| 117 | C22×D4 | — | 2 | 1 | no | — |
| 118 | C23×C4 | — | 4 | 1 | no | — |
| 119 | C23×C4 | — | 4 | 2 | no | — |
| 120 | C2×Q16 | — | 2 | 4 | no | — |
| 121 | C2×Q16 | — | 2 | 2 | no | — |
| 122 | C2×Q16 | C24 | 1 | 4 | yes | — |
| 123 | C2×D8 | — | 2 | 4 | no | — |
| 124 | C2×D8 | — | 2 | 1 | no | — |
| 125 | C2×D8 | C24 | 1 | 2 | yes | — |
| 126 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 127 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 128 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 129 | C4⋊Q8 | — | 2 | 4 | no | — |
| 130 | C4⋊Q8 | — | 2 | 4 | no | — |
| 131 | C4⋊Q8 | — | 2 | 2 | no | — |
| 132 | C4⋊Q8 | — | 2 | 4 | no | — |
| 133 | C4⋊Q8 | — | 2 | 2 | no | — |
| 134 | C4⋊Q8 | — | 4 | 4 | no | — |
| 135 | C4⋊Q8 | — | 4 | 8 | no | — |
| 136 | C4⋊Q8 | — | 4 | 2 | no | — |
| 137 | #125 | — | 2 | 8 | no | — |
| 138 | #125 | — | 4 | 4 | no | — |
| 139 | #125 | — | 4 | 8 | no | — |
| 140 | #125 | — | 4 | 4 | no | — |
| 141 | #125 | — | 4 | 4 | no | — |
| 142 | #125 | — | 4 | 2 | no | — |
| 143 | C22⋊Q8 | — | 2 | 4 | no | — |
| 144 | C22⋊Q8 | — | 4 | 8 | no | — |
| 145 | C22⋊Q8 | — | 4 | 2 | no | — |
| 146 | C22⋊Q8 | — | 4 | 4 | no | — |
| 147 | C22⋊Q8 | — | 4 | 16 | no | — |
| 148 | C4×Q8 | — | 2 | 8 | no | — |
| 149 | C4×Q8 | — | 2 | 4 | no | — |
| 150 | C4×Q8 | — | 4 | 4 | no | — |
| 151 | C4×Q8 | — | 4 | 4 | no | — |
| 152 | C4×Q8 | — | 4 | 4 | no | — |
| 153 | C4×Q8 | — | 4 | 2 | no | — |
| 154 | C4×Q8 | D8 | 1 | 4 | yes | — |
| 155 | C4×Q8 | C2×D4 | 1 | 4 | yes | — |
| 156 | C4×D4 | — | 2 | 4 | no | — |
| 157 | C4×D4 | — | 2 | 8 | no | — |
| 158 | C4×D4 | — | 4 | 4 | no | — |
| 159 | C4×D4 | — | 4 | 2 | no | — |
| 160 | #117 | — | 4 | 4 | no | — |
| 161 | #117 | — | 4 | 2 | no | — |
| 162 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 163 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 164 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 165 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 166 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 167 | C2×C4⋊C4 | — | 4 | 8 | no | — |
| 168 | C2×C4⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 169 | C2×C4⋊C4 | D8 | 1 | 2 | yes | — |
| 170 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 171 | C2×C22⋊C4 | — | 4 | 2 | no | — |
| 172 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 173 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 174 | SD32 | — | 2 | 16 | no | — |
| 175 | C2×C16 | C24 | 1 | 8 | yes | — |
| 176 | #107 | — | 2 | 4 | no | — |
| 177 | #107 | — | 2 | 1 | no | — |
| 178 | #107 | — | 2 | 2 | no | — |
| 179 | #107 | — | 2 | 4 | no | — |
| 180 | #107 | — | 2 | 2 | no | — |
| 181 | #107 | C24 | 1 | 4 | yes | — |
| 182 | #107 | C24 | 1 | 4 | yes | — |
| 183 | #107 | C24 | 1 | 2 | yes | — |
| 184 | #106 | — | 2 | 8 | no | — |
| 185 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 186 | C4⋊C8 | C2×D4 | 1 | 8 | yes | — |
| 187 | #103 | — | 2 | 4 | no | — |
| 188 | #103 | — | 2 | 8 | no | — |
| 189 | #103 | — | 2 | 4 | no | — |
| 190 | #102 | — | 2 | 8 | no | — |
| 191 | #102 | — | 2 | 4 | no | — |
| 192 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 193 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 194 | C4×C8 | C24 | 1 | 2 | yes | — |
| 195 | C4×C8 | C24 | 1 | 4 | yes | — |
| 196 | C24 | — | 4 | 1 | no | — |
| 197 | C2×Q8 | — | 2 | 4 | no | — |
| 198 | C2×Q8 | — | 2 | 4 | no | — |
| 199 | C2×Q8 | — | 4 | 2 | no | — |
| 200 | C2×Q8 | — | 4 | 4 | no | — |
| 201 | C2×Q8 | — | 4 | 4 | no | — |
| 202 | C2×Q8 | — | 16 | 2 | no | — |
| 203 | C2×Q8 | C2×D8 | 1 | 4 | yes | — |
| 204 | C2×D4 | — | 2 | 4 | no | — |
| 205 | C2×D4 | — | 2 | 2 | no | — |
| 206 | C2×D4 | — | 4 | 2 | no | — |
| 207 | C2×D4 | — | 4 | 1 | no | — |
| 208 | C22×C4 | — | 4 | 1 | no | — |
| 209 | C22×C4 | — | 4 | 2 | no | — |
| 210 | C22×C4 | — | 4 | 4 | no | — |
| 211 | C22×C4 | — | 4 | 8 | no | — |
| 212 | C22×C4 | — | 8 | 4 | no | — |
| 213 | C22×C4 | — | 16 | 4 | no | — |
| 214 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 215 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 217 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 218 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 219 | Q16 | — | 2 | 8 | no | — |
| 220 | Q16 | — | 8 | 4 | no | — |
| 221 | Q16 | — | 8 | 2 | no | — |
| 222 | D8 | — | 2 | 4 | no | — |
| 223 | D8 | — | 2 | 2 | no | — |
| 224 | D8 | 2−1+4 | 1 | 4 | yes | — |
| 225 | C2×C8 | — | 2 | 2 | no | — |
| 226 | C2×C8 | — | 4 | 2 | no | — |
| 227 | C2×C8 | — | 4 | 1 | no | — |
| 228 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 229 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 230 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 231 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 232 | C4⋊C4 | — | 2 | 2 | no | — |
| 233 | C4⋊C4 | — | 2 | 4 | no | — |
| 234 | C4⋊C4 | — | 2 | 4 | no | — |
| 235 | C4⋊C4 | — | 4 | 1 | no | — |
| 236 | C4⋊C4 | — | 4 | 2 | no | — |
| 237 | C4⋊C4 | — | 4 | 8 | no | — |
| 238 | C4⋊C4 | — | 4 | 4 | no | — |
| 239 | C4⋊C4 | — | 4 | 8 | no | — |
| 240 | C4⋊C4 | — | 4 | 4 | no | — |
| 241 | C4⋊C4 | — | 4 | 2 | no | — |
| 242 | C4⋊C4 | — | 4 | 4 | no | — |
| 243 | C4⋊C4 | — | 4 | 2 | no | — |
| 244 | C4⋊C4 | — | 4 | 4 | no | — |
| 245 | C4⋊C4 | — | 4 | 8 | no | — |
| 246 | C4⋊C4 | — | 4 | 4 | no | — |
| 247 | C4⋊C4 | — | 4 | 2 | no | — |
| 248 | C4⋊C4 | — | 4 | 4 | no | — |
| 249 | C4⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 250 | C4⋊C4 | C2×D8 | 1 | 8 | yes | — |
| 251 | C22⋊C4 | — | 4 | 4 | no | — |
| 252 | C22⋊C4 | — | 4 | 4 | no | — |
| 253 | C22⋊C4 | — | 4 | 8 | no | — |
| 254 | C42 | — | 4 | 8 | no | — |
| 255 | C42 | — | 4 | 4 | no | — |
| 256 | C42 | C2×D8 | 1 | 4 | yes | — |
| 257 | C42 | C2×D8 | 1 | 2 | yes | — |
| 258 | C42 | C22×D4 | 1 | 4 | yes | — |
| 259 | C42 | C22×D4 | 1 | 2 | yes | — |
| 260 | C16 | — | 2 | 8 | no | — |
| 261 | C23 | — | 4 | 4 | no | — |
| 262 | C23 | — | 4 | 2 | no | — |
| 263 | C23 | — | 8 | 2 | no | — |
| 264 | C23 | — | 8 | 2 | no | — |
| 265 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 266 | Q8 | — | 2 | 8 | no | — |
| 267 | Q8 | — | 4 | 4 | no | — |
| 268 | Q8 | — | 4 | 4 | no | — |
| 269 | Q8 | — | 16 | 4 | no | — |
| 270 | Q8 | — | 16 | 2 | no | — |
| 271 | Q8 | — | 16 | 4 | no | — |
| 272 | D4 | — | 2 | 4 | no | — |
| 273 | D4 | — | 4 | 4 | no | — |
| 274 | D4 | — | 4 | 2 | no | — |
| 275 | C2×C4 | — | 2 | 4 | no | — |
| 276 | C2×C4 | — | 2 | 1 | no | — |
| 277 | C2×C4 | — | 2 | 1 | no | — |
| 278 | C2×C4 | — | 4 | 2 | no | — |
| 279 | C2×C4 | — | 4 | 1 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 4 | 4 | no | — |
| 282 | C2×C4 | — | 4 | 2 | no | — |
| 283 | C2×C4 | — | 4 | 4 | no | — |
| 284 | C2×C4 | — | 8 | 4 | no | — |
| 285 | C2×C4 | — | 8 | 2 | no | — |
| 286 | C2×C4 | — | 8 | 8 | no | — |
| 287 | C2×C4 | — | 16 | 8 | no | — |
| 288 | C2×C4 | #510 | 1 | 4 | yes | — |
| 289 | C2×C4 | C22×D8 | 1 | 4 | yes | — |
| 290 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 292 | C2×C4 | C22×D8 | 1 | 4 | yes | — |
| 293 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 294 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 2 | 4 | no | — |
| 296 | C8 | — | 2 | 2 | no | — |
| 297 | C8 | — | 2 | 1 | no | — |
| 298 | C8 | C2×2−1+4 | 1 | 2 | yes | — |
| 299 | C22 | — | 2 | 1 | no | — |
| 300 | C22 | — | 2 | 1 | no | — |
| 301 | C22 | — | 4 | 4 | no | — |
| 302 | C22 | — | 8 | 8 | no | — |
| 303 | C22 | — | 8 | 2 | no | — |
| 304 | C22 | — | 8 | 2 | no | — |
| 305 | C22 | C2×#510 | 1 | 2 | yes | — |
| 306 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 307 | C4 | — | 2 | 1 | no | — |
| 308 | C4 | — | 2 | 4 | no | — |
| 309 | C4 | — | 2 | 4 | no | — |
| 310 | C4 | — | 2 | 2 | no | — |
| 311 | C4 | — | 8 | 4 | no | — |
| 312 | C4 | — | 8 | 2 | no | — |
| 313 | C4 | — | 8 | 4 | no | — |
| 314 | C4 | — | 8 | 4 | no | — |
| 315 | C4 | C2×#510 | 1 | 4 | yes | — |
| 316 | C4 | #3486 | 1 | 1 | yes | characteristic |
| 317 | C4 | #3586 | 1 | 1 | yes | characteristic |
| 318 | C2 | — | 2 | 2 | no | — |
| 319 | C2 | — | 8 | 4 | no | — |
| 320 | C2 | #61817 | 1 | 1 | yes | characteristic |
| 321 | C2 | C22×#510 | 1 | 1 | yes | characteristic |
| 322 | C2 | #62636 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #62360 | C2 | semidirect |
| 3. #33952 | C2 | semidirect |
| 4. #33939 | C2 | semidirect |
| 5. #33802 | C2 | semidirect |
| 6. #33796 | C2 | semidirect |
| 9. #33497 | C2 | semidirect |
| 12. #3395 | C22 | semidirect |
| 13. Q8×Q16 | C22 | semidirect |
| 28. #2255 | C22 | semidirect |
| 29. #2252 | C22 | semidirect |
| 30. #2245 | C22 | semidirect |
| 31. #2241 | C22 | semidirect |
| 34. #2229 | C22 | semidirect |
| 35. #2227 | C22 | semidirect |
| 41. #2183 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #62360 | 2 |
| C2×#3380 | 2 |
| #33952 | 2 |
| #33939 | 2 |
| #33802 | 2 |
| #33796 | 2 |
| #33670 | 2 |
| #33661 | 2 |
| #33497 | 2 |