Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1258 conjugacy classes of subgroups.
The subgroup lattice has 313 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #61289 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33922 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #33919 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #33904 | C2 | 1 | 8 | yes | maximal, complemented |
| 5 | C2×#2253 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #33652 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #33630 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #33611 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C4×#508 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #33434 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | #33411 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #3512 | — | 2 | 2 | no | — |
| 13 | C2×#508 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#504 | C22 | 1 | 2 | yes | — |
| 15 | #3351 | — | 2 | 8 | no | — |
| 16 | #3339 | C22 | 1 | 8 | yes | — |
| 17 | #3336 | C22 | 1 | 8 | yes | — |
| 18 | C2×#499 | C22 | 1 | 2 | yes | — |
| 19 | #3099 | — | 2 | 4 | no | — |
| 20 | #3080 | — | 2 | 2 | no | — |
| 21 | #3061 | C22 | 1 | 4 | yes | — |
| 22 | C4×#135 | C22 | 1 | 2 | yes | — |
| 23 | #2931 | C22 | 1 | 1 | yes | characteristic |
| 24 | #2913 | C22 | 1 | 4 | yes | — |
| 25 | #2253 | C22 | 1 | 4 | yes | — |
| 26 | #2253 | C22 | 1 | 4 | yes | — |
| 27 | #2237 | C22 | 1 | 8 | yes | — |
| 28 | #2236 | C22 | 1 | 8 | yes | — |
| 29 | #2234 | C22 | 1 | 8 | yes | — |
| 30 | #2232 | C22 | 1 | 8 | yes | — |
| 31 | #2229 | C22 | 1 | 8 | yes | — |
| 32 | #2228 | C22 | 1 | 8 | yes | — |
| 33 | #2219 | C22 | 1 | 4 | yes | — |
| 34 | #2218 | C22 | 1 | 4 | yes | — |
| 35 | #2217 | C22 | 1 | 8 | yes | — |
| 36 | C4×Q32 | C22 | 1 | 2 | yes | — |
| 37 | C4×SD32 | C22 | 1 | 4 | yes | — |
| 38 | C4×D16 | C22 | 1 | 2 | yes | — |
| 39 | #2169 | C22 | 1 | 4 | yes | — |
| 40 | D4×C16 | C22 | 1 | 4 | yes | — |
| 41 | #2159 | C22 | 1 | 4 | yes | — |
| 42 | #2159 | C22 | 1 | 8 | yes | — |
| 43 | #2158 | C22 | 1 | 4 | yes | — |
| 44 | #2158 | C22 | 1 | 1 | yes | characteristic |
| 45 | C2×#367 | C22 | 1 | 2 | yes | — |
| 46 | C2×#367 | C22 | 1 | 4 | yes | — |
| 47 | C2×#366 | C22 | 1 | 2 | yes | — |
| 48 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 49 | #2139 | C22 | 1 | 4 | yes | — |
| 50 | #2138 | C22 | 1 | 2 | yes | — |
| 51 | #2113 | C22 | 1 | 2 | yes | — |
| 52 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 53 | C2×#135 | C23 | 1 | 2 | yes | — |
| 54 | C2×#131 | C23 | 1 | 2 | yes | — |
| 55 | #553 | — | 2 | 8 | no | — |
| 56 | #553 | — | 4 | 8 | no | — |
| 57 | #552 | — | 2 | 8 | no | — |
| 58 | #541 | — | 4 | 2 | no | — |
| 59 | #540 | — | 4 | 2 | no | — |
| 60 | #537 | — | 2 | 4 | no | — |
| 61 | #532 | D4 | 1 | 1 | yes | characteristic |
| 62 | #529 | — | 4 | 4 | no | — |
| 63 | C2×#125 | — | 2 | 2 | no | — |
| 64 | #520 | — | 2 | 4 | no | — |
| 65 | C4×#41 | — | 2 | 2 | no | — |
| 66 | #508 | — | 2 | 4 | no | — |
| 67 | C2×Q32 | C23 | 1 | 1 | yes | characteristic |
| 68 | C2×SD32 | C23 | 1 | 2 | yes | — |
| 69 | C2×D16 | C23 | 1 | 1 | yes | characteristic |
| 70 | #504 | — | 2 | 8 | no | — |
| 71 | C2×M32 | C23 | 1 | 2 | yes | — |
| 72 | C2×M32 | C23 | 1 | 4 | yes | — |
| 73 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 74 | C22×C16 | C23 | 1 | 4 | yes | — |
| 75 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 76 | #499 | — | 2 | 4 | no | — |
| 77 | #499 | C23 | 1 | 8 | yes | — |
| 78 | #485 | — | 2 | 8 | no | — |
| 79 | #483 | — | 2 | 8 | no | — |
| 80 | #482 | — | 2 | 8 | no | — |
| 81 | #481 | — | 2 | 8 | no | — |
| 82 | #479 | — | 2 | 8 | no | — |
| 83 | #478 | — | 2 | 8 | no | — |
| 84 | #468 | — | 2 | 8 | no | — |
| 85 | #467 | C23 | 1 | 8 | yes | — |
| 86 | #466 | C23 | 1 | 8 | yes | — |
| 87 | C4×Q16 | C23 | 1 | 4 | yes | — |
| 88 | C4×SD16 | — | 2 | 4 | no | — |
| 89 | C4×D8 | C23 | 1 | 4 | yes | — |
| 90 | #436 | D4 | 1 | 4 | yes | — |
| 91 | C8×D4 | C23 | 1 | 4 | yes | — |
| 92 | #428 | — | 2 | 4 | no | — |
| 93 | #428 | — | 2 | 8 | no | — |
| 94 | #427 | C23 | 1 | 8 | yes | — |
| 95 | #427 | C23 | 1 | 2 | yes | — |
| 96 | C2×#107 | C23 | 1 | 8 | yes | — |
| 97 | C2×#107 | C23 | 1 | 4 | yes | — |
| 98 | C2×#106 | — | 2 | 2 | no | — |
| 99 | #423 | D4 | 1 | 1 | yes | characteristic |
| 100 | #417 | — | 2 | 4 | no | — |
| 101 | #416 | — | 2 | 2 | no | — |
| 102 | #408 | D4 | 1 | 2 | yes | — |
| 103 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 104 | #367 | C23 | 1 | 2 | yes | — |
| 105 | #367 | C23 | 1 | 4 | yes | — |
| 106 | #367 | C23 | 1 | 4 | yes | — |
| 107 | #367 | C23 | 1 | 8 | yes | — |
| 108 | #366 | C23 | 1 | 4 | yes | — |
| 109 | #366 | C23 | 1 | 4 | yes | — |
| 110 | #366 | C23 | 1 | 2 | yes | — |
| 111 | #366 | C23 | 1 | 8 | yes | — |
| 112 | C4⋊C16 | C23 | 1 | 4 | yes | — |
| 113 | #358 | C23 | 1 | 4 | yes | — |
| 114 | #358 | C23 | 1 | 8 | yes | — |
| 115 | #357 | C23 | 1 | 4 | yes | — |
| 116 | #357 | C23 | 1 | 8 | yes | — |
| 117 | C22⋊C16 | C23 | 1 | 8 | yes | — |
| 118 | C4×C16 | C23 | 1 | 2 | yes | — |
| 119 | C4×C16 | C23 | 1 | 2 | yes | — |
| 120 | C2×#41 | — | 2 | 2 | no | — |
| 121 | C2×#41 | C2×D4 | 1 | 2 | yes | — |
| 122 | #135 | — | 2 | 4 | no | — |
| 123 | #135 | — | 2 | 4 | no | — |
| 124 | C2×Q16 | C24 | 1 | 2 | yes | — |
| 125 | C2×SD16 | — | 2 | 2 | no | — |
| 126 | C2×D8 | C24 | 1 | 2 | yes | — |
| 127 | #131 | — | 2 | 4 | no | — |
| 128 | #131 | — | 2 | 4 | no | — |
| 129 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 130 | C2×M16 | C2×D4 | 1 | 4 | yes | — |
| 131 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 132 | C22×C8 | C24 | 1 | 4 | yes | — |
| 133 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 134 | C4⋊Q8 | D8 | 1 | 1 | yes | characteristic |
| 135 | #127 | D8 | 1 | 1 | yes | characteristic |
| 136 | #126 | — | 4 | 8 | no | — |
| 137 | #126 | — | 4 | 8 | no | — |
| 138 | #125 | — | 2 | 8 | no | — |
| 139 | #125 | — | 4 | 8 | no | — |
| 140 | #125 | — | 4 | 2 | no | — |
| 141 | #125 | — | 4 | 2 | no | — |
| 142 | #124 | D8 | 1 | 2 | yes | — |
| 143 | #123 | — | 4 | 8 | no | — |
| 144 | #123 | — | 4 | 8 | no | — |
| 145 | C22⋊Q8 | — | 2 | 8 | no | — |
| 146 | C22⋊Q8 | — | 4 | 4 | no | — |
| 147 | #121 | — | 2 | 8 | no | — |
| 148 | #121 | — | 4 | 4 | no | — |
| 149 | #121 | — | 4 | 8 | no | — |
| 150 | #121 | — | 4 | 8 | no | — |
| 151 | #121 | D8 | 1 | 4 | yes | — |
| 152 | C4×Q8 | — | 2 | 4 | no | — |
| 153 | C4×Q8 | — | 4 | 4 | no | — |
| 154 | C4×D4 | — | 2 | 4 | no | — |
| 155 | C4×D4 | — | 4 | 4 | no | — |
| 156 | C4×D4 | — | 4 | 8 | no | — |
| 157 | C4×D4 | — | 4 | 4 | no | — |
| 158 | C4×D4 | — | 4 | 8 | no | — |
| 159 | C4×D4 | C2×D4 | 1 | 4 | yes | — |
| 160 | #117 | — | 2 | 2 | no | — |
| 161 | #117 | — | 2 | 8 | no | — |
| 162 | #117 | — | 4 | 8 | no | — |
| 163 | #117 | — | 4 | 2 | no | — |
| 164 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 165 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 166 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 167 | C2×C42 | — | 4 | 2 | no | — |
| 168 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | Q32 | — | 2 | 2 | no | — |
| 170 | SD32 | — | 2 | 4 | no | — |
| 171 | D16 | — | 2 | 2 | no | — |
| 172 | M32 | — | 2 | 4 | no | — |
| 173 | M32 | — | 2 | 8 | no | — |
| 174 | C2×C16 | — | 2 | 2 | no | — |
| 175 | C2×C16 | — | 2 | 8 | no | — |
| 176 | C2×C16 | #41 | 1 | 4 | yes | — |
| 177 | C2×C16 | C24 | 1 | 2 | yes | — |
| 178 | C2×C16 | C24 | 1 | 4 | yes | — |
| 179 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 180 | C2×C16 | C24 | 1 | 1 | yes | characteristic |
| 181 | #107 | — | 2 | 4 | no | — |
| 182 | #107 | — | 2 | 8 | no | — |
| 183 | #107 | C24 | 1 | 4 | yes | — |
| 184 | #107 | C24 | 1 | 8 | yes | — |
| 185 | #106 | — | 2 | 8 | no | — |
| 186 | #106 | — | 2 | 2 | no | — |
| 187 | #106 | — | 2 | 8 | no | — |
| 188 | C4⋊C8 | C2×D4 | 1 | 4 | yes | — |
| 189 | #103 | — | 2 | 4 | no | — |
| 190 | #103 | — | 2 | 8 | no | — |
| 191 | #102 | — | 2 | 4 | no | — |
| 192 | #102 | — | 2 | 8 | no | — |
| 193 | C22⋊C8 | C2×D4 | 1 | 8 | yes | — |
| 194 | C4×C8 | C2×D4 | 1 | 2 | yes | — |
| 195 | C4×C8 | C24 | 1 | 2 | yes | — |
| 196 | #41 | — | 2 | 4 | no | — |
| 197 | #41 | — | 2 | 4 | no | — |
| 198 | #41 | — | 4 | 4 | no | — |
| 199 | #41 | — | 4 | 4 | no | — |
| 200 | C2×Q8 | — | 2 | 2 | no | — |
| 201 | C2×Q8 | C2×D8 | 1 | 2 | yes | — |
| 202 | C2×D4 | — | 2 | 2 | no | — |
| 203 | C2×D4 | — | 4 | 2 | no | — |
| 204 | C2×D4 | — | 4 | 8 | no | — |
| 205 | C2×D4 | C2×D8 | 1 | 4 | yes | — |
| 206 | C2×D4 | C2×D8 | 1 | 2 | yes | — |
| 207 | C2×D4 | C2×D8 | 1 | 4 | yes | — |
| 208 | C22×C4 | — | 4 | 4 | no | — |
| 209 | C22×C4 | — | 4 | 8 | no | — |
| 210 | C22×C4 | — | 4 | 2 | no | — |
| 211 | C22×C4 | — | 4 | 4 | no | — |
| 212 | C22×C4 | C2×D8 | 1 | 2 | yes | — |
| 213 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 214 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 215 | Q16 | — | 2 | 2 | no | — |
| 216 | Q16 | — | 2 | 2 | no | — |
| 217 | SD16 | — | 4 | 4 | no | — |
| 218 | D8 | — | 2 | 2 | no | — |
| 219 | D8 | — | 2 | 2 | no | — |
| 220 | M16 | — | 2 | 4 | no | — |
| 221 | M16 | — | 2 | 8 | no | — |
| 222 | C2×C8 | — | 2 | 2 | no | — |
| 223 | C2×C8 | — | 2 | 4 | no | — |
| 224 | C2×C8 | — | 2 | 4 | no | — |
| 225 | C2×C8 | C22×D4 | 1 | 2 | yes | — |
| 226 | C2×C8 | C22×D4 | 1 | 4 | yes | — |
| 227 | C2×C8 | C22×D4 | 1 | 1 | yes | characteristic |
| 228 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 229 | C2×C8 | 2−1+4 | 1 | 2 | yes | — |
| 230 | C2×C8 | C25 | 1 | 1 | yes | characteristic |
| 231 | C4⋊C4 | — | 2 | 4 | no | — |
| 232 | C4⋊C4 | — | 2 | 8 | no | — |
| 233 | C4⋊C4 | — | 4 | 4 | no | — |
| 234 | C4⋊C4 | — | 4 | 8 | no | — |
| 235 | C4⋊C4 | — | 4 | 8 | no | — |
| 236 | C4⋊C4 | — | 4 | 8 | no | — |
| 237 | C4⋊C4 | — | 4 | 4 | no | — |
| 238 | C4⋊C4 | — | 4 | 8 | no | — |
| 239 | C4⋊C4 | C2×D8 | 1 | 4 | yes | — |
| 240 | C22⋊C4 | — | 4 | 2 | no | — |
| 241 | C22⋊C4 | — | 4 | 8 | no | — |
| 242 | C22⋊C4 | — | 4 | 8 | no | — |
| 243 | C22⋊C4 | — | 4 | 4 | no | — |
| 244 | C22⋊C4 | — | 4 | 8 | no | — |
| 245 | C22⋊C4 | C2×D8 | 1 | 8 | yes | — |
| 246 | C42 | — | 4 | 2 | no | — |
| 247 | C42 | — | 4 | 4 | no | — |
| 248 | C42 | — | 4 | 8 | no | — |
| 249 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 250 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 251 | C42 | C22×D4 | 1 | 2 | yes | — |
| 252 | C16 | — | 2 | 4 | no | — |
| 253 | C16 | — | 2 | 2 | no | — |
| 254 | C16 | C2×#41 | 1 | 2 | yes | — |
| 255 | C16 | C2×#41 | 1 | 2 | yes | — |
| 256 | C23 | — | 4 | 2 | no | — |
| 257 | C23 | C22×D8 | 1 | 4 | yes | — |
| 258 | C23 | C22×D8 | 1 | 1 | yes | characteristic |
| 259 | Q8 | — | 2 | 4 | no | — |
| 260 | Q8 | — | 4 | 2 | no | — |
| 261 | Q8 | — | 4 | 2 | no | — |
| 262 | D4 | — | 2 | 8 | no | — |
| 263 | D4 | — | 2 | 4 | no | — |
| 264 | D4 | — | 4 | 4 | no | — |
| 265 | D4 | — | 4 | 2 | no | — |
| 266 | D4 | — | 4 | 2 | no | — |
| 267 | D4 | — | 8 | 4 | no | — |
| 268 | D4 | — | 16 | 8 | no | — |
| 269 | C2×C4 | — | 2 | 4 | no | — |
| 270 | C2×C4 | — | 2 | 4 | no | — |
| 271 | C2×C4 | — | 4 | 2 | no | — |
| 272 | C2×C4 | — | 4 | 4 | no | — |
| 273 | C2×C4 | — | 4 | 2 | no | — |
| 274 | C2×C4 | — | 4 | 8 | no | — |
| 275 | C2×C4 | — | 8 | 4 | no | — |
| 276 | C2×C4 | — | 16 | 4 | no | — |
| 277 | C2×C4 | — | 16 | 8 | no | — |
| 278 | C2×C4 | — | 16 | 4 | no | — |
| 279 | C2×C4 | #547 | 1 | 2 | yes | — |
| 280 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 281 | C2×C4 | C22×D8 | 1 | 4 | yes | — |
| 282 | C2×C4 | C22×D8 | 1 | 2 | yes | — |
| 283 | C2×C4 | C22×D8 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | #578 | 1 | 2 | yes | — |
| 285 | C2×C4 | C23×D4 | 1 | 1 | yes | characteristic |
| 286 | C8 | — | 2 | 4 | no | — |
| 287 | C8 | — | 2 | 2 | no | — |
| 288 | C8 | #547 | 1 | 2 | yes | — |
| 289 | C8 | C22×#41 | 1 | 1 | yes | characteristic |
| 290 | C8 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 291 | C22 | — | 2 | 2 | no | — |
| 292 | C22 | — | 2 | 4 | no | — |
| 293 | C22 | — | 2 | 8 | no | — |
| 294 | C22 | — | 8 | 2 | no | — |
| 295 | C22 | — | 8 | 4 | no | — |
| 296 | C22 | #3334 | 1 | 2 | yes | — |
| 297 | C22 | C23×D8 | 1 | 1 | yes | characteristic |
| 298 | C4 | — | 2 | 4 | no | — |
| 299 | C4 | — | 2 | 2 | no | — |
| 300 | C4 | — | 2 | 2 | no | — |
| 301 | C4 | — | 8 | 2 | no | — |
| 302 | C4 | — | 8 | 4 | no | — |
| 303 | C4 | — | 8 | 8 | no | — |
| 304 | C4 | #3334 | 1 | 2 | yes | — |
| 305 | C4 | C2×#547 | 1 | 1 | yes | characteristic |
| 306 | C4 | C2×#578 | 1 | 1 | yes | characteristic |
| 307 | C2 | — | 2 | 2 | no | — |
| 308 | C2 | — | 4 | 4 | no | — |
| 309 | C2 | — | 16 | 2 | no | — |
| 310 | C2 | C2×#3334 | 1 | 1 | yes | characteristic |
| 311 | C2 | #62635 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #61289 | C2 | semidirect |
| 2. #33922 | C2 | semidirect |
| 3. #33919 | C2 | semidirect |
| 4. #33904 | C2 | semidirect |
| 5. C2×#2253 | C2 | semidirect |
| 6. #33652 | C2 | semidirect |
| 7. #33630 | C2 | semidirect |
| 8. #33611 | C2 | semidirect |
| 9. C4×#508 | C2 | semidirect |
| 10. #33434 | C2 | semidirect |
| 11. #33411 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #61289 | 2 |
| #33922 | 2 |
| #33919 | 2 |
| #33904 | 2 |
| C2×#2253 | 2 |
| #33652 | 2 |
| #33630 | 2 |
| #33611 | 2 |
| C4×#508 | 2 |
| #33434 | 2 |
| #33411 | 2 |