Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1735 conjugacy classes of subgroups.
The subgroup lattice has 317 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #60648 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #23360 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #23200 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #23196 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22460 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #22457 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #22285 | C2 | 1 | 8 | yes | maximal, complemented |
| 8 | #21783 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #20787 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | C2×#578 | C22 | 1 | 4 | yes | — |
| 11 | #3499 | C22 | 1 | 2 | yes | — |
| 12 | #3156 | C4 | 1 | 8 | yes | complemented |
| 13 | #3156 | C22 | 1 | 4 | yes | — |
| 14 | #3152 | C4 | 1 | 4 | yes | complemented |
| 15 | #3152 | C22 | 1 | 2 | yes | — |
| 16 | #3147 | C4 | 1 | 2 | yes | complemented |
| 17 | #3147 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#494 | C4 | 1 | 2 | yes | complemented |
| 19 | C2×#494 | C22 | 1 | 1 | yes | characteristic |
| 20 | #2957 | C22 | 1 | 8 | yes | — |
| 21 | #2950 | C22 | 1 | 8 | yes | — |
| 22 | #2889 | C22 | 1 | 4 | yes | — |
| 23 | #2884 | C22 | 1 | 4 | yes | — |
| 24 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 25 | #1979 | C22 | 1 | 8 | yes | — |
| 26 | #1976 | C22 | 1 | 8 | yes | — |
| 27 | #1912 | C22 | 1 | 8 | yes | — |
| 28 | #1909 | C22 | 1 | 8 | yes | — |
| 29 | #1905 | C22 | 1 | 16 | yes | — |
| 30 | #1902 | C22 | 1 | 16 | yes | — |
| 31 | #1886 | C22 | 1 | 8 | yes | — |
| 32 | #1884 | C22 | 1 | 8 | yes | — |
| 33 | #1841 | C22 | 1 | 4 | yes | — |
| 34 | #1833 | C22 | 1 | 4 | yes | — |
| 35 | #1827 | C22 | 1 | 4 | yes | — |
| 36 | #1813 | C22 | 1 | 2 | yes | — |
| 37 | #1812 | C22 | 1 | 2 | yes | — |
| 38 | #1796 | C22 | 1 | 8 | yes | — |
| 39 | #1772 | C22 | 1 | 4 | yes | — |
| 40 | #1757 | C22 | 1 | 4 | yes | — |
| 41 | #1747 | C22 | 1 | 4 | yes | — |
| 42 | C2×2−1+4 | C23 | 1 | 4 | yes | — |
| 43 | #578 | — | 4 | 8 | no | — |
| 44 | #578 | — | 4 | 8 | no | — |
| 45 | C2×#137 | C2×C4 | 1 | 16 | yes | — |
| 46 | C2×#137 | C23 | 1 | 8 | yes | — |
| 47 | C2×#135 | C2×C4 | 1 | 8 | yes | — |
| 48 | C2×#135 | C23 | 1 | 4 | yes | — |
| 49 | C22×Q16 | C2×C4 | 1 | 8 | yes | — |
| 50 | C22×Q16 | C23 | 1 | 4 | yes | — |
| 51 | #568 | C23 | 1 | 8 | yes | — |
| 52 | C2×#131 | C23 | 1 | 2 | yes | — |
| 53 | D4×Q8 | — | 4 | 2 | no | — |
| 54 | #547 | — | 4 | 2 | no | — |
| 55 | #547 | — | 4 | 4 | no | — |
| 56 | #536 | C2×C4 | 1 | 8 | yes | — |
| 57 | #536 | C23 | 1 | 4 | yes | — |
| 58 | #532 | C2×C4 | 1 | 4 | yes | — |
| 59 | #532 | C23 | 1 | 2 | yes | — |
| 60 | C2×C4⋊Q8 | C2×C4 | 1 | 4 | yes | — |
| 61 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 62 | #519 | C23 | 1 | 8 | yes | — |
| 63 | C4×#41 | C23 | 1 | 1 | yes | characteristic |
| 64 | #497 | C2×C4 | 1 | 16 | yes | — |
| 65 | #497 | C2×C4 | 1 | 8 | yes | — |
| 66 | #495 | C2×C4 | 1 | 8 | yes | — |
| 67 | #495 | C2×C4 | 1 | 4 | yes | — |
| 68 | #494 | C2×C4 | 1 | 1 | yes | characteristic |
| 69 | #494 | C2×C4 | 1 | 2 | yes | — |
| 70 | #494 | C2×C4 | 1 | 4 | yes | — |
| 71 | #494 | C2×C4 | 1 | 8 | yes | — |
| 72 | #494 | C2×C4 | 1 | 2 | yes | — |
| 73 | #494 | C2×C4 | 1 | 4 | yes | — |
| 74 | #493 | C2×C4 | 1 | 1 | yes | characteristic |
| 75 | #492 | C2×C4 | 1 | 4 | yes | — |
| 76 | #492 | C2×C4 | 1 | 2 | yes | — |
| 77 | #444 | — | 4 | 8 | no | — |
| 78 | #443 | — | 4 | 8 | no | — |
| 79 | #441 | — | 4 | 8 | no | — |
| 80 | C4×Q16 | — | 4 | 8 | no | — |
| 81 | C2×#108 | C23 | 1 | 4 | yes | — |
| 82 | #427 | C23 | 1 | 4 | yes | — |
| 83 | #421 | C23 | 1 | 16 | yes | — |
| 84 | #419 | C23 | 1 | 16 | yes | — |
| 85 | #413 | — | 4 | 4 | no | — |
| 86 | #413 | C23 | 1 | 4 | yes | — |
| 87 | C2×#101 | — | 4 | 4 | no | — |
| 88 | C2×#101 | C23 | 1 | 8 | yes | — |
| 89 | #410 | — | 4 | 8 | no | — |
| 90 | #410 | C23 | 1 | 4 | yes | — |
| 91 | #410 | C23 | 1 | 8 | yes | — |
| 92 | #405 | C2×C4 | 1 | 4 | yes | — |
| 93 | #405 | C23 | 1 | 8 | yes | — |
| 94 | #405 | C23 | 1 | 2 | yes | — |
| 95 | C4×M16 | C2×C4 | 1 | 2 | yes | — |
| 96 | C4×M16 | C23 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C8 | C2×C4 | 1 | 2 | yes | — |
| 98 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 99 | #344 | C23 | 1 | 8 | yes | — |
| 100 | #339 | C23 | 1 | 8 | yes | — |
| 101 | #338 | C23 | 1 | 4 | yes | — |
| 102 | H(3,ℤ/4) | C23 | 1 | 4 | yes | — |
| 103 | H(3,ℤ/4) | C23 | 1 | 8 | yes | — |
| 104 | 2−1+4 | — | 4 | 16 | no | — |
| 105 | C2×#41 | — | 2 | 8 | no | — |
| 106 | C2×#41 | — | 2 | 4 | no | — |
| 107 | C2×#41 | C22×C4 | 1 | 8 | yes | — |
| 108 | C2×#41 | C24 | 1 | 4 | yes | — |
| 109 | C2×#41 | C24 | 1 | 1 | yes | characteristic |
| 110 | C22×Q8 | — | 2 | 4 | no | — |
| 111 | C22×Q8 | C22×C4 | 1 | 8 | yes | — |
| 112 | C22×Q8 | C24 | 1 | 4 | yes | — |
| 113 | #137 | — | 4 | 32 | no | — |
| 114 | #137 | — | 4 | 16 | no | — |
| 115 | #135 | — | 2 | 8 | no | — |
| 116 | #135 | — | 2 | 8 | no | — |
| 117 | #135 | — | 4 | 8 | no | — |
| 118 | #135 | — | 4 | 8 | no | — |
| 119 | C2×Q16 | — | 2 | 8 | no | — |
| 120 | C2×Q16 | — | 2 | 8 | no | — |
| 121 | C2×Q16 | — | 4 | 8 | no | — |
| 122 | C2×Q16 | — | 4 | 8 | no | — |
| 123 | C2×Q16 | C22×C4 | 1 | 4 | yes | — |
| 124 | C2×Q16 | C22×C4 | 1 | 8 | yes | — |
| 125 | C2×Q16 | C22×C4 | 1 | 8 | yes | — |
| 126 | C2×Q16 | C22×C4 | 1 | 16 | yes | — |
| 127 | C2×SD16 | C22×C4 | 1 | 8 | yes | — |
| 128 | C2×SD16 | C22×C4 | 1 | 16 | yes | — |
| 129 | C2×D8 | C22×C4 | 1 | 4 | yes | — |
| 130 | #131 | — | 2 | 4 | no | — |
| 131 | #131 | — | 2 | 4 | no | — |
| 132 | #131 | — | 4 | 16 | no | — |
| 133 | C2×M16 | — | 4 | 8 | no | — |
| 134 | C2×M16 | C22×C4 | 1 | 4 | yes | — |
| 135 | C2×M16 | C24 | 1 | 8 | yes | — |
| 136 | C2×M16 | C24 | 1 | 2 | yes | — |
| 137 | C22×C8 | C22×C4 | 1 | 4 | yes | — |
| 138 | C22×C8 | C24 | 1 | 2 | yes | — |
| 139 | C4⋊Q8 | — | 4 | 4 | no | — |
| 140 | C4⋊Q8 | — | 4 | 2 | no | — |
| 141 | C4⋊Q8 | C22×C4 | 1 | 2 | yes | — |
| 142 | C4⋊Q8 | C22×C4 | 1 | 4 | yes | — |
| 143 | C4⋊Q8 | C22×C4 | 1 | 8 | yes | — |
| 144 | C4⋊Q8 | C22×C4 | 1 | 4 | yes | — |
| 145 | #127 | C22×C4 | 1 | 2 | yes | — |
| 146 | #124 | C22×C4 | 1 | 8 | yes | — |
| 147 | #124 | C22×C4 | 1 | 4 | yes | — |
| 148 | #123 | — | 4 | 8 | no | — |
| 149 | #123 | — | 4 | 4 | no | — |
| 150 | C22⋊Q8 | — | 4 | 8 | no | — |
| 151 | C22⋊Q8 | — | 4 | 4 | no | — |
| 152 | #121 | — | 4 | 4 | no | — |
| 153 | #121 | — | 4 | 2 | no | — |
| 154 | C4×Q8 | — | 2 | 1 | no | — |
| 155 | C4×Q8 | — | 4 | 16 | no | — |
| 156 | C4×D4 | — | 2 | 2 | no | — |
| 157 | C4×D4 | — | 2 | 1 | no | — |
| 158 | C4×D4 | — | 4 | 4 | no | — |
| 159 | C4×D4 | — | 4 | 2 | no | — |
| 160 | #117 | — | 4 | 8 | no | — |
| 161 | #117 | C22×C4 | 1 | 2 | yes | — |
| 162 | #117 | C24 | 1 | 8 | yes | — |
| 163 | #117 | C24 | 1 | 1 | yes | characteristic |
| 164 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 165 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 166 | C2×C42 | C22×C4 | 1 | 2 | yes | — |
| 167 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 168 | #108 | — | 2 | 4 | no | — |
| 169 | #108 | — | 2 | 4 | no | — |
| 170 | #107 | — | 2 | 4 | no | — |
| 171 | #106 | — | 2 | 4 | no | — |
| 172 | #104 | — | 4 | 16 | no | — |
| 173 | #103 | — | 4 | 16 | no | — |
| 174 | #101 | — | 2 | 4 | no | — |
| 175 | #101 | — | 4 | 8 | no | — |
| 176 | #100 | — | 2 | 4 | no | — |
| 177 | #99 | — | 2 | 8 | no | — |
| 178 | #99 | — | 4 | 8 | no | — |
| 179 | #97 | — | 2 | 8 | no | — |
| 180 | #97 | C22×C4 | 1 | 4 | yes | — |
| 181 | #97 | C22×C4 | 1 | 2 | yes | — |
| 182 | C4×C8 | — | 2 | 8 | no | — |
| 183 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 184 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 185 | C4×C8 | C22×C4 | 1 | 4 | yes | — |
| 186 | C4×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 187 | C4×C8 | C22×C4 | 1 | 2 | yes | — |
| 188 | #41 | — | 2 | 4 | no | — |
| 189 | #41 | — | 2 | 16 | no | — |
| 190 | #41 | — | 4 | 16 | no | — |
| 191 | #41 | — | 8 | 16 | no | — |
| 192 | #41 | — | 8 | 8 | no | — |
| 193 | C2×Q8 | — | 2 | 8 | no | — |
| 194 | C2×Q8 | — | 2 | 4 | no | — |
| 195 | C2×Q8 | — | 2 | 16 | no | — |
| 196 | C2×Q8 | — | 4 | 16 | no | — |
| 197 | C2×Q8 | — | 8 | 8 | no | — |
| 198 | C2×Q8 | C23×C4 | 1 | 4 | yes | — |
| 199 | C2×Q8 | C23×C4 | 1 | 8 | yes | — |
| 200 | C2×Q8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 201 | C2×D4 | — | 2 | 4 | no | — |
| 202 | C2×D4 | — | 2 | 8 | no | — |
| 203 | C2×D4 | — | 4 | 2 | no | — |
| 204 | C2×D4 | C23×C4 | 1 | 4 | yes | — |
| 205 | C2×D4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 206 | C2×D4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 207 | C22×C4 | — | 2 | 2 | no | — |
| 208 | C22×C4 | — | 2 | 1 | no | — |
| 209 | C22×C4 | — | 2 | 4 | no | — |
| 210 | C22×C4 | — | 2 | 8 | no | — |
| 211 | C22×C4 | — | 2 | 4 | no | — |
| 212 | C22×C4 | — | 2 | 8 | no | — |
| 213 | C22×C4 | C23×C4 | 1 | 2 | yes | — |
| 214 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 215 | Q16 | — | 2 | 8 | no | — |
| 216 | Q16 | — | 2 | 8 | no | — |
| 217 | Q16 | — | 2 | 16 | no | — |
| 218 | Q16 | — | 2 | 4 | no | — |
| 219 | Q16 | — | 2 | 4 | no | — |
| 220 | Q16 | — | 4 | 16 | no | — |
| 221 | SD16 | — | 2 | 16 | no | — |
| 222 | SD16 | — | 4 | 16 | no | — |
| 223 | D8 | — | 2 | 4 | no | — |
| 224 | D8 | — | 2 | 4 | no | — |
| 225 | M16 | — | 2 | 4 | no | — |
| 226 | M16 | — | 2 | 8 | no | — |
| 227 | M16 | — | 2 | 8 | no | — |
| 228 | M16 | — | 2 | 8 | no | — |
| 229 | M16 | — | 8 | 8 | no | — |
| 230 | C2×C8 | — | 2 | 4 | no | — |
| 231 | C2×C8 | — | 2 | 4 | no | — |
| 232 | C2×C8 | — | 2 | 8 | no | — |
| 233 | C2×C8 | — | 8 | 8 | no | — |
| 234 | C2×C8 | C23×C4 | 1 | 4 | yes | — |
| 235 | C2×C8 | C23×C4 | 1 | 2 | yes | — |
| 236 | C2×C8 | C23×C4 | 1 | 2 | yes | — |
| 237 | C2×C8 | H(5,𝔽2) | 1 | 4 | yes | — |
| 238 | C2×C8 | H(5,𝔽2) | 1 | 4 | yes | — |
| 239 | C4⋊C4 | — | 2 | 2 | no | — |
| 240 | C4⋊C4 | — | 2 | 1 | no | — |
| 241 | C4⋊C4 | — | 2 | 8 | no | — |
| 242 | C4⋊C4 | — | 4 | 2 | no | — |
| 243 | C4⋊C4 | — | 4 | 4 | no | — |
| 244 | C4⋊C4 | — | 4 | 8 | no | — |
| 245 | C4⋊C4 | — | 4 | 4 | no | — |
| 246 | C4⋊C4 | — | 8 | 8 | no | — |
| 247 | C22⋊C4 | — | 2 | 2 | no | — |
| 248 | C22⋊C4 | — | 2 | 1 | no | — |
| 249 | C22⋊C4 | — | 2 | 8 | no | — |
| 250 | C22⋊C4 | — | 4 | 8 | no | — |
| 251 | C22⋊C4 | — | 4 | 4 | no | — |
| 252 | C42 | — | 2 | 2 | no | — |
| 253 | C42 | — | 2 | 1 | no | — |
| 254 | C42 | — | 2 | 8 | no | — |
| 255 | C42 | — | 8 | 8 | no | — |
| 256 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 257 | C42 | C23×C4 | 1 | 2 | yes | — |
| 258 | C42 | C23×C4 | 1 | 1 | yes | characteristic |
| 259 | C23 | — | 2 | 4 | no | — |
| 260 | C23 | #518 | 1 | 2 | yes | — |
| 261 | C23 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 262 | Q8 | — | 2 | 2 | no | — |
| 263 | Q8 | — | 2 | 8 | no | — |
| 264 | Q8 | — | 2 | 16 | no | — |
| 265 | Q8 | — | 4 | 8 | no | — |
| 266 | Q8 | — | 8 | 8 | no | — |
| 267 | D4 | — | 2 | 4 | no | — |
| 268 | D4 | — | 2 | 8 | no | — |
| 269 | D4 | — | 2 | 2 | no | — |
| 270 | D4 | — | 4 | 8 | no | — |
| 271 | D4 | — | 8 | 4 | no | — |
| 272 | D4 | — | 8 | 8 | no | — |
| 273 | C2×C4 | — | 2 | 1 | no | — |
| 274 | C2×C4 | — | 2 | 4 | no | — |
| 275 | C2×C4 | — | 2 | 2 | no | — |
| 276 | C2×C4 | — | 2 | 1 | no | — |
| 277 | C2×C4 | — | 2 | 4 | no | — |
| 278 | C2×C4 | — | 2 | 8 | no | — |
| 279 | C2×C4 | — | 4 | 2 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 4 | 8 | no | — |
| 282 | C2×C4 | — | 4 | 8 | no | — |
| 283 | C2×C4 | — | 4 | 8 | no | — |
| 284 | C2×C4 | — | 8 | 8 | no | — |
| 285 | C2×C4 | — | 8 | 8 | no | — |
| 286 | C2×C4 | #518 | 1 | 2 | yes | — |
| 287 | C2×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 289 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 290 | C8 | — | 2 | 4 | no | — |
| 291 | C8 | — | 4 | 8 | no | — |
| 292 | C8 | #518 | 1 | 2 | yes | — |
| 293 | C8 | #518 | 1 | 2 | yes | — |
| 294 | C8 | #518 | 1 | 4 | yes | — |
| 295 | C22 | — | 2 | 2 | no | — |
| 296 | C22 | — | 4 | 1 | no | — |
| 297 | C22 | — | 4 | 2 | no | — |
| 298 | C22 | — | 4 | 4 | no | — |
| 299 | C22 | — | 8 | 4 | no | — |
| 300 | C22 | C2×#518 | 1 | 1 | yes | characteristic |
| 301 | C22 | #3543 | 1 | 2 | yes | — |
| 302 | C4 | — | 2 | 2 | no | — |
| 303 | C4 | — | 2 | 2 | no | — |
| 304 | C4 | — | 4 | 2 | no | — |
| 305 | C4 | — | 4 | 1 | no | — |
| 306 | C4 | — | 4 | 4 | no | — |
| 307 | C4 | — | 4 | 8 | no | — |
| 308 | C4 | — | 8 | 8 | no | — |
| 309 | C4 | C2×#518 | 1 | 2 | yes | — |
| 310 | C4 | #3543 | 1 | 2 | yes | — |
| 311 | C2 | — | 2 | 1 | no | — |
| 312 | C2 | — | 2 | 2 | no | — |
| 313 | C2 | — | 4 | 2 | no | — |
| 314 | C2 | — | 8 | 4 | no | — |
| 315 | C2 | #34364 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #34364 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60648 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23360 | C2 | semidirect |
| 3. #23200 | C2 | semidirect |
| 4. #23196 | C2 | semidirect |
| 5. #22460 | C2 | semidirect |
| 6. #22457 | C2 | semidirect |
| 7. #22285 | C2 | semidirect |
| 8. #21783 | C2 | semidirect |
| 9. #20787 | C2 | semidirect |
| 12. #3156 | C4 | semidirect |
| 14. #3152 | C4 | semidirect |
| 16. #3147 | C4 | semidirect |
| 18. C2×#494 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60648 | 2 |
| #23360 | 2 |
| #23200 | 2 |
| #23196 | 2 |
| #22460 | 2 |
| #22457 | 2 |
| #22285 | 2 |
| #21783 | 2 |
| #20787 | 2 |