Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1580 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60461 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #60113 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33659 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #33657 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #33656 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×#2238 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #33404 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #33392 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #33337 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | C2×#576 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#568 | — | 2 | 1 | no | — |
| 12 | C2×#568 | C22 | 1 | 1 | yes | characteristic |
| 13 | #3488 | — | 2 | 1 | no | — |
| 14 | #3407 | C22 | 1 | 2 | yes | — |
| 15 | #3062 | — | 2 | 3 | no | — |
| 16 | #3060 | C22 | 1 | 3 | yes | — |
| 17 | #3058 | C22 | 1 | 3 | yes | — |
| 18 | #3055 | — | 2 | 3 | no | — |
| 19 | #3054 | C22 | 1 | 3 | yes | — |
| 20 | C2×#469 | C22 | 1 | 3 | yes | — |
| 21 | #2916 | — | 2 | 3 | no | — |
| 22 | #2916 | — | 2 | 1 | no | — |
| 23 | #2916 | C22 | 1 | 2 | yes | — |
| 24 | #2916 | C22 | 1 | 6 | yes | complemented |
| 25 | C2×#430 | — | 2 | 3 | no | — |
| 26 | C2×#430 | C22 | 1 | 6 | yes | complemented |
| 27 | #2912 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2897 | — | 2 | 1 | no | — |
| 29 | #2239 | C22 | 1 | 4 | yes | complemented |
| 30 | #2239 | C22 | 1 | 12 | yes | complemented |
| 31 | #2238 | C22 | 1 | 12 | yes | complemented |
| 32 | #2238 | C22 | 1 | 24 | yes | complemented |
| 33 | #2238 | C22 | 1 | 12 | yes | complemented |
| 34 | #2155 | C22 | 1 | 12 | yes | complemented |
| 35 | C2×#365 | C22 | 1 | 12 | yes | complemented |
| 36 | #2148 | C22 | 1 | 12 | yes | complemented |
| 37 | C2×#361 | C22 | 1 | 12 | yes | complemented |
| 38 | #2115 | C22 | 1 | 6 | yes | complemented |
| 39 | C2×#349 | C22 | 1 | 6 | yes | complemented |
| 40 | C2×H(5,𝔽2) | — | 2 | 1 | no | — |
| 41 | C22×#41 | D4 | 1 | 1 | yes | characteristic |
| 42 | #576 | — | 2 | 8 | no | — |
| 43 | C2×AGL(1,ℤ/8) | — | 2 | 3 | no | — |
| 44 | C2×#135 | C23 | 1 | 3 | yes | — |
| 45 | C22×D8 | C23 | 1 | 3 | yes | — |
| 46 | #568 | — | 2 | 8 | no | — |
| 47 | #568 | — | 2 | 6 | no | — |
| 48 | #568 | — | 2 | 2 | no | — |
| 49 | #568 | — | 4 | 4 | no | — |
| 50 | C2×#131 | — | 4 | 4 | no | — |
| 51 | C2×#131 | D4 | 1 | 3 | yes | — |
| 52 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 53 | C2×#131 | C23 | 1 | 3 | yes | — |
| 54 | C2×#131 | C23 | 1 | 1 | yes | characteristic |
| 55 | C22×M16 | — | 2 | 6 | no | — |
| 56 | C22×M16 | D4 | 1 | 3 | yes | — |
| 57 | C22×M16 | C23 | 1 | 3 | yes | — |
| 58 | #550 | — | 4 | 1 | no | — |
| 59 | #548 | — | 4 | 1 | no | — |
| 60 | #546 | — | 4 | 3 | no | — |
| 61 | D42 | — | 4 | 3 | no | — |
| 62 | #537 | — | 2 | 3 | no | — |
| 63 | #535 | — | 2 | 3 | no | — |
| 64 | #525 | — | 2 | 3 | no | — |
| 65 | C2×#121 | — | 2 | 3 | no | — |
| 66 | #520 | — | 2 | 1 | no | — |
| 67 | #504 | — | 2 | 8 | no | — |
| 68 | C2×M32 | C23 | 1 | 12 | yes | — |
| 69 | #470 | C23 | 1 | 1 | yes | characteristic |
| 70 | #470 | C23 | 1 | 3 | yes | — |
| 71 | #469 | — | 2 | 3 | no | — |
| 72 | #469 | — | 2 | 3 | no | — |
| 73 | #469 | C23 | 1 | 3 | yes | — |
| 74 | #469 | C23 | 1 | 6 | yes | — |
| 75 | #469 | C23 | 1 | 3 | yes | — |
| 76 | #468 | — | 2 | 3 | no | — |
| 77 | #468 | — | 2 | 3 | no | — |
| 78 | #466 | — | 2 | 6 | no | — |
| 79 | #465 | — | 2 | 6 | no | — |
| 80 | #430 | — | 2 | 6 | no | — |
| 81 | #430 | — | 2 | 6 | no | — |
| 82 | #430 | — | 2 | 6 | no | — |
| 83 | #430 | — | 2 | 6 | no | — |
| 84 | #430 | C23 | 1 | 4 | yes | — |
| 85 | #430 | C23 | 1 | 12 | yes | — |
| 86 | C2×#108 | — | 2 | 12 | no | — |
| 87 | C2×#108 | — | 2 | 6 | no | — |
| 88 | C2×#108 | C23 | 1 | 12 | yes | — |
| 89 | #428 | — | 2 | 3 | no | — |
| 90 | #427 | C23 | 1 | 3 | yes | — |
| 91 | C2×#106 | C23 | 1 | 3 | yes | — |
| 92 | #418 | — | 2 | 3 | no | — |
| 93 | #417 | — | 2 | 3 | no | — |
| 94 | #416 | — | 2 | 3 | no | — |
| 95 | C2×#102 | — | 2 | 3 | no | — |
| 96 | #365 | C23 | 1 | 24 | yes | — |
| 97 | #365 | C23 | 1 | 8 | yes | — |
| 98 | #362 | C23 | 1 | 8 | yes | — |
| 99 | #361 | C23 | 1 | 24 | yes | — |
| 100 | #349 | C23 | 1 | 4 | yes | — |
| 101 | #349 | C23 | 1 | 12 | yes | — |
| 102 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 103 | H(5,𝔽2) | — | 4 | 4 | no | — |
| 104 | C2×#41 | — | 2 | 4 | no | — |
| 105 | C2×#41 | — | 2 | 4 | no | — |
| 106 | C2×#41 | — | 2 | 3 | no | — |
| 107 | C2×#41 | — | 4 | 1 | no | — |
| 108 | C2×#41 | SD16 | 1 | 3 | yes | — |
| 109 | C2×#41 | SD16 | 1 | 1 | yes | characteristic |
| 110 | C2×#41 | C2×D4 | 1 | 3 | yes | — |
| 111 | C2×#41 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C22×Q8 | SD16 | 1 | 1 | yes | characteristic |
| 113 | C22×D4 | — | 2 | 3 | no | — |
| 114 | C22×D4 | — | 4 | 3 | no | — |
| 115 | C22×D4 | — | 4 | 3 | no | — |
| 116 | C22×D4 | SD16 | 1 | 3 | yes | — |
| 117 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 118 | AGL(1,ℤ/8) | — | 4 | 12 | no | — |
| 119 | #135 | — | 2 | 12 | no | — |
| 120 | C2×Q16 | C24 | 1 | 1 | yes | characteristic |
| 121 | C2×SD16 | — | 2 | 3 | no | — |
| 122 | C2×D8 | — | 2 | 3 | no | — |
| 123 | C2×D8 | — | 2 | 12 | no | — |
| 124 | C2×D8 | C24 | 1 | 3 | yes | — |
| 125 | #131 | — | 2 | 4 | no | — |
| 126 | #131 | — | 2 | 12 | no | — |
| 127 | #131 | — | 2 | 12 | no | — |
| 128 | #131 | — | 4 | 24 | no | — |
| 129 | #131 | — | 4 | 8 | no | — |
| 130 | #131 | #41 | 1 | 8 | yes | — |
| 131 | C2×M16 | — | 2 | 12 | no | — |
| 132 | C2×M16 | — | 2 | 12 | no | — |
| 133 | C2×M16 | — | 2 | 4 | no | — |
| 134 | C2×M16 | — | 2 | 12 | no | — |
| 135 | C2×M16 | — | 2 | 12 | no | — |
| 136 | C2×M16 | — | 4 | 6 | no | — |
| 137 | C2×M16 | — | 4 | 6 | no | — |
| 138 | C2×M16 | — | 4 | 6 | no | — |
| 139 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 140 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 141 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 142 | C2×M16 | C2×D4 | 1 | 3 | yes | — |
| 143 | C2×M16 | C24 | 1 | 1 | yes | characteristic |
| 144 | C2×M16 | C24 | 1 | 3 | yes | — |
| 145 | C22×C8 | — | 4 | 6 | no | — |
| 146 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 147 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 148 | C22×C8 | C24 | 1 | 3 | yes | — |
| 149 | #127 | — | 4 | 3 | no | — |
| 150 | #124 | — | 4 | 3 | no | — |
| 151 | #123 | — | 4 | 3 | no | — |
| 152 | C22⋊Q8 | — | 2 | 1 | no | — |
| 153 | C22⋊Q8 | — | 2 | 3 | no | — |
| 154 | C22⋊Q8 | — | 4 | 3 | no | — |
| 155 | #121 | — | 2 | 3 | no | — |
| 156 | #121 | — | 2 | 6 | no | — |
| 157 | #121 | — | 2 | 3 | no | — |
| 158 | #121 | — | 4 | 6 | no | — |
| 159 | #121 | — | 4 | 3 | no | — |
| 160 | #121 | — | 4 | 3 | no | — |
| 161 | #121 | — | 4 | 3 | no | — |
| 162 | #120 | — | 4 | 3 | no | — |
| 163 | #120 | — | 4 | 6 | no | — |
| 164 | C4×Q8 | — | 4 | 1 | no | — |
| 165 | C4×D4 | — | 4 | 3 | no | — |
| 166 | C4×D4 | — | 4 | 3 | no | — |
| 167 | C4×D4 | — | 4 | 3 | no | — |
| 168 | #117 | — | 2 | 3 | no | — |
| 169 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 170 | C2×C22⋊C4 | — | 4 | 3 | no | — |
| 171 | M32 | — | 2 | 12 | no | — |
| 172 | M32 | C24 | 1 | 8 | yes | — |
| 173 | C2×C16 | — | 2 | 12 | no | — |
| 174 | #108 | — | 2 | 12 | no | — |
| 175 | #108 | — | 2 | 4 | no | — |
| 176 | #108 | — | 2 | 12 | no | — |
| 177 | #108 | C24 | 1 | 8 | yes | — |
| 178 | #107 | — | 2 | 3 | no | — |
| 179 | #106 | — | 2 | 3 | no | — |
| 180 | #106 | C24 | 1 | 3 | yes | — |
| 181 | #106 | C24 | 1 | 1 | yes | characteristic |
| 182 | #103 | — | 2 | 1 | no | — |
| 183 | #103 | — | 2 | 3 | no | — |
| 184 | #102 | — | 2 | 3 | no | — |
| 185 | #102 | — | 2 | 6 | no | — |
| 186 | #102 | — | 2 | 3 | no | — |
| 187 | C24 | — | 4 | 3 | no | — |
| 188 | C24 | C2×SD16 | 1 | 3 | yes | — |
| 189 | #41 | — | 2 | 4 | no | — |
| 190 | #41 | — | 2 | 12 | no | — |
| 191 | #41 | — | 2 | 6 | no | — |
| 192 | #41 | — | 2 | 6 | no | — |
| 193 | #41 | — | 4 | 6 | no | — |
| 194 | #41 | — | 4 | 6 | no | — |
| 195 | #41 | — | 8 | 4 | no | — |
| 196 | #41 | #123 | 1 | 4 | yes | — |
| 197 | #41 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 198 | C2×Q8 | — | 2 | 4 | no | — |
| 199 | C2×Q8 | — | 2 | 1 | no | — |
| 200 | C2×Q8 | C2×SD16 | 1 | 3 | yes | — |
| 201 | C2×Q8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 202 | C2×D4 | — | 2 | 12 | no | — |
| 203 | C2×D4 | — | 2 | 3 | no | — |
| 204 | C2×D4 | — | 4 | 3 | no | — |
| 205 | C2×D4 | — | 4 | 3 | no | — |
| 206 | C2×D4 | — | 4 | 3 | no | — |
| 207 | C2×D4 | — | 4 | 6 | no | — |
| 208 | C2×D4 | — | 4 | 6 | no | — |
| 209 | C2×D4 | — | 4 | 3 | no | — |
| 210 | C2×D4 | — | 4 | 6 | no | — |
| 211 | C2×D4 | — | 8 | 12 | no | — |
| 212 | C2×D4 | — | 16 | 6 | no | — |
| 213 | C2×D4 | C2×SD16 | 1 | 6 | yes | — |
| 214 | C2×D4 | C2×SD16 | 1 | 3 | yes | — |
| 215 | C2×D4 | C2×SD16 | 1 | 3 | yes | — |
| 216 | C22×C4 | — | 2 | 6 | no | — |
| 217 | C22×C4 | — | 2 | 6 | no | — |
| 218 | C22×C4 | — | 4 | 3 | no | — |
| 219 | C22×C4 | — | 4 | 3 | no | — |
| 220 | C22×C4 | C2×SD16 | 1 | 3 | yes | — |
| 221 | C22×C4 | C2×SD16 | 1 | 3 | yes | — |
| 222 | C22×C4 | C2×SD16 | 1 | 3 | yes | — |
| 223 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 224 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 225 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 226 | Q16 | — | 2 | 2 | no | — |
| 227 | SD16 | — | 4 | 6 | no | — |
| 228 | D8 | — | 2 | 6 | no | — |
| 229 | D8 | — | 4 | 6 | no | — |
| 230 | M16 | — | 2 | 6 | no | — |
| 231 | M16 | — | 2 | 6 | no | — |
| 232 | M16 | — | 2 | 6 | no | — |
| 233 | M16 | — | 2 | 2 | no | — |
| 234 | M16 | — | 2 | 6 | no | — |
| 235 | M16 | — | 2 | 8 | no | — |
| 236 | M16 | — | 4 | 4 | no | — |
| 237 | M16 | — | 4 | 12 | no | — |
| 238 | M16 | — | 4 | 12 | no | — |
| 239 | M16 | #123 | 1 | 12 | yes | — |
| 240 | C2×C8 | — | 2 | 6 | no | — |
| 241 | C2×C8 | — | 2 | 12 | no | — |
| 242 | C2×C8 | — | 4 | 4 | no | — |
| 243 | C2×C8 | — | 4 | 12 | no | — |
| 244 | C2×C8 | — | 4 | 12 | no | — |
| 245 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 246 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 247 | C2×C8 | C22×D4 | 1 | 3 | yes | — |
| 248 | C2×C8 | C2×#41 | 1 | 12 | yes | — |
| 249 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 250 | C4⋊C4 | — | 2 | 3 | no | — |
| 251 | C4⋊C4 | — | 2 | 1 | no | — |
| 252 | C4⋊C4 | — | 4 | 3 | no | — |
| 253 | C4⋊C4 | — | 4 | 1 | no | — |
| 254 | C4⋊C4 | — | 4 | 3 | no | — |
| 255 | C22⋊C4 | — | 4 | 3 | no | — |
| 256 | C22⋊C4 | — | 4 | 1 | no | — |
| 257 | C22⋊C4 | — | 4 | 3 | no | — |
| 258 | C22⋊C4 | — | 4 | 3 | no | — |
| 259 | C22⋊C4 | — | 4 | 6 | no | — |
| 260 | C22⋊C4 | — | 4 | 3 | no | — |
| 261 | C42 | — | 4 | 3 | no | — |
| 262 | C16 | — | 2 | 8 | no | — |
| 263 | C23 | — | 2 | 6 | no | — |
| 264 | C23 | — | 4 | 3 | no | — |
| 265 | C23 | — | 8 | 3 | no | — |
| 266 | C23 | — | 8 | 6 | no | — |
| 267 | C23 | — | 16 | 3 | no | — |
| 268 | C23 | — | 16 | 3 | no | — |
| 269 | C23 | C22×SD16 | 1 | 1 | yes | characteristic |
| 270 | C23 | C22×SD16 | 1 | 3 | yes | — |
| 271 | C23 | C22×SD16 | 1 | 3 | yes | — |
| 272 | Q8 | — | 2 | 6 | no | — |
| 273 | Q8 | — | 4 | 1 | no | — |
| 274 | Q8 | — | 4 | 1 | no | — |
| 275 | Q8 | #481 | 1 | 4 | yes | — |
| 276 | D4 | — | 2 | 12 | no | — |
| 277 | D4 | — | 2 | 6 | no | — |
| 278 | D4 | — | 4 | 3 | no | — |
| 279 | D4 | — | 4 | 3 | no | — |
| 280 | D4 | — | 8 | 6 | no | — |
| 281 | D4 | — | 8 | 6 | no | — |
| 282 | D4 | — | 8 | 6 | no | — |
| 283 | D4 | — | 16 | 3 | no | — |
| 284 | D4 | — | 16 | 6 | no | — |
| 285 | D4 | #481 | 1 | 12 | yes | — |
| 286 | C2×C4 | — | 2 | 6 | no | — |
| 287 | C2×C4 | — | 2 | 6 | no | — |
| 288 | C2×C4 | — | 2 | 6 | no | — |
| 289 | C2×C4 | — | 2 | 3 | no | — |
| 290 | C2×C4 | — | 2 | 1 | no | — |
| 291 | C2×C4 | — | 2 | 1 | no | — |
| 292 | C2×C4 | — | 2 | 3 | no | — |
| 293 | C2×C4 | — | 4 | 1 | no | — |
| 294 | C2×C4 | — | 4 | 3 | no | — |
| 295 | C2×C4 | — | 4 | 1 | no | — |
| 296 | C2×C4 | — | 8 | 6 | no | — |
| 297 | C2×C4 | — | 16 | 3 | no | — |
| 298 | C2×C4 | C2×#123 | 1 | 6 | yes | — |
| 299 | C2×C4 | C22×SD16 | 1 | 1 | yes | characteristic |
| 300 | C2×C4 | C22×SD16 | 1 | 3 | yes | — |
| 301 | C2×C4 | C22×SD16 | 1 | 3 | yes | — |
| 302 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 6 | yes | — |
| 303 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 304 | C8 | — | 2 | 1 | no | — |
| 305 | C8 | — | 2 | 3 | no | — |
| 306 | C8 | — | 4 | 8 | no | — |
| 307 | C8 | C2×#123 | 1 | 6 | yes | — |
| 308 | C8 | C22×#41 | 1 | 2 | yes | — |
| 309 | C22 | — | 2 | 1 | no | — |
| 310 | C22 | — | 2 | 3 | no | — |
| 311 | C22 | — | 4 | 3 | no | — |
| 312 | C22 | — | 4 | 1 | no | — |
| 313 | C22 | — | 4 | 3 | no | — |
| 314 | C22 | — | 8 | 6 | no | — |
| 315 | C22 | — | 8 | 3 | no | — |
| 316 | C22 | — | 8 | 3 | no | — |
| 317 | C22 | — | 8 | 3 | no | — |
| 318 | C22 | — | 32 | 6 | no | — |
| 319 | C22 | C2×#481 | 1 | 6 | yes | — |
| 320 | C22 | C23×SD16 | 1 | 1 | yes | characteristic |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 2 | 3 | no | — |
| 323 | C4 | — | 8 | 3 | no | — |
| 324 | C4 | — | 8 | 1 | no | — |
| 325 | C4 | — | 8 | 1 | no | — |
| 326 | C4 | C2×#481 | 1 | 6 | yes | — |
| 327 | C4 | C22×#123 | 1 | 1 | yes | characteristic |
| 328 | C4 | C22×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 2 | 6 | no | — |
| 331 | C2 | — | 4 | 3 | no | — |
| 332 | C2 | — | 4 | 1 | no | — |
| 333 | C2 | — | 16 | 3 | no | — |
| 334 | C2 | C22×#481 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#481 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #60461 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #60113 | C2 | semidirect |
| 3. #33659 | C2 | semidirect |
| 4. #33657 | C2 | semidirect |
| 5. #33656 | C2 | semidirect |
| 6. C2×#2238 | C2 | semidirect |
| 7. #33404 | C2 | semidirect |
| 8. #33392 | C2 | semidirect |
| 9. #33337 | C2 | semidirect |
| 24. #2916 | C22 | semidirect |
| 26. C2×#430 | C22 | semidirect |
| 29. #2239 | C22 | semidirect |
| 30. #2239 | C22 | semidirect |
| 31. #2238 | C22 | semidirect |
| 32. #2238 | C22 | semidirect |
| 33. #2238 | C22 | semidirect |
| 34. #2155 | C22 | semidirect |
| 35. C2×#365 | C22 | semidirect |
| 36. #2148 | C22 | semidirect |
| 37. C2×#361 | C22 | semidirect |
| 38. #2115 | C22 | semidirect |
| 39. C2×#349 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60461 | 2 |
| #60113 | 2 |
| #33659 | 2 |
| #33657 | 2 |
| #33656 | 2 |
| C2×#2238 | 2 |
| #33404 | 2 |
| #33392 | 2 |
| #33337 | 2 |