Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1070 conjugacy classes of subgroups.
The subgroup lattice has 322 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #29232 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #29230 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #29154 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #28249 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #27833 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #24985 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #23833 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #21732 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #21558 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #20787 | C2 | 1 | 4 | yes | maximal, complemented |
| 11 | C4×#443 | C2 | 1 | 2 | yes | maximal, complemented |
| 12 | #3160 | C22 | 1 | 1 | yes | characteristic |
| 13 | #3067 | C22 | 1 | 4 | yes | — |
| 14 | #2960 | C22 | 1 | 2 | yes | — |
| 15 | #2955 | C22 | 1 | 4 | yes | — |
| 16 | C2×#443 | C22 | 1 | 2 | yes | — |
| 17 | C2×#443 | C22 | 1 | 2 | yes | — |
| 18 | C4×#135 | C22 | 1 | 2 | yes | — |
| 19 | C4×#131 | C22 | 1 | 4 | yes | — |
| 20 | #2103 | C22 | 1 | 4 | yes | — |
| 21 | #2102 | C22 | 1 | 4 | yes | — |
| 22 | #2080 | C22 | 1 | 8 | yes | — |
| 23 | #2053 | C22 | 1 | 4 | yes | — |
| 24 | #2050 | C22 | 1 | 4 | yes | — |
| 25 | #2031 | C22 | 1 | 8 | yes | — |
| 26 | #1994 | C22 | 1 | 2 | yes | — |
| 27 | #1976 | C22 | 1 | 4 | yes | — |
| 28 | #1953 | C22 | 1 | 4 | yes | — |
| 29 | #1945 | C22 | 1 | 4 | yes | — |
| 30 | #1932 | C22 | 1 | 8 | yes | — |
| 31 | #1906 | C22 | 1 | 8 | yes | — |
| 32 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1841 | C22 | 1 | 8 | yes | — |
| 34 | #1833 | C22 | 1 | 4 | yes | — |
| 35 | #1833 | C22 | 1 | 4 | yes | — |
| 36 | #1827 | C22 | 1 | 8 | yes | — |
| 37 | #1814 | C22 | 1 | 4 | yes | — |
| 38 | #1813 | C22 | 1 | 4 | yes | — |
| 39 | #1812 | C22 | 1 | 4 | yes | — |
| 40 | C4×#108 | C22 | 1 | 2 | yes | — |
| 41 | #1772 | C22 | 1 | 2 | yes | — |
| 42 | #1772 | C22 | 1 | 2 | yes | — |
| 43 | #1772 | C22 | 1 | 4 | yes | — |
| 44 | #1764 | C22 | 1 | 4 | yes | — |
| 45 | #1764 | C22 | 1 | 4 | yes | — |
| 46 | C4×#104 | C22 | 1 | 4 | yes | — |
| 47 | #1757 | C22 | 1 | 8 | yes | — |
| 48 | #1747 | C22 | 1 | 4 | yes | — |
| 49 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 50 | #575 | C23 | 1 | 4 | yes | — |
| 51 | C2×#135 | C23 | 1 | 2 | yes | — |
| 52 | C2×#131 | C23 | 1 | 4 | yes | — |
| 53 | C2×#131 | C23 | 1 | 4 | yes | — |
| 54 | #533 | C23 | 1 | 2 | yes | — |
| 55 | C4×#41 | C23 | 1 | 4 | yes | — |
| 56 | #500 | — | 2 | 1 | no | — |
| 57 | #499 | — | 2 | 2 | no | — |
| 58 | #498 | — | 2 | 1 | no | — |
| 59 | #473 | — | 2 | 4 | no | — |
| 60 | #472 | — | 2 | 4 | no | — |
| 61 | #471 | — | 2 | 8 | no | — |
| 62 | #446 | — | 4 | 2 | no | — |
| 63 | C8×Q8 | — | 4 | 2 | no | — |
| 64 | #444 | — | 4 | 4 | no | — |
| 65 | #443 | — | 2 | 2 | no | — |
| 66 | #443 | — | 2 | 2 | no | — |
| 67 | #443 | — | 2 | 2 | no | — |
| 68 | #443 | — | 2 | 2 | no | — |
| 69 | #443 | — | 2 | 2 | no | — |
| 70 | #443 | — | 2 | 2 | no | — |
| 71 | #443 | — | 2 | 2 | no | — |
| 72 | #443 | — | 2 | 2 | no | — |
| 73 | #443 | — | 4 | 4 | no | — |
| 74 | C4×Q16 | — | 2 | 2 | no | — |
| 75 | C4×SD16 | — | 2 | 4 | no | — |
| 76 | C4×D8 | — | 2 | 2 | no | — |
| 77 | #430 | C23 | 1 | 4 | yes | — |
| 78 | C2×#108 | C23 | 1 | 2 | yes | — |
| 79 | C2×#108 | C23 | 1 | 4 | yes | — |
| 80 | C2×#108 | C23 | 1 | 2 | yes | — |
| 81 | #427 | C23 | 1 | 2 | yes | — |
| 82 | #427 | C23 | 1 | 4 | yes | — |
| 83 | #424 | C23 | 1 | 4 | yes | — |
| 84 | #423 | C23 | 1 | 2 | yes | — |
| 85 | #421 | C23 | 1 | 8 | yes | — |
| 86 | C2×#104 | C23 | 1 | 4 | yes | — |
| 87 | C2×#104 | C23 | 1 | 4 | yes | — |
| 88 | #416 | C23 | 1 | 4 | yes | — |
| 89 | #413 | C23 | 1 | 8 | yes | — |
| 90 | #410 | C23 | 1 | 8 | yes | — |
| 91 | #405 | — | 2 | 4 | no | — |
| 92 | #405 | — | 2 | 4 | no | — |
| 93 | #405 | C23 | 1 | 4 | yes | — |
| 94 | #405 | C23 | 1 | 4 | yes | — |
| 95 | #405 | C23 | 1 | 4 | yes | — |
| 96 | C4×M16 | — | 2 | 4 | no | — |
| 97 | C4×M16 | C23 | 1 | 4 | yes | — |
| 98 | C2×C4×C8 | — | 2 | 4 | no | — |
| 99 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 100 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 101 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 102 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 103 | C43 | C23 | 1 | 1 | yes | characteristic |
| 104 | #344 | C23 | 1 | 8 | yes | — |
| 105 | #344 | C23 | 1 | 8 | yes | — |
| 106 | #341 | C23 | 1 | 4 | yes | — |
| 107 | #339 | C23 | 1 | 4 | yes | — |
| 108 | #339 | C23 | 1 | 8 | yes | — |
| 109 | #339 | C23 | 1 | 4 | yes | — |
| 110 | #338 | C23 | 1 | 4 | yes | — |
| 111 | H(3,ℤ/4) | C23 | 1 | 4 | yes | — |
| 112 | H(3,ℤ/4) | C23 | 1 | 8 | yes | — |
| 113 | C2×#41 | C24 | 1 | 4 | yes | — |
| 114 | #137 | — | 2 | 8 | no | — |
| 115 | AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 116 | #135 | — | 2 | 2 | no | — |
| 117 | #135 | — | 2 | 2 | no | — |
| 118 | #135 | — | 2 | 2 | no | — |
| 119 | #135 | — | 2 | 2 | no | — |
| 120 | #135 | — | 4 | 4 | no | — |
| 121 | C2×Q16 | #41 | 1 | 2 | yes | — |
| 122 | C2×SD16 | #41 | 1 | 4 | yes | — |
| 123 | C2×D8 | #41 | 1 | 2 | yes | — |
| 124 | #131 | — | 2 | 4 | no | — |
| 125 | #131 | — | 2 | 4 | no | — |
| 126 | #131 | — | 2 | 4 | no | — |
| 127 | #131 | — | 2 | 4 | no | — |
| 128 | #131 | — | 2 | 4 | no | — |
| 129 | #131 | — | 2 | 4 | no | — |
| 130 | #131 | — | 2 | 4 | no | — |
| 131 | #131 | — | 2 | 4 | no | — |
| 132 | C2×M16 | — | 2 | 4 | no | — |
| 133 | C2×M16 | — | 2 | 4 | no | — |
| 134 | C2×M16 | C24 | 1 | 4 | yes | — |
| 135 | C2×M16 | C24 | 1 | 4 | yes | — |
| 136 | C2×M16 | C24 | 1 | 4 | yes | — |
| 137 | C22×C8 | — | 2 | 4 | no | — |
| 138 | C22×C8 | — | 2 | 4 | no | — |
| 139 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 140 | C22×C8 | C24 | 1 | 2 | yes | — |
| 141 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 142 | C4⋊Q8 | — | 2 | 2 | no | — |
| 143 | #125 | — | 2 | 2 | no | — |
| 144 | C22⋊Q8 | — | 4 | 2 | no | — |
| 145 | C4×Q8 | — | 2 | 4 | no | — |
| 146 | C4×Q8 | — | 4 | 2 | no | — |
| 147 | C4×D4 | — | 2 | 4 | no | — |
| 148 | C4×D4 | — | 4 | 4 | no | — |
| 149 | #117 | — | 2 | 4 | no | — |
| 150 | #117 | C24 | 1 | 4 | yes | — |
| 151 | #117 | C24 | 1 | 4 | yes | — |
| 152 | C2×C42 | — | 2 | 4 | no | — |
| 153 | C2×C42 | — | 4 | 1 | no | — |
| 154 | C2×C42 | C24 | 1 | 2 | yes | — |
| 155 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 156 | #108 | — | 2 | 2 | no | — |
| 157 | #108 | — | 2 | 2 | no | — |
| 158 | #108 | — | 2 | 2 | no | — |
| 159 | #108 | — | 2 | 2 | no | — |
| 160 | #108 | — | 2 | 4 | no | — |
| 161 | #108 | — | 2 | 4 | no | — |
| 162 | #108 | — | 4 | 4 | no | — |
| 163 | #107 | — | 2 | 2 | no | — |
| 164 | #107 | — | 2 | 4 | no | — |
| 165 | #106 | — | 2 | 2 | no | — |
| 166 | #106 | — | 2 | 4 | no | — |
| 167 | C4⋊C8 | — | 4 | 2 | no | — |
| 168 | C4⋊C8 | — | 4 | 4 | no | — |
| 169 | #104 | — | 2 | 4 | no | — |
| 170 | #104 | — | 2 | 4 | no | — |
| 171 | #104 | — | 2 | 4 | no | — |
| 172 | #104 | — | 2 | 4 | no | — |
| 173 | #104 | — | 4 | 8 | no | — |
| 174 | #103 | — | 2 | 4 | no | — |
| 175 | #102 | — | 2 | 4 | no | — |
| 176 | #101 | — | 2 | 8 | no | — |
| 177 | #100 | — | 2 | 8 | no | — |
| 178 | #99 | — | 4 | 8 | no | — |
| 179 | #97 | — | 2 | 4 | no | — |
| 180 | #97 | — | 2 | 4 | no | — |
| 181 | #97 | — | 2 | 4 | no | — |
| 182 | #97 | — | 2 | 4 | no | — |
| 183 | C4×C8 | — | 2 | 4 | no | — |
| 184 | C4×C8 | — | 2 | 4 | no | — |
| 185 | C4×C8 | — | 2 | 2 | no | — |
| 186 | C4×C8 | — | 2 | 2 | no | — |
| 187 | C4×C8 | — | 2 | 1 | no | — |
| 188 | C4×C8 | — | 2 | 1 | no | — |
| 189 | C4×C8 | — | 2 | 4 | no | — |
| 190 | C4×C8 | — | 2 | 4 | no | — |
| 191 | C4×C8 | #41 | 1 | 2 | yes | — |
| 192 | C4×C8 | #41 | 1 | 2 | yes | — |
| 193 | C4×C8 | #41 | 1 | 2 | yes | — |
| 194 | C4×C8 | #41 | 1 | 2 | yes | — |
| 195 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 196 | C4×C8 | #41 | 1 | 2 | yes | — |
| 197 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 198 | #41 | — | 2 | 4 | no | — |
| 199 | #41 | — | 2 | 4 | no | — |
| 200 | #41 | — | 2 | 4 | no | — |
| 201 | #41 | — | 2 | 4 | no | — |
| 202 | C2×Q8 | — | 4 | 2 | no | — |
| 203 | C2×Q8 | C2×#41 | 1 | 4 | yes | — |
| 204 | C2×D4 | — | 2 | 4 | no | — |
| 205 | C2×D4 | C2×#41 | 1 | 4 | yes | — |
| 206 | C22×C4 | — | 2 | 2 | no | — |
| 207 | C22×C4 | — | 2 | 1 | no | — |
| 208 | C22×C4 | — | 2 | 4 | no | — |
| 209 | C22×C4 | — | 4 | 4 | no | — |
| 210 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 211 | Q16 | — | 2 | 2 | no | — |
| 212 | Q16 | — | 2 | 2 | no | — |
| 213 | Q16 | — | 4 | 4 | no | — |
| 214 | SD16 | — | 2 | 4 | no | — |
| 215 | SD16 | — | 2 | 4 | no | — |
| 216 | SD16 | — | 4 | 8 | no | — |
| 217 | D8 | — | 2 | 2 | no | — |
| 218 | D8 | — | 2 | 2 | no | — |
| 219 | D8 | — | 4 | 4 | no | — |
| 220 | M16 | — | 2 | 4 | no | — |
| 221 | M16 | — | 2 | 8 | no | — |
| 222 | M16 | — | 2 | 4 | no | — |
| 223 | M16 | — | 2 | 4 | no | — |
| 224 | M16 | — | 2 | 4 | no | — |
| 225 | M16 | — | 4 | 4 | no | — |
| 226 | M16 | — | 4 | 4 | no | — |
| 227 | M16 | — | 4 | 4 | no | — |
| 228 | M16 | — | 4 | 4 | no | — |
| 229 | C2×C8 | — | 2 | 4 | no | — |
| 230 | C2×C8 | — | 2 | 4 | no | — |
| 231 | C2×C8 | — | 2 | 2 | no | — |
| 232 | C2×C8 | — | 2 | 1 | no | — |
| 233 | C2×C8 | — | 2 | 4 | no | — |
| 234 | C2×C8 | — | 4 | 4 | no | — |
| 235 | C2×C8 | — | 4 | 4 | no | — |
| 236 | C2×C8 | — | 4 | 4 | no | — |
| 237 | C2×C8 | — | 4 | 4 | no | — |
| 238 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 239 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 240 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 241 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 242 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 243 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 244 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 245 | C2×C8 | C2×#41 | 1 | 2 | yes | — |
| 246 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 247 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 248 | C4⋊C4 | — | 2 | 4 | no | — |
| 249 | C4⋊C4 | — | 2 | 4 | no | — |
| 250 | C4⋊C4 | — | 4 | 4 | no | — |
| 251 | C4⋊C4 | — | 4 | 4 | no | — |
| 252 | C4⋊C4 | — | 4 | 2 | no | — |
| 253 | C22⋊C4 | — | 2 | 4 | no | — |
| 254 | C22⋊C4 | — | 2 | 4 | no | — |
| 255 | C22⋊C4 | — | 4 | 4 | no | — |
| 256 | C42 | — | 2 | 2 | no | — |
| 257 | C42 | — | 2 | 1 | no | — |
| 258 | C42 | — | 2 | 4 | no | — |
| 259 | C42 | — | 2 | 4 | no | — |
| 260 | C42 | — | 4 | 4 | no | — |
| 261 | C42 | — | 4 | 4 | no | — |
| 262 | C42 | C2×#41 | 1 | 2 | yes | — |
| 263 | C42 | C2×#41 | 1 | 2 | yes | — |
| 264 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 265 | C42 | C2×#41 | 1 | 1 | yes | characteristic |
| 266 | C23 | — | 2 | 4 | no | — |
| 267 | C23 | #585 | 1 | 1 | yes | characteristic |
| 268 | Q8 | — | 2 | 4 | no | — |
| 269 | Q8 | — | 2 | 4 | no | — |
| 270 | Q8 | — | 8 | 4 | no | — |
| 271 | D4 | — | 2 | 4 | no | — |
| 272 | D4 | — | 2 | 4 | no | — |
| 273 | D4 | — | 4 | 4 | no | — |
| 274 | D4 | — | 4 | 4 | no | — |
| 275 | C2×C4 | — | 2 | 1 | no | — |
| 276 | C2×C4 | — | 2 | 2 | no | — |
| 277 | C2×C4 | — | 2 | 1 | no | — |
| 278 | C2×C4 | — | 2 | 2 | no | — |
| 279 | C2×C4 | — | 2 | 4 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 4 | 2 | no | — |
| 282 | C2×C4 | — | 4 | 4 | no | — |
| 283 | C2×C4 | — | 4 | 4 | no | — |
| 284 | C2×C4 | — | 4 | 4 | no | — |
| 285 | C2×C4 | — | 4 | 4 | no | — |
| 286 | C2×C4 | — | 8 | 8 | no | — |
| 287 | C2×C4 | C22×#41 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 289 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 290 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 291 | C8 | — | 4 | 4 | no | — |
| 292 | C8 | — | 4 | 4 | no | — |
| 293 | C8 | — | 4 | 1 | no | — |
| 294 | C8 | — | 4 | 4 | no | — |
| 295 | C8 | #529 | 1 | 2 | yes | — |
| 296 | C8 | #529 | 1 | 2 | yes | — |
| 297 | C8 | #529 | 1 | 2 | yes | — |
| 298 | C8 | #529 | 1 | 2 | yes | — |
| 299 | C8 | #555 | 1 | 2 | yes | — |
| 300 | C8 | #555 | 1 | 1 | yes | characteristic |
| 301 | C8 | #555 | 1 | 1 | yes | characteristic |
| 302 | C22 | — | 4 | 1 | no | — |
| 303 | C22 | — | 4 | 4 | no | — |
| 304 | C22 | — | 8 | 4 | no | — |
| 305 | C22 | #3460 | 1 | 1 | yes | characteristic |
| 306 | C22 | #3475 | 1 | 2 | yes | — |
| 307 | C4 | — | 2 | 4 | no | — |
| 308 | C4 | — | 2 | 2 | no | — |
| 309 | C4 | — | 4 | 1 | no | — |
| 310 | C4 | — | 4 | 2 | no | — |
| 311 | C4 | — | 4 | 4 | no | — |
| 312 | C4 | — | 8 | 4 | no | — |
| 313 | C4 | — | 8 | 4 | no | — |
| 314 | C4 | C2×#529 | 1 | 2 | yes | — |
| 315 | C4 | C2×#555 | 1 | 1 | yes | characteristic |
| 316 | C4 | #3481 | 1 | 1 | yes | characteristic |
| 317 | C2 | — | 2 | 1 | no | — |
| 318 | C2 | — | 2 | 2 | no | — |
| 319 | C2 | — | 8 | 4 | no | — |
| 320 | C2 | #34943 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C8 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #34943 |
| prime radical | #28249 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #29232 | C2 | semidirect |
| 2. #29230 | C2 | semidirect |
| 3. #29154 | C2 | semidirect |
| 5. #27833 | C2 | semidirect |
| 6. #24985 | C2 | semidirect |
| 7. #23833 | C2 | semidirect |
| 8. #21732 | C2 | semidirect |
| 9. #21558 | C2 | semidirect |
| 10. #20787 | C2 | semidirect |
| 11. C4×#443 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #29232 | 2 |
| #29230 | 2 |
| #29154 | 2 |
| #28249 | 2 |
| #27833 | 2 |
| #24985 | 2 |
| #23833 | 2 |
| #21732 | 2 |
| #21558 | 2 |
| #20787 | 2 |
| C4×#443 | 2 |