Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1634 conjugacy classes of subgroups.
The subgroup lattice has 312 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60664 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×#3163 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #60193 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #28703 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | #28702 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #28700 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #23674 | C2 | 1 | 2 | yes | maximal |
| 8 | C2×#2090 | C2 | 1 | 2 | yes | maximal |
| 9 | #21338 | C2 | 1 | 2 | yes | maximal |
| 10 | #21182 | C2 | 1 | 4 | yes | maximal |
| 11 | #20459 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #3458 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#544 | C22 | 1 | 2 | yes | — |
| 14 | #3166 | C22 | 1 | 8 | yes | — |
| 15 | #3165 | C22 | 1 | 8 | yes | — |
| 16 | #3164 | — | 2 | 8 | no | — |
| 17 | #3163 | — | 2 | 8 | no | — |
| 18 | #3163 | — | 2 | 8 | no | — |
| 19 | C2×#501 | C22 | 1 | 4 | yes | — |
| 20 | C2×#499 | C22 | 1 | 2 | yes | — |
| 21 | C2×#498 | C22 | 1 | 2 | yes | — |
| 22 | #2962 | — | 2 | 8 | no | — |
| 23 | #2961 | C22 | 1 | 4 | yes | — |
| 24 | #2960 | C22 | 1 | 4 | yes | — |
| 25 | C2×#430 | C22 | 1 | 2 | yes | — |
| 26 | #2911 | C22 | 1 | 4 | yes | — |
| 27 | C2×#428 | C22 | 1 | 4 | yes | — |
| 28 | C2×#428 | C22 | 1 | 2 | yes | — |
| 29 | C2×#427 | C22 | 1 | 1 | yes | characteristic |
| 30 | C22×#106 | C22 | 1 | 1 | yes | characteristic |
| 31 | #2905 | C22 | 1 | 1 | yes | characteristic |
| 32 | C2×#424 | C22 | 1 | 2 | yes | — |
| 33 | #2873 | C22 | 1 | 2 | yes | — |
| 34 | #2873 | C22 | 1 | 2 | yes | — |
| 35 | C2×#405 | C22 | 1 | 1 | yes | characteristic |
| 36 | #2090 | C22 | 1 | 4 | yes | — |
| 37 | #2090 | C22 | 1 | 8 | yes | — |
| 38 | #2090 | C22 | 1 | 4 | yes | — |
| 39 | #2090 | C22 | 1 | 8 | yes | — |
| 40 | #2089 | C22 | 1 | 8 | yes | — |
| 41 | #1998 | C22 | 1 | 8 | yes | — |
| 42 | #1997 | C22 | 1 | 8 | yes | — |
| 43 | #1952 | C22 | 1 | 16 | yes | — |
| 44 | #1951 | C22 | 1 | 16 | yes | — |
| 45 | #1778 | C22 | 1 | 4 | yes | — |
| 46 | C4×#108 | C22 | 1 | 4 | yes | — |
| 47 | #1742 | C22 | 1 | 4 | yes | — |
| 48 | C2×#343 | C22 | 1 | 4 | yes | — |
| 49 | C2×#341 | C22 | 1 | 2 | yes | — |
| 50 | C22×M16 | C23 | 1 | 2 | yes | — |
| 51 | C22×M16 | C23 | 1 | 4 | yes | — |
| 52 | C22×M16 | C23 | 1 | 1 | yes | characteristic |
| 53 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 54 | #544 | — | 2 | 8 | no | — |
| 55 | #544 | — | 2 | 8 | no | — |
| 56 | #544 | — | 2 | 2 | no | — |
| 57 | #544 | D4 | 1 | 8 | yes | — |
| 58 | #543 | — | 2 | 2 | no | — |
| 59 | #533 | — | 2 | 4 | no | — |
| 60 | #533 | C23 | 1 | 4 | yes | — |
| 61 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 62 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 63 | C2×#125 | — | 2 | 2 | no | — |
| 64 | C2×#125 | C23 | 1 | 4 | yes | — |
| 65 | C2×C22⋊Q8 | — | 2 | 4 | no | — |
| 66 | C2×#117 | C23 | 1 | 2 | yes | — |
| 67 | C2×#117 | C23 | 1 | 2 | yes | — |
| 68 | C2×#117 | C23 | 1 | 1 | yes | characteristic |
| 69 | C22×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 70 | #501 | — | 2 | 8 | no | — |
| 71 | #501 | — | 2 | 8 | no | — |
| 72 | #501 | — | 2 | 16 | no | — |
| 73 | #501 | — | 2 | 16 | no | — |
| 74 | #500 | — | 2 | 8 | no | — |
| 75 | #499 | — | 2 | 16 | no | — |
| 76 | #499 | — | 2 | 8 | no | — |
| 77 | #498 | — | 2 | 8 | no | — |
| 78 | #498 | — | 2 | 8 | no | — |
| 79 | #446 | — | 2 | 16 | no | — |
| 80 | #446 | — | 2 | 8 | no | — |
| 81 | C8×Q8 | — | 2 | 8 | no | — |
| 82 | #430 | Q8 | 1 | 16 | yes | complemented |
| 83 | C2×#108 | C23 | 1 | 8 | yes | — |
| 84 | #428 | — | 2 | 8 | no | — |
| 85 | #428 | — | 2 | 8 | no | — |
| 86 | #428 | — | 2 | 4 | no | — |
| 87 | #428 | — | 2 | 4 | no | — |
| 88 | #428 | C23 | 1 | 8 | yes | — |
| 89 | #428 | C23 | 1 | 8 | yes | — |
| 90 | #428 | C23 | 1 | 16 | yes | — |
| 91 | #427 | — | 2 | 4 | no | — |
| 92 | #427 | — | 2 | 8 | no | — |
| 93 | C2×#107 | — | 2 | 4 | no | — |
| 94 | C2×#107 | C23 | 1 | 2 | yes | — |
| 95 | C2×#107 | C23 | 1 | 4 | yes | — |
| 96 | C2×#106 | — | 2 | 4 | no | — |
| 97 | C2×#106 | — | 2 | 4 | no | — |
| 98 | C2×#106 | C23 | 1 | 1 | yes | characteristic |
| 99 | C2×#106 | C23 | 1 | 2 | yes | — |
| 100 | C2×#106 | C23 | 1 | 1 | yes | characteristic |
| 101 | C2×#106 | C23 | 1 | 2 | yes | — |
| 102 | C2×#106 | C23 | 1 | 4 | yes | — |
| 103 | #424 | — | 2 | 4 | no | — |
| 104 | #424 | — | 2 | 4 | no | — |
| 105 | #424 | C23 | 1 | 4 | yes | — |
| 106 | #424 | C23 | 1 | 4 | yes | — |
| 107 | #423 | C23 | 1 | 4 | yes | — |
| 108 | C2×C4⋊C8 | — | 2 | 4 | no | — |
| 109 | #405 | — | 2 | 4 | no | — |
| 110 | #405 | — | 2 | 8 | no | — |
| 111 | #405 | D4 | 1 | 8 | yes | — |
| 112 | #405 | C23 | 1 | 8 | yes | — |
| 113 | C4×M16 | — | 2 | 4 | no | — |
| 114 | C4×M16 | C23 | 1 | 8 | yes | — |
| 115 | C2×#97 | C23 | 1 | 2 | yes | — |
| 116 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 117 | #343 | C23 | 1 | 8 | yes | — |
| 118 | #343 | C23 | 1 | 8 | yes | — |
| 119 | #343 | C23 | 1 | 16 | yes | — |
| 120 | #341 | C23 | 1 | 4 | yes | — |
| 121 | #341 | C23 | 1 | 4 | yes | — |
| 122 | C22×Q8 | — | 2 | 4 | no | — |
| 123 | C23×C4 | — | 2 | 2 | no | — |
| 124 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 125 | C2×M16 | — | 2 | 4 | no | — |
| 126 | C2×M16 | — | 2 | 8 | no | — |
| 127 | C2×M16 | — | 2 | 2 | no | — |
| 128 | C2×M16 | — | 2 | 8 | no | — |
| 129 | C2×M16 | C2×D4 | 1 | 4 | yes | — |
| 130 | C2×M16 | C2×Q8 | 1 | 16 | yes | — |
| 131 | C2×M16 | C2×Q8 | 1 | 8 | yes | — |
| 132 | C2×M16 | C24 | 1 | 4 | yes | — |
| 133 | C2×M16 | C24 | 1 | 8 | yes | — |
| 134 | C22×C8 | — | 2 | 4 | no | — |
| 135 | C22×C8 | — | 2 | 4 | no | — |
| 136 | C22×C8 | — | 2 | 2 | no | — |
| 137 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 138 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 139 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 140 | C22×C8 | C24 | 1 | 2 | yes | — |
| 141 | C4⋊Q8 | — | 2 | 8 | no | — |
| 142 | C4⋊Q8 | — | 2 | 8 | no | — |
| 143 | C4⋊Q8 | — | 2 | 8 | no | — |
| 144 | C4⋊Q8 | — | 4 | 8 | no | — |
| 145 | C4⋊Q8 | C2×D4 | 1 | 8 | yes | — |
| 146 | #125 | — | 2 | 8 | no | — |
| 147 | #125 | — | 2 | 8 | no | — |
| 148 | #125 | — | 2 | 8 | no | — |
| 149 | #125 | — | 4 | 8 | no | — |
| 150 | #125 | C2×D4 | 1 | 8 | yes | — |
| 151 | C22⋊Q8 | — | 2 | 8 | no | — |
| 152 | C22⋊Q8 | — | 2 | 16 | no | — |
| 153 | C22⋊Q8 | — | 4 | 16 | no | — |
| 154 | C4×Q8 | — | 2 | 8 | no | — |
| 155 | #117 | — | 2 | 2 | no | — |
| 156 | #117 | — | 2 | 2 | no | — |
| 157 | #117 | — | 2 | 4 | no | — |
| 158 | #117 | — | 2 | 4 | no | — |
| 159 | #117 | C2×D4 | 1 | 4 | yes | — |
| 160 | #117 | C2×D4 | 1 | 4 | yes | — |
| 161 | #117 | C2×D4 | 1 | 4 | yes | — |
| 162 | #117 | C24 | 1 | 4 | yes | — |
| 163 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 164 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 165 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 166 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 167 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 168 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 169 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 170 | C2×C4⋊C4 | #41 | 1 | 2 | yes | — |
| 171 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 172 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 173 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 174 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 175 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 176 | C2×C22⋊C4 | #41 | 1 | 2 | yes | — |
| 177 | C2×C42 | — | 2 | 2 | no | — |
| 178 | C2×C42 | — | 2 | 2 | no | — |
| 179 | C2×C42 | C24 | 1 | 2 | yes | — |
| 180 | #108 | C2×Q8 | 1 | 32 | yes | — |
| 181 | #107 | — | 2 | 4 | no | — |
| 182 | #107 | — | 2 | 8 | no | — |
| 183 | #107 | — | 2 | 16 | no | — |
| 184 | #107 | — | 2 | 8 | no | — |
| 185 | #107 | — | 2 | 8 | no | — |
| 186 | #106 | — | 2 | 4 | no | — |
| 187 | #106 | — | 2 | 4 | no | — |
| 188 | #106 | — | 2 | 2 | no | — |
| 189 | #106 | — | 2 | 2 | no | — |
| 190 | #106 | — | 2 | 8 | no | — |
| 191 | #106 | — | 2 | 16 | no | — |
| 192 | #106 | — | 2 | 8 | no | — |
| 193 | #106 | — | 2 | 8 | no | — |
| 194 | C4⋊C8 | — | 2 | 8 | no | — |
| 195 | C4⋊C8 | — | 2 | 8 | no | — |
| 196 | C4⋊C8 | — | 2 | 8 | no | — |
| 197 | #97 | — | 2 | 8 | no | — |
| 198 | #97 | — | 2 | 8 | no | — |
| 199 | #97 | C2×D4 | 1 | 8 | yes | — |
| 200 | C4×C8 | — | 2 | 8 | no | — |
| 201 | C4×C8 | — | 2 | 8 | no | — |
| 202 | C4×C8 | C2×D4 | 1 | 8 | yes | — |
| 203 | C24 | C2×#41 | 1 | 1 | yes | characteristic |
| 204 | C2×Q8 | — | 2 | 8 | no | — |
| 205 | C2×Q8 | — | 4 | 8 | no | — |
| 206 | C2×Q8 | — | 16 | 4 | no | — |
| 207 | C22×C4 | — | 2 | 4 | no | — |
| 208 | C22×C4 | — | 2 | 2 | no | — |
| 209 | C22×C4 | — | 2 | 2 | no | — |
| 210 | C22×C4 | — | 2 | 4 | no | — |
| 211 | C22×C4 | — | 4 | 4 | no | — |
| 212 | C22×C4 | — | 16 | 2 | no | — |
| 213 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 214 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 215 | C22×C4 | C22×Q8 | 1 | 4 | yes | — |
| 216 | C22×C4 | C2×#41 | 1 | 4 | yes | — |
| 217 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 218 | C22×C4 | 2−1+4 | 1 | 4 | yes | — |
| 219 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 220 | M16 | — | 2 | 16 | no | — |
| 221 | M16 | — | 2 | 8 | no | — |
| 222 | M16 | — | 2 | 8 | no | — |
| 223 | M16 | — | 4 | 4 | no | — |
| 224 | M16 | C4⋊Q8 | 1 | 16 | yes | — |
| 225 | M16 | C22×Q8 | 1 | 16 | yes | — |
| 226 | C2×C8 | — | 2 | 8 | no | — |
| 227 | C2×C8 | — | 2 | 16 | no | — |
| 228 | C2×C8 | — | 2 | 8 | no | — |
| 229 | C2×C8 | — | 2 | 2 | no | — |
| 230 | C2×C8 | — | 2 | 2 | no | — |
| 231 | C2×C8 | — | 2 | 4 | no | — |
| 232 | C2×C8 | — | 4 | 4 | no | — |
| 233 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 234 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 235 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 236 | C2×C8 | C22×Q8 | 1 | 8 | yes | — |
| 237 | C4⋊C4 | — | 2 | 4 | no | — |
| 238 | C4⋊C4 | — | 2 | 8 | no | — |
| 239 | C4⋊C4 | — | 2 | 16 | no | — |
| 240 | C4⋊C4 | — | 2 | 4 | no | — |
| 241 | C4⋊C4 | — | 2 | 2 | no | — |
| 242 | C4⋊C4 | — | 2 | 4 | no | — |
| 243 | C4⋊C4 | — | 2 | 2 | no | — |
| 244 | C4⋊C4 | — | 2 | 2 | no | — |
| 245 | C4⋊C4 | — | 2 | 2 | no | — |
| 246 | C4⋊C4 | — | 4 | 4 | no | — |
| 247 | C4⋊C4 | — | 4 | 8 | no | — |
| 248 | C4⋊C4 | — | 4 | 8 | no | — |
| 249 | C4⋊C4 | — | 4 | 8 | no | — |
| 250 | C4⋊C4 | #121 | 1 | 8 | yes | — |
| 251 | C4⋊C4 | C22×D4 | 1 | 8 | yes | — |
| 252 | C22⋊C4 | — | 2 | 4 | no | — |
| 253 | C22⋊C4 | — | 2 | 4 | no | — |
| 254 | C22⋊C4 | — | 4 | 4 | no | — |
| 255 | C22⋊C4 | #121 | 1 | 8 | yes | — |
| 256 | C42 | — | 2 | 4 | no | — |
| 257 | C42 | — | 2 | 4 | no | — |
| 258 | C42 | — | 4 | 4 | no | — |
| 259 | C42 | C22×D4 | 1 | 4 | yes | — |
| 260 | C42 | C22×D4 | 1 | 4 | yes | — |
| 261 | C23 | — | 4 | 2 | no | — |
| 262 | C23 | C2×#121 | 1 | 2 | yes | — |
| 263 | C23 | #557 | 1 | 4 | yes | — |
| 264 | C23 | C22×#41 | 1 | 1 | yes | characteristic |
| 265 | Q8 | — | 8 | 8 | no | — |
| 266 | Q8 | — | 16 | 8 | no | — |
| 267 | C2×C4 | — | 2 | 8 | no | — |
| 268 | C2×C4 | — | 4 | 4 | no | — |
| 269 | C2×C4 | — | 4 | 2 | no | — |
| 270 | C2×C4 | — | 4 | 2 | no | — |
| 271 | C2×C4 | — | 4 | 4 | no | — |
| 272 | C2×C4 | — | 8 | 4 | no | — |
| 273 | C2×C4 | — | 8 | 2 | no | — |
| 274 | C2×C4 | — | 8 | 4 | no | — |
| 275 | C2×C4 | — | 8 | 2 | no | — |
| 276 | C2×C4 | — | 16 | 4 | no | — |
| 277 | C2×C4 | C2×#121 | 1 | 8 | yes | — |
| 278 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 279 | C2×C4 | C2×C4⋊Q8 | 1 | 4 | yes | — |
| 280 | C2×C4 | #536 | 1 | 4 | yes | — |
| 281 | C2×C4 | D4×Q8 | 1 | 8 | yes | — |
| 282 | C2×C4 | #557 | 1 | 4 | yes | — |
| 283 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 285 | C2×C4 | C23×Q8 | 1 | 2 | yes | — |
| 286 | C2×C4 | C2×2−1+4 | 1 | 2 | yes | — |
| 287 | C8 | — | 2 | 2 | no | — |
| 288 | C8 | — | 2 | 2 | no | — |
| 289 | C8 | — | 2 | 16 | no | — |
| 290 | C8 | — | 4 | 4 | no | — |
| 291 | C8 | C2×C4⋊Q8 | 1 | 8 | yes | — |
| 292 | C22 | — | 2 | 2 | no | — |
| 293 | C22 | — | 2 | 4 | no | — |
| 294 | C22 | — | 4 | 4 | no | — |
| 295 | C22 | #2546 | 1 | 4 | yes | — |
| 296 | C22 | C22×#121 | 1 | 1 | yes | characteristic |
| 297 | C22 | C2×#557 | 1 | 2 | yes | — |
| 298 | C4 | — | 2 | 8 | no | — |
| 299 | C4 | — | 4 | 8 | no | — |
| 300 | C4 | — | 8 | 2 | no | — |
| 301 | C4 | — | 8 | 4 | no | — |
| 302 | C4 | — | 8 | 2 | no | — |
| 303 | C4 | #2546 | 1 | 4 | yes | — |
| 304 | C4 | C22×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 305 | C4 | C2×#536 | 1 | 1 | yes | characteristic |
| 306 | C4 | C2×D4×Q8 | 1 | 2 | yes | — |
| 307 | C2 | — | 2 | 2 | no | — |
| 308 | C2 | — | 2 | 4 | no | — |
| 309 | C2 | C2×#2546 | 1 | 1 | yes | characteristic |
| 310 | C2 | #60470 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #28703 | C2 | semidirect |
| 5. #28702 | C2 | semidirect |
| 6. #28700 | C2 | semidirect |
| 82. #430 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60664 | 2 |
| C2×#3163 | 2 |
| #60193 | 2 |
| #28703 | 2 |
| #28702 | 2 |
| #28700 | 2 |
| #23674 | 2 |
| C2×#2090 | 2 |
| #21338 | 2 |
| #21182 | 2 |
| #20459 | 2 |