Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1340 conjugacy classes of subgroups.
The subgroup lattice has 323 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #59871 | C2 | 1 | 2 | yes | maximal |
| 2 | #32685 | C2 | 1 | 4 | yes | maximal |
| 3 | #29290 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #29286 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #28811 | C2 | 1 | 4 | yes | maximal |
| 6 | #28678 | C2 | 1 | 4 | yes | maximal |
| 7 | #26445 | C2 | 1 | 4 | yes | maximal |
| 8 | #26443 | C2 | 1 | 4 | yes | maximal |
| 9 | #23156 | C2 | 1 | 4 | yes | maximal |
| 10 | C4×#498 | C2 | 1 | 2 | yes | maximal |
| 11 | C8×C4⋊Q8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×#499 | C22 | 1 | 4 | yes | — |
| 13 | C2×#498 | C22 | 1 | 2 | yes | — |
| 14 | C2×C8×Q8 | C22 | 1 | 2 | yes | — |
| 15 | #2806 | — | 2 | 4 | no | — |
| 16 | #2584 | C22 | 1 | 8 | yes | — |
| 17 | #2582 | — | 2 | 4 | no | — |
| 18 | #2567 | — | 2 | 2 | no | — |
| 19 | #2552 | C22 | 1 | 4 | yes | — |
| 20 | #2552 | C22 | 1 | 4 | yes | — |
| 21 | #2507 | — | 2 | 4 | no | — |
| 22 | #2369 | C22 | 1 | 4 | yes | — |
| 23 | #2365 | C22 | 1 | 4 | yes | — |
| 24 | Q8×C4⋊C4 | — | 2 | 2 | no | — |
| 25 | C4×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 26 | C4×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 27 | C4×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 28 | #2087 | C22 | 1 | 4 | yes | — |
| 29 | #2087 | C22 | 1 | 4 | yes | — |
| 30 | #2086 | C22 | 1 | 4 | yes | — |
| 31 | #2086 | C22 | 1 | 4 | yes | — |
| 32 | #2059 | C22 | 1 | 8 | yes | — |
| 33 | #2059 | C22 | 1 | 8 | yes | — |
| 34 | #1990 | C22 | 1 | 2 | yes | — |
| 35 | #1990 | C22 | 1 | 2 | yes | — |
| 36 | #1989 | C22 | 1 | 8 | yes | — |
| 37 | #1988 | C22 | 1 | 4 | yes | — |
| 38 | #1987 | C22 | 1 | 4 | yes | — |
| 39 | #1984 | C22 | 1 | 2 | yes | — |
| 40 | #1947 | C22 | 1 | 4 | yes | — |
| 41 | #1947 | C22 | 1 | 4 | yes | — |
| 42 | #1941 | C22 | 1 | 4 | yes | — |
| 43 | #1941 | C22 | 1 | 8 | yes | — |
| 44 | #1941 | C22 | 1 | 4 | yes | — |
| 45 | #1939 | C22 | 1 | 4 | yes | — |
| 46 | #1920 | C22 | 1 | 8 | yes | — |
| 47 | #1919 | C22 | 1 | 8 | yes | — |
| 48 | #1844 | C22 | 1 | 2 | yes | — |
| 49 | #1842 | C22 | 1 | 1 | yes | characteristic |
| 50 | C4×#106 | C22 | 1 | 4 | yes | — |
| 51 | C4×#106 | C22 | 1 | 4 | yes | — |
| 52 | C4×#106 | C22 | 1 | 4 | yes | — |
| 53 | C8×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 54 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 55 | C2×C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 56 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 57 | C2×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 58 | C2×#125 | C23 | 1 | 8 | yes | — |
| 59 | C2×C4×Q8 | — | 2 | 4 | no | — |
| 60 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 61 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 62 | C2×C4×Q8 | C23 | 1 | 2 | yes | — |
| 63 | #499 | — | 4 | 8 | no | — |
| 64 | #498 | Q8 | 1 | 16 | yes | complemented |
| 65 | C8×Q8 | — | 4 | 4 | no | — |
| 66 | C2×#107 | C23 | 1 | 4 | yes | — |
| 67 | C2×#107 | C23 | 1 | 4 | yes | — |
| 68 | C2×#106 | C23 | 1 | 4 | yes | — |
| 69 | C2×#106 | C23 | 1 | 4 | yes | — |
| 70 | C2×#106 | C23 | 1 | 4 | yes | — |
| 71 | C2×#106 | C23 | 1 | 4 | yes | — |
| 72 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 73 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 74 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 75 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 76 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 77 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 78 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×C8 | C23 | 1 | 2 | yes | — |
| 80 | #398 | — | 2 | 8 | no | — |
| 81 | #398 | — | 2 | 8 | no | — |
| 82 | #398 | — | 2 | 8 | no | — |
| 83 | #395 | — | 2 | 8 | no | — |
| 84 | #391 | — | 2 | 4 | no | — |
| 85 | #391 | — | 2 | 2 | no | — |
| 86 | #391 | — | 2 | 2 | no | — |
| 87 | #391 | — | 2 | 8 | no | — |
| 88 | #391 | D4 | 1 | 2 | yes | — |
| 89 | #389 | — | 2 | 4 | no | — |
| 90 | #389 | — | 2 | 4 | no | — |
| 91 | #389 | — | 2 | 8 | no | — |
| 92 | #389 | — | 2 | 4 | no | — |
| 93 | #389 | — | 2 | 8 | no | — |
| 94 | #389 | — | 2 | 2 | no | — |
| 95 | #389 | — | 2 | 2 | no | — |
| 96 | #389 | D4 | 1 | 4 | yes | — |
| 97 | #389 | C23 | 1 | 4 | yes | — |
| 98 | #389 | C23 | 1 | 4 | yes | — |
| 99 | #389 | C23 | 1 | 8 | yes | — |
| 100 | #389 | C23 | 1 | 8 | yes | — |
| 101 | #387 | — | 2 | 8 | no | — |
| 102 | #384 | — | 2 | 4 | no | — |
| 103 | #384 | — | 2 | 4 | no | — |
| 104 | #384 | D4 | 1 | 1 | yes | characteristic |
| 105 | #384 | C23 | 1 | 2 | yes | — |
| 106 | #384 | C23 | 1 | 2 | yes | — |
| 107 | C4×C4⋊C4 | — | 2 | 4 | no | — |
| 108 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 109 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 110 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 111 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 112 | C4×C4⋊C4 | C23 | 1 | 8 | yes | — |
| 113 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 114 | C43 | C23 | 1 | 1 | yes | characteristic |
| 115 | #340 | C23 | 1 | 4 | yes | — |
| 116 | #340 | C23 | 1 | 4 | yes | — |
| 117 | #340 | C23 | 1 | 8 | yes | — |
| 118 | #340 | C23 | 1 | 8 | yes | — |
| 119 | #340 | C23 | 1 | 4 | yes | — |
| 120 | #340 | C23 | 1 | 4 | yes | — |
| 121 | #340 | C23 | 1 | 4 | yes | — |
| 122 | #340 | C23 | 1 | 4 | yes | — |
| 123 | #336 | C23 | 1 | 8 | yes | — |
| 124 | C22×Q8 | — | 2 | 2 | no | — |
| 125 | C22×Q8 | — | 2 | 2 | no | — |
| 126 | C22×Q8 | C2×D4 | 1 | 2 | yes | — |
| 127 | C22×C8 | C24 | 1 | 4 | yes | — |
| 128 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 129 | C22×C8 | C24 | 1 | 2 | yes | — |
| 130 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 131 | C4⋊Q8 | — | 2 | 4 | no | — |
| 132 | C4⋊Q8 | — | 2 | 4 | no | — |
| 133 | C4⋊Q8 | — | 2 | 4 | no | — |
| 134 | C4⋊Q8 | C2×Q8 | 1 | 8 | yes | — |
| 135 | C4⋊Q8 | C2×Q8 | 1 | 8 | yes | — |
| 136 | #125 | — | 4 | 8 | no | — |
| 137 | #125 | — | 4 | 8 | no | — |
| 138 | C4×Q8 | — | 4 | 2 | no | — |
| 139 | C4×Q8 | — | 4 | 2 | no | — |
| 140 | C4×Q8 | — | 4 | 8 | no | — |
| 141 | C4×Q8 | — | 8 | 4 | no | — |
| 142 | C4×Q8 | — | 8 | 8 | no | — |
| 143 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 144 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 145 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 146 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 147 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 148 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 149 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 150 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 151 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 152 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 153 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 154 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 155 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 156 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 157 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 158 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 159 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 160 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 161 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 162 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 163 | C2×C42 | — | 2 | 4 | no | — |
| 164 | C2×C42 | — | 2 | 4 | no | — |
| 165 | C2×C42 | — | 2 | 8 | no | — |
| 166 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 167 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 168 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 170 | C2×C42 | C24 | 1 | 4 | yes | — |
| 171 | C2×C42 | C24 | 1 | 2 | yes | — |
| 172 | #107 | — | 4 | 4 | no | — |
| 173 | #107 | — | 4 | 4 | no | — |
| 174 | #106 | — | 2 | 8 | no | — |
| 175 | #106 | — | 4 | 4 | no | — |
| 176 | #106 | C2×Q8 | 1 | 16 | yes | — |
| 177 | #106 | C2×Q8 | 1 | 16 | yes | — |
| 178 | C4⋊C8 | — | 2 | 8 | no | — |
| 179 | C4⋊C8 | — | 4 | 2 | no | — |
| 180 | C4⋊C8 | — | 4 | 4 | no | — |
| 181 | C4×C8 | — | 2 | 2 | no | — |
| 182 | C4×C8 | — | 2 | 8 | no | — |
| 183 | C4×C8 | — | 2 | 4 | no | — |
| 184 | C4×C8 | — | 4 | 1 | no | — |
| 185 | C4×C8 | C2×Q8 | 1 | 8 | yes | — |
| 186 | #95 | — | 2 | 4 | no | — |
| 187 | #95 | — | 2 | 4 | no | — |
| 188 | #95 | — | 2 | 8 | no | — |
| 189 | #95 | — | 2 | 4 | no | — |
| 190 | #95 | — | 2 | 8 | no | — |
| 191 | #95 | — | 2 | 4 | no | — |
| 192 | #95 | — | 2 | 8 | no | — |
| 193 | #95 | C2×D4 | 1 | 8 | yes | — |
| 194 | C2×Q8 | — | 2 | 8 | no | — |
| 195 | C2×Q8 | — | 2 | 8 | no | — |
| 196 | C2×Q8 | — | 2 | 4 | no | — |
| 197 | C2×Q8 | — | 4 | 2 | no | — |
| 198 | C2×Q8 | — | 4 | 2 | no | — |
| 199 | C2×Q8 | — | 4 | 4 | no | — |
| 200 | C2×Q8 | — | 4 | 4 | no | — |
| 201 | C2×Q8 | — | 8 | 2 | no | — |
| 202 | C2×Q8 | — | 8 | 2 | no | — |
| 203 | C22×C4 | — | 2 | 4 | no | — |
| 204 | C22×C4 | — | 2 | 4 | no | — |
| 205 | C22×C4 | — | 2 | 4 | no | — |
| 206 | C22×C4 | — | 2 | 4 | no | — |
| 207 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 208 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 209 | C22×C4 | C22×D4 | 1 | 4 | yes | — |
| 210 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 211 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 212 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 213 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 214 | C2×C8 | — | 2 | 2 | no | — |
| 215 | C2×C8 | — | 2 | 2 | no | — |
| 216 | C2×C8 | — | 2 | 1 | no | — |
| 217 | C2×C8 | — | 2 | 4 | no | — |
| 218 | C2×C8 | — | 2 | 2 | no | — |
| 219 | C2×C8 | — | 2 | 8 | no | — |
| 220 | C2×C8 | C22×Q8 | 1 | 4 | yes | — |
| 221 | C2×C8 | C22×Q8 | 1 | 4 | yes | — |
| 222 | C2×C8 | H(5,𝔽2) | 1 | 2 | yes | — |
| 223 | C4⋊C4 | — | 2 | 8 | no | — |
| 224 | C4⋊C4 | — | 2 | 8 | no | — |
| 225 | C4⋊C4 | — | 2 | 8 | no | — |
| 226 | C4⋊C4 | — | 2 | 8 | no | — |
| 227 | C4⋊C4 | — | 2 | 4 | no | — |
| 228 | C4⋊C4 | — | 2 | 4 | no | — |
| 229 | C4⋊C4 | — | 2 | 4 | no | — |
| 230 | C4⋊C4 | — | 2 | 4 | no | — |
| 231 | C4⋊C4 | — | 2 | 4 | no | — |
| 232 | C4⋊C4 | — | 2 | 4 | no | — |
| 233 | C4⋊C4 | — | 4 | 2 | no | — |
| 234 | C4⋊C4 | — | 4 | 4 | no | — |
| 235 | C4⋊C4 | — | 4 | 8 | no | — |
| 236 | C4⋊C4 | — | 4 | 4 | no | — |
| 237 | C4⋊C4 | — | 4 | 4 | no | — |
| 238 | C4⋊C4 | — | 8 | 2 | no | — |
| 239 | C4⋊C4 | — | 8 | 8 | no | — |
| 240 | C4⋊C4 | — | 8 | 2 | no | — |
| 241 | C4⋊C4 | — | 8 | 4 | no | — |
| 242 | C4⋊C4 | — | 8 | 8 | no | — |
| 243 | C4⋊C4 | — | 8 | 8 | no | — |
| 244 | C4⋊C4 | — | 8 | 8 | no | — |
| 245 | C4⋊C4 | C22×Q8 | 1 | 8 | yes | — |
| 246 | C4⋊C4 | C22×Q8 | 1 | 8 | yes | — |
| 247 | C42 | — | 2 | 4 | no | — |
| 248 | C42 | — | 2 | 1 | no | — |
| 249 | C42 | — | 2 | 4 | no | — |
| 250 | C42 | — | 2 | 1 | no | — |
| 251 | C42 | — | 2 | 4 | no | — |
| 252 | C42 | — | 4 | 1 | no | — |
| 253 | C42 | — | 4 | 8 | no | — |
| 254 | C42 | — | 4 | 8 | no | — |
| 255 | C42 | — | 8 | 8 | no | — |
| 256 | C42 | #135 | 1 | 4 | yes | — |
| 257 | C42 | C22×Q8 | 1 | 4 | yes | — |
| 258 | C42 | 2−1+4 | 1 | 4 | yes | — |
| 259 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 260 | Q8 | — | 4 | 8 | no | — |
| 261 | Q8 | — | 4 | 16 | no | — |
| 262 | Q8 | — | 8 | 8 | no | — |
| 263 | C2×C4 | — | 2 | 2 | no | — |
| 264 | C2×C4 | — | 2 | 1 | no | — |
| 265 | C2×C4 | — | 2 | 1 | no | — |
| 266 | C2×C4 | — | 2 | 2 | no | — |
| 267 | C2×C4 | — | 2 | 4 | no | — |
| 268 | C2×C4 | — | 2 | 8 | no | — |
| 269 | C2×C4 | — | 2 | 8 | no | — |
| 270 | C2×C4 | — | 2 | 1 | no | — |
| 271 | C2×C4 | — | 2 | 4 | no | — |
| 272 | C2×C4 | — | 2 | 4 | no | — |
| 273 | C2×C4 | — | 2 | 4 | no | — |
| 274 | C2×C4 | — | 2 | 2 | no | — |
| 275 | C2×C4 | — | 2 | 8 | no | — |
| 276 | C2×C4 | — | 4 | 4 | no | — |
| 277 | C2×C4 | — | 4 | 4 | no | — |
| 278 | C2×C4 | — | 4 | 4 | no | — |
| 279 | C2×C4 | — | 4 | 4 | no | — |
| 280 | C2×C4 | — | 4 | 4 | no | — |
| 281 | C2×C4 | — | 4 | 4 | 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 | D4×Q8 | 1 | 4 | yes | — |
| 287 | C2×C4 | D4×Q8 | 1 | 4 | yes | — |
| 288 | C2×C4 | C2×#135 | 1 | 2 | yes | — |
| 289 | C2×C4 | C2×#135 | 1 | 4 | yes | — |
| 290 | C2×C4 | #577 | 1 | 4 | yes | — |
| 291 | C2×C4 | C23×Q8 | 1 | 2 | yes | — |
| 292 | C2×C4 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C2×2−1+4 | 1 | 2 | yes | — |
| 294 | C2×C4 | #585 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 2 | 4 | no | — |
| 296 | C8 | — | 4 | 1 | no | — |
| 297 | C8 | — | 4 | 4 | no | — |
| 298 | C8 | Q82 | 1 | 4 | yes | — |
| 299 | C22 | C2×D4×Q8 | 1 | 2 | yes | — |
| 300 | C22 | #3496 | 1 | 1 | yes | characteristic |
| 301 | C22 | C22×#135 | 1 | 1 | yes | characteristic |
| 302 | C22 | C2×#577 | 1 | 2 | yes | — |
| 303 | C22 | #3584 | 1 | 1 | yes | characteristic |
| 304 | C4 | — | 2 | 2 | no | — |
| 305 | C4 | — | 2 | 4 | no | — |
| 306 | C4 | — | 4 | 4 | no | — |
| 307 | C4 | — | 4 | 1 | no | — |
| 308 | C4 | — | 4 | 4 | no | — |
| 309 | C4 | — | 4 | 8 | no | — |
| 310 | C4 | — | 4 | 8 | no | — |
| 311 | C4 | — | 8 | 4 | no | — |
| 312 | C4 | — | 8 | 4 | no | — |
| 313 | C4 | #3381 | 1 | 8 | yes | — |
| 314 | C4 | C2×Q82 | 1 | 1 | yes | characteristic |
| 315 | C4 | #3502 | 1 | 2 | yes | — |
| 316 | C4 | #3550 | 1 | 1 | yes | characteristic |
| 317 | C2 | #59832 | 1 | 1 | yes | characteristic |
| 318 | C2 | C2×#3381 | 1 | 2 | yes | — |
| 319 | C2 | #61371 | 1 | 2 | yes | — |
| 320 | C2 | #62248 | 1 | 1 | yes | characteristic |
| 321 | C2 | #62331 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 64. #498 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #59871 | 2 |
| #32685 | 2 |
| #29290 | 2 |
| #29286 | 2 |
| #28811 | 2 |
| #28678 | 2 |
| #26445 | 2 |
| #26443 | 2 |
| #23156 | 2 |
| C4×#498 | 2 |
| C8×C4⋊Q8 | 2 |