← #498106
Subgroups — #498106
Order 512.
722 conjugacy classes of subgroups.
The subgroup lattice has 351 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #33442 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33340 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #21477 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #13192 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #13123 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #12987 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #12982 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #12918 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #12615 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #3573 | C22 | 1 | 1 | yes | characteristic |
| 11 | #3407 | C22 | 1 | 2 | yes | — |
| 12 | #3407 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#504 | C22 | 1 | 2 | yes | — |
| 14 | #3018 | C4 | 1 | 1 | yes | characteristic |
| 15 | #2955 | C22 | 1 | 2 | yes | — |
| 16 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2171 | — | 2 | 4 | no | — |
| 18 | #2170 | — | 2 | 4 | no | — |
| 19 | #2153 | C22 | 1 | 2 | yes | complemented |
| 20 | #2150 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | #2117 | — | 2 | 4 | no | — |
| 22 | #2117 | C22 | 1 | 2 | yes | — |
| 23 | #2117 | C22 | 1 | 2 | yes | — |
| 24 | C2×#350 | — | 2 | 4 | no | — |
| 25 | C2×#350 | C22 | 1 | 2 | yes | — |
| 26 | C2×#350 | C22 | 1 | 2 | yes | — |
| 27 | C2×#350 | C22 | 1 | 2 | yes | complemented |
| 28 | #2115 | C22 | 1 | 1 | yes | characteristic, complemented |
| 29 | #2108 | C22 | 1 | 2 | yes | complemented |
| 30 | C2×#346 | C22 | 1 | 1 | yes | characteristic, complemented |
| 31 | #1884 | C4 | 1 | 1 | yes | characteristic |
| 32 | #1881 | C4 | 1 | 2 | yes | — |
| 33 | #1861 | C4 | 1 | 2 | yes | — |
| 34 | #1858 | C4 | 1 | 1 | yes | characteristic |
| 35 | #1794 | C4 | 1 | 1 | yes | characteristic |
| 36 | #1764 | C22 | 1 | 2 | yes | — |
| 37 | #1757 | C22 | 1 | 1 | yes | characteristic |
| 38 | #1379 | C22 | 1 | 2 | yes | complemented |
| 39 | #1378 | C22 | 1 | 2 | yes | complemented |
| 40 | #1376 | C22 | 1 | 2 | yes | complemented |
| 41 | #585 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | #575 | C2×C4 | 1 | 2 | yes | — |
| 43 | #568 | — | 2 | 4 | no | — |
| 44 | #568 | — | 4 | 1 | no | — |
| 45 | #568 | C2×C4 | 1 | 2 | yes | — |
| 46 | #568 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C2×#131 | — | 2 | 4 | no | — |
| 48 | C2×#131 | — | 4 | 1 | no | — |
| 49 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 51 | C2×#131 | C23 | 1 | 2 | yes | — |
| 52 | #532 | C2×C4 | 1 | 1 | yes | characteristic |
| 53 | #504 | — | 2 | 4 | no | — |
| 54 | #504 | — | 2 | 4 | no | — |
| 55 | #504 | — | 2 | 4 | no | — |
| 56 | #504 | — | 4 | 2 | no | — |
| 57 | #504 | — | 4 | 2 | no | — |
| 58 | C2×M32 | — | 2 | 2 | no | — |
| 59 | C2×M32 | — | 2 | 4 | no | — |
| 60 | C2×M32 | — | 4 | 1 | no | — |
| 61 | C2×M32 | C23 | 1 | 2 | yes | — |
| 62 | C2×M32 | C23 | 1 | 2 | yes | — |
| 63 | C2×M32 | C23 | 1 | 2 | yes | — |
| 64 | C22×C16 | — | 2 | 2 | no | — |
| 65 | C22×C16 | C23 | 1 | 2 | yes | — |
| 66 | #456 | C8 | 1 | 2 | yes | — |
| 67 | #455 | C8 | 1 | 2 | yes | — |
| 68 | #454 | C8 | 1 | 2 | yes | — |
| 69 | #453 | C8 | 1 | 2 | yes | — |
| 70 | #444 | — | 2 | 4 | no | — |
| 71 | #443 | — | 2 | 4 | no | — |
| 72 | #436 | — | 4 | 1 | no | — |
| 73 | C8×D4 | — | 4 | 1 | no | — |
| 74 | #430 | C2×C4 | 1 | 2 | yes | — |
| 75 | #423 | C2×C4 | 1 | 1 | yes | characteristic |
| 76 | #421 | C2×C4 | 1 | 2 | yes | — |
| 77 | #421 | C2×C4 | 1 | 2 | yes | — |
| 78 | C2×#104 | C2×C4 | 1 | 2 | yes | — |
| 79 | #416 | C2×C4 | 1 | 2 | yes | — |
| 80 | #413 | C2×C4 | 1 | 1 | yes | characteristic |
| 81 | #410 | C2×C4 | 1 | 1 | yes | characteristic |
| 82 | #408 | C2×C4 | 1 | 1 | yes | characteristic |
| 83 | #408 | C2×C4 | 1 | 1 | yes | characteristic |
| 84 | #405 | C23 | 1 | 2 | yes | — |
| 85 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 86 | #364 | — | 2 | 4 | no | — |
| 87 | C4⋊C16 | — | 4 | 1 | no | — |
| 88 | #350 | — | 2 | 4 | no | — |
| 89 | #350 | — | 2 | 4 | no | — |
| 90 | #350 | — | 2 | 2 | no | — |
| 91 | #350 | — | 2 | 2 | no | — |
| 92 | #350 | — | 2 | 2 | no | — |
| 93 | #350 | — | 2 | 2 | no | — |
| 94 | #350 | D4 | 1 | 4 | yes | complemented |
| 95 | #350 | D4 | 1 | 4 | yes | complemented |
| 96 | #349 | — | 4 | 1 | no | — |
| 97 | #346 | — | 2 | 2 | no | — |
| 98 | #346 | — | 4 | 1 | no | — |
| 99 | C4×C16 | — | 2 | 2 | no | — |
| 100 | #344 | C2×C4 | 1 | 2 | yes | — |
| 101 | #341 | C2×C4 | 1 | 2 | yes | — |
| 102 | #339 | C2×C4 | 1 | 2 | yes | — |
| 103 | 2−1+4 | — | 2 | 2 | no | — |
| 104 | 2−1+4 | C2×C8 | 1 | 2 | yes | — |
| 105 | H(5,𝔽2) | — | 2 | 2 | no | — |
| 106 | H(5,𝔽2) | — | 4 | 1 | no | — |
| 107 | H(5,𝔽2) | C2×C8 | 1 | 2 | yes | — |
| 108 | C2×#41 | — | 2 | 4 | no | — |
| 109 | C2×#41 | — | 4 | 1 | no | — |
| 110 | C2×#41 | C22×C4 | 1 | 1 | yes | characteristic |
| 111 | C2×#41 | C22×C4 | 1 | 1 | yes | characteristic |
| 112 | C2×#41 | C22×C4 | 1 | 2 | yes | — |
| 113 | #137 | — | 2 | 2 | no | — |
| 114 | #137 | C2×C8 | 1 | 4 | yes | — |
| 115 | AGL(1,ℤ/8) | — | 2 | 2 | no | — |
| 116 | AGL(1,ℤ/8) | C2×C8 | 1 | 4 | yes | — |
| 117 | #135 | — | 2 | 4 | no | — |
| 118 | #131 | — | 2 | 2 | no | — |
| 119 | #131 | — | 2 | 2 | no | — |
| 120 | #131 | — | 2 | 4 | no | — |
| 121 | #131 | — | 2 | 4 | no | — |
| 122 | #131 | — | 2 | 2 | no | — |
| 123 | #131 | — | 2 | 2 | no | — |
| 124 | #131 | — | 2 | 2 | no | — |
| 125 | #131 | — | 2 | 2 | no | — |
| 126 | #131 | — | 4 | 2 | no | — |
| 127 | #131 | — | 4 | 2 | no | — |
| 128 | #131 | — | 4 | 4 | no | — |
| 129 | #131 | — | 4 | 2 | no | — |
| 130 | #131 | C2×D4 | 1 | 4 | yes | — |
| 131 | #131 | #41 | 1 | 4 | yes | — |
| 132 | C2×M16 | — | 2 | 2 | no | — |
| 133 | C2×M16 | — | 2 | 1 | no | — |
| 134 | C2×M16 | — | 2 | 1 | no | — |
| 135 | C2×M16 | — | 2 | 4 | no | — |
| 136 | C2×M16 | — | 4 | 4 | no | — |
| 137 | C2×M16 | — | 4 | 1 | no | — |
| 138 | C2×M16 | — | 4 | 2 | no | — |
| 139 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 140 | C2×M16 | C22×C4 | 1 | 1 | yes | characteristic |
| 141 | C2×M16 | C22×C4 | 1 | 2 | yes | — |
| 142 | C2×M16 | C22×C4 | 1 | 2 | yes | — |
| 143 | C22×C8 | — | 2 | 2 | no | — |
| 144 | C22×C8 | — | 2 | 1 | no | — |
| 145 | C22×C8 | — | 2 | 1 | no | — |
| 146 | C22×C8 | — | 4 | 2 | no | — |
| 147 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 148 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 149 | C4⋊Q8 | C2×C8 | 1 | 1 | yes | characteristic |
| 150 | #127 | C2×C8 | 1 | 1 | yes | characteristic |
| 151 | #124 | C2×C8 | 1 | 1 | yes | characteristic |
| 152 | #124 | C2×C8 | 1 | 1 | yes | characteristic |
| 153 | #121 | — | 4 | 1 | no | — |
| 154 | C4×D4 | — | 4 | 1 | no | — |
| 155 | #117 | C22×C4 | 1 | 2 | yes | — |
| 156 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 157 | M32 | — | 2 | 2 | no | — |
| 158 | M32 | — | 2 | 2 | no | — |
| 159 | M32 | — | 2 | 2 | no | — |
| 160 | M32 | — | 2 | 2 | no | — |
| 161 | M32 | — | 2 | 2 | no | — |
| 162 | M32 | — | 4 | 2 | no | — |
| 163 | M32 | — | 4 | 4 | no | — |
| 164 | M32 | — | 4 | 4 | no | — |
| 165 | M32 | C2×D4 | 1 | 4 | yes | — |
| 166 | C2×C16 | — | 2 | 2 | no | — |
| 167 | C2×C16 | — | 2 | 2 | no | — |
| 168 | C2×C16 | — | 2 | 2 | no | — |
| 169 | C2×C16 | — | 2 | 2 | no | — |
| 170 | C2×C16 | — | 2 | 2 | no | — |
| 171 | C2×C16 | — | 4 | 2 | no | — |
| 172 | C2×C16 | — | 4 | 4 | no | — |
| 173 | C2×C16 | — | 4 | 4 | no | — |
| 174 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 175 | C2×C16 | C2×D4 | 1 | 2 | yes | — |
| 176 | #108 | — | 2 | 4 | no | — |
| 177 | C4⋊C8 | — | 4 | 1 | no | — |
| 178 | #104 | — | 2 | 4 | no | — |
| 179 | #104 | — | 2 | 4 | no | — |
| 180 | #104 | C2×C8 | 1 | 4 | yes | — |
| 181 | #104 | C2×C8 | 1 | 4 | yes | — |
| 182 | #103 | — | 2 | 2 | no | — |
| 183 | #102 | — | 2 | 2 | no | — |
| 184 | #101 | C2×C8 | 1 | 2 | yes | — |
| 185 | #100 | C2×C8 | 1 | 2 | yes | — |
| 186 | #99 | — | 4 | 1 | no | — |
| 187 | C22⋊C8 | — | 4 | 1 | no | — |
| 188 | C22⋊C8 | — | 4 | 1 | no | — |
| 189 | #97 | — | 2 | 2 | no | — |
| 190 | C4×C8 | — | 2 | 2 | no | — |
| 191 | C4×C8 | — | 2 | 1 | no | — |
| 192 | C4×C8 | — | 2 | 1 | no | — |
| 193 | #41 | — | 2 | 2 | no | — |
| 194 | #41 | — | 2 | 2 | no | — |
| 195 | #41 | — | 2 | 4 | no | — |
| 196 | #41 | — | 2 | 4 | no | — |
| 197 | #41 | — | 2 | 2 | no | — |
| 198 | #41 | — | 2 | 2 | no | — |
| 199 | #41 | — | 2 | 2 | no | — |
| 200 | #41 | — | 2 | 2 | no | — |
| 201 | #41 | — | 4 | 2 | no | — |
| 202 | #41 | — | 4 | 2 | no | — |
| 203 | #41 | — | 4 | 4 | no | — |
| 204 | #41 | — | 4 | 2 | no | — |
| 205 | #41 | C4×D4 | 1 | 4 | yes | — |
| 206 | #41 | C22×C8 | 1 | 4 | yes | — |
| 207 | C2×Q8 | — | 2 | 4 | no | — |
| 208 | C2×Q8 | — | 4 | 1 | no | — |
| 209 | C2×Q8 | C22×C8 | 1 | 1 | yes | characteristic |
| 210 | C2×Q8 | C22×C8 | 1 | 1 | yes | characteristic |
| 211 | C2×Q8 | #131 | 1 | 2 | yes | — |
| 212 | C2×D4 | — | 2 | 2 | no | — |
| 213 | C2×D4 | — | 2 | 4 | no | — |
| 214 | C2×D4 | — | 2 | 1 | no | — |
| 215 | C2×D4 | — | 2 | 1 | no | — |
| 216 | C2×D4 | — | 4 | 1 | no | — |
| 217 | C2×D4 | — | 4 | 2 | no | — |
| 218 | C2×D4 | — | 4 | 1 | no | — |
| 219 | C2×D4 | — | 4 | 4 | no | — |
| 220 | C2×D4 | C22×C8 | 1 | 1 | yes | characteristic |
| 221 | C2×D4 | C22×C8 | 1 | 1 | yes | characteristic |
| 222 | C2×D4 | #131 | 1 | 2 | yes | — |
| 223 | C22×C4 | — | 2 | 2 | no | — |
| 224 | C22×C4 | — | 2 | 1 | no | — |
| 225 | C22×C4 | — | 2 | 1 | no | — |
| 226 | C22×C4 | — | 2 | 1 | no | — |
| 227 | C22×C4 | — | 4 | 2 | no | — |
| 228 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 229 | Q16 | — | 2 | 4 | no | — |
| 230 | SD16 | — | 2 | 4 | no | — |
| 231 | SD16 | — | 2 | 4 | no | — |
| 232 | D8 | — | 2 | 4 | no | — |
| 233 | M16 | — | 2 | 2 | no | — |
| 234 | M16 | — | 2 | 2 | no | — |
| 235 | M16 | — | 2 | 4 | no | — |
| 236 | M16 | — | 2 | 4 | no | — |
| 237 | M16 | — | 2 | 2 | no | — |
| 238 | M16 | — | 2 | 2 | no | — |
| 239 | M16 | — | 4 | 2 | no | — |
| 240 | M16 | — | 4 | 4 | no | — |
| 241 | M16 | — | 4 | 2 | no | — |
| 242 | M16 | — | 4 | 2 | no | — |
| 243 | M16 | C4×D4 | 1 | 4 | yes | — |
| 244 | M16 | C22×C8 | 1 | 4 | yes | — |
| 245 | C2×C8 | — | 2 | 2 | no | — |
| 246 | C2×C8 | — | 2 | 2 | no | — |
| 247 | C2×C8 | — | 2 | 4 | no | — |
| 248 | C2×C8 | — | 2 | 4 | no | — |
| 249 | C2×C8 | — | 2 | 1 | no | — |
| 250 | C2×C8 | — | 2 | 1 | no | — |
| 251 | C2×C8 | — | 2 | 1 | no | — |
| 252 | C2×C8 | — | 2 | 2 | no | — |
| 253 | C2×C8 | — | 4 | 2 | no | — |
| 254 | C2×C8 | — | 4 | 4 | no | — |
| 255 | C2×C8 | — | 4 | 1 | no | — |
| 256 | C2×C8 | — | 4 | 1 | no | — |
| 257 | C2×C8 | — | 4 | 1 | no | — |
| 258 | C2×C8 | — | 4 | 1 | no | — |
| 259 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 260 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 261 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 262 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 263 | C4⋊C4 | — | 2 | 2 | no | — |
| 264 | C4⋊C4 | — | 4 | 1 | no | — |
| 265 | C22⋊C4 | — | 2 | 2 | no | — |
| 266 | C22⋊C4 | — | 4 | 1 | no | — |
| 267 | C22⋊C4 | — | 4 | 1 | no | — |
| 268 | C42 | — | 2 | 1 | no | — |
| 269 | C42 | — | 2 | 2 | no | — |
| 270 | C42 | C22×C8 | 1 | 1 | yes | characteristic |
| 271 | C42 | C22×C8 | 1 | 1 | yes | characteristic |
| 272 | C16 | — | 2 | 4 | no | — |
| 273 | C16 | — | 2 | 2 | no | — |
| 274 | C16 | — | 2 | 2 | no | — |
| 275 | C16 | — | 4 | 2 | no | — |
| 276 | C16 | — | 4 | 2 | no | — |
| 277 | C23 | — | 2 | 2 | no | — |
| 278 | C23 | — | 2 | 1 | no | — |
| 279 | C23 | — | 2 | 1 | no | — |
| 280 | C23 | — | 4 | 2 | no | — |
| 281 | C23 | #568 | 1 | 1 | yes | characteristic |
| 282 | Q8 | — | 2 | 2 | no | — |
| 283 | Q8 | — | 2 | 2 | no | — |
| 284 | Q8 | — | 2 | 2 | no | — |
| 285 | Q8 | — | 4 | 2 | no | — |
| 286 | Q8 | C8×D4 | 1 | 4 | yes | — |
| 287 | D4 | — | 2 | 4 | no | — |
| 288 | D4 | — | 2 | 4 | no | — |
| 289 | D4 | — | 2 | 2 | no | — |
| 290 | D4 | — | 2 | 2 | no | — |
| 291 | D4 | — | 2 | 2 | no | — |
| 292 | D4 | — | 4 | 4 | no | — |
| 293 | D4 | — | 4 | 2 | no | — |
| 294 | D4 | — | 4 | 2 | no | — |
| 295 | D4 | — | 4 | 2 | no | — |
| 296 | D4 | — | 8 | 2 | no | — |
| 297 | D4 | C8×D4 | 1 | 4 | yes | — |
| 298 | C2×C4 | — | 2 | 2 | no | — |
| 299 | C2×C4 | — | 2 | 4 | no | — |
| 300 | C2×C4 | — | 2 | 4 | no | — |
| 301 | C2×C4 | — | 2 | 1 | no | — |
| 302 | C2×C4 | — | 2 | 1 | no | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 1 | no | — |
| 305 | C2×C4 | — | 4 | 2 | no | — |
| 306 | C2×C4 | — | 4 | 4 | no | — |
| 307 | C2×C4 | — | 4 | 1 | no | — |
| 308 | C2×C4 | — | 4 | 1 | no | — |
| 309 | C2×C4 | — | 4 | 1 | no | — |
| 310 | C2×C4 | — | 4 | 1 | no | — |
| 311 | C2×C4 | — | 4 | 2 | no | — |
| 312 | C2×C4 | — | 8 | 1 | no | — |
| 313 | C2×C4 | C2×C4×D4 | 1 | 2 | yes | — |
| 314 | C2×C4 | #518 | 1 | 1 | yes | characteristic |
| 315 | C2×C4 | C23×C8 | 1 | 1 | yes | characteristic |
| 316 | C2×C4 | C2×#131 | 1 | 2 | yes | — |
| 317 | C8 | — | 2 | 4 | no | — |
| 318 | C8 | — | 2 | 4 | no | — |
| 319 | C8 | — | 4 | 1 | no | — |
| 320 | C8 | — | 4 | 1 | no | — |
| 321 | C8 | — | 4 | 1 | no | — |
| 322 | C8 | D42 | 1 | 1 | yes | characteristic |
| 323 | C8 | #546 | 1 | 2 | yes | — |
| 324 | C8 | #551 | 1 | 1 | yes | characteristic |
| 325 | C22 | — | 2 | 4 | no | — |
| 326 | C22 | — | 4 | 1 | no | — |
| 327 | C22 | — | 4 | 1 | no | — |
| 328 | C22 | — | 4 | 1 | no | — |
| 329 | C22 | — | 8 | 2 | no | — |
| 330 | C22 | — | 8 | 1 | no | — |
| 331 | C22 | — | 8 | 1 | no | — |
| 332 | C22 | — | 16 | 2 | no | — |
| 333 | C22 | #2933 | 1 | 2 | yes | — |
| 334 | C22 | #2972 | 1 | 1 | yes | characteristic |
| 335 | C4 | — | 2 | 4 | no | — |
| 336 | C4 | — | 4 | 1 | no | — |
| 337 | C4 | — | 4 | 1 | no | — |
| 338 | C4 | — | 4 | 1 | no | — |
| 339 | C4 | — | 4 | 1 | no | — |
| 340 | C4 | — | 8 | 2 | no | — |
| 341 | C4 | #2358 | 1 | 1 | yes | characteristic |
| 342 | C4 | C2×C8×D4 | 1 | 2 | yes | — |
| 343 | C4 | #2977 | 1 | 1 | yes | characteristic |
| 344 | C2 | — | 2 | 1 | no | — |
| 345 | C2 | — | 2 | 2 | no | — |
| 346 | C2 | — | 4 | 4 | no | — |
| 347 | C2 | — | 8 | 1 | no | — |
| 348 | C2 | — | 8 | 1 | no | — |
| 349 | C2 | #22842 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta