Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1473 conjugacy classes of subgroups.
The subgroup lattice has 308 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #36585 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #33266 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #33079 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #28013 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #27837 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #27564 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #27532 | C2 | 1 | 6 | yes | maximal, complemented |
| 8 | #22685 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | #22451 | C2 | 1 | 3 | yes | maximal |
| 10 | C8×#126 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | C42×Q16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 12 | C22×#135 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×Q16 | C22 | 1 | 3 | yes | — |
| 14 | C2×C4×Q16 | C22 | 1 | 3 | yes | — |
| 15 | #2860 | — | 2 | 2 | no | — |
| 16 | #2860 | C22 | 1 | 2 | yes | — |
| 17 | #2513 | — | 2 | 6 | no | — |
| 18 | #2513 | C22 | 1 | 6 | yes | — |
| 19 | #2494 | — | 2 | 6 | no | — |
| 20 | #2494 | C22 | 1 | 6 | yes | — |
| 21 | #2351 | C22 | 1 | 6 | yes | — |
| 22 | #2341 | C22 | 1 | 6 | yes | — |
| 23 | C4×#126 | — | 2 | 2 | no | — |
| 24 | C4×#126 | C22 | 1 | 1 | yes | characteristic |
| 25 | C42×Q8 | C22 | 1 | 2 | yes | — |
| 26 | #2101 | C22 | 1 | 6 | yes | — |
| 27 | #2077 | C22 | 1 | 6 | yes | — |
| 28 | #2073 | C22 | 1 | 6 | yes | — |
| 29 | #2065 | C22 | 1 | 6 | yes | — |
| 30 | #1970 | C22 | 1 | 3 | yes | — |
| 31 | #1970 | C22 | 1 | 3 | yes | — |
| 32 | #1942 | C22 | 1 | 3 | yes | — |
| 33 | #1942 | C22 | 1 | 3 | yes | — |
| 34 | #1934 | C22 | 1 | 6 | yes | — |
| 35 | #1926 | C22 | 1 | 6 | yes | — |
| 36 | #1923 | C22 | 1 | 3 | yes | — |
| 37 | #1919 | C22 | 1 | 6 | yes | — |
| 38 | #1916 | C22 | 1 | 3 | yes | — |
| 39 | #1880 | C22 | 1 | 6 | yes | — |
| 40 | #1880 | C22 | 1 | 6 | yes | — |
| 41 | #1865 | C22 | 1 | 6 | yes | — |
| 42 | #1865 | C22 | 1 | 6 | yes | — |
| 43 | #1846 | C22 | 1 | 3 | yes | — |
| 44 | #1839 | C22 | 1 | 1 | yes | characteristic |
| 45 | #1808 | C22 | 1 | 3 | yes | — |
| 46 | #1805 | C22 | 1 | 6 | yes | — |
| 47 | #1788 | C22 | 1 | 6 | yes | — |
| 48 | C4×#107 | C22 | 1 | 3 | yes | — |
| 49 | C8×C4⋊C4 | C22 | 1 | 3 | yes | — |
| 50 | C4×#103 | C22 | 1 | 6 | yes | — |
| 51 | C8×C22⋊C4 | C22 | 1 | 3 | yes | — |
| 52 | C42×C8 | C22 | 1 | 1 | yes | characteristic |
| 53 | C22×#41 | C23 | 1 | 2 | yes | — |
| 54 | C2×#135 | — | 2 | 12 | no | — |
| 55 | C22×Q16 | C23 | 1 | 1 | yes | characteristic |
| 56 | C22×SD16 | C23 | 1 | 2 | yes | — |
| 57 | C22×D8 | C23 | 1 | 1 | yes | characteristic |
| 58 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 59 | C2×#126 | — | 2 | 2 | no | — |
| 60 | C2×#126 | D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×Q8 | C23 | 1 | 6 | yes | — |
| 62 | C2×C4×Q8 | C23 | 1 | 6 | yes | — |
| 63 | C4×Q16 | — | 2 | 12 | no | — |
| 64 | C4×Q16 | — | 2 | 12 | no | — |
| 65 | C2×#107 | C23 | 1 | 3 | yes | — |
| 66 | C2×#107 | C23 | 1 | 3 | yes | — |
| 67 | C2×#106 | C23 | 1 | 6 | yes | — |
| 68 | C2×C4⋊C8 | C23 | 1 | 3 | yes | — |
| 69 | C2×#103 | C23 | 1 | 6 | yes | — |
| 70 | C2×#103 | C23 | 1 | 6 | yes | — |
| 71 | C2×#102 | C23 | 1 | 6 | yes | — |
| 72 | C2×#102 | C23 | 1 | 6 | yes | — |
| 73 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 74 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 75 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 76 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 77 | #401 | — | 2 | 6 | no | — |
| 78 | #397 | — | 2 | 6 | no | — |
| 79 | #397 | — | 2 | 6 | no | — |
| 80 | #397 | — | 2 | 6 | no | — |
| 81 | #390 | — | 2 | 6 | no | — |
| 82 | #390 | — | 2 | 6 | no | — |
| 83 | #390 | C23 | 1 | 6 | yes | — |
| 84 | #390 | C23 | 1 | 6 | yes | — |
| 85 | #389 | — | 2 | 6 | no | — |
| 86 | #389 | — | 2 | 6 | no | — |
| 87 | #389 | C23 | 1 | 6 | yes | — |
| 88 | #389 | C23 | 1 | 6 | yes | — |
| 89 | #388 | — | 2 | 6 | no | — |
| 90 | #388 | — | 2 | 6 | no | — |
| 91 | #388 | — | 2 | 6 | no | — |
| 92 | #388 | — | 2 | 6 | no | — |
| 93 | #388 | D4 | 1 | 3 | yes | — |
| 94 | #387 | — | 2 | 6 | no | — |
| 95 | #387 | D4 | 1 | 3 | yes | — |
| 96 | #383 | D4 | 1 | 1 | yes | characteristic |
| 97 | #382 | — | 2 | 6 | no | — |
| 98 | #382 | C23 | 1 | 6 | yes | — |
| 99 | #380 | — | 2 | 6 | no | — |
| 100 | #380 | C23 | 1 | 6 | yes | — |
| 101 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 102 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 103 | C4×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 104 | C4×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 105 | C4×C22⋊C4 | — | 2 | 6 | no | — |
| 106 | C4×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 107 | C43 | — | 2 | 2 | no | — |
| 108 | C43 | C23 | 1 | 1 | yes | characteristic |
| 109 | #340 | C23 | 1 | 6 | yes | — |
| 110 | #340 | C23 | 1 | 6 | yes | — |
| 111 | #340 | C23 | 1 | 6 | yes | — |
| 112 | #340 | C23 | 1 | 6 | yes | — |
| 113 | #336 | C23 | 1 | 3 | yes | — |
| 114 | #336 | C23 | 1 | 3 | yes | — |
| 115 | #336 | C23 | 1 | 3 | yes | — |
| 116 | #336 | C23 | 1 | 3 | yes | — |
| 117 | C2×#41 | — | 2 | 12 | no | — |
| 118 | C2×#41 | — | 4 | 6 | no | — |
| 119 | C22×Q8 | C24 | 1 | 2 | yes | — |
| 120 | C22×D4 | — | 2 | 2 | no | — |
| 121 | C22×D4 | C24 | 1 | 2 | yes | — |
| 122 | C23×C4 | — | 2 | 2 | no | — |
| 123 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 124 | #135 | — | 4 | 16 | no | — |
| 125 | C2×Q16 | #41 | 1 | 12 | yes | — |
| 126 | C2×SD16 | — | 2 | 12 | no | — |
| 127 | C2×D8 | — | 2 | 6 | no | — |
| 128 | C22×C8 | — | 2 | 6 | no | — |
| 129 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 130 | C22×C8 | C24 | 1 | 3 | yes | — |
| 131 | C22×C8 | C24 | 1 | 3 | yes | — |
| 132 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 133 | #126 | — | 2 | 4 | no | — |
| 134 | #126 | — | 4 | 8 | no | — |
| 135 | C4×Q8 | — | 2 | 12 | no | — |
| 136 | C4×Q8 | — | 2 | 12 | no | — |
| 137 | C4×Q8 | — | 4 | 6 | no | — |
| 138 | C4×Q8 | — | 4 | 6 | no | — |
| 139 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 140 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 141 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 142 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 143 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 144 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 145 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 146 | C2×C4⋊C4 | C24 | 1 | 6 | yes | — |
| 147 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 148 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 149 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 150 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 151 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 152 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 153 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 154 | C2×C42 | — | 2 | 6 | no | — |
| 155 | C2×C42 | — | 2 | 6 | no | — |
| 156 | C2×C42 | — | 2 | 6 | no | — |
| 157 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 158 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 159 | C2×C42 | C24 | 1 | 3 | yes | — |
| 160 | C2×C42 | C24 | 1 | 3 | yes | — |
| 161 | #107 | — | 2 | 6 | no | — |
| 162 | #107 | — | 2 | 6 | no | — |
| 163 | #106 | — | 4 | 6 | no | — |
| 164 | C4⋊C8 | — | 4 | 3 | no | — |
| 165 | #103 | — | 2 | 12 | no | — |
| 166 | #103 | — | 2 | 12 | no | — |
| 167 | #102 | — | 4 | 6 | no | — |
| 168 | #102 | — | 4 | 6 | no | — |
| 169 | C22⋊C8 | — | 4 | 3 | no | — |
| 170 | C4×C8 | — | 2 | 6 | no | — |
| 171 | C4×C8 | — | 2 | 6 | no | — |
| 172 | C4×C8 | — | 4 | 3 | no | — |
| 173 | #95 | — | 2 | 6 | no | — |
| 174 | #95 | — | 2 | 6 | no | — |
| 175 | #95 | — | 2 | 6 | no | — |
| 176 | #95 | — | 2 | 6 | no | — |
| 177 | #95 | — | 2 | 6 | no | — |
| 178 | #95 | — | 2 | 6 | no | — |
| 179 | #95 | C2×D4 | 1 | 3 | yes | — |
| 180 | #95 | C2×D4 | 1 | 3 | yes | — |
| 181 | #95 | C2×D4 | 1 | 3 | yes | — |
| 182 | #95 | C2×D4 | 1 | 3 | yes | — |
| 183 | C24 | — | 2 | 2 | no | — |
| 184 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 185 | #41 | — | 4 | 8 | no | — |
| 186 | #41 | — | 8 | 12 | no | — |
| 187 | C2×Q8 | — | 2 | 6 | no | — |
| 188 | C2×Q8 | C2×#41 | 1 | 12 | yes | — |
| 189 | C2×D4 | — | 2 | 6 | no | — |
| 190 | C2×D4 | — | 2 | 6 | no | — |
| 191 | C2×D4 | — | 4 | 12 | no | — |
| 192 | C22×C4 | — | 2 | 2 | no | — |
| 193 | C22×C4 | — | 2 | 3 | no | — |
| 194 | C22×C4 | — | 2 | 3 | no | — |
| 195 | C22×C4 | — | 2 | 6 | no | — |
| 196 | C22×C4 | — | 2 | 6 | no | — |
| 197 | C22×C4 | — | 4 | 12 | no | — |
| 198 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 199 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 200 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 201 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 202 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 203 | C22×C4 | C25 | 1 | 1 | yes | characteristic |
| 204 | Q16 | #126 | 1 | 16 | yes | complemented |
| 205 | SD16 | — | 4 | 8 | no | — |
| 206 | D8 | — | 4 | 4 | no | — |
| 207 | C2×C8 | — | 2 | 3 | no | — |
| 208 | C2×C8 | — | 2 | 3 | no | — |
| 209 | C2×C8 | — | 2 | 3 | no | — |
| 210 | C2×C8 | — | 2 | 3 | no | — |
| 211 | C2×C8 | — | 2 | 3 | no | — |
| 212 | C2×C8 | — | 2 | 3 | no | — |
| 213 | C2×C8 | — | 2 | 3 | no | — |
| 214 | C2×C8 | — | 4 | 4 | no | — |
| 215 | C2×C8 | C2×#41 | 1 | 6 | yes | — |
| 216 | C4⋊C4 | — | 2 | 6 | no | — |
| 217 | C4⋊C4 | — | 2 | 6 | no | — |
| 218 | C4⋊C4 | — | 2 | 6 | no | — |
| 219 | C4⋊C4 | — | 2 | 6 | no | — |
| 220 | C4⋊C4 | — | 2 | 6 | no | — |
| 221 | C4⋊C4 | — | 4 | 3 | no | — |
| 222 | C4⋊C4 | — | 4 | 12 | no | — |
| 223 | C4⋊C4 | — | 4 | 12 | no | — |
| 224 | C4⋊C4 | — | 4 | 12 | no | — |
| 225 | C4⋊C4 | — | 8 | 6 | no | — |
| 226 | C22⋊C4 | — | 2 | 6 | no | — |
| 227 | C22⋊C4 | — | 4 | 3 | no | — |
| 228 | C22⋊C4 | — | 4 | 12 | no | — |
| 229 | C22⋊C4 | — | 8 | 6 | no | — |
| 230 | C22⋊C4 | — | 8 | 6 | no | — |
| 231 | C22⋊C4 | — | 8 | 6 | no | — |
| 232 | C22⋊C4 | — | 8 | 6 | no | — |
| 233 | C42 | — | 2 | 3 | no | — |
| 234 | C42 | — | 2 | 3 | no | — |
| 235 | C42 | — | 2 | 3 | no | — |
| 236 | C42 | — | 2 | 3 | no | — |
| 237 | C42 | — | 4 | 3 | no | — |
| 238 | C42 | — | 4 | 12 | no | — |
| 239 | C42 | — | 4 | 12 | no | — |
| 240 | C42 | — | 8 | 6 | no | — |
| 241 | C42 | #135 | 1 | 4 | yes | complemented |
| 242 | C23 | — | 2 | 3 | no | — |
| 243 | C23 | — | 2 | 3 | no | — |
| 244 | C23 | — | 2 | 1 | no | — |
| 245 | C23 | — | 4 | 6 | no | — |
| 246 | C23 | — | 4 | 6 | no | — |
| 247 | C23 | — | 4 | 2 | no | — |
| 248 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 249 | Q8 | — | 2 | 12 | no | — |
| 250 | Q8 | C2×#126 | 1 | 8 | yes | — |
| 251 | D4 | — | 4 | 2 | no | — |
| 252 | D4 | — | 4 | 6 | no | — |
| 253 | D4 | — | 8 | 8 | no | — |
| 254 | C2×C4 | — | 2 | 3 | no | — |
| 255 | C2×C4 | — | 2 | 3 | no | — |
| 256 | C2×C4 | — | 2 | 12 | no | — |
| 257 | C2×C4 | — | 2 | 3 | no | — |
| 258 | C2×C4 | — | 2 | 3 | no | — |
| 259 | C2×C4 | — | 2 | 3 | no | — |
| 260 | C2×C4 | — | 2 | 3 | no | — |
| 261 | C2×C4 | — | 2 | 3 | no | — |
| 262 | C2×C4 | — | 2 | 3 | no | — |
| 263 | C2×C4 | — | 2 | 3 | no | — |
| 264 | C2×C4 | — | 2 | 3 | no | — |
| 265 | C2×C4 | — | 2 | 3 | no | — |
| 266 | C2×C4 | — | 2 | 3 | no | — |
| 267 | C2×C4 | — | 4 | 1 | no | — |
| 268 | C2×C4 | — | 4 | 3 | no | — |
| 269 | C2×C4 | — | 4 | 6 | no | — |
| 270 | C2×C4 | — | 4 | 6 | no | — |
| 271 | C2×C4 | — | 4 | 6 | no | — |
| 272 | C2×C4 | — | 4 | 6 | no | — |
| 273 | C2×C4 | — | 4 | 6 | no | — |
| 274 | C2×C4 | — | 4 | 6 | no | — |
| 275 | C2×C4 | — | 8 | 8 | no | — |
| 276 | C2×C4 | C2×#135 | 1 | 6 | yes | — |
| 277 | C2×C4 | C22×#41 | 1 | 3 | yes | — |
| 278 | C2×C4 | #585 | 1 | 3 | yes | — |
| 279 | C8 | — | 4 | 1 | no | — |
| 280 | C8 | — | 4 | 3 | no | — |
| 281 | C8 | — | 4 | 3 | no | — |
| 282 | C8 | C2×#126 | 1 | 4 | yes | — |
| 283 | C22 | — | 4 | 1 | no | — |
| 284 | C22 | — | 4 | 3 | no | — |
| 285 | C22 | — | 4 | 3 | no | — |
| 286 | C22 | — | 8 | 2 | no | — |
| 287 | C22 | — | 8 | 6 | no | — |
| 288 | C22 | — | 8 | 6 | no | — |
| 289 | C22 | D4×#41 | 1 | 3 | yes | — |
| 290 | C22 | C22×#135 | 1 | 1 | yes | characteristic |
| 291 | C22 | #3584 | 1 | 3 | yes | — |
| 292 | C4 | — | 2 | 8 | no | — |
| 293 | C4 | — | 2 | 6 | no | — |
| 294 | C4 | — | 4 | 1 | no | — |
| 295 | C4 | — | 4 | 3 | no | — |
| 296 | C4 | — | 4 | 3 | no | — |
| 297 | C4 | — | 4 | 3 | no | — |
| 298 | C4 | — | 8 | 6 | no | — |
| 299 | C4 | — | 8 | 6 | no | — |
| 300 | C4 | C22×#126 | 1 | 1 | yes | characteristic |
| 301 | C4 | #3521 | 1 | 3 | yes | — |
| 302 | C2 | — | 8 | 1 | no | — |
| 303 | C2 | — | 16 | 2 | no | — |
| 304 | C2 | D4×#126 | 1 | 1 | yes | characteristic |
| 305 | C2 | #61430 | 1 | 3 | yes | — |
| 306 | C2 | #61488 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#126 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #36585 | C2 | semidirect |
| 2. #33266 | C2 | semidirect |
| 3. #33079 | C2 | semidirect |
| 4. #28013 | C2 | semidirect |
| 5. #27837 | C2 | semidirect |
| 6. #27564 | C2 | semidirect |
| 7. #27532 | C2 | semidirect |
| 8. #22685 | C2 | semidirect |
| 10. C8×#126 | C2 | semidirect |
| 11. C42×Q16 | C2 | semidirect |
| 204. Q16 | #126 | semidirect |
| 241. C42 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #36585 | 2 |
| #33266 | 2 |
| #33079 | 2 |
| #28013 | 2 |
| #27837 | 2 |
| #27564 | 2 |
| #27532 | 2 |
| #22685 | 2 |
| #22451 | 2 |
| C8×#126 | 2 |
| C42×Q16 | 2 |