Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
594 conjugacy classes of subgroups.
The subgroup lattice has 345 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #33650 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #28504 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22691 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13393 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13387 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #13261 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #13206 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #13196 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #13180 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #12996 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #12671 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | #3155 | — | 2 | 1 | no | — |
| 13 | C2×#493 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#472 | C22 | 1 | 2 | yes | — |
| 15 | #3058 | C22 | 1 | 2 | yes | — |
| 16 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2952 | — | 2 | 1 | no | — |
| 18 | #2913 | C22 | 1 | 2 | yes | — |
| 19 | #2234 | — | 2 | 2 | no | — |
| 20 | #2234 | — | 2 | 2 | no | — |
| 21 | #2233 | — | 2 | 2 | no | — |
| 22 | #2232 | — | 2 | 2 | no | — |
| 23 | C2×#368 | C22 | 1 | 1 | yes | characteristic |
| 24 | #2159 | C22 | 1 | 2 | yes | — |
| 25 | #2158 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#361 | C22 | 1 | 2 | yes | — |
| 28 | #2140 | C22 | 1 | 2 | yes | — |
| 29 | C2×#357 | C22 | 1 | 2 | yes | — |
| 30 | C2×#357 | C22 | 1 | 2 | yes | — |
| 31 | C2×#350 | C22 | 1 | 2 | yes | — |
| 32 | #2113 | C22 | 1 | 2 | yes | — |
| 33 | C2×#346 | C22 | 1 | 1 | yes | characteristic |
| 34 | #2002 | — | 2 | 2 | no | — |
| 35 | #1978 | — | 2 | 1 | no | — |
| 36 | #1951 | — | 2 | 1 | no | — |
| 37 | #1860 | — | 2 | 2 | no | — |
| 38 | #1846 | C22 | 1 | 1 | yes | characteristic |
| 39 | #1816 | C22 | 1 | 2 | yes | — |
| 40 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 41 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 42 | #1387 | C22 | 1 | 2 | yes | — |
| 43 | #1382 | C22 | 1 | 1 | yes | characteristic |
| 44 | #1382 | C22 | 1 | 1 | yes | characteristic |
| 45 | C2×AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 46 | C2×#135 | D4 | 1 | 1 | yes | characteristic |
| 47 | C22×D8 | — | 2 | 1 | no | — |
| 48 | C22×D8 | C23 | 1 | 2 | yes | — |
| 49 | C2×#131 | C23 | 1 | 2 | yes | — |
| 50 | #535 | — | 2 | 2 | no | — |
| 51 | C2×#127 | — | 2 | 1 | no | — |
| 52 | #520 | — | 2 | 2 | no | — |
| 53 | C2×M32 | C23 | 1 | 2 | yes | — |
| 54 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 55 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 56 | #496 | — | 4 | 2 | no | — |
| 57 | #495 | — | 4 | 1 | no | — |
| 58 | #493 | — | 2 | 2 | no | — |
| 59 | #493 | — | 2 | 2 | no | — |
| 60 | #493 | — | 4 | 1 | no | — |
| 61 | #472 | — | 2 | 4 | no | — |
| 62 | #472 | — | 4 | 2 | no | — |
| 63 | #469 | — | 4 | 2 | no | — |
| 64 | #466 | — | 2 | 2 | no | — |
| 65 | #466 | — | 2 | 2 | no | — |
| 66 | #443 | — | 2 | 2 | no | — |
| 67 | #443 | — | 2 | 2 | no | — |
| 68 | #442 | — | 4 | 1 | no | — |
| 69 | #441 | — | 4 | 1 | no | — |
| 70 | C4×SD16 | — | 4 | 2 | no | — |
| 71 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 72 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 73 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 74 | #428 | — | 2 | 2 | no | — |
| 75 | #428 | — | 4 | 2 | no | — |
| 76 | #427 | — | 2 | 2 | no | — |
| 77 | #427 | C23 | 1 | 1 | yes | characteristic |
| 78 | #427 | C23 | 1 | 1 | yes | characteristic |
| 79 | C2×#107 | — | 2 | 2 | no | — |
| 80 | C2×#107 | C23 | 1 | 2 | yes | — |
| 81 | C2×#106 | — | 2 | 1 | no | — |
| 82 | C2×#104 | D4 | 1 | 2 | yes | — |
| 83 | #418 | — | 2 | 2 | no | — |
| 84 | #417 | — | 2 | 2 | no | — |
| 85 | C2×#100 | — | 2 | 2 | no | — |
| 86 | #405 | — | 2 | 1 | no | — |
| 87 | #405 | — | 2 | 1 | no | — |
| 88 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | #382 | — | 2 | 1 | no | — |
| 90 | #368 | — | 2 | 2 | no | — |
| 91 | #367 | — | 2 | 2 | no | — |
| 92 | #367 | D4 | 1 | 2 | yes | complemented |
| 93 | #366 | — | 2 | 2 | no | — |
| 94 | #366 | D4 | 1 | 2 | yes | complemented |
| 95 | #364 | — | 2 | 2 | no | — |
| 96 | #361 | — | 4 | 2 | no | — |
| 97 | #357 | — | 2 | 2 | no | — |
| 98 | #357 | — | 2 | 2 | no | — |
| 99 | #357 | — | 2 | 2 | no | — |
| 100 | #357 | — | 2 | 2 | no | — |
| 101 | #357 | — | 4 | 2 | no | — |
| 102 | #350 | — | 4 | 2 | no | — |
| 103 | C22⋊C16 | — | 2 | 2 | no | — |
| 104 | C22⋊C16 | — | 2 | 2 | no | — |
| 105 | #346 | — | 2 | 2 | no | — |
| 106 | #343 | — | 2 | 2 | no | — |
| 107 | #339 | — | 2 | 1 | no | — |
| 108 | #339 | — | 2 | 1 | no | — |
| 109 | C2×#41 | C2×D4 | 1 | 2 | yes | — |
| 110 | C22×D4 | — | 2 | 2 | no | — |
| 111 | C22×D4 | — | 8 | 1 | no | — |
| 112 | AGL(1,ℤ/8) | — | 4 | 4 | no | — |
| 113 | AGL(1,ℤ/8) | — | 8 | 2 | no | — |
| 114 | #135 | — | 2 | 2 | no | — |
| 115 | #135 | — | 2 | 2 | no | — |
| 116 | C2×Q16 | D8 | 1 | 1 | yes | characteristic |
| 117 | C2×SD16 | — | 4 | 2 | no | — |
| 118 | C2×SD16 | D8 | 1 | 2 | yes | — |
| 119 | C2×D8 | — | 2 | 2 | no | — |
| 120 | C2×D8 | — | 2 | 2 | no | — |
| 121 | C2×D8 | — | 2 | 2 | no | — |
| 122 | C2×D8 | — | 2 | 2 | no | — |
| 123 | C2×D8 | — | 2 | 2 | no | — |
| 124 | C2×D8 | — | 2 | 2 | no | — |
| 125 | C2×D8 | — | 4 | 2 | no | — |
| 126 | C2×D8 | — | 4 | 2 | no | — |
| 127 | C2×D8 | — | 4 | 1 | no | — |
| 128 | C2×D8 | — | 4 | 1 | no | — |
| 129 | C2×D8 | — | 4 | 2 | no | — |
| 130 | C2×D8 | D8 | 1 | 1 | yes | characteristic |
| 131 | #131 | — | 2 | 2 | no | — |
| 132 | #131 | — | 2 | 2 | no | — |
| 133 | #131 | — | 2 | 2 | no | — |
| 134 | #131 | — | 2 | 2 | no | — |
| 135 | C2×M16 | — | 2 | 2 | no | — |
| 136 | C2×M16 | — | 2 | 1 | no | — |
| 137 | C2×M16 | — | 2 | 1 | no | — |
| 138 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 139 | C22×C8 | — | 2 | 2 | no | — |
| 140 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 141 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 142 | #127 | — | 4 | 2 | no | — |
| 143 | #127 | — | 4 | 2 | no | — |
| 144 | #127 | — | 8 | 1 | no | — |
| 145 | #124 | — | 4 | 2 | no | — |
| 146 | #121 | — | 4 | 2 | no | — |
| 147 | #121 | — | 4 | 2 | no | — |
| 148 | #120 | — | 8 | 2 | no | — |
| 149 | C4×Q8 | — | 4 | 2 | no | — |
| 150 | C4×D4 | — | 4 | 2 | no | — |
| 151 | C4×D4 | — | 8 | 2 | no | — |
| 152 | #117 | — | 2 | 1 | no | — |
| 153 | #117 | — | 2 | 1 | no | — |
| 154 | #117 | — | 4 | 2 | no | — |
| 155 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 156 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 157 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 158 | M32 | — | 2 | 2 | no | — |
| 159 | M32 | — | 2 | 2 | no | — |
| 160 | C2×C16 | — | 2 | 1 | no | — |
| 161 | C2×C16 | — | 2 | 1 | no | — |
| 162 | C2×C16 | — | 2 | 2 | no | — |
| 163 | C2×C16 | — | 2 | 1 | no | — |
| 164 | C2×C16 | — | 2 | 1 | no | — |
| 165 | C2×C16 | — | 2 | 1 | no | — |
| 166 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 167 | C2×C16 | C2×D4 | 1 | 1 | yes | characteristic |
| 168 | #108 | — | 2 | 2 | no | — |
| 169 | #108 | — | 2 | 1 | no | — |
| 170 | #108 | — | 2 | 1 | no | — |
| 171 | #108 | — | 2 | 1 | no | — |
| 172 | #108 | #41 | 1 | 2 | yes | — |
| 173 | #107 | — | 2 | 2 | no | — |
| 174 | #107 | — | 2 | 2 | no | — |
| 175 | #107 | — | 4 | 2 | no | — |
| 176 | #107 | C2×D4 | 1 | 2 | yes | — |
| 177 | #107 | C2×D4 | 1 | 2 | yes | — |
| 178 | #106 | — | 2 | 1 | no | — |
| 179 | #106 | — | 2 | 1 | no | — |
| 180 | #106 | — | 4 | 2 | no | — |
| 181 | #106 | — | 4 | 1 | no | — |
| 182 | #106 | — | 4 | 1 | no | — |
| 183 | #104 | — | 2 | 4 | no | — |
| 184 | #103 | — | 4 | 2 | no | — |
| 185 | #102 | — | 4 | 2 | no | — |
| 186 | #102 | — | 4 | 2 | no | — |
| 187 | #102 | — | 4 | 2 | no | — |
| 188 | #100 | — | 4 | 4 | no | — |
| 189 | #97 | — | 4 | 1 | no | — |
| 190 | #97 | — | 4 | 1 | no | — |
| 191 | C4×C8 | — | 4 | 1 | no | — |
| 192 | C4×C8 | — | 4 | 1 | no | — |
| 193 | C4×C8 | #41 | 1 | 2 | yes | — |
| 194 | C4×C8 | #41 | 1 | 2 | yes | — |
| 195 | #95 | — | 8 | 1 | no | — |
| 196 | C24 | — | 4 | 2 | no | — |
| 197 | #41 | — | 2 | 2 | no | — |
| 198 | #41 | — | 2 | 2 | no | — |
| 199 | #41 | — | 2 | 2 | no | — |
| 200 | #41 | — | 2 | 2 | no | — |
| 201 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 202 | C2×D4 | — | 2 | 2 | no | — |
| 203 | C2×D4 | — | 4 | 2 | no | — |
| 204 | C2×D4 | — | 4 | 2 | no | — |
| 205 | C2×D4 | — | 4 | 2 | no | — |
| 206 | C2×D4 | — | 4 | 2 | no | — |
| 207 | C2×D4 | — | 4 | 2 | no | — |
| 208 | C2×D4 | — | 4 | 2 | no | — |
| 209 | C2×D4 | — | 8 | 4 | no | — |
| 210 | C2×D4 | — | 8 | 2 | no | — |
| 211 | C2×D4 | — | 8 | 2 | no | — |
| 212 | C2×D4 | — | 16 | 4 | no | — |
| 213 | C2×D4 | C2×D8 | 1 | 2 | yes | — |
| 214 | C22×C4 | — | 2 | 1 | no | — |
| 215 | C22×C4 | — | 2 | 2 | no | — |
| 216 | C22×C4 | — | 4 | 2 | no | — |
| 217 | C22×C4 | — | 8 | 2 | no | — |
| 218 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 219 | Q16 | — | 2 | 2 | no | — |
| 220 | SD16 | — | 2 | 4 | no | — |
| 221 | SD16 | — | 8 | 4 | no | — |
| 222 | D8 | — | 2 | 2 | no | — |
| 223 | D8 | — | 4 | 2 | no | — |
| 224 | D8 | — | 4 | 2 | no | — |
| 225 | D8 | — | 4 | 2 | no | — |
| 226 | D8 | — | 4 | 2 | no | — |
| 227 | D8 | — | 8 | 4 | no | — |
| 228 | D8 | — | 8 | 2 | no | — |
| 229 | D8 | — | 8 | 2 | no | — |
| 230 | M16 | — | 2 | 2 | no | — |
| 231 | M16 | — | 2 | 2 | no | — |
| 232 | M16 | — | 2 | 2 | no | — |
| 233 | M16 | — | 4 | 1 | no | — |
| 234 | M16 | — | 4 | 2 | no | — |
| 235 | M16 | — | 4 | 2 | no | — |
| 236 | M16 | — | 4 | 1 | no | — |
| 237 | M16 | — | 4 | 1 | no | — |
| 238 | C2×C8 | — | 2 | 2 | no | — |
| 239 | C2×C8 | — | 4 | 1 | no | — |
| 240 | C2×C8 | — | 4 | 2 | no | — |
| 241 | C2×C8 | — | 4 | 2 | no | — |
| 242 | C2×C8 | — | 4 | 1 | no | — |
| 243 | C2×C8 | #123 | 1 | 2 | yes | — |
| 244 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 245 | C2×C8 | C2×D8 | 1 | 1 | yes | characteristic |
| 246 | C2×C8 | #137 | 1 | 2 | yes | — |
| 247 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 248 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 249 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 251 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 252 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 253 | C4⋊C4 | — | 2 | 2 | no | — |
| 254 | C4⋊C4 | — | 2 | 2 | no | — |
| 255 | C4⋊C4 | — | 4 | 2 | no | — |
| 256 | C4⋊C4 | — | 4 | 2 | no | — |
| 257 | C4⋊C4 | — | 8 | 2 | no | — |
| 258 | C22⋊C4 | — | 4 | 1 | no | — |
| 259 | C22⋊C4 | — | 4 | 1 | no | — |
| 260 | C22⋊C4 | — | 8 | 2 | no | — |
| 261 | C22⋊C4 | — | 8 | 2 | no | — |
| 262 | C22⋊C4 | — | 8 | 2 | no | — |
| 263 | C42 | — | 2 | 1 | no | — |
| 264 | C42 | — | 4 | 1 | no | — |
| 265 | C42 | — | 4 | 1 | no | — |
| 266 | C42 | — | 8 | 2 | no | — |
| 267 | C42 | #123 | 1 | 2 | yes | — |
| 268 | C16 | — | 2 | 1 | no | — |
| 269 | C16 | — | 2 | 1 | no | — |
| 270 | C16 | — | 4 | 1 | no | — |
| 271 | C16 | — | 4 | 2 | no | — |
| 272 | C23 | — | 2 | 2 | no | — |
| 273 | C23 | — | 8 | 2 | no | — |
| 274 | C23 | — | 8 | 2 | no | — |
| 275 | C23 | — | 8 | 2 | no | — |
| 276 | C23 | — | 16 | 2 | no | — |
| 277 | C23 | — | 16 | 2 | no | — |
| 278 | C23 | #576 | 1 | 1 | yes | characteristic |
| 279 | Q8 | — | 2 | 2 | no | — |
| 280 | Q8 | — | 2 | 2 | no | — |
| 281 | D4 | — | 2 | 2 | no | — |
| 282 | D4 | — | 2 | 2 | no | — |
| 283 | D4 | — | 4 | 2 | no | — |
| 284 | D4 | — | 4 | 2 | no | — |
| 285 | D4 | — | 8 | 2 | no | — |
| 286 | D4 | — | 8 | 2 | no | — |
| 287 | D4 | — | 8 | 2 | no | — |
| 288 | D4 | — | 8 | 2 | no | — |
| 289 | D4 | — | 16 | 8 | no | — |
| 290 | D4 | — | 32 | 2 | no | — |
| 291 | D4 | — | 32 | 2 | no | — |
| 292 | C2×C4 | — | 2 | 2 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 4 | 2 | no | — |
| 296 | C2×C4 | — | 4 | 2 | no | — |
| 297 | C2×C4 | — | 4 | 2 | no | — |
| 298 | C2×C4 | — | 4 | 2 | no | — |
| 299 | C2×C4 | — | 8 | 2 | no | — |
| 300 | C2×C4 | — | 16 | 2 | no | — |
| 301 | C2×C4 | — | 32 | 2 | no | — |
| 302 | C2×C4 | C2×#123 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | C2×#137 | 1 | 1 | yes | characteristic |
| 307 | C2×C4 | #577 | 1 | 1 | yes | characteristic |
| 308 | C8 | — | 4 | 2 | no | — |
| 309 | C8 | — | 4 | 2 | no | — |
| 310 | C8 | — | 8 | 1 | no | — |
| 311 | C8 | #480 | 1 | 2 | yes | — |
| 312 | C8 | #480 | 1 | 2 | yes | — |
| 313 | C8 | #532 | 1 | 1 | yes | characteristic |
| 314 | C8 | #540 | 1 | 1 | yes | characteristic |
| 315 | C8 | #550 | 1 | 1 | yes | characteristic |
| 316 | C8 | #559 | 1 | 1 | yes | characteristic |
| 317 | C22 | — | 2 | 2 | no | — |
| 318 | C22 | — | 4 | 2 | no | — |
| 319 | C22 | — | 4 | 2 | no | — |
| 320 | C22 | — | 4 | 2 | no | — |
| 321 | C22 | — | 16 | 4 | no | — |
| 322 | C22 | — | 16 | 2 | no | — |
| 323 | C22 | — | 16 | 2 | no | — |
| 324 | C22 | — | 16 | 2 | no | — |
| 325 | C22 | — | 32 | 2 | no | — |
| 326 | C22 | #3096 | 1 | 1 | yes | characteristic |
| 327 | C22 | D4×D8 | 1 | 1 | yes | characteristic |
| 328 | C22 | #3333 | 1 | 1 | yes | characteristic |
| 329 | C4 | — | 4 | 2 | no | — |
| 330 | C4 | — | 4 | 2 | no | — |
| 331 | C4 | — | 8 | 2 | no | — |
| 332 | C4 | — | 8 | 2 | no | — |
| 333 | C4 | — | 16 | 2 | no | — |
| 334 | C4 | #2445 | 1 | 1 | yes | characteristic |
| 335 | C4 | C2×#480 | 1 | 1 | yes | characteristic |
| 336 | C4 | #3278 | 1 | 1 | yes | characteristic |
| 337 | C4 | #3294 | 1 | 1 | yes | characteristic |
| 338 | C2 | — | 2 | 1 | no | — |
| 339 | C2 | — | 2 | 1 | no | — |
| 340 | C2 | — | 8 | 2 | no | — |
| 341 | C2 | — | 32 | 2 | no | — |
| 342 | C2 | #26230 | 1 | 1 | yes | characteristic |
| 343 | C2 | #33854 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #28504 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#443 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33650 | C2 | semidirect |
| 3. #22691 | C2 | semidirect |
| 4. #13393 | C2 | semidirect |
| 5. #13387 | C2 | semidirect |
| 6. #13261 | C2 | semidirect |
| 7. #13206 | C2 | semidirect |
| 8. #13196 | C2 | semidirect |
| 9. #13180 | C2 | semidirect |
| 10. #12996 | C2 | semidirect |
| 11. #12671 | C2 | semidirect |
| 92. #367 | D4 | semidirect |
| 94. #366 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33650 | 2 |
| #28504 | 2 |
| #22691 | 2 |
| #13393 | 2 |
| #13387 | 2 |
| #13261 | 2 |
| #13206 | 2 |
| #13196 | 2 |
| #13180 | 2 |
| #12996 | 2 |
| #12671 | 2 |