Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
985 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3522 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #3497 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #3399 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #3391 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #3384 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | Q8×SD16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #3373 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #3366 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #3362 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #3358 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #3253 | C2 | 1 | 3 | yes | maximal, complemented |
| 12 | #3251 | C2 | 1 | 3 | yes | maximal, complemented |
| 13 | #3184 | C2 | 1 | 3 | yes | maximal, complemented |
| 14 | #3171 | C2 | 1 | 3 | yes | maximal, complemented |
| 15 | Q8×M16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 16 | C2×2−1+4 | — | 2 | 1 | no | — |
| 17 | #559 | C22 | 1 | 3 | yes | — |
| 18 | Q82 | C22 | 1 | 1 | yes | characteristic |
| 19 | #554 | C22 | 1 | 3 | yes | — |
| 20 | #550 | C22 | 1 | 1 | yes | characteristic |
| 21 | #550 | C22 | 1 | 1 | yes | characteristic |
| 22 | #550 | C22 | 1 | 3 | yes | — |
| 23 | D4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 24 | D4×Q8 | C22 | 1 | 3 | yes | — |
| 25 | #548 | — | 2 | 1 | no | — |
| 26 | #548 | — | 2 | 1 | no | — |
| 27 | #548 | — | 2 | 3 | no | — |
| 28 | #547 | — | 2 | 3 | no | — |
| 29 | #542 | — | 2 | 3 | no | — |
| 30 | #537 | — | 2 | 3 | no | — |
| 31 | #532 | C22 | 1 | 3 | yes | — |
| 32 | #532 | C22 | 1 | 3 | yes | — |
| 33 | #532 | C22 | 1 | 3 | yes | — |
| 34 | #529 | C22 | 1 | 3 | yes | — |
| 35 | #520 | — | 2 | 1 | no | — |
| 36 | C2×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 37 | #501 | C22 | 1 | 3 | yes | — |
| 38 | #498 | C22 | 1 | 3 | yes | — |
| 39 | #496 | C22 | 1 | 3 | yes | — |
| 40 | #493 | C22 | 1 | 3 | yes | — |
| 41 | #490 | C22 | 1 | 3 | yes | — |
| 42 | #489 | C22 | 1 | 3 | yes | — |
| 43 | #486 | C22 | 1 | 3 | yes | — |
| 44 | #486 | C22 | 1 | 3 | yes | — |
| 45 | #478 | C22 | 1 | 6 | yes | — |
| 46 | #476 | C22 | 1 | 3 | yes | — |
| 47 | #476 | C22 | 1 | 3 | yes | — |
| 48 | #476 | C22 | 1 | 6 | yes | — |
| 49 | #475 | C22 | 1 | 3 | yes | — |
| 50 | #474 | C22 | 1 | 3 | yes | — |
| 51 | #469 | C22 | 1 | 3 | yes | — |
| 52 | #469 | C22 | 1 | 3 | yes | — |
| 53 | #469 | C22 | 1 | 6 | yes | — |
| 54 | #468 | C22 | 1 | 1 | yes | characteristic |
| 55 | #468 | C22 | 1 | 3 | yes | — |
| 56 | #463 | C22 | 1 | 6 | yes | — |
| 57 | #460 | C22 | 1 | 3 | yes | — |
| 58 | #459 | C22 | 1 | 3 | yes | — |
| 59 | #459 | C22 | 1 | 3 | yes | — |
| 60 | #459 | C22 | 1 | 6 | yes | — |
| 61 | #459 | C22 | 1 | 3 | yes | — |
| 62 | #446 | C22 | 1 | 3 | yes | — |
| 63 | #446 | C22 | 1 | 3 | yes | — |
| 64 | C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 65 | C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 66 | #442 | C22 | 1 | 6 | yes | — |
| 67 | #442 | C22 | 1 | 6 | yes | — |
| 68 | C4×SD16 | C22 | 1 | 3 | yes | — |
| 69 | C4×SD16 | C22 | 1 | 3 | yes | — |
| 70 | C4×D8 | C22 | 1 | 3 | yes | — |
| 71 | C4×D8 | C22 | 1 | 3 | yes | — |
| 72 | #423 | C22 | 1 | 3 | yes | — |
| 73 | C4×M16 | C22 | 1 | 3 | yes | — |
| 74 | 2−1+4 | — | 4 | 8 | no | — |
| 75 | C2×#41 | — | 2 | 1 | no | — |
| 76 | C2×#41 | — | 2 | 3 | no | — |
| 77 | C2×#41 | — | 2 | 1 | no | — |
| 78 | C2×#41 | — | 2 | 3 | no | — |
| 79 | C2×#41 | — | 2 | 3 | no | — |
| 80 | C2×#41 | — | 2 | 3 | no | — |
| 81 | C22×Q8 | — | 2 | 3 | no | — |
| 82 | C22×Q8 | — | 2 | 1 | no | — |
| 83 | C22×Q8 | D4 | 1 | 1 | yes | characteristic |
| 84 | C2×SD16 | C23 | 1 | 1 | yes | characteristic |
| 85 | C2×SD16 | C23 | 1 | 3 | yes | — |
| 86 | C2×D8 | C23 | 1 | 3 | yes | — |
| 87 | C2×D8 | C23 | 1 | 3 | yes | — |
| 88 | C2×D8 | C23 | 1 | 6 | yes | — |
| 89 | C2×M16 | C23 | 1 | 1 | yes | characteristic |
| 90 | C2×M16 | C23 | 1 | 3 | yes | — |
| 91 | C4⋊Q8 | — | 2 | 3 | no | — |
| 92 | C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 93 | C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 94 | C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 95 | #127 | C23 | 1 | 3 | yes | — |
| 96 | #127 | C23 | 1 | 3 | yes | — |
| 97 | #127 | C23 | 1 | 3 | yes | — |
| 98 | #126 | — | 2 | 3 | no | — |
| 99 | #125 | C23 | 1 | 3 | yes | — |
| 100 | #124 | — | 2 | 3 | no | — |
| 101 | #124 | — | 2 | 3 | no | — |
| 102 | #124 | — | 2 | 3 | no | — |
| 103 | #124 | — | 2 | 3 | no | — |
| 104 | #124 | C23 | 1 | 3 | yes | — |
| 105 | #123 | — | 2 | 3 | no | — |
| 106 | #123 | — | 2 | 6 | no | — |
| 107 | C22⋊Q8 | — | 2 | 3 | no | — |
| 108 | C22⋊Q8 | — | 2 | 3 | no | — |
| 109 | C22⋊Q8 | — | 2 | 3 | no | — |
| 110 | C22⋊Q8 | — | 2 | 6 | no | — |
| 111 | C22⋊Q8 | — | 2 | 6 | no | — |
| 112 | C22⋊Q8 | C23 | 1 | 1 | yes | characteristic |
| 113 | #121 | — | 2 | 3 | no | — |
| 114 | #121 | — | 2 | 3 | no | — |
| 115 | #121 | — | 2 | 3 | no | — |
| 116 | #121 | — | 2 | 3 | no | — |
| 117 | #121 | — | 2 | 3 | no | — |
| 118 | #121 | — | 2 | 6 | no | — |
| 119 | #121 | — | 2 | 3 | no | — |
| 120 | #121 | C23 | 1 | 3 | yes | — |
| 121 | #121 | C23 | 1 | 1 | yes | characteristic |
| 122 | #121 | C23 | 1 | 3 | yes | — |
| 123 | C4×Q8 | — | 2 | 1 | no | — |
| 124 | C4×Q8 | D4 | 1 | 1 | yes | characteristic |
| 125 | C4×Q8 | D4 | 1 | 3 | yes | — |
| 126 | C4×Q8 | C23 | 1 | 3 | yes | — |
| 127 | C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 128 | C4×Q8 | C23 | 1 | 3 | yes | — |
| 129 | C4×D4 | — | 2 | 3 | no | — |
| 130 | C4×D4 | — | 2 | 3 | no | — |
| 131 | C4×D4 | — | 2 | 3 | no | — |
| 132 | C4×D4 | — | 2 | 3 | no | — |
| 133 | C4×D4 | — | 2 | 3 | no | — |
| 134 | C4×D4 | C23 | 1 | 3 | yes | — |
| 135 | C4×D4 | C23 | 1 | 3 | yes | — |
| 136 | C4×D4 | C23 | 1 | 3 | yes | — |
| 137 | C4×D4 | C23 | 1 | 6 | yes | — |
| 138 | C4×D4 | C23 | 1 | 6 | yes | — |
| 139 | #117 | — | 2 | 3 | no | — |
| 140 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 141 | C2×C4⋊C4 | D4 | 1 | 3 | yes | — |
| 142 | C2×C42 | C23 | 1 | 3 | yes | — |
| 143 | #107 | C23 | 1 | 1 | yes | characteristic |
| 144 | #107 | C23 | 1 | 3 | yes | — |
| 145 | #106 | C23 | 1 | 3 | yes | — |
| 146 | #106 | C23 | 1 | 3 | yes | — |
| 147 | #106 | C23 | 1 | 6 | yes | — |
| 148 | C4⋊C8 | C23 | 1 | 3 | yes | — |
| 149 | C4⋊C8 | C23 | 1 | 3 | yes | — |
| 150 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 151 | #103 | C23 | 1 | 1 | yes | characteristic |
| 152 | #103 | C23 | 1 | 3 | yes | — |
| 153 | #102 | C23 | 1 | 1 | yes | characteristic |
| 154 | #102 | C23 | 1 | 3 | yes | — |
| 155 | #102 | C23 | 1 | 3 | yes | — |
| 156 | #102 | C23 | 1 | 3 | yes | — |
| 157 | #102 | C23 | 1 | 3 | yes | — |
| 158 | #102 | C23 | 1 | 3 | yes | — |
| 159 | #102 | C23 | 1 | 6 | yes | — |
| 160 | #102 | C23 | 1 | 6 | yes | — |
| 161 | #97 | C23 | 1 | 3 | yes | — |
| 162 | #97 | C23 | 1 | 3 | yes | — |
| 163 | C4×C8 | C23 | 1 | 3 | yes | — |
| 164 | C4×C8 | C23 | 1 | 3 | yes | — |
| 165 | #41 | — | 4 | 12 | no | — |
| 166 | #41 | — | 4 | 12 | no | — |
| 167 | #41 | — | 4 | 12 | no | — |
| 168 | #41 | — | 4 | 4 | no | — |
| 169 | #41 | — | 4 | 4 | no | — |
| 170 | #41 | — | 8 | 6 | no | — |
| 171 | C2×Q8 | — | 2 | 4 | no | — |
| 172 | C2×Q8 | — | 2 | 3 | no | — |
| 173 | C2×Q8 | — | 2 | 3 | no | — |
| 174 | C2×Q8 | — | 2 | 3 | no | — |
| 175 | C2×Q8 | — | 4 | 12 | no | — |
| 176 | C2×Q8 | — | 4 | 4 | no | — |
| 177 | C2×Q8 | C2×D4 | 1 | 3 | yes | — |
| 178 | C2×Q8 | C2×D4 | 1 | 1 | yes | characteristic |
| 179 | C2×Q8 | C24 | 1 | 1 | yes | characteristic |
| 180 | C2×D4 | — | 2 | 3 | no | — |
| 181 | C2×D4 | — | 2 | 1 | no | — |
| 182 | C2×D4 | — | 2 | 3 | no | — |
| 183 | C2×D4 | — | 2 | 3 | no | — |
| 184 | C2×D4 | — | 2 | 3 | no | — |
| 185 | C2×D4 | — | 2 | 3 | no | — |
| 186 | C2×D4 | — | 2 | 3 | no | — |
| 187 | C2×D4 | — | 2 | 3 | no | — |
| 188 | C2×D4 | — | 2 | 3 | no | — |
| 189 | C2×D4 | C24 | 1 | 3 | yes | — |
| 190 | C2×D4 | C24 | 1 | 3 | yes | — |
| 191 | C2×D4 | C24 | 1 | 1 | yes | characteristic |
| 192 | C22×C4 | — | 2 | 3 | no | — |
| 193 | C22×C4 | — | 2 | 6 | no | — |
| 194 | C22×C4 | — | 2 | 3 | no | — |
| 195 | C22×C4 | — | 2 | 3 | no | — |
| 196 | C22×C4 | — | 2 | 6 | no | — |
| 197 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 198 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 199 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 200 | SD16 | — | 2 | 2 | no | — |
| 201 | SD16 | — | 4 | 3 | no | — |
| 202 | D8 | — | 2 | 6 | no | — |
| 203 | D8 | — | 4 | 6 | no | — |
| 204 | D8 | — | 4 | 3 | no | — |
| 205 | M16 | — | 2 | 2 | no | — |
| 206 | M16 | — | 4 | 3 | no | — |
| 207 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 208 | C2×C8 | C24 | 1 | 3 | yes | — |
| 209 | C2×C8 | C24 | 1 | 3 | yes | — |
| 210 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 211 | C4⋊C4 | — | 2 | 3 | no | — |
| 212 | C4⋊C4 | — | 2 | 3 | no | — |
| 213 | C4⋊C4 | — | 2 | 1 | no | — |
| 214 | C4⋊C4 | — | 2 | 3 | no | — |
| 215 | C4⋊C4 | — | 2 | 3 | no | — |
| 216 | C4⋊C4 | — | 2 | 3 | no | — |
| 217 | C4⋊C4 | — | 2 | 3 | no | — |
| 218 | C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 219 | C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 220 | C4⋊C4 | C2×D4 | 1 | 6 | yes | — |
| 221 | C4⋊C4 | C24 | 1 | 3 | yes | — |
| 222 | C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 223 | C4⋊C4 | C24 | 1 | 3 | yes | — |
| 224 | C4⋊C4 | C24 | 1 | 1 | yes | characteristic |
| 225 | C22⋊C4 | — | 2 | 1 | no | — |
| 226 | C22⋊C4 | — | 2 | 3 | no | — |
| 227 | C22⋊C4 | — | 2 | 3 | no | — |
| 228 | C22⋊C4 | — | 2 | 1 | no | — |
| 229 | C22⋊C4 | — | 2 | 3 | no | — |
| 230 | C22⋊C4 | — | 2 | 3 | no | — |
| 231 | C22⋊C4 | — | 2 | 6 | no | — |
| 232 | C22⋊C4 | — | 2 | 3 | no | — |
| 233 | C22⋊C4 | — | 2 | 3 | no | — |
| 234 | C22⋊C4 | — | 2 | 3 | no | — |
| 235 | C22⋊C4 | — | 2 | 6 | no | — |
| 236 | C42 | — | 2 | 3 | no | — |
| 237 | C42 | C2×D4 | 1 | 3 | yes | — |
| 238 | C42 | C2×D4 | 1 | 3 | yes | — |
| 239 | C42 | C24 | 1 | 3 | yes | — |
| 240 | C42 | C24 | 1 | 3 | yes | — |
| 241 | C23 | — | 2 | 3 | no | — |
| 242 | C23 | — | 2 | 3 | no | — |
| 243 | C23 | — | 2 | 1 | no | — |
| 244 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 245 | Q8 | — | 2 | 6 | no | — |
| 246 | Q8 | — | 2 | 1 | no | — |
| 247 | Q8 | — | 2 | 1 | no | — |
| 248 | Q8 | — | 4 | 6 | no | — |
| 249 | Q8 | — | 4 | 6 | no | — |
| 250 | Q8 | — | 8 | 3 | no | — |
| 251 | Q8 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 252 | D4 | — | 2 | 3 | no | — |
| 253 | D4 | — | 2 | 3 | no | — |
| 254 | D4 | — | 2 | 1 | no | — |
| 255 | D4 | — | 2 | 3 | no | — |
| 256 | D4 | — | 2 | 1 | no | — |
| 257 | D4 | — | 2 | 3 | no | — |
| 258 | D4 | — | 4 | 6 | no | — |
| 259 | D4 | — | 4 | 6 | no | — |
| 260 | D4 | — | 4 | 2 | no | — |
| 261 | D4 | — | 4 | 6 | no | — |
| 262 | D4 | — | 4 | 6 | no | — |
| 263 | D4 | — | 4 | 6 | no | — |
| 264 | D4 | — | 8 | 3 | no | — |
| 265 | D4 | — | 8 | 3 | no | — |
| 266 | D4 | — | 8 | 3 | no | — |
| 267 | C2×C4 | — | 2 | 6 | no | — |
| 268 | C2×C4 | — | 2 | 1 | no | — |
| 269 | C2×C4 | — | 2 | 1 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 3 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 2 | 3 | no | — |
| 274 | C2×C4 | — | 2 | 1 | no | — |
| 275 | C2×C4 | — | 4 | 3 | no | — |
| 276 | C2×C4 | — | 4 | 12 | no | — |
| 277 | C2×C4 | — | 4 | 6 | no | — |
| 278 | C2×C4 | — | 4 | 6 | no | — |
| 279 | C2×C4 | — | 4 | 6 | no | — |
| 280 | C2×C4 | — | 8 | 6 | no | — |
| 281 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 283 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 284 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 285 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 286 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 287 | C8 | — | 2 | 1 | no | — |
| 288 | C8 | — | 2 | 3 | no | — |
| 289 | C8 | — | 2 | 3 | no | — |
| 290 | C8 | 2−1+4 | 1 | 2 | yes | complemented |
| 291 | C22 | — | 2 | 1 | no | — |
| 292 | C22 | — | 2 | 1 | no | — |
| 293 | C22 | — | 2 | 1 | no | — |
| 294 | C22 | — | 4 | 3 | no | — |
| 295 | C22 | — | 4 | 1 | no | — |
| 296 | C22 | — | 4 | 3 | no | — |
| 297 | C22 | — | 4 | 3 | no | — |
| 298 | C22 | — | 4 | 3 | no | — |
| 299 | C22 | — | 4 | 1 | no | — |
| 300 | C22 | — | 4 | 3 | no | — |
| 301 | C22 | — | 4 | 1 | no | — |
| 302 | C22 | — | 4 | 3 | no | — |
| 303 | C22 | C23×D4 | 1 | 1 | yes | characteristic |
| 304 | C4 | — | 2 | 3 | no | — |
| 305 | C4 | — | 2 | 3 | no | — |
| 306 | C4 | — | 2 | 1 | no | — |
| 307 | C4 | — | 2 | 3 | no | — |
| 308 | C4 | — | 4 | 3 | no | — |
| 309 | C4 | — | 4 | 1 | no | — |
| 310 | C4 | — | 4 | 1 | no | — |
| 311 | C4 | — | 4 | 3 | no | — |
| 312 | C4 | — | 4 | 1 | no | — |
| 313 | C4 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 314 | C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 315 | C4 | #585 | 1 | 1 | yes | characteristic |
| 316 | C2 | — | 4 | 1 | no | — |
| 317 | C2 | — | 8 | 3 | no | — |
| 318 | C2 | — | 8 | 3 | no | — |
| 319 | C2 | — | 8 | 1 | no | — |
| 320 | C2 | #3497 | 1 | 1 | yes | characteristic |
| 321 | C2 | C22×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 322 | C2 | #3585 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3522 | C2 | semidirect |
| 2. #3497 | C2 | semidirect |
| 3. #3399 | C2 | semidirect |
| 4. #3391 | C2 | semidirect |
| 5. #3384 | C2 | semidirect |
| 6. Q8×SD16 | C2 | semidirect |
| 7. #3373 | C2 | semidirect |
| 8. #3366 | C2 | semidirect |
| 9. #3362 | C2 | semidirect |
| 10. #3358 | C2 | semidirect |
| 11. #3253 | C2 | semidirect |
| 12. #3251 | C2 | semidirect |
| 13. #3184 | C2 | semidirect |
| 14. #3171 | C2 | semidirect |
| 15. Q8×M16 | C2 | semidirect |
| 290. C8 | 2−1+4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3522 | 2 |
| #3497 | 2 |
| #3399 | 2 |
| #3391 | 2 |
| #3384 | 2 |
| Q8×SD16 | 2 |
| #3373 | 2 |
| #3366 | 2 |
| #3362 | 2 |
| #3358 | 2 |
| #3253 | 2 |
| #3251 | 2 |
| #3184 | 2 |
| #3171 | 2 |
| Q8×M16 | 2 |