Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1500 conjugacy classes of subgroups.
The subgroup lattice has 311 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #35181 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#2568 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #29106 | C2 | 1 | 4 | yes | maximal |
| 4 | #27964 | C2 | 1 | 4 | yes | maximal |
| 5 | #27816 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | #27730 | C2 | 1 | 8 | yes | maximal, complemented |
| 7 | #21833 | C2 | 1 | 2 | yes | maximal |
| 8 | #20734 | C2 | 1 | 2 | yes | maximal |
| 9 | #20533 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | C22×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C22×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#431 | C22 | 1 | 2 | yes | — |
| 13 | #2851 | C22 | 1 | 4 | yes | — |
| 14 | #2646 | — | 2 | 2 | no | — |
| 15 | #2571 | C22 | 1 | 4 | yes | — |
| 16 | #2570 | — | 2 | 4 | no | — |
| 17 | #2568 | — | 2 | 4 | no | — |
| 18 | #2568 | C22 | 1 | 8 | yes | — |
| 19 | #2552 | C22 | 1 | 4 | yes | — |
| 20 | C2×#400 | — | 2 | 2 | no | — |
| 21 | #2317 | — | 2 | 2 | no | — |
| 22 | C2×#391 | — | 2 | 2 | no | — |
| 23 | C2×#391 | C22 | 1 | 2 | yes | — |
| 24 | #2285 | C22 | 1 | 2 | yes | — |
| 25 | C2×#382 | C22 | 1 | 2 | yes | — |
| 26 | C2×C4×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 27 | #2101 | C22 | 1 | 16 | yes | — |
| 28 | #2074 | C22 | 1 | 8 | yes | — |
| 29 | #2026 | C22 | 1 | 8 | yes | — |
| 30 | #1991 | C22 | 1 | 8 | yes | — |
| 31 | #1990 | C22 | 1 | 4 | yes | — |
| 32 | #1990 | C22 | 1 | 4 | yes | — |
| 33 | #1988 | C22 | 1 | 8 | yes | — |
| 34 | #1933 | C22 | 1 | 16 | yes | — |
| 35 | #1835 | C22 | 1 | 8 | yes | — |
| 36 | #1824 | C22 | 1 | 4 | yes | — |
| 37 | #1823 | C22 | 1 | 8 | yes | — |
| 38 | #1803 | C22 | 1 | 4 | yes | — |
| 39 | #1803 | C22 | 1 | 4 | yes | — |
| 40 | #1787 | C22 | 1 | 4 | yes | — |
| 41 | C8×C4⋊C4 | C22 | 1 | 8 | yes | — |
| 42 | C4×#103 | C22 | 1 | 8 | yes | — |
| 43 | C4×C22⋊C8 | C22 | 1 | 4 | yes | — |
| 44 | C23×Q8 | — | 2 | 1 | no | — |
| 45 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 46 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 47 | C2×C4⋊Q8 | — | 2 | 8 | no | — |
| 48 | C2×C4⋊Q8 | C23 | 1 | 4 | yes | — |
| 49 | C2×#125 | C23 | 1 | 4 | yes | — |
| 50 | C2×C4×Q8 | — | 2 | 2 | no | — |
| 51 | C2×C4×Q8 | — | 2 | 8 | no | — |
| 52 | C2×C4×Q8 | C23 | 1 | 4 | yes | — |
| 53 | C2×#117 | — | 2 | 2 | no | — |
| 54 | C22×C4⋊C4 | — | 2 | 1 | no | — |
| 55 | C22×C4⋊C4 | — | 2 | 1 | no | — |
| 56 | C22×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 57 | C22×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 58 | C22×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 59 | C22×C42 | — | 2 | 1 | no | — |
| 60 | C22×C42 | C23 | 1 | 2 | yes | — |
| 61 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 62 | #431 | — | 2 | 8 | no | — |
| 63 | C2×C4⋊C8 | C23 | 1 | 4 | yes | — |
| 64 | C2×#103 | C23 | 1 | 8 | yes | — |
| 65 | C2×#103 | C23 | 1 | 8 | yes | — |
| 66 | C2×C22⋊C8 | C23 | 1 | 4 | yes | — |
| 67 | C2×C4×C8 | C23 | 1 | 4 | yes | — |
| 68 | C2×C4×C8 | C23 | 1 | 8 | yes | — |
| 69 | #400 | — | 2 | 16 | no | — |
| 70 | #400 | — | 2 | 8 | no | — |
| 71 | #398 | — | 2 | 8 | no | — |
| 72 | #391 | — | 2 | 8 | no | — |
| 73 | #391 | — | 2 | 8 | no | — |
| 74 | #391 | — | 2 | 4 | no | — |
| 75 | #391 | C23 | 1 | 8 | yes | — |
| 76 | #389 | — | 2 | 4 | no | — |
| 77 | #389 | — | 2 | 4 | no | — |
| 78 | #389 | C23 | 1 | 8 | yes | — |
| 79 | #387 | — | 2 | 8 | no | — |
| 80 | #385 | — | 2 | 8 | no | — |
| 81 | #384 | — | 2 | 4 | no | — |
| 82 | #384 | C23 | 1 | 2 | yes | — |
| 83 | #384 | C23 | 1 | 2 | yes | — |
| 84 | #382 | — | 2 | 4 | no | — |
| 85 | #382 | — | 2 | 4 | no | — |
| 86 | #382 | — | 2 | 2 | no | — |
| 87 | #382 | C23 | 1 | 8 | yes | — |
| 88 | #382 | C23 | 1 | 4 | yes | — |
| 89 | #380 | — | 2 | 2 | no | — |
| 90 | #380 | C23 | 1 | 4 | yes | — |
| 91 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 92 | C4×C4⋊C4 | D4 | 1 | 4 | yes | — |
| 93 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 94 | C4×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 95 | C2×#95 | — | 2 | 4 | no | — |
| 96 | C2×#95 | — | 2 | 4 | no | — |
| 97 | C2×#95 | D4 | 1 | 2 | yes | — |
| 98 | #340 | C23 | 1 | 8 | yes | — |
| 99 | #340 | C23 | 1 | 8 | yes | — |
| 100 | #340 | C23 | 1 | 16 | yes | — |
| 101 | #340 | C23 | 1 | 16 | yes | — |
| 102 | #336 | C23 | 1 | 8 | yes | — |
| 103 | C22×Q8 | — | 2 | 4 | no | — |
| 104 | C22×Q8 | — | 2 | 1 | no | — |
| 105 | C22×Q8 | — | 2 | 4 | no | — |
| 106 | C22×Q8 | — | 2 | 4 | no | — |
| 107 | C22×Q8 | — | 4 | 2 | no | — |
| 108 | C22×Q8 | — | 4 | 4 | no | — |
| 109 | C22×Q8 | — | 4 | 8 | no | — |
| 110 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 111 | C23×C4 | — | 2 | 1 | no | — |
| 112 | C23×C4 | — | 2 | 2 | no | — |
| 113 | C23×C4 | — | 2 | 1 | no | — |
| 114 | C23×C4 | C2×D4 | 1 | 4 | yes | — |
| 115 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 118 | C22×C8 | C24 | 1 | 8 | yes | — |
| 119 | C4⋊Q8 | — | 2 | 8 | no | — |
| 120 | C4⋊Q8 | — | 2 | 8 | no | — |
| 121 | C4⋊Q8 | — | 4 | 8 | no | — |
| 122 | #125 | — | 4 | 8 | no | — |
| 123 | C4×Q8 | — | 2 | 8 | no | — |
| 124 | C4×Q8 | — | 2 | 8 | no | — |
| 125 | C4×Q8 | — | 4 | 8 | no | — |
| 126 | #117 | — | 4 | 8 | no | — |
| 127 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 128 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 129 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 130 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 131 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 132 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 133 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 134 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 135 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 136 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 137 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 138 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 139 | C2×C4⋊C4 | — | 2 | 8 | no | — |
| 140 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 141 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 142 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 143 | C2×C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 144 | C2×C4⋊C4 | C24 | 1 | 2 | yes | — |
| 145 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 146 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 147 | C2×C4⋊C4 | C24 | 1 | 4 | yes | — |
| 148 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 149 | C2×C42 | — | 2 | 4 | no | — |
| 150 | C2×C42 | — | 2 | 4 | no | — |
| 151 | C2×C42 | — | 4 | 4 | no | — |
| 152 | C2×C42 | — | 4 | 4 | no | — |
| 153 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 154 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 155 | C2×C42 | #41 | 1 | 8 | yes | — |
| 156 | C2×C42 | C24 | 1 | 4 | yes | — |
| 157 | C2×C42 | C24 | 1 | 2 | yes | — |
| 158 | C4⋊C8 | — | 2 | 8 | no | — |
| 159 | #103 | — | 2 | 16 | no | — |
| 160 | #103 | — | 4 | 8 | no | — |
| 161 | C22⋊C8 | — | 2 | 8 | no | — |
| 162 | C4×C8 | — | 2 | 8 | no | — |
| 163 | C4×C8 | — | 4 | 8 | no | — |
| 164 | #95 | — | 2 | 16 | no | — |
| 165 | #95 | — | 2 | 16 | no | — |
| 166 | #95 | — | 2 | 4 | no | — |
| 167 | #95 | — | 2 | 4 | no | — |
| 168 | #95 | — | 2 | 8 | no | — |
| 169 | #95 | — | 2 | 8 | no | — |
| 170 | #95 | C2×D4 | 1 | 8 | yes | — |
| 171 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 172 | C2×Q8 | — | 2 | 4 | no | — |
| 173 | C2×Q8 | — | 2 | 2 | no | — |
| 174 | C2×Q8 | — | 2 | 4 | no | — |
| 175 | C2×Q8 | — | 2 | 4 | no | — |
| 176 | C2×Q8 | — | 4 | 16 | no | — |
| 177 | C2×Q8 | — | 4 | 2 | no | — |
| 178 | C2×Q8 | — | 4 | 2 | no | — |
| 179 | C2×Q8 | — | 4 | 8 | no | — |
| 180 | C2×Q8 | — | 4 | 8 | no | — |
| 181 | C2×Q8 | — | 4 | 8 | no | — |
| 182 | C2×Q8 | — | 8 | 16 | no | — |
| 183 | C2×Q8 | 2−1+4 | 1 | 4 | yes | — |
| 184 | C22×C4 | — | 2 | 2 | no | — |
| 185 | C22×C4 | — | 2 | 2 | no | — |
| 186 | C22×C4 | — | 2 | 4 | no | — |
| 187 | C22×C4 | — | 2 | 2 | no | — |
| 188 | C22×C4 | — | 2 | 8 | no | — |
| 189 | C22×C4 | — | 2 | 2 | no | — |
| 190 | C22×C4 | — | 2 | 4 | no | — |
| 191 | C22×C4 | — | 2 | 4 | no | — |
| 192 | C22×C4 | — | 2 | 4 | no | — |
| 193 | C22×C4 | — | 4 | 2 | no | — |
| 194 | C22×C4 | — | 4 | 4 | no | — |
| 195 | C22×C4 | — | 4 | 4 | no | — |
| 196 | C22×C4 | — | 4 | 2 | no | — |
| 197 | C22×C4 | — | 4 | 2 | no | — |
| 198 | C22×C4 | — | 4 | 4 | no | — |
| 199 | C22×C4 | — | 4 | 2 | no | — |
| 200 | C22×C4 | — | 4 | 4 | no | — |
| 201 | C22×C4 | — | 4 | 8 | no | — |
| 202 | C22×C4 | #124 | 1 | 4 | yes | — |
| 203 | C22×C4 | #124 | 1 | 4 | yes | — |
| 204 | C22×C4 | #135 | 1 | 8 | yes | — |
| 205 | C22×C4 | C22×D4 | 1 | 8 | yes | — |
| 206 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 207 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 208 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 209 | C22×C4 | C2×#41 | 1 | 4 | yes | — |
| 210 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 211 | C2×C8 | — | 2 | 8 | no | — |
| 212 | C2×C8 | — | 2 | 8 | no | — |
| 213 | C2×C8 | — | 2 | 8 | no | — |
| 214 | C4⋊C4 | — | 2 | 8 | no | — |
| 215 | C4⋊C4 | — | 2 | 8 | no | — |
| 216 | C4⋊C4 | — | 2 | 4 | no | — |
| 217 | C4⋊C4 | — | 2 | 8 | no | — |
| 218 | C4⋊C4 | — | 2 | 8 | no | — |
| 219 | C4⋊C4 | — | 2 | 8 | no | — |
| 220 | C4⋊C4 | — | 4 | 2 | no | — |
| 221 | C4⋊C4 | — | 4 | 4 | no | — |
| 222 | C4⋊C4 | — | 4 | 8 | no | — |
| 223 | C4⋊C4 | — | 4 | 8 | no | — |
| 224 | C4⋊C4 | — | 4 | 4 | no | — |
| 225 | C4⋊C4 | — | 4 | 4 | no | — |
| 226 | C4⋊C4 | — | 8 | 2 | no | — |
| 227 | C4⋊C4 | — | 8 | 2 | no | — |
| 228 | C4⋊C4 | — | 8 | 8 | no | — |
| 229 | C4⋊C4 | — | 8 | 8 | no | — |
| 230 | C22⋊C4 | — | 4 | 8 | no | — |
| 231 | C42 | — | 2 | 8 | no | — |
| 232 | C42 | — | 4 | 4 | no | — |
| 233 | C42 | — | 4 | 8 | no | — |
| 234 | C42 | — | 4 | 8 | no | — |
| 235 | C42 | #124 | 1 | 8 | yes | — |
| 236 | C42 | C2×#41 | 1 | 4 | yes | — |
| 237 | C42 | C2×#41 | 1 | 4 | yes | — |
| 238 | C23 | — | 2 | 2 | no | — |
| 239 | C23 | — | 2 | 1 | no | — |
| 240 | C23 | — | 2 | 1 | no | — |
| 241 | C23 | C2×#124 | 1 | 2 | yes | — |
| 242 | C23 | C2×#135 | 1 | 4 | yes | — |
| 243 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 2 | 8 | no | — |
| 245 | Q8 | — | 4 | 8 | no | — |
| 246 | Q8 | — | 4 | 4 | no | — |
| 247 | Q8 | — | 8 | 16 | no | — |
| 248 | C2×C4 | — | 2 | 2 | no | — |
| 249 | C2×C4 | — | 2 | 2 | no | — |
| 250 | C2×C4 | — | 2 | 1 | no | — |
| 251 | C2×C4 | — | 2 | 1 | no | — |
| 252 | C2×C4 | — | 2 | 8 | no | — |
| 253 | C2×C4 | — | 2 | 8 | no | — |
| 254 | C2×C4 | — | 2 | 8 | no | — |
| 255 | C2×C4 | — | 2 | 4 | no | — |
| 256 | C2×C4 | — | 2 | 8 | no | — |
| 257 | C2×C4 | — | 4 | 4 | no | — |
| 258 | C2×C4 | — | 4 | 8 | no | — |
| 259 | C2×C4 | — | 4 | 2 | no | — |
| 260 | C2×C4 | — | 4 | 2 | no | — |
| 261 | C2×C4 | — | 4 | 4 | no | — |
| 262 | C2×C4 | — | 4 | 4 | no | — |
| 263 | C2×C4 | — | 4 | 4 | no | — |
| 264 | C2×C4 | — | 4 | 2 | no | — |
| 265 | C2×C4 | — | 4 | 4 | no | — |
| 266 | C2×C4 | — | 4 | 4 | no | — |
| 267 | C2×C4 | — | 4 | 4 | no | — |
| 268 | C2×C4 | — | 4 | 4 | no | — |
| 269 | C2×C4 | — | 4 | 4 | no | — |
| 270 | C2×C4 | — | 4 | 8 | no | — |
| 271 | C2×C4 | — | 8 | 8 | no | — |
| 272 | C2×C4 | — | 8 | 4 | no | — |
| 273 | C2×C4 | — | 8 | 8 | no | — |
| 274 | C2×C4 | #488 | 1 | 16 | yes | — |
| 275 | C2×C4 | C2×#124 | 1 | 2 | yes | — |
| 276 | C2×C4 | C2×#124 | 1 | 4 | yes | — |
| 277 | C2×C4 | C2×#124 | 1 | 4 | yes | — |
| 278 | C2×C4 | C2×#135 | 1 | 4 | yes | — |
| 279 | C2×C4 | C2×#135 | 1 | 4 | yes | — |
| 280 | C2×C4 | C22×#41 | 1 | 2 | yes | — |
| 281 | C2×C4 | C2×2−1+4 | 1 | 2 | yes | — |
| 282 | C2×C4 | #585 | 1 | 2 | yes | — |
| 283 | C8 | — | 4 | 8 | no | — |
| 284 | C22 | — | 2 | 4 | no | — |
| 285 | C22 | — | 2 | 2 | no | — |
| 286 | C22 | — | 2 | 4 | no | — |
| 287 | C22 | — | 2 | 2 | no | — |
| 288 | C22 | C2×#488 | 1 | 4 | yes | — |
| 289 | C22 | C22×#124 | 1 | 1 | yes | characteristic |
| 290 | C22 | #3485 | 1 | 1 | yes | characteristic |
| 291 | C22 | #3486 | 1 | 1 | yes | characteristic |
| 292 | C22 | C22×#135 | 1 | 2 | yes | — |
| 293 | C22 | #3586 | 1 | 2 | yes | — |
| 294 | C4 | — | 2 | 2 | no | — |
| 295 | C4 | — | 2 | 4 | no | — |
| 296 | C4 | — | 4 | 8 | no | — |
| 297 | C4 | — | 4 | 4 | no | — |
| 298 | C4 | — | 8 | 4 | no | — |
| 299 | C4 | — | 8 | 4 | no | — |
| 300 | C4 | — | 8 | 4 | no | — |
| 301 | C4 | — | 8 | 2 | no | — |
| 302 | C4 | C2×#488 | 1 | 8 | yes | — |
| 303 | C4 | #3495 | 1 | 4 | yes | — |
| 304 | C2 | — | 2 | 4 | no | — |
| 305 | C2 | #35644 | 1 | 1 | yes | characteristic |
| 306 | C2 | C22×#488 | 1 | 1 | yes | characteristic |
| 307 | C2 | #60620 | 1 | 1 | yes | characteristic |
| 308 | C2 | #61701 | 1 | 2 | yes | — |
| 309 | C2 | #61822 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #27816 | C2 | semidirect |
| 6. #27730 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #35181 | 2 |
| C2×#2568 | 2 |
| #29106 | 2 |
| #27964 | 2 |
| #27816 | 2 |
| #27730 | 2 |
| #21833 | 2 |
| #20734 | 2 |
| #20533 | 2 |