Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
748 conjugacy classes of subgroups.
The subgroup lattice has 346 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #33191 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #33189 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#1722 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #19301 | C2 | 1 | 2 | yes | maximal |
| 5 | #19296 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #17336 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #9808 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #9804 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #9772 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #9768 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #8825 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C2×#500 | C22 | 1 | 2 | yes | — |
| 13 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#492 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#486 | C22 | 1 | 2 | yes | — |
| 16 | C2×#473 | — | 2 | 2 | no | — |
| 17 | C2×#471 | — | 2 | 2 | no | — |
| 18 | #3039 | C22 | 1 | 1 | yes | characteristic |
| 19 | #3039 | C22 | 1 | 2 | yes | — |
| 20 | #3038 | C22 | 1 | 1 | yes | characteristic |
| 21 | #2965 | C22 | 1 | 2 | yes | — |
| 22 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 23 | #1979 | — | 2 | 2 | no | — |
| 24 | #1977 | — | 2 | 2 | no | — |
| 25 | #1966 | C22 | 1 | 2 | yes | — |
| 26 | #1966 | C22 | 1 | 1 | yes | characteristic |
| 27 | #1965 | C22 | 1 | 1 | yes | characteristic |
| 28 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 30 | #1722 | — | 2 | 2 | no | — |
| 31 | #1722 | — | 2 | 1 | no | — |
| 32 | #1722 | — | 2 | 1 | no | — |
| 33 | #1702 | — | 2 | 4 | no | — |
| 34 | #1701 | — | 2 | 4 | no | — |
| 35 | #1694 | — | 2 | 4 | no | — |
| 36 | #1693 | — | 2 | 4 | no | — |
| 37 | #1566 | C22 | 1 | 2 | yes | — |
| 38 | #1566 | C22 | 1 | 2 | yes | — |
| 39 | #1546 | C22 | 1 | 2 | yes | — |
| 40 | #1546 | C22 | 1 | 2 | yes | — |
| 41 | #1546 | C22 | 1 | 2 | yes | — |
| 42 | #1545 | C22 | 1 | 2 | yes | — |
| 43 | #1455 | — | 2 | 1 | no | — |
| 44 | #1453 | — | 2 | 1 | no | — |
| 45 | #1451 | — | 2 | 2 | no | — |
| 46 | #1451 | C22 | 1 | 2 | yes | — |
| 47 | #1450 | C22 | 1 | 1 | yes | characteristic |
| 48 | C2×#322 | C22 | 1 | 1 | yes | characteristic |
| 49 | C2×#137 | — | 2 | 1 | no | — |
| 50 | C2×#137 | — | 2 | 2 | no | — |
| 51 | C2×AGL(1,ℤ/8) | — | 2 | 1 | no | — |
| 52 | C22×Q16 | — | 2 | 2 | no | — |
| 53 | C22×SD16 | — | 2 | 2 | no | — |
| 54 | C2×#131 | D4 | 1 | 2 | yes | — |
| 55 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 56 | C2×C4⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 57 | C2×#127 | C23 | 1 | 1 | yes | characteristic |
| 58 | C4×#41 | C23 | 1 | 1 | yes | characteristic |
| 59 | #500 | — | 2 | 4 | no | — |
| 60 | #500 | — | 2 | 2 | no | — |
| 61 | #500 | — | 2 | 2 | no | — |
| 62 | #494 | — | 2 | 1 | no | — |
| 63 | #494 | — | 2 | 1 | no | — |
| 64 | #494 | — | 2 | 2 | no | — |
| 65 | #492 | — | 2 | 1 | no | — |
| 66 | #492 | — | 2 | 1 | no | — |
| 67 | #492 | — | 2 | 2 | no | — |
| 68 | #486 | — | 2 | 4 | no | — |
| 69 | #486 | — | 2 | 2 | no | — |
| 70 | #486 | — | 2 | 2 | no | — |
| 71 | #473 | — | 4 | 8 | no | — |
| 72 | #471 | — | 4 | 8 | no | — |
| 73 | #462 | — | 2 | 2 | no | — |
| 74 | #462 | — | 2 | 2 | no | — |
| 75 | #462 | — | 2 | 2 | no | — |
| 76 | #461 | — | 2 | 2 | no | — |
| 77 | #461 | — | 2 | 2 | no | — |
| 78 | #461 | — | 2 | 2 | no | — |
| 79 | #460 | — | 2 | 2 | no | — |
| 80 | #459 | — | 2 | 2 | no | — |
| 81 | #446 | D4 | 1 | 4 | yes | — |
| 82 | #436 | D4 | 1 | 4 | yes | — |
| 83 | #435 | — | 2 | 4 | no | — |
| 84 | #431 | — | 2 | 2 | no | — |
| 85 | #431 | C23 | 1 | 2 | yes | — |
| 86 | C2×#108 | — | 2 | 2 | no | — |
| 87 | C2×#108 | — | 2 | 2 | no | — |
| 88 | C2×#107 | — | 2 | 4 | no | — |
| 89 | #423 | C23 | 1 | 2 | yes | — |
| 90 | #423 | C23 | 1 | 1 | yes | characteristic |
| 91 | #423 | C23 | 1 | 1 | yes | characteristic |
| 92 | #419 | — | 2 | 1 | no | — |
| 93 | #419 | — | 2 | 2 | no | — |
| 94 | #418 | — | 2 | 1 | no | — |
| 95 | C2×#102 | — | 2 | 4 | no | — |
| 96 | C2×#101 | — | 2 | 2 | no | — |
| 97 | C2×#101 | — | 2 | 4 | no | — |
| 98 | C2×#100 | — | 2 | 2 | no | — |
| 99 | #405 | — | 2 | 2 | no | — |
| 100 | #405 | D4 | 1 | 2 | yes | — |
| 101 | C4×M16 | — | 2 | 1 | no | — |
| 102 | C4×M16 | — | 2 | 2 | no | — |
| 103 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C8 | — | 2 | 1 | no | — |
| 105 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 106 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 107 | #334 | — | 2 | 4 | no | — |
| 108 | #334 | — | 2 | 4 | no | — |
| 109 | #332 | — | 2 | 4 | no | — |
| 110 | #332 | — | 2 | 4 | no | — |
| 111 | #332 | — | 2 | 4 | no | — |
| 112 | #331 | — | 2 | 4 | no | — |
| 113 | #322 | — | 2 | 1 | no | — |
| 114 | #322 | — | 2 | 2 | no | — |
| 115 | #322 | — | 2 | 2 | no | — |
| 116 | #322 | D4 | 1 | 2 | yes | — |
| 117 | #322 | D4 | 1 | 1 | yes | characteristic |
| 118 | #322 | D4 | 1 | 1 | yes | characteristic |
| 119 | C82 | — | 2 | 1 | no | — |
| 120 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 121 | C22×Q8 | — | 2 | 1 | no | — |
| 122 | C22×Q8 | — | 2 | 2 | no | — |
| 123 | C22×D4 | — | 2 | 1 | no | — |
| 124 | C22×D4 | — | 2 | 2 | no | — |
| 125 | #137 | — | 4 | 4 | no | — |
| 126 | #137 | — | 4 | 8 | no | — |
| 127 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 128 | C2×Q16 | — | 4 | 1 | no | — |
| 129 | C2×Q16 | — | 4 | 2 | no | — |
| 130 | C2×Q16 | — | 4 | 2 | no | — |
| 131 | C2×Q16 | — | 4 | 2 | no | — |
| 132 | C2×Q16 | — | 4 | 4 | no | — |
| 133 | C2×Q16 | — | 4 | 4 | no | — |
| 134 | C2×SD16 | — | 4 | 1 | no | — |
| 135 | C2×SD16 | — | 4 | 1 | no | — |
| 136 | C2×SD16 | — | 4 | 2 | no | — |
| 137 | C2×SD16 | — | 4 | 2 | no | — |
| 138 | C2×SD16 | — | 4 | 2 | no | — |
| 139 | C2×SD16 | — | 4 | 4 | no | — |
| 140 | C2×SD16 | — | 4 | 4 | no | — |
| 141 | C2×D8 | — | 4 | 1 | no | — |
| 142 | #131 | — | 2 | 4 | no | — |
| 143 | #131 | — | 2 | 4 | no | — |
| 144 | C2×M16 | — | 2 | 2 | no | — |
| 145 | C2×M16 | — | 2 | 4 | no | — |
| 146 | C2×M16 | — | 2 | 1 | no | — |
| 147 | C2×M16 | — | 2 | 1 | no | — |
| 148 | C2×M16 | — | 2 | 2 | no | — |
| 149 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 150 | C22×C8 | — | 2 | 2 | no | — |
| 151 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 152 | C22×C8 | #41 | 1 | 4 | yes | — |
| 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 | 1 | no | — |
| 158 | C4⋊Q8 | — | 2 | 1 | no | — |
| 159 | C4⋊Q8 | — | 2 | 2 | no | — |
| 160 | #127 | — | 2 | 2 | no | — |
| 161 | #127 | — | 2 | 1 | no | — |
| 162 | #127 | — | 2 | 1 | no | — |
| 163 | C4×Q8 | C2×D4 | 1 | 2 | yes | — |
| 164 | C4×D4 | — | 2 | 2 | no | — |
| 165 | C4×D4 | C2×D4 | 1 | 2 | yes | — |
| 166 | #117 | — | 2 | 1 | no | — |
| 167 | #117 | C2×D4 | 1 | 1 | yes | characteristic |
| 168 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 169 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 170 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 171 | C2×C42 | — | 2 | 1 | no | — |
| 172 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 173 | #108 | — | 4 | 4 | no | — |
| 174 | #108 | — | 4 | 4 | no | — |
| 175 | #107 | — | 4 | 4 | no | — |
| 176 | #107 | — | 4 | 4 | no | — |
| 177 | C4⋊C8 | — | 2 | 4 | no | — |
| 178 | C4⋊C8 | — | 2 | 2 | no | — |
| 179 | C4⋊C8 | — | 2 | 2 | no | — |
| 180 | C4⋊C8 | — | 2 | 2 | no | — |
| 181 | C4⋊C8 | — | 2 | 2 | no | — |
| 182 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 183 | #103 | — | 4 | 2 | no | — |
| 184 | #103 | — | 4 | 2 | no | — |
| 185 | #103 | — | 4 | 2 | no | — |
| 186 | #102 | — | 4 | 4 | no | — |
| 187 | #102 | — | 4 | 4 | no | — |
| 188 | #102 | — | 4 | 2 | no | — |
| 189 | #101 | — | 4 | 4 | no | — |
| 190 | #101 | — | 4 | 8 | no | — |
| 191 | #100 | — | 4 | 4 | no | — |
| 192 | C22⋊C8 | — | 2 | 2 | no | — |
| 193 | C22⋊C8 | — | 2 | 4 | no | — |
| 194 | #97 | — | 2 | 2 | no | — |
| 195 | #97 | — | 2 | 4 | no | — |
| 196 | #97 | — | 2 | 1 | no | — |
| 197 | C4×C8 | — | 2 | 2 | no | — |
| 198 | C4×C8 | — | 2 | 2 | no | — |
| 199 | C4×C8 | — | 2 | 1 | no | — |
| 200 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 201 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 202 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 203 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 204 | C4×C8 | C2×D4 | 1 | 4 | yes | — |
| 205 | C4×C8 | #41 | 1 | 2 | yes | — |
| 206 | C24 | — | 4 | 1 | no | — |
| 207 | #41 | — | 2 | 2 | no | — |
| 208 | #41 | — | 2 | 2 | no | — |
| 209 | C2×Q8 | — | 4 | 1 | no | — |
| 210 | C2×Q8 | — | 4 | 2 | no | — |
| 211 | C2×Q8 | — | 4 | 2 | no | — |
| 212 | C2×Q8 | — | 4 | 1 | no | — |
| 213 | C2×Q8 | — | 4 | 2 | no | — |
| 214 | C2×Q8 | — | 4 | 4 | no | — |
| 215 | C2×Q8 | — | 4 | 2 | no | — |
| 216 | C2×Q8 | — | 4 | 4 | no | — |
| 217 | C2×Q8 | #127 | 1 | 1 | yes | characteristic |
| 218 | C2×D4 | — | 2 | 1 | no | — |
| 219 | C2×D4 | — | 4 | 2 | no | — |
| 220 | C2×D4 | — | 4 | 1 | no | — |
| 221 | C2×D4 | — | 4 | 1 | no | — |
| 222 | C2×D4 | — | 4 | 2 | no | — |
| 223 | C2×D4 | — | 4 | 2 | no | — |
| 224 | C2×D4 | — | 4 | 2 | no | — |
| 225 | C2×D4 | — | 8 | 4 | no | — |
| 226 | C2×D4 | #127 | 1 | 1 | yes | characteristic |
| 227 | C22×C4 | — | 2 | 2 | no | — |
| 228 | C22×C4 | — | 2 | 1 | no | — |
| 229 | C22×C4 | — | 4 | 2 | no | — |
| 230 | C22×C4 | — | 4 | 1 | no | — |
| 231 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 232 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 233 | Q16 | — | 8 | 2 | no | — |
| 234 | Q16 | — | 8 | 4 | no | — |
| 235 | Q16 | — | 8 | 4 | no | — |
| 236 | Q16 | — | 8 | 4 | no | — |
| 237 | SD16 | — | 8 | 2 | no | — |
| 238 | SD16 | — | 8 | 4 | no | — |
| 239 | SD16 | — | 8 | 2 | no | — |
| 240 | SD16 | — | 8 | 4 | no | — |
| 241 | SD16 | — | 8 | 4 | no | — |
| 242 | D8 | — | 8 | 2 | no | — |
| 243 | M16 | — | 2 | 2 | no | — |
| 244 | M16 | — | 2 | 2 | no | — |
| 245 | M16 | — | 4 | 4 | no | — |
| 246 | M16 | — | 4 | 2 | no | — |
| 247 | M16 | — | 4 | 2 | no | — |
| 248 | M16 | — | 4 | 4 | no | — |
| 249 | M16 | — | 8 | 4 | no | — |
| 250 | C2×C8 | — | 2 | 2 | no | — |
| 251 | C2×C8 | — | 2 | 4 | no | — |
| 252 | C2×C8 | — | 4 | 4 | no | — |
| 253 | C2×C8 | — | 4 | 4 | no | — |
| 254 | C2×C8 | — | 4 | 1 | no | — |
| 255 | C2×C8 | — | 4 | 2 | no | — |
| 256 | C2×C8 | — | 4 | 1 | no | — |
| 257 | C2×C8 | #121 | 1 | 4 | yes | — |
| 258 | C2×C8 | #121 | 1 | 4 | yes | — |
| 259 | C2×C8 | #121 | 1 | 2 | yes | — |
| 260 | C2×C8 | #121 | 1 | 2 | yes | — |
| 261 | C2×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 262 | C2×C8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 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 | #127 | 1 | 2 | yes | — |
| 271 | C22⋊C4 | — | 2 | 1 | no | — |
| 272 | C22⋊C4 | — | 2 | 2 | no | — |
| 273 | C42 | — | 2 | 1 | no | — |
| 274 | C42 | — | 2 | 2 | no | — |
| 275 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 276 | C42 | C22×D4 | 1 | 2 | yes | — |
| 277 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 278 | C23 | — | 2 | 1 | no | — |
| 279 | C23 | — | 8 | 1 | no | — |
| 280 | C23 | — | 8 | 2 | no | — |
| 281 | C23 | — | 16 | 2 | no | — |
| 282 | C23 | #532 | 1 | 1 | yes | characteristic |
| 283 | Q8 | — | 2 | 1 | no | — |
| 284 | Q8 | — | 2 | 1 | no | — |
| 285 | Q8 | — | 8 | 1 | no | — |
| 286 | Q8 | — | 8 | 2 | no | — |
| 287 | Q8 | — | 8 | 1 | no | — |
| 288 | Q8 | — | 8 | 2 | no | — |
| 289 | Q8 | — | 8 | 2 | no | — |
| 290 | Q8 | — | 8 | 4 | no | — |
| 291 | D4 | — | 2 | 1 | no | — |
| 292 | D4 | — | 2 | 1 | no | — |
| 293 | D4 | — | 4 | 2 | no | — |
| 294 | D4 | — | 8 | 1 | no | — |
| 295 | D4 | — | 8 | 1 | no | — |
| 296 | D4 | — | 8 | 2 | no | — |
| 297 | D4 | — | 8 | 4 | no | — |
| 298 | D4 | — | 16 | 2 | no | — |
| 299 | C2×C4 | — | 2 | 4 | no | — |
| 300 | C2×C4 | — | 2 | 2 | no | — |
| 301 | C2×C4 | — | 2 | 1 | no | — |
| 302 | C2×C4 | — | 4 | 2 | no | — |
| 303 | C2×C4 | — | 4 | 4 | no | — |
| 304 | C2×C4 | — | 8 | 2 | no | — |
| 305 | C2×C4 | — | 8 | 1 | no | — |
| 306 | C2×C4 | — | 8 | 2 | no | — |
| 307 | C2×C4 | — | 8 | 4 | no | — |
| 308 | C2×C4 | C2×#121 | 1 | 1 | yes | characteristic |
| 309 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 310 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 311 | C2×C4 | #532 | 1 | 2 | yes | — |
| 312 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 313 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 314 | C8 | — | 2 | 4 | no | — |
| 315 | C8 | — | 2 | 4 | no | — |
| 316 | C8 | — | 4 | 2 | no | — |
| 317 | C8 | — | 4 | 4 | no | — |
| 318 | C8 | — | 8 | 1 | no | — |
| 319 | C8 | — | 8 | 1 | no | — |
| 320 | C8 | — | 8 | 2 | no | — |
| 321 | C8 | #469 | 1 | 4 | yes | — |
| 322 | C8 | #469 | 1 | 4 | yes | — |
| 323 | C22 | — | 2 | 2 | no | — |
| 324 | C22 | — | 4 | 1 | no | — |
| 325 | C22 | — | 4 | 2 | no | — |
| 326 | C22 | — | 16 | 1 | no | — |
| 327 | C22 | — | 16 | 2 | no | — |
| 328 | C22 | — | 32 | 2 | no | — |
| 329 | C22 | #2856 | 1 | 1 | yes | characteristic |
| 330 | C22 | #3184 | 1 | 2 | yes | — |
| 331 | C4 | — | 2 | 2 | no | — |
| 332 | C4 | — | 4 | 1 | no | — |
| 333 | C4 | — | 4 | 2 | no | — |
| 334 | C4 | — | 16 | 1 | no | — |
| 335 | C4 | — | 16 | 2 | no | — |
| 336 | C4 | C2×#469 | 1 | 2 | yes | — |
| 337 | C4 | #3069 | 1 | 4 | yes | — |
| 338 | C4 | #3151 | 1 | 1 | yes | characteristic |
| 339 | C4 | #3151 | 1 | 1 | yes | characteristic |
| 340 | C2 | — | 2 | 2 | no | — |
| 341 | C2 | — | 8 | 1 | no | — |
| 342 | C2 | — | 32 | 1 | no | — |
| 343 | C2 | #33175 | 1 | 1 | yes | characteristic |
| 344 | C2 | #33194 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2856 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3038 | 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. #33191 | C2 | semidirect |
| 3. C2×#1722 | C2 | semidirect |
| 5. #19296 | C2 | semidirect |
| 6. #17336 | C2 | semidirect |
| 7. #9808 | C2 | semidirect |
| 8. #9804 | C2 | semidirect |
| 9. #9772 | C2 | semidirect |
| 10. #9768 | C2 | semidirect |
| 11. #8825 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33191 | 2 |
| #33189 | 2 |
| C2×#1722 | 2 |
| #19301 | 2 |
| #19296 | 2 |
| #17336 | 2 |
| #9808 | 2 |
| #9804 | 2 |
| #9772 | 2 |
| #9768 | 2 |
| #8825 | 2 |