Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
336 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #30302 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11724 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #11723 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11297 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #9193 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #9191 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #7188 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #2905 | C22 | 1 | 1 | yes | characteristic |
| 9 | #2628 | C22 | 1 | 1 | yes | characteristic |
| 10 | #2628 | C22 | 1 | 1 | yes | characteristic |
| 11 | #2085 | — | 2 | 1 | no | — |
| 12 | #2085 | — | 2 | 1 | no | — |
| 13 | #2058 | — | 2 | 1 | no | — |
| 14 | #2058 | — | 2 | 1 | no | — |
| 15 | #1809 | — | 2 | 1 | no | — |
| 16 | #1807 | — | 2 | 1 | no | — |
| 17 | #1742 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1738 | — | 2 | 1 | no | — |
| 19 | C2×#341 | — | 2 | 1 | no | — |
| 20 | #1729 | C22 | 1 | 1 | yes | characteristic |
| 21 | #1549 | — | 2 | 1 | no | — |
| 22 | #1548 | — | 2 | 1 | no | — |
| 23 | #1458 | C22 | 1 | 1 | yes | characteristic |
| 24 | #1458 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1354 | — | 2 | 2 | no | — |
| 26 | #1353 | — | 2 | 2 | no | — |
| 27 | #1286 | — | 2 | 1 | no | — |
| 28 | #1286 | — | 2 | 1 | no | — |
| 29 | C22×M16 | — | 2 | 1 | no | — |
| 30 | C22×M16 | — | 2 | 1 | no | — |
| 31 | C22×M16 | D4 | 1 | 1 | yes | characteristic |
| 32 | C23×C8 | D4 | 1 | 1 | yes | characteristic |
| 33 | C2×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 35 | C2×#117 | C23 | 1 | 1 | yes | characteristic |
| 36 | #431 | — | 2 | 1 | no | — |
| 37 | #431 | — | 2 | 1 | no | — |
| 38 | C2×#107 | — | 2 | 1 | no | — |
| 39 | C2×#106 | — | 2 | 1 | no | — |
| 40 | C2×#106 | — | 2 | 1 | no | — |
| 41 | C2×#106 | — | 2 | 1 | no | — |
| 42 | #424 | — | 4 | 2 | no | — |
| 43 | #423 | — | 2 | 1 | no | — |
| 44 | #423 | — | 2 | 1 | no | — |
| 45 | #400 | — | 2 | 1 | no | — |
| 46 | #400 | — | 2 | 1 | no | — |
| 47 | #399 | — | 2 | 1 | no | — |
| 48 | #399 | — | 2 | 1 | no | — |
| 49 | #395 | — | 2 | 1 | no | — |
| 50 | #395 | — | 2 | 1 | no | — |
| 51 | #393 | — | 2 | 1 | no | — |
| 52 | #393 | — | 2 | 1 | no | — |
| 53 | #389 | — | 2 | 1 | no | — |
| 54 | #389 | — | 2 | 1 | no | — |
| 55 | #382 | D4 | 1 | 1 | yes | characteristic |
| 56 | #382 | D4 | 1 | 1 | yes | characteristic |
| 57 | #380 | — | 2 | 1 | no | — |
| 58 | #380 | — | 2 | 1 | no | — |
| 59 | #343 | — | 2 | 1 | no | — |
| 60 | #343 | D4 | 1 | 2 | yes | — |
| 61 | #341 | — | 4 | 2 | no | — |
| 62 | #341 | — | 4 | 2 | no | — |
| 63 | #340 | — | 2 | 1 | no | — |
| 64 | #340 | — | 2 | 1 | no | — |
| 65 | #340 | — | 2 | 1 | no | — |
| 66 | #340 | — | 2 | 1 | no | — |
| 67 | #336 | — | 2 | 1 | no | — |
| 68 | #336 | — | 2 | 1 | no | — |
| 69 | #333 | — | 4 | 2 | no | — |
| 70 | #332 | — | 4 | 2 | no | — |
| 71 | #324 | — | 2 | 1 | no | — |
| 72 | #324 | — | 2 | 1 | no | — |
| 73 | #323 | — | 2 | 1 | no | — |
| 74 | #323 | — | 2 | 1 | no | — |
| 75 | C22×Q8 | — | 2 | 1 | no | — |
| 76 | C22×Q8 | — | 2 | 1 | no | — |
| 77 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C2×M16 | — | 2 | 1 | no | — |
| 79 | C2×M16 | — | 2 | 2 | no | — |
| 80 | C2×M16 | — | 2 | 2 | no | — |
| 81 | C2×M16 | — | 4 | 1 | no | — |
| 82 | C2×M16 | — | 4 | 1 | no | — |
| 83 | C2×M16 | — | 4 | 1 | no | — |
| 84 | C2×M16 | — | 4 | 1 | no | — |
| 85 | C2×M16 | — | 4 | 1 | no | — |
| 86 | C2×M16 | — | 4 | 1 | no | — |
| 87 | C2×M16 | — | 4 | 1 | no | — |
| 88 | C2×M16 | — | 4 | 1 | no | — |
| 89 | C2×M16 | — | 4 | 1 | no | — |
| 90 | C2×M16 | — | 4 | 1 | no | — |
| 91 | C2×M16 | — | 4 | 1 | no | — |
| 92 | C2×M16 | — | 4 | 1 | no | — |
| 93 | C2×M16 | Q16 | 1 | 2 | yes | — |
| 94 | C22×C8 | — | 2 | 1 | no | — |
| 95 | C22×C8 | — | 2 | 1 | no | — |
| 96 | C22×C8 | — | 2 | 1 | no | — |
| 97 | C22×C8 | — | 4 | 1 | no | — |
| 98 | C22×C8 | — | 4 | 1 | no | — |
| 99 | C22×C8 | — | 4 | 1 | no | — |
| 100 | C22×C8 | — | 4 | 1 | no | — |
| 101 | C22×C8 | SD16 | 1 | 2 | yes | — |
| 102 | C4⋊Q8 | — | 4 | 1 | no | — |
| 103 | C4⋊Q8 | — | 4 | 1 | no | — |
| 104 | C4⋊Q8 | — | 4 | 1 | no | — |
| 105 | C4⋊Q8 | — | 4 | 1 | no | — |
| 106 | #117 | — | 4 | 1 | no | — |
| 107 | #117 | — | 4 | 1 | no | — |
| 108 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 109 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 110 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 111 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 112 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 113 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 114 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 115 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 116 | C2×C4⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | C2×C4⋊C4 | #41 | 1 | 1 | yes | characteristic |
| 118 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 119 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 120 | C2×C22⋊C4 | #41 | 1 | 1 | yes | characteristic |
| 121 | C2×C22⋊C4 | #41 | 1 | 1 | yes | characteristic |
| 122 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 124 | #107 | — | 8 | 1 | no | — |
| 125 | #106 | — | 8 | 1 | no | — |
| 126 | #106 | — | 8 | 1 | no | — |
| 127 | #106 | — | 8 | 1 | no | — |
| 128 | C4⋊C8 | — | 2 | 2 | no | — |
| 129 | C4⋊C8 | — | 2 | 2 | no | — |
| 130 | C4⋊C8 | — | 4 | 1 | no | — |
| 131 | C4⋊C8 | — | 4 | 1 | no | — |
| 132 | C4⋊C8 | — | 4 | 1 | no | — |
| 133 | C4⋊C8 | — | 4 | 1 | no | — |
| 134 | C22⋊C8 | — | 4 | 1 | no | — |
| 135 | C22⋊C8 | — | 4 | 1 | no | — |
| 136 | C4×C8 | — | 4 | 1 | no | — |
| 137 | C4×C8 | — | 4 | 1 | no | — |
| 138 | #95 | — | 2 | 1 | no | — |
| 139 | #95 | — | 2 | 1 | no | — |
| 140 | #95 | — | 2 | 1 | no | — |
| 141 | #95 | — | 2 | 1 | no | — |
| 142 | #95 | — | 4 | 1 | no | — |
| 143 | #95 | — | 4 | 1 | no | — |
| 144 | #95 | — | 4 | 1 | no | — |
| 145 | #95 | — | 4 | 1 | no | — |
| 146 | C24 | #124 | 1 | 1 | yes | characteristic |
| 147 | C2×Q8 | — | 4 | 2 | no | — |
| 148 | C2×Q8 | — | 8 | 1 | no | — |
| 149 | C2×Q8 | — | 16 | 1 | no | — |
| 150 | C2×Q8 | — | 16 | 1 | no | — |
| 151 | C22×C4 | — | 2 | 1 | no | — |
| 152 | C22×C4 | — | 2 | 1 | no | — |
| 153 | C22×C4 | — | 2 | 1 | no | — |
| 154 | C22×C4 | — | 2 | 1 | no | — |
| 155 | C22×C4 | — | 4 | 1 | no | — |
| 156 | C22×C4 | — | 4 | 1 | no | — |
| 157 | C22×C4 | — | 4 | 1 | no | — |
| 158 | C22×C4 | — | 4 | 1 | no | — |
| 159 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 160 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 161 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 162 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 163 | C22×C4 | #123 | 1 | 1 | yes | characteristic |
| 164 | C22×C4 | #123 | 1 | 1 | yes | characteristic |
| 165 | C22×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 166 | C22×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 167 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 168 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 169 | M16 | — | 4 | 2 | no | — |
| 170 | M16 | — | 4 | 2 | no | — |
| 171 | M16 | — | 4 | 2 | no | — |
| 172 | M16 | — | 4 | 2 | no | — |
| 173 | M16 | — | 4 | 2 | no | — |
| 174 | M16 | — | 8 | 1 | no | — |
| 175 | M16 | — | 8 | 1 | no | — |
| 176 | M16 | — | 8 | 1 | no | — |
| 177 | M16 | — | 8 | 1 | no | — |
| 178 | M16 | — | 8 | 1 | no | — |
| 179 | C2×C8 | — | 2 | 2 | no | — |
| 180 | C2×C8 | — | 2 | 2 | no | — |
| 181 | C2×C8 | — | 4 | 1 | no | — |
| 182 | C2×C8 | — | 4 | 1 | no | — |
| 183 | C2×C8 | — | 4 | 1 | no | — |
| 184 | C2×C8 | — | 4 | 1 | no | — |
| 185 | C2×C8 | — | 4 | 2 | no | — |
| 186 | C2×C8 | — | 4 | 2 | no | — |
| 187 | C2×C8 | — | 4 | 2 | no | — |
| 188 | C2×C8 | — | 8 | 1 | no | — |
| 189 | C2×C8 | — | 8 | 1 | no | — |
| 190 | C2×C8 | — | 8 | 1 | no | — |
| 191 | C2×C8 | — | 8 | 1 | no | — |
| 192 | C2×C8 | — | 8 | 1 | no | — |
| 193 | C2×C8 | — | 8 | 1 | no | — |
| 194 | C2×C8 | — | 8 | 1 | no | — |
| 195 | C4⋊C4 | — | 4 | 1 | no | — |
| 196 | C4⋊C4 | — | 4 | 1 | no | — |
| 197 | C4⋊C4 | — | 4 | 2 | no | — |
| 198 | C4⋊C4 | — | 4 | 1 | no | — |
| 199 | C4⋊C4 | — | 4 | 1 | no | — |
| 200 | C4⋊C4 | — | 4 | 1 | no | — |
| 201 | C4⋊C4 | — | 4 | 1 | no | — |
| 202 | C4⋊C4 | — | 4 | 1 | no | — |
| 203 | C4⋊C4 | — | 4 | 1 | no | — |
| 204 | C4⋊C4 | — | 4 | 1 | no | — |
| 205 | C4⋊C4 | — | 4 | 1 | no | — |
| 206 | C4⋊C4 | — | 8 | 1 | no | — |
| 207 | C4⋊C4 | — | 16 | 1 | no | — |
| 208 | C4⋊C4 | — | 16 | 1 | no | — |
| 209 | C22⋊C4 | — | 4 | 1 | no | — |
| 210 | C22⋊C4 | — | 4 | 1 | no | — |
| 211 | C22⋊C4 | — | 16 | 1 | no | — |
| 212 | C22⋊C4 | — | 16 | 1 | no | — |
| 213 | C42 | — | 2 | 1 | no | — |
| 214 | C42 | — | 2 | 1 | no | — |
| 215 | C42 | — | 2 | 1 | no | — |
| 216 | C42 | — | 2 | 1 | no | — |
| 217 | C23 | — | 2 | 1 | no | — |
| 218 | C23 | — | 2 | 1 | no | — |
| 219 | C23 | — | 4 | 1 | no | — |
| 220 | C23 | — | 4 | 1 | no | — |
| 221 | C23 | #394 | 1 | 1 | yes | characteristic |
| 222 | C23 | #487 | 1 | 1 | yes | characteristic |
| 223 | C23 | #491 | 1 | 1 | yes | characteristic |
| 224 | Q8 | — | 16 | 2 | no | — |
| 225 | Q8 | — | 16 | 2 | no | — |
| 226 | C2×C4 | — | 2 | 1 | no | — |
| 227 | C2×C4 | — | 2 | 1 | no | — |
| 228 | C2×C4 | — | 2 | 1 | no | — |
| 229 | C2×C4 | — | 2 | 1 | no | — |
| 230 | C2×C4 | — | 2 | 1 | no | — |
| 231 | C2×C4 | — | 2 | 1 | no | — |
| 232 | C2×C4 | — | 2 | 1 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 2 | 1 | no | — |
| 235 | C2×C4 | — | 2 | 1 | no | — |
| 236 | C2×C4 | — | 2 | 1 | no | — |
| 237 | C2×C4 | — | 2 | 1 | no | — |
| 238 | C2×C4 | — | 2 | 1 | no | — |
| 239 | C2×C4 | — | 2 | 1 | no | — |
| 240 | C2×C4 | — | 4 | 1 | no | — |
| 241 | C2×C4 | — | 4 | 1 | no | — |
| 242 | C2×C4 | — | 4 | 1 | no | — |
| 243 | C2×C4 | — | 4 | 1 | no | — |
| 244 | C2×C4 | — | 8 | 1 | no | — |
| 245 | C2×C4 | — | 8 | 1 | no | — |
| 246 | C2×C4 | — | 8 | 1 | no | — |
| 247 | C2×C4 | — | 8 | 1 | no | — |
| 248 | C2×C4 | — | 8 | 1 | no | — |
| 249 | C2×C4 | — | 8 | 1 | no | — |
| 250 | C2×C4 | — | 8 | 1 | no | — |
| 251 | C2×C4 | — | 8 | 1 | no | — |
| 252 | C2×C4 | — | 8 | 1 | no | — |
| 253 | C2×C4 | — | 8 | 1 | no | — |
| 254 | C2×C4 | — | 8 | 1 | no | — |
| 255 | C2×C4 | — | 8 | 1 | no | — |
| 256 | C2×C4 | #448 | 1 | 1 | yes | characteristic |
| 257 | C2×C4 | #451 | 1 | 1 | yes | characteristic |
| 258 | C2×C4 | #454 | 1 | 1 | yes | characteristic |
| 259 | C2×C4 | #455 | 1 | 1 | yes | characteristic |
| 260 | C2×C4 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 261 | C2×C4 | #458 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | #461 | 1 | 1 | yes | characteristic |
| 263 | C2×C4 | #462 | 1 | 1 | yes | characteristic |
| 264 | C8 | — | 4 | 2 | no | — |
| 265 | C8 | — | 4 | 2 | no | — |
| 266 | C8 | — | 8 | 2 | no | — |
| 267 | C8 | — | 8 | 2 | no | — |
| 268 | C8 | — | 16 | 1 | no | — |
| 269 | C8 | — | 16 | 1 | no | — |
| 270 | C22 | — | 2 | 1 | no | — |
| 271 | C22 | — | 2 | 1 | no | — |
| 272 | C22 | — | 2 | 1 | no | — |
| 273 | C22 | — | 2 | 1 | no | — |
| 274 | C22 | — | 4 | 1 | no | — |
| 275 | C22 | — | 4 | 1 | no | — |
| 276 | C22 | — | 4 | 1 | no | — |
| 277 | C22 | — | 4 | 1 | no | — |
| 278 | C22 | — | 8 | 1 | no | — |
| 279 | C22 | #2040 | 1 | 1 | yes | characteristic |
| 280 | C22 | #2043 | 1 | 1 | yes | characteristic |
| 281 | C22 | #2044 | 1 | 1 | yes | characteristic |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 2 | 1 | no | — |
| 284 | C4 | — | 4 | 1 | no | — |
| 285 | C4 | — | 4 | 1 | no | — |
| 286 | C4 | — | 4 | 1 | no | — |
| 287 | C4 | — | 4 | 1 | no | — |
| 288 | C4 | — | 16 | 1 | no | — |
| 289 | C4 | — | 16 | 1 | no | — |
| 290 | C4 | — | 16 | 1 | no | — |
| 291 | C4 | — | 16 | 1 | no | — |
| 292 | C4 | #1599 | 1 | 1 | yes | characteristic |
| 293 | C4 | #1602 | 1 | 1 | yes | characteristic |
| 294 | C4 | #1608 | 1 | 1 | yes | characteristic |
| 295 | C4 | #1617 | 1 | 1 | yes | characteristic |
| 296 | C2 | — | 2 | 1 | no | — |
| 297 | C2 | — | 2 | 1 | no | — |
| 298 | C2 | — | 4 | 1 | no | — |
| 299 | C2 | — | 4 | 1 | no | — |
| 300 | C2 | #10407 | 1 | 1 | yes | characteristic |
| 301 | C2 | #13430 | 1 | 1 | yes | characteristic |
| 302 | C2 | #13437 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator, Frattini, cosocle | C2×#117 | C23 |
| prime radical | C24 | #124 |
| LCS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #30302 | 2 |
| #11724 | 2 |
| #11723 | 2 |
| #11297 | 2 |
| #9193 | 2 |
| #9191 | 2 |
| #7188 | 2 |