Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1109 conjugacy classes of subgroups.
The subgroup lattice has 307 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#563 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#536 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #2734 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #2717 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #2698 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #2680 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #2660 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #2611 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #2553 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #2440 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | #2434 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #2406 | C2 | 1 | 2 | yes | maximal |
| 13 | #2372 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 14 | C22×#41 | C22 | 1 | 1 | yes | characteristic |
| 15 | C23×Q8 | C22 | 1 | 1 | yes | characteristic |
| 16 | #563 | — | 2 | 8 | no | — |
| 17 | #536 | — | 2 | 8 | no | — |
| 18 | C2×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 21 | C2×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 22 | C2×#126 | C22 | 1 | 4 | yes | — |
| 23 | C2×#125 | C22 | 1 | 2 | yes | — |
| 24 | C2×#124 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#124 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#123 | C22 | 1 | 2 | yes | — |
| 27 | C2×#123 | C22 | 1 | 4 | yes | — |
| 28 | C2×#123 | C22 | 1 | 2 | yes | — |
| 29 | C2×C22⋊Q8 | C22 | 1 | 4 | yes | — |
| 30 | C2×C22⋊Q8 | C22 | 1 | 4 | yes | — |
| 31 | C2×C22⋊Q8 | C22 | 1 | 2 | yes | — |
| 32 | C2×#117 | C22 | 1 | 1 | yes | characteristic |
| 33 | C22×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 34 | #400 | C22 | 1 | 2 | yes | — |
| 35 | #399 | C22 | 1 | 2 | yes | — |
| 36 | #398 | C22 | 1 | 2 | yes | — |
| 37 | #398 | C22 | 1 | 4 | yes | — |
| 38 | #398 | C22 | 1 | 4 | yes | — |
| 39 | #398 | C22 | 1 | 4 | yes | — |
| 40 | #397 | C22 | 1 | 2 | yes | — |
| 41 | #397 | C22 | 1 | 4 | yes | — |
| 42 | #397 | C22 | 1 | 4 | yes | — |
| 43 | #395 | C22 | 1 | 4 | yes | — |
| 44 | #395 | C22 | 1 | 4 | yes | — |
| 45 | #395 | C22 | 1 | 4 | yes | — |
| 46 | #395 | C22 | 1 | 4 | yes | — |
| 47 | #394 | C22 | 1 | 4 | yes | — |
| 48 | #391 | C22 | 1 | 2 | yes | — |
| 49 | #391 | C22 | 1 | 2 | yes | — |
| 50 | #389 | C22 | 1 | 4 | yes | — |
| 51 | #389 | C22 | 1 | 4 | yes | — |
| 52 | #388 | C22 | 1 | 4 | yes | — |
| 53 | #388 | C22 | 1 | 4 | yes | — |
| 54 | #387 | C22 | 1 | 4 | yes | — |
| 55 | #387 | C22 | 1 | 4 | yes | — |
| 56 | #386 | C22 | 1 | 2 | yes | — |
| 57 | #386 | C22 | 1 | 2 | yes | — |
| 58 | #385 | C22 | 1 | 4 | yes | — |
| 59 | #385 | C22 | 1 | 4 | yes | — |
| 60 | #385 | C22 | 1 | 4 | yes | — |
| 61 | #385 | C22 | 1 | 4 | yes | — |
| 62 | #384 | C22 | 1 | 4 | yes | — |
| 63 | #382 | C22 | 1 | 4 | yes | — |
| 64 | #381 | C22 | 1 | 2 | yes | — |
| 65 | #380 | C22 | 1 | 4 | yes | — |
| 66 | C4×C22⋊C4 | C22 | 1 | 4 | yes | — |
| 67 | C4×C22⋊C4 | C22 | 1 | 2 | yes | — |
| 68 | #376 | C22 | 1 | 2 | yes | — |
| 69 | C2×#41 | — | 2 | 4 | no | — |
| 70 | C2×#41 | — | 4 | 2 | no | — |
| 71 | C2×#41 | — | 4 | 2 | no | — |
| 72 | C22×Q8 | — | 2 | 4 | no | — |
| 73 | C22×Q8 | — | 4 | 2 | no | — |
| 74 | C22×Q8 | — | 4 | 2 | no | — |
| 75 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 76 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 77 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 78 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 79 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 80 | C22×D4 | C23 | 1 | 2 | yes | — |
| 81 | C22×D4 | C23 | 1 | 1 | yes | characteristic |
| 82 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 83 | C23×C4 | C23 | 1 | 2 | yes | — |
| 84 | C23×C4 | C23 | 1 | 2 | yes | — |
| 85 | C23×C4 | C23 | 1 | 2 | yes | — |
| 86 | C23×C4 | C23 | 1 | 2 | yes | — |
| 87 | C4⋊Q8 | — | 2 | 4 | no | — |
| 88 | C4⋊Q8 | — | 2 | 8 | no | — |
| 89 | C4⋊Q8 | — | 2 | 4 | no | — |
| 90 | C4⋊Q8 | — | 4 | 4 | no | — |
| 91 | #126 | — | 2 | 16 | no | — |
| 92 | #125 | — | 2 | 8 | no | — |
| 93 | #124 | — | 2 | 4 | no | — |
| 94 | #124 | D4 | 1 | 8 | yes | — |
| 95 | #123 | — | 2 | 8 | no | — |
| 96 | #123 | — | 2 | 16 | no | — |
| 97 | #123 | — | 4 | 4 | no | — |
| 98 | C22⋊Q8 | — | 2 | 16 | no | — |
| 99 | C22⋊Q8 | — | 2 | 16 | no | — |
| 100 | C22⋊Q8 | — | 4 | 4 | no | — |
| 101 | #117 | — | 2 | 4 | no | — |
| 102 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 103 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 104 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 105 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 106 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 107 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 108 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 109 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 110 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 111 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 112 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 113 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 114 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 115 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 116 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 117 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 118 | C2×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 119 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 120 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 121 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 122 | C2×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 123 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 124 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 125 | C2×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 126 | C2×C22⋊C4 | C23 | 1 | 4 | yes | — |
| 127 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 128 | C2×C42 | C23 | 1 | 2 | yes | — |
| 129 | C2×C42 | C23 | 1 | 4 | yes | — |
| 130 | C2×C42 | C23 | 1 | 2 | yes | — |
| 131 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 132 | #95 | C23 | 1 | 2 | yes | — |
| 133 | #95 | C23 | 1 | 4 | yes | — |
| 134 | #95 | C23 | 1 | 2 | yes | — |
| 135 | #95 | C23 | 1 | 4 | yes | — |
| 136 | #95 | C23 | 1 | 4 | yes | — |
| 137 | #95 | C23 | 1 | 4 | yes | — |
| 138 | #95 | C23 | 1 | 4 | yes | — |
| 139 | #95 | C23 | 1 | 4 | yes | — |
| 140 | #95 | C23 | 1 | 4 | yes | — |
| 141 | #95 | C23 | 1 | 4 | yes | — |
| 142 | #95 | C23 | 1 | 4 | yes | — |
| 143 | #95 | C23 | 1 | 4 | yes | — |
| 144 | C24 | C24 | 1 | 2 | yes | — |
| 145 | C24 | C24 | 1 | 1 | yes | characteristic |
| 146 | #41 | — | 4 | 16 | no | — |
| 147 | C2×Q8 | — | 2 | 4 | no | — |
| 148 | C2×Q8 | — | 2 | 2 | no | — |
| 149 | C2×Q8 | — | 2 | 2 | no | — |
| 150 | C2×Q8 | — | 2 | 4 | no | — |
| 151 | C2×Q8 | — | 2 | 4 | no | — |
| 152 | C2×Q8 | — | 4 | 1 | no | — |
| 153 | C2×Q8 | — | 4 | 1 | no | — |
| 154 | C2×Q8 | — | 4 | 2 | no | — |
| 155 | C2×Q8 | — | 4 | 1 | no | — |
| 156 | C2×Q8 | — | 4 | 1 | no | — |
| 157 | C2×Q8 | — | 4 | 1 | no | — |
| 158 | C2×Q8 | — | 4 | 1 | no | — |
| 159 | C2×Q8 | — | 4 | 2 | no | — |
| 160 | C2×Q8 | — | 4 | 2 | no | — |
| 161 | C2×Q8 | — | 4 | 16 | no | — |
| 162 | C2×Q8 | C2×D4 | 1 | 4 | yes | — |
| 163 | C2×D4 | — | 2 | 4 | no | — |
| 164 | C2×D4 | — | 2 | 4 | no | — |
| 165 | C2×D4 | — | 4 | 1 | no | — |
| 166 | C2×D4 | — | 4 | 1 | no | — |
| 167 | C2×D4 | — | 4 | 2 | no | — |
| 168 | C2×D4 | C2×D4 | 1 | 4 | yes | — |
| 169 | C22×C4 | — | 2 | 2 | no | — |
| 170 | C22×C4 | — | 2 | 2 | no | — |
| 171 | C22×C4 | — | 2 | 2 | no | — |
| 172 | C22×C4 | — | 2 | 4 | no | — |
| 173 | C22×C4 | — | 2 | 4 | no | — |
| 174 | C22×C4 | — | 2 | 4 | no | — |
| 175 | C22×C4 | — | 2 | 4 | no | — |
| 176 | C22×C4 | — | 2 | 4 | no | — |
| 177 | C22×C4 | — | 2 | 8 | no | — |
| 178 | C22×C4 | — | 4 | 2 | no | — |
| 179 | C22×C4 | — | 4 | 2 | no | — |
| 180 | C22×C4 | — | 4 | 2 | no | — |
| 181 | C22×C4 | — | 4 | 2 | no | — |
| 182 | C22×C4 | — | 4 | 2 | no | — |
| 183 | C22×C4 | C24 | 1 | 4 | yes | — |
| 184 | C22×C4 | C24 | 1 | 2 | yes | — |
| 185 | C22×C4 | C24 | 1 | 4 | yes | — |
| 186 | C22×C4 | C24 | 1 | 2 | yes | — |
| 187 | C22×C4 | C24 | 1 | 2 | yes | — |
| 188 | C22×C4 | C24 | 1 | 4 | yes | — |
| 189 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 190 | C22×C4 | C24 | 1 | 2 | yes | — |
| 191 | C22×C4 | C24 | 1 | 2 | yes | — |
| 192 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 193 | C22×C4 | C24 | 1 | 4 | yes | — |
| 194 | C4⋊C4 | — | 2 | 4 | no | — |
| 195 | C4⋊C4 | — | 2 | 4 | no | — |
| 196 | C4⋊C4 | — | 2 | 8 | no | — |
| 197 | C4⋊C4 | — | 2 | 8 | no | — |
| 198 | C4⋊C4 | — | 2 | 8 | no | — |
| 199 | C4⋊C4 | — | 2 | 8 | no | — |
| 200 | C4⋊C4 | — | 2 | 8 | no | — |
| 201 | C4⋊C4 | — | 2 | 4 | no | — |
| 202 | C4⋊C4 | — | 2 | 8 | no | — |
| 203 | C4⋊C4 | — | 2 | 8 | no | — |
| 204 | C4⋊C4 | — | 4 | 4 | no | — |
| 205 | C4⋊C4 | — | 4 | 2 | no | — |
| 206 | C4⋊C4 | — | 4 | 4 | no | — |
| 207 | C4⋊C4 | — | 4 | 4 | no | — |
| 208 | C4⋊C4 | — | 4 | 4 | no | — |
| 209 | C22⋊C4 | — | 2 | 8 | no | — |
| 210 | C22⋊C4 | — | 2 | 8 | no | — |
| 211 | C22⋊C4 | — | 2 | 4 | no | — |
| 212 | C22⋊C4 | — | 2 | 8 | no | — |
| 213 | C22⋊C4 | — | 2 | 4 | no | — |
| 214 | C22⋊C4 | — | 2 | 8 | no | — |
| 215 | C22⋊C4 | — | 2 | 8 | no | — |
| 216 | C22⋊C4 | — | 4 | 2 | no | — |
| 217 | C22⋊C4 | C2×D4 | 1 | 16 | yes | — |
| 218 | C42 | — | 2 | 2 | no | — |
| 219 | C42 | — | 2 | 4 | no | — |
| 220 | C42 | — | 2 | 8 | no | — |
| 221 | C42 | — | 4 | 2 | no | — |
| 222 | C42 | C2×D4 | 1 | 4 | yes | — |
| 223 | C23 | — | 2 | 1 | no | — |
| 224 | C23 | — | 2 | 2 | no | — |
| 225 | C23 | — | 2 | 1 | no | — |
| 226 | C23 | — | 2 | 2 | no | — |
| 227 | C23 | — | 2 | 4 | no | — |
| 228 | C23 | — | 2 | 4 | no | — |
| 229 | C23 | — | 2 | 4 | no | — |
| 230 | C23 | C22×D4 | 1 | 4 | yes | — |
| 231 | C23 | H(5,𝔽2) | 1 | 2 | yes | — |
| 232 | C23 | C25 | 1 | 1 | yes | characteristic |
| 233 | Q8 | — | 4 | 4 | no | — |
| 234 | Q8 | — | 4 | 8 | no | — |
| 235 | Q8 | — | 4 | 4 | no | — |
| 236 | Q8 | — | 4 | 4 | no | — |
| 237 | Q8 | — | 8 | 4 | no | — |
| 238 | D4 | — | 4 | 4 | no | — |
| 239 | D4 | — | 4 | 8 | no | — |
| 240 | C2×C4 | — | 2 | 8 | no | — |
| 241 | C2×C4 | — | 2 | 1 | no | — |
| 242 | C2×C4 | — | 2 | 4 | no | — |
| 243 | C2×C4 | — | 2 | 2 | no | — |
| 244 | C2×C4 | — | 2 | 2 | no | — |
| 245 | C2×C4 | — | 2 | 1 | no | — |
| 246 | C2×C4 | — | 2 | 1 | no | — |
| 247 | C2×C4 | — | 2 | 1 | no | — |
| 248 | C2×C4 | — | 2 | 4 | no | — |
| 249 | C2×C4 | — | 2 | 4 | no | — |
| 250 | C2×C4 | — | 2 | 2 | no | — |
| 251 | C2×C4 | — | 2 | 4 | no | — |
| 252 | C2×C4 | — | 2 | 2 | no | — |
| 253 | C2×C4 | — | 2 | 4 | no | — |
| 254 | C2×C4 | — | 2 | 2 | no | — |
| 255 | C2×C4 | — | 2 | 4 | no | — |
| 256 | C2×C4 | — | 2 | 4 | no | — |
| 257 | C2×C4 | — | 2 | 4 | no | — |
| 258 | C2×C4 | — | 2 | 2 | no | — |
| 259 | C2×C4 | — | 2 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 2 | no | — |
| 261 | C2×C4 | — | 2 | 2 | no | — |
| 262 | C2×C4 | — | 2 | 4 | no | — |
| 263 | C2×C4 | — | 2 | 4 | no | — |
| 264 | C2×C4 | — | 2 | 4 | no | — |
| 265 | C2×C4 | — | 2 | 4 | no | — |
| 266 | C2×C4 | — | 2 | 4 | no | — |
| 267 | C2×C4 | — | 4 | 8 | no | — |
| 268 | C2×C4 | — | 4 | 8 | no | — |
| 269 | C2×C4 | — | 4 | 8 | no | — |
| 270 | C2×C4 | — | 8 | 4 | no | — |
| 271 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 272 | C2×C4 | C22×D4 | 1 | 4 | yes | — |
| 273 | C2×C4 | C22×D4 | 1 | 4 | yes | — |
| 274 | C2×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 275 | C2×C4 | 2−1+4 | 1 | 8 | yes | — |
| 276 | C22 | — | 2 | 2 | no | — |
| 277 | C22 | — | 2 | 4 | no | — |
| 278 | C22 | — | 2 | 8 | no | — |
| 279 | C22 | — | 2 | 4 | no | — |
| 280 | C22 | — | 4 | 2 | no | — |
| 281 | C22 | — | 4 | 2 | no | — |
| 282 | C22 | — | 4 | 2 | no | — |
| 283 | C22 | — | 4 | 2 | no | — |
| 284 | C22 | — | 4 | 4 | no | — |
| 285 | C22 | C23×D4 | 1 | 1 | yes | characteristic |
| 286 | C22 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 287 | C22 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 288 | C22 | C2×2−1+4 | 1 | 4 | yes | — |
| 289 | C4 | — | 2 | 2 | no | — |
| 290 | C4 | — | 2 | 2 | no | — |
| 291 | C4 | — | 2 | 4 | no | — |
| 292 | C4 | — | 2 | 4 | no | — |
| 293 | C4 | — | 2 | 8 | no | — |
| 294 | C4 | — | 4 | 2 | no | — |
| 295 | C4 | — | 4 | 2 | no | — |
| 296 | C4 | — | 4 | 4 | no | — |
| 297 | C4 | — | 4 | 2 | no | — |
| 298 | C4 | — | 4 | 4 | no | — |
| 299 | C4 | — | 4 | 4 | no | — |
| 300 | C2 | — | 4 | 2 | no | — |
| 301 | C2 | — | 4 | 4 | no | — |
| 302 | C2 | #3484 | 1 | 1 | yes | characteristic |
| 303 | C2 | #3499 | 1 | 2 | yes | — |
| 304 | C2 | #3538 | 1 | 2 | yes | — |
| 305 | C2 | #3544 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#563 | C2 | semidirect |
| 3. #2734 | C2 | semidirect |
| 4. #2717 | C2 | semidirect |
| 5. #2698 | C2 | semidirect |
| 6. #2680 | C2 | semidirect |
| 7. #2660 | C2 | semidirect |
| 8. #2611 | C2 | semidirect |
| 9. #2553 | C2 | semidirect |
| 10. #2440 | C2 | semidirect |
| 11. #2434 | C2 | semidirect |
| 13. #2372 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#563 | 2 |
| C2×#536 | 2 |
| #2734 | 2 |
| #2717 | 2 |
| #2698 | 2 |
| #2680 | 2 |
| #2660 | 2 |
| #2611 | 2 |
| #2553 | 2 |
| #2440 | 2 |
| #2434 | 2 |
| #2406 | 2 |
| #2372 | 2 |