← #62274
Subgroups — #62274
Order 256.
906 conjugacy classes of subgroups.
The subgroup lattice has 308 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #3539 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #3337 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #3315 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #3309 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #3284 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #3261 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #3259 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3203 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #3197 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #3174 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | #3060 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #3050 | C2 | 1 | 2 | yes | maximal, complemented |
| 13 | #2929 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | C2×#135 | C22 | 1 | 2 | yes | — |
| 15 | C2×#135 | C22 | 1 | 2 | yes | complemented |
| 16 | C2×#131 | C22 | 1 | 1 | yes | characteristic |
| 17 | C23×C8 | C22 | 1 | 1 | yes | characteristic |
| 18 | #560 | — | 2 | 4 | no | — |
| 19 | #553 | — | 2 | 4 | no | — |
| 20 | #553 | C22 | 1 | 4 | yes | — |
| 21 | #552 | C22 | 1 | 4 | yes | — |
| 22 | #550 | C22 | 1 | 4 | yes | — |
| 23 | #547 | C22 | 1 | 4 | yes | — |
| 24 | #541 | C22 | 1 | 2 | yes | — |
| 25 | #540 | — | 2 | 4 | no | — |
| 26 | #540 | C22 | 1 | 2 | yes | — |
| 27 | #539 | — | 2 | 4 | no | — |
| 28 | #537 | C22 | 1 | 4 | yes | — |
| 29 | #525 | C22 | 1 | 4 | yes | — |
| 30 | #525 | C22 | 1 | 1 | yes | characteristic |
| 31 | #499 | C22 | 1 | 2 | yes | complemented |
| 32 | #492 | C22 | 1 | 2 | yes | complemented |
| 33 | #491 | C22 | 1 | 2 | yes | complemented |
| 34 | #490 | C22 | 1 | 2 | yes | complemented |
| 35 | #485 | C22 | 1 | 4 | yes | — |
| 36 | #484 | C22 | 1 | 4 | yes | complemented |
| 37 | #482 | C22 | 1 | 4 | yes | — |
| 38 | #480 | C22 | 1 | 4 | yes | complemented |
| 39 | #479 | C22 | 1 | 4 | yes | complemented |
| 40 | #478 | C22 | 1 | 4 | yes | complemented |
| 41 | #470 | C22 | 1 | 2 | yes | — |
| 42 | #470 | C22 | 1 | 2 | yes | complemented |
| 43 | #469 | C22 | 1 | 2 | yes | — |
| 44 | #469 | C22 | 1 | 2 | yes | complemented |
| 45 | #468 | C22 | 1 | 4 | yes | — |
| 46 | #467 | C22 | 1 | 2 | yes | — |
| 47 | #467 | C22 | 1 | 2 | yes | — |
| 48 | #466 | C22 | 1 | 2 | yes | — |
| 49 | #466 | C22 | 1 | 2 | yes | — |
| 50 | #465 | C22 | 1 | 4 | yes | — |
| 51 | #465 | C22 | 1 | 4 | yes | — |
| 52 | #465 | C22 | 1 | 4 | yes | complemented |
| 53 | #462 | C22 | 1 | 4 | yes | complemented |
| 54 | #459 | C22 | 1 | 4 | yes | complemented |
| 55 | #452 | C22 | 1 | 4 | yes | — |
| 56 | #452 | C22 | 1 | 4 | yes | complemented |
| 57 | #449 | C22 | 1 | 4 | yes | — |
| 58 | #449 | C22 | 1 | 4 | yes | complemented |
| 59 | #440 | C22 | 1 | 4 | yes | complemented |
| 60 | C4×SD16 | C22 | 1 | 4 | yes | complemented |
| 61 | #435 | C22 | 1 | 4 | yes | — |
| 62 | C8×D4 | C22 | 1 | 4 | yes | — |
| 63 | #427 | C22 | 1 | 2 | yes | — |
| 64 | C2×#106 | C22 | 1 | 2 | yes | complemented |
| 65 | #424 | C22 | 1 | 1 | yes | characteristic |
| 66 | #417 | C22 | 1 | 4 | yes | complemented |
| 67 | #416 | C22 | 1 | 4 | yes | — |
| 68 | #408 | C22 | 1 | 1 | yes | characteristic |
| 69 | #407 | C22 | 1 | 1 | yes | characteristic |
| 70 | #405 | C22 | 1 | 1 | yes | characteristic |
| 71 | C2×#41 | C23 | 1 | 4 | yes | — |
| 72 | C2×#41 | C23 | 1 | 4 | yes | — |
| 73 | C2×#41 | C23 | 1 | 1 | yes | characteristic |
| 74 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 75 | #135 | — | 4 | 4 | no | — |
| 76 | #135 | — | 4 | 4 | no | — |
| 77 | C2×Q16 | C23 | 1 | 2 | yes | — |
| 78 | C2×Q16 | C23 | 1 | 2 | yes | — |
| 79 | C2×SD16 | C23 | 1 | 4 | yes | — |
| 80 | C2×SD16 | C23 | 1 | 4 | yes | — |
| 81 | C2×D8 | C23 | 1 | 2 | yes | — |
| 82 | C2×D8 | C23 | 1 | 2 | yes | — |
| 83 | #131 | — | 4 | 2 | no | — |
| 84 | C2×M16 | C23 | 1 | 1 | yes | characteristic |
| 85 | C2×M16 | C23 | 1 | 2 | yes | — |
| 86 | C2×M16 | C23 | 1 | 1 | yes | characteristic |
| 87 | C22×C8 | — | 4 | 2 | no | — |
| 88 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 89 | C22×C8 | C23 | 1 | 2 | yes | — |
| 90 | C22×C8 | C23 | 1 | 2 | yes | — |
| 91 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 92 | C22×C8 | C23 | 1 | 2 | yes | — |
| 93 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 94 | #127 | — | 2 | 4 | no | — |
| 95 | #127 | C23 | 1 | 2 | yes | — |
| 96 | #126 | — | 2 | 4 | no | — |
| 97 | #126 | — | 2 | 4 | no | — |
| 98 | #126 | — | 2 | 4 | no | — |
| 99 | #125 | — | 2 | 4 | no | — |
| 100 | #125 | — | 2 | 4 | no | — |
| 101 | #125 | C23 | 1 | 2 | yes | — |
| 102 | #125 | C23 | 1 | 2 | yes | — |
| 103 | #123 | — | 2 | 4 | no | — |
| 104 | #123 | — | 2 | 4 | no | — |
| 105 | #123 | — | 2 | 4 | no | — |
| 106 | #123 | — | 2 | 4 | no | — |
| 107 | #123 | — | 2 | 4 | no | — |
| 108 | #123 | — | 2 | 4 | no | — |
| 109 | #123 | D4 | 1 | 2 | yes | — |
| 110 | C22⋊Q8 | — | 2 | 4 | no | — |
| 111 | C22⋊Q8 | D4 | 1 | 2 | yes | — |
| 112 | C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 113 | C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 114 | C22⋊Q8 | C23 | 1 | 4 | yes | — |
| 115 | #121 | — | 2 | 4 | no | — |
| 116 | #121 | — | 2 | 4 | no | — |
| 117 | #121 | — | 2 | 4 | no | — |
| 118 | #121 | — | 2 | 4 | no | — |
| 119 | #121 | — | 2 | 4 | no | — |
| 120 | #121 | — | 2 | 4 | no | — |
| 121 | #121 | — | 2 | 4 | no | — |
| 122 | #121 | — | 2 | 4 | no | — |
| 123 | #121 | D4 | 1 | 2 | yes | — |
| 124 | #121 | C23 | 1 | 4 | yes | — |
| 125 | #121 | C23 | 1 | 4 | yes | — |
| 126 | #121 | C23 | 1 | 4 | yes | — |
| 127 | #120 | — | 2 | 4 | no | — |
| 128 | #120 | D4 | 1 | 2 | yes | — |
| 129 | C4×Q8 | C23 | 1 | 4 | yes | — |
| 130 | C4×D4 | — | 2 | 4 | no | — |
| 131 | C4×D4 | — | 2 | 4 | no | — |
| 132 | C4×D4 | — | 2 | 4 | no | — |
| 133 | C4×D4 | — | 2 | 4 | no | — |
| 134 | C4×D4 | C23 | 1 | 4 | yes | — |
| 135 | C4×D4 | C23 | 1 | 4 | yes | — |
| 136 | #117 | — | 2 | 4 | no | — |
| 137 | #117 | C23 | 1 | 4 | yes | — |
| 138 | #117 | C23 | 1 | 1 | yes | characteristic |
| 139 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 140 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 141 | #107 | C23 | 1 | 2 | yes | — |
| 142 | #107 | C23 | 1 | 2 | yes | — |
| 143 | #107 | C23 | 1 | 4 | yes | — |
| 144 | #106 | C23 | 1 | 4 | yes | — |
| 145 | #106 | C23 | 1 | 2 | yes | — |
| 146 | #106 | C23 | 1 | 2 | yes | — |
| 147 | C4⋊C8 | C23 | 1 | 4 | yes | — |
| 148 | #103 | C23 | 1 | 4 | yes | — |
| 149 | #103 | C23 | 1 | 4 | yes | — |
| 150 | #103 | C23 | 1 | 4 | yes | — |
| 151 | #103 | C23 | 1 | 4 | yes | — |
| 152 | #102 | C23 | 1 | 4 | yes | — |
| 153 | #102 | C23 | 1 | 4 | yes | — |
| 154 | #102 | C23 | 1 | 4 | yes | — |
| 155 | #102 | C23 | 1 | 4 | yes | — |
| 156 | C22⋊C8 | C23 | 1 | 4 | yes | — |
| 157 | C22⋊C8 | C23 | 1 | 4 | yes | — |
| 158 | #97 | C23 | 1 | 2 | yes | — |
| 159 | C4×C8 | C23 | 1 | 2 | yes | — |
| 160 | C24 | C2×D4 | 1 | 1 | yes | characteristic |
| 161 | #41 | — | 4 | 1 | no | — |
| 162 | #41 | — | 4 | 1 | no | — |
| 163 | #41 | — | 4 | 4 | no | — |
| 164 | #41 | — | 4 | 4 | no | — |
| 165 | #41 | — | 4 | 4 | no | — |
| 166 | #41 | — | 4 | 4 | no | — |
| 167 | C2×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 168 | C2×Q8 | C24 | 1 | 4 | yes | — |
| 169 | C2×D4 | — | 2 | 4 | no | — |
| 170 | C2×D4 | — | 2 | 4 | no | — |
| 171 | C2×D4 | — | 2 | 4 | no | — |
| 172 | C2×D4 | — | 2 | 4 | no | — |
| 173 | C2×D4 | — | 2 | 4 | no | — |
| 174 | C2×D4 | — | 2 | 4 | no | — |
| 175 | C2×D4 | C2×D4 | 1 | 4 | yes | — |
| 176 | C2×D4 | C2×D4 | 1 | 2 | yes | — |
| 177 | C2×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 178 | C2×D4 | C24 | 1 | 4 | yes | — |
| 179 | C22×C4 | — | 2 | 4 | no | — |
| 180 | C22×C4 | — | 2 | 4 | no | — |
| 181 | C22×C4 | — | 2 | 4 | no | — |
| 182 | C22×C4 | — | 2 | 4 | no | — |
| 183 | C22×C4 | — | 4 | 1 | no | — |
| 184 | C22×C4 | — | 4 | 1 | no | — |
| 185 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 186 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 187 | C22×C4 | C24 | 1 | 2 | yes | — |
| 188 | C22×C4 | C24 | 1 | 2 | yes | — |
| 189 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 190 | Q16 | — | 4 | 2 | no | — |
| 191 | Q16 | — | 4 | 2 | no | — |
| 192 | SD16 | — | 2 | 8 | no | — |
| 193 | SD16 | — | 4 | 4 | no | — |
| 194 | D8 | — | 4 | 2 | no | — |
| 195 | D8 | — | 4 | 2 | no | — |
| 196 | M16 | — | 4 | 1 | no | — |
| 197 | M16 | — | 4 | 2 | no | — |
| 198 | M16 | — | 4 | 1 | no | — |
| 199 | C2×C8 | — | 2 | 2 | no | — |
| 200 | C2×C8 | — | 2 | 4 | no | — |
| 201 | C2×C8 | — | 2 | 4 | no | — |
| 202 | C2×C8 | — | 4 | 2 | no | — |
| 203 | C2×C8 | — | 4 | 1 | no | — |
| 204 | C2×C8 | C24 | 1 | 2 | yes | — |
| 205 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 206 | C2×C8 | C24 | 1 | 2 | yes | — |
| 207 | C2×C8 | C24 | 1 | 2 | yes | — |
| 208 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 209 | C4⋊C4 | — | 2 | 4 | no | — |
| 210 | C4⋊C4 | — | 2 | 4 | no | — |
| 211 | C4⋊C4 | — | 2 | 4 | no | — |
| 212 | C4⋊C4 | — | 2 | 4 | no | — |
| 213 | C4⋊C4 | — | 2 | 4 | no | — |
| 214 | C4⋊C4 | — | 2 | 4 | no | — |
| 215 | C4⋊C4 | — | 2 | 4 | no | — |
| 216 | C4⋊C4 | — | 2 | 4 | no | — |
| 217 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 218 | C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 219 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 220 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 221 | C22⋊C4 | — | 2 | 4 | no | — |
| 222 | C22⋊C4 | — | 2 | 4 | no | — |
| 223 | C22⋊C4 | — | 2 | 4 | no | — |
| 224 | C22⋊C4 | — | 2 | 4 | no | — |
| 225 | C22⋊C4 | — | 2 | 4 | no | — |
| 226 | C22⋊C4 | — | 2 | 4 | no | — |
| 227 | C22⋊C4 | — | 2 | 4 | no | — |
| 228 | C22⋊C4 | — | 2 | 4 | no | — |
| 229 | C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 230 | C22⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 231 | C22⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 232 | C42 | — | 2 | 4 | no | — |
| 233 | C42 | — | 2 | 4 | no | — |
| 234 | C42 | C24 | 1 | 2 | yes | — |
| 235 | C23 | — | 2 | 4 | no | — |
| 236 | C23 | — | 4 | 1 | no | — |
| 237 | C23 | — | 4 | 2 | no | — |
| 238 | C23 | C22×D4 | 1 | 2 | yes | — |
| 239 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 240 | C23 | C22×D4 | 1 | 2 | yes | — |
| 241 | Q8 | — | 2 | 4 | no | — |
| 242 | Q8 | — | 2 | 4 | no | — |
| 243 | Q8 | — | 4 | 1 | no | — |
| 244 | D4 | — | 2 | 4 | no | — |
| 245 | D4 | — | 2 | 4 | no | — |
| 246 | D4 | — | 4 | 4 | no | — |
| 247 | D4 | — | 4 | 1 | no | — |
| 248 | D4 | — | 4 | 2 | no | — |
| 249 | D4 | — | 8 | 4 | no | — |
| 250 | D4 | — | 8 | 4 | no | — |
| 251 | D4 | — | 8 | 4 | no | — |
| 252 | D4 | — | 8 | 4 | no | — |
| 253 | D4 | — | 8 | 4 | no | — |
| 254 | D4 | — | 8 | 4 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 2 | 2 | no | — |
| 257 | C2×C4 | — | 2 | 2 | no | — |
| 258 | C2×C4 | — | 2 | 1 | no | — |
| 259 | C2×C4 | — | 2 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 2 | no | — |
| 261 | C2×C4 | — | 2 | 4 | no | — |
| 262 | C2×C4 | — | 2 | 4 | no | — |
| 263 | C2×C4 | — | 2 | 4 | no | — |
| 264 | C2×C4 | — | 4 | 1 | no | — |
| 265 | C2×C4 | — | 4 | 2 | no | — |
| 266 | C2×C4 | — | 8 | 4 | no | — |
| 267 | C2×C4 | — | 8 | 4 | no | — |
| 268 | C2×C4 | — | 8 | 4 | no | — |
| 269 | C2×C4 | — | 8 | 4 | no | — |
| 270 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 271 | C2×C4 | C22×D4 | 1 | 4 | yes | — |
| 272 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 274 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 2 | 2 | no | — |
| 277 | C8 | — | 2 | 2 | no | — |
| 278 | C8 | — | 2 | 2 | no | — |
| 279 | C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 280 | C22 | — | 2 | 4 | no | — |
| 281 | C22 | — | 2 | 2 | no | — |
| 282 | C22 | — | 2 | 2 | no | — |
| 283 | C22 | — | 2 | 2 | no | — |
| 284 | C22 | — | 2 | 1 | no | — |
| 285 | C22 | — | 2 | 2 | no | — |
| 286 | C22 | — | 2 | 2 | no | — |
| 287 | C22 | — | 4 | 4 | no | — |
| 288 | C22 | — | 4 | 4 | no | — |
| 289 | C22 | — | 4 | 4 | no | — |
| 290 | C22 | — | 4 | 4 | no | — |
| 291 | C22 | C23×D4 | 1 | 1 | yes | characteristic |
| 292 | C4 | — | 2 | 4 | no | — |
| 293 | C4 | — | 2 | 2 | no | — |
| 294 | C4 | — | 2 | 1 | no | — |
| 295 | C4 | — | 2 | 2 | no | — |
| 296 | C4 | — | 4 | 4 | no | — |
| 297 | C4 | — | 4 | 4 | no | — |
| 298 | C4 | — | 4 | 4 | no | — |
| 299 | C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 300 | C4 | #585 | 1 | 1 | yes | characteristic |
| 301 | C2 | — | 4 | 2 | no | — |
| 302 | C2 | — | 4 | 2 | no | — |
| 303 | C2 | — | 4 | 1 | no | — |
| 304 | C2 | — | 8 | 4 | no | — |
| 305 | C2 | #3496 | 1 | 1 | yes | characteristic |
| 306 | C2 | #3584 | 1 | 2 | yes | — |
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