← #503824
Subgroups — #503824
Order 512.
401 conjugacy classes of subgroups.
The subgroup lattice has 276 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #29230 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22694 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #13544 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #13520 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #13182 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #13161 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #12942 | C2 | 1 | 2 | yes | maximal |
| 8 | #12891 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #12829 | C2 | 1 | 2 | yes | maximal |
| 10 | #12825 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #12641 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #2957 | C22 | 1 | 2 | yes | — |
| 13 | C2×#443 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2916 | — | 2 | 1 | no | — |
| 15 | #2161 | C22 | 1 | 2 | yes | — |
| 16 | C2×#364 | C22 | 1 | 1 | yes | characteristic |
| 17 | #2151 | C22 | 1 | 2 | yes | — |
| 18 | #2148 | C22 | 1 | 2 | yes | — |
| 19 | C2×#359 | C22 | 1 | 1 | yes | characteristic |
| 20 | #2138 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 22 | #2102 | — | 2 | 2 | no | — |
| 23 | #1953 | — | 2 | 1 | no | — |
| 24 | #1953 | C22 | 1 | 2 | yes | — |
| 25 | #1953 | C22 | 1 | 2 | yes | — |
| 26 | #1861 | — | 2 | 2 | no | — |
| 27 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 28 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1833 | — | 2 | 1 | no | — |
| 30 | #1833 | C22 | 1 | 2 | yes | — |
| 31 | #1814 | — | 2 | 1 | no | — |
| 32 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 33 | #1772 | C22 | 1 | 1 | yes | characteristic |
| 34 | #1389 | C22 | 1 | 2 | yes | — |
| 35 | #1385 | C22 | 1 | 2 | yes | — |
| 36 | #1382 | C22 | 1 | 1 | yes | characteristic |
| 37 | #1381 | C22 | 1 | 2 | yes | — |
| 38 | #1381 | C22 | 1 | 2 | yes | — |
| 39 | #1381 | C22 | 1 | 1 | yes | characteristic |
| 40 | #1379 | C22 | 1 | 2 | yes | — |
| 41 | C2×#135 | C23 | 1 | 1 | yes | characteristic |
| 42 | #568 | — | 2 | 2 | no | — |
| 43 | C2×#131 | — | 2 | 1 | no | — |
| 44 | C2×M32 | C23 | 1 | 2 | yes | — |
| 45 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 46 | C22×C16 | C23 | 1 | 1 | yes | characteristic |
| 47 | #444 | — | 2 | 4 | no | — |
| 48 | #443 | — | 2 | 1 | no | — |
| 49 | #443 | — | 2 | 1 | no | — |
| 50 | #443 | — | 4 | 2 | no | — |
| 51 | #443 | Q8 | 1 | 2 | yes | — |
| 52 | #443 | Q8 | 1 | 2 | yes | — |
| 53 | #430 | — | 2 | 2 | no | — |
| 54 | #430 | — | 2 | 2 | no | — |
| 55 | #430 | — | 4 | 2 | no | — |
| 56 | C2×#108 | — | 2 | 1 | no | — |
| 57 | C2×#108 | — | 2 | 1 | no | — |
| 58 | C2×#108 | — | 4 | 2 | no | — |
| 59 | C2×#108 | D4 | 1 | 1 | yes | characteristic |
| 60 | C2×#108 | C23 | 1 | 2 | yes | — |
| 61 | C2×#108 | C23 | 1 | 2 | yes | — |
| 62 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 63 | C2×#108 | C23 | 1 | 1 | yes | characteristic |
| 64 | #427 | D4 | 1 | 1 | yes | characteristic |
| 65 | #427 | C23 | 1 | 1 | yes | characteristic |
| 66 | #423 | — | 2 | 1 | no | — |
| 67 | #423 | — | 2 | 1 | no | — |
| 68 | #421 | — | 2 | 2 | no | — |
| 69 | C2×#104 | — | 2 | 1 | no | — |
| 70 | #405 | — | 2 | 1 | no | — |
| 71 | #405 | C23 | 1 | 2 | yes | — |
| 72 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 73 | #368 | — | 4 | 2 | no | — |
| 74 | #364 | — | 2 | 1 | no | — |
| 75 | #364 | — | 2 | 1 | no | — |
| 76 | C4⋊C16 | — | 4 | 2 | no | — |
| 77 | #362 | — | 2 | 2 | no | — |
| 78 | #361 | — | 2 | 2 | no | — |
| 79 | #359 | — | 2 | 2 | no | — |
| 80 | #358 | — | 2 | 1 | no | — |
| 81 | #357 | — | 2 | 1 | no | — |
| 82 | C4×C16 | — | 2 | 1 | no | — |
| 83 | C4×C16 | — | 2 | 1 | no | — |
| 84 | #344 | — | 2 | 2 | no | — |
| 85 | #344 | — | 2 | 2 | no | — |
| 86 | #344 | — | 2 | 2 | no | — |
| 87 | #339 | — | 2 | 1 | no | — |
| 88 | #339 | D4 | 1 | 2 | yes | — |
| 89 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 90 | C2×#41 | — | 2 | 1 | no | — |
| 91 | #135 | — | 2 | 1 | no | — |
| 92 | #135 | — | 2 | 1 | no | — |
| 93 | #135 | C2×Q8 | 1 | 2 | yes | — |
| 94 | #135 | #41 | 1 | 2 | yes | — |
| 95 | C2×Q16 | #41 | 1 | 1 | yes | characteristic |
| 96 | C2×SD16 | — | 2 | 1 | no | — |
| 97 | C2×D8 | #41 | 1 | 1 | yes | characteristic |
| 98 | #131 | — | 2 | 2 | no | — |
| 99 | #131 | — | 2 | 2 | no | — |
| 100 | #131 | — | 4 | 4 | no | — |
| 101 | #131 | — | 4 | 1 | no | — |
| 102 | #131 | — | 4 | 1 | no | — |
| 103 | #131 | — | 8 | 2 | no | — |
| 104 | C2×M16 | — | 2 | 1 | no | — |
| 105 | C2×M16 | — | 2 | 2 | no | — |
| 106 | C2×M16 | — | 2 | 2 | no | — |
| 107 | C2×M16 | — | 2 | 1 | no | — |
| 108 | C2×M16 | — | 4 | 1 | no | — |
| 109 | C2×M16 | — | 4 | 4 | no | — |
| 110 | C2×M16 | C2×D4 | 1 | 2 | yes | — |
| 111 | C22×C8 | — | 4 | 1 | no | — |
| 112 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | C22×C8 | C24 | 1 | 1 | yes | characteristic |
| 114 | #117 | — | 2 | 1 | no | — |
| 115 | #117 | C2×D4 | 1 | 2 | yes | — |
| 116 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | M32 | — | 2 | 2 | no | — |
| 118 | M32 | — | 2 | 2 | no | — |
| 119 | C2×C16 | — | 2 | 1 | no | — |
| 120 | C2×C16 | — | 2 | 1 | no | — |
| 121 | C2×C16 | — | 2 | 1 | no | — |
| 122 | C2×C16 | — | 2 | 1 | no | — |
| 123 | C2×C16 | — | 2 | 1 | no | — |
| 124 | C2×C16 | — | 2 | 2 | no | — |
| 125 | C2×C16 | — | 2 | 1 | no | — |
| 126 | #108 | — | 2 | 1 | no | — |
| 127 | #108 | — | 2 | 1 | no | — |
| 128 | #108 | — | 2 | 1 | no | — |
| 129 | #108 | — | 2 | 1 | no | — |
| 130 | #108 | — | 2 | 2 | no | — |
| 131 | #108 | — | 2 | 2 | no | — |
| 132 | #108 | — | 2 | 1 | no | — |
| 133 | #108 | — | 2 | 2 | no | — |
| 134 | #108 | — | 2 | 2 | no | — |
| 135 | #108 | — | 4 | 4 | no | — |
| 136 | #108 | — | 4 | 1 | no | — |
| 137 | #108 | — | 4 | 1 | no | — |
| 138 | #108 | — | 4 | 1 | no | — |
| 139 | #108 | — | 4 | 1 | no | — |
| 140 | #108 | — | 8 | 2 | no | — |
| 141 | #108 | C2×Q8 | 1 | 2 | yes | — |
| 142 | #107 | — | 2 | 1 | no | — |
| 143 | #107 | — | 2 | 1 | no | — |
| 144 | #106 | — | 2 | 1 | no | — |
| 145 | #106 | — | 2 | 1 | no | — |
| 146 | C4⋊C8 | — | 8 | 1 | no | — |
| 147 | C4⋊C8 | — | 8 | 1 | no | — |
| 148 | #104 | — | 2 | 2 | no | — |
| 149 | #104 | — | 2 | 2 | no | — |
| 150 | #104 | — | 4 | 4 | no | — |
| 151 | #97 | — | 2 | 2 | no | — |
| 152 | #97 | — | 4 | 1 | no | — |
| 153 | C4×C8 | — | 2 | 2 | no | — |
| 154 | C4×C8 | — | 4 | 1 | no | — |
| 155 | C4×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 156 | C4×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 157 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 158 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 159 | #41 | — | 2 | 2 | no | — |
| 160 | #41 | — | 2 | 2 | no | — |
| 161 | #41 | — | 4 | 1 | no | — |
| 162 | #41 | — | 4 | 1 | no | — |
| 163 | C2×Q8 | — | 2 | 1 | no | — |
| 164 | C2×D4 | — | 2 | 1 | no | — |
| 165 | C2×D4 | — | 4 | 1 | no | — |
| 166 | C22×C4 | — | 2 | 1 | no | — |
| 167 | C22×C4 | — | 4 | 1 | no | — |
| 168 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 169 | Q16 | — | 2 | 1 | no | — |
| 170 | Q16 | #125 | 1 | 2 | yes | — |
| 171 | SD16 | — | 2 | 2 | no | — |
| 172 | SD16 | — | 2 | 2 | no | — |
| 173 | D8 | — | 2 | 1 | no | — |
| 174 | D8 | #125 | 1 | 2 | yes | — |
| 175 | M16 | — | 2 | 2 | no | — |
| 176 | M16 | — | 2 | 2 | no | — |
| 177 | M16 | — | 2 | 2 | no | — |
| 178 | M16 | — | 4 | 1 | no | — |
| 179 | M16 | — | 4 | 2 | no | — |
| 180 | M16 | — | 4 | 2 | no | — |
| 181 | M16 | — | 4 | 2 | no | — |
| 182 | M16 | — | 4 | 1 | no | — |
| 183 | M16 | — | 4 | 2 | no | — |
| 184 | M16 | — | 4 | 2 | no | — |
| 185 | M16 | — | 4 | 2 | no | — |
| 186 | M16 | — | 4 | 1 | no | — |
| 187 | M16 | — | 8 | 4 | no | — |
| 188 | C2×C8 | — | 2 | 2 | no | — |
| 189 | C2×C8 | — | 4 | 1 | no | — |
| 190 | C2×C8 | — | 4 | 2 | no | — |
| 191 | C2×C8 | — | 4 | 2 | no | — |
| 192 | C2×C8 | — | 4 | 1 | no | — |
| 193 | C2×C8 | — | 4 | 2 | no | — |
| 194 | C2×C8 | — | 4 | 2 | no | — |
| 195 | C2×C8 | — | 8 | 4 | no | — |
| 196 | C2×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 197 | C2×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 198 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 199 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 200 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 201 | C2×C8 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 202 | C2×C8 | C22×Q8 | 1 | 1 | yes | characteristic |
| 203 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 204 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 205 | C2×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 206 | C2×C8 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 207 | C2×C8 | 2−1+4 | 1 | 1 | yes | characteristic |
| 208 | C4⋊C4 | — | 2 | 2 | no | — |
| 209 | C4⋊C4 | — | 4 | 1 | no | — |
| 210 | C22⋊C4 | — | 2 | 2 | no | — |
| 211 | C22⋊C4 | — | 4 | 1 | no | — |
| 212 | C42 | — | 2 | 1 | no | — |
| 213 | C42 | — | 2 | 2 | no | — |
| 214 | C42 | — | 4 | 1 | no | — |
| 215 | C42 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 216 | C42 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 217 | C16 | — | 4 | 1 | no | — |
| 218 | C16 | — | 4 | 1 | no | — |
| 219 | C16 | — | 4 | 2 | no | — |
| 220 | C23 | — | 4 | 1 | no | — |
| 221 | C23 | #578 | 1 | 1 | yes | characteristic |
| 222 | Q8 | — | 2 | 2 | no | — |
| 223 | Q8 | — | 4 | 1 | no | — |
| 224 | D4 | — | 2 | 2 | no | — |
| 225 | D4 | — | 4 | 2 | no | — |
| 226 | D4 | — | 4 | 2 | no | — |
| 227 | D4 | — | 4 | 1 | no | — |
| 228 | C2×C4 | — | 2 | 1 | no | — |
| 229 | C2×C4 | — | 2 | 1 | no | — |
| 230 | C2×C4 | — | 4 | 2 | no | — |
| 231 | C2×C4 | — | 4 | 1 | no | — |
| 232 | C2×C4 | — | 4 | 2 | no | — |
| 233 | C2×C4 | — | 4 | 2 | no | — |
| 234 | C2×C4 | — | 4 | 2 | no | — |
| 235 | C2×C4 | — | 8 | 1 | no | — |
| 236 | C2×C4 | C2×C22⋊Q8 | 1 | 1 | yes | characteristic |
| 237 | C2×C4 | #546 | 1 | 1 | yes | characteristic |
| 238 | C2×C4 | #548 | 1 | 1 | yes | characteristic |
| 239 | C2×C4 | C2×#135 | 1 | 1 | yes | characteristic |
| 240 | C2×C4 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 241 | C2×C4 | #576 | 1 | 1 | yes | characteristic |
| 242 | C8 | — | 4 | 2 | no | — |
| 243 | C8 | — | 4 | 2 | no | — |
| 244 | C8 | — | 8 | 2 | no | — |
| 245 | C8 | — | 8 | 2 | no | — |
| 246 | C8 | — | 8 | 1 | no | — |
| 247 | C8 | #478 | 1 | 1 | yes | characteristic |
| 248 | C8 | #478 | 1 | 1 | yes | characteristic |
| 249 | C8 | #478 | 1 | 1 | yes | characteristic |
| 250 | C8 | #478 | 1 | 1 | yes | characteristic |
| 251 | C8 | C2×#125 | 1 | 1 | yes | characteristic |
| 252 | C8 | #544 | 1 | 1 | yes | characteristic |
| 253 | C8 | #552 | 1 | 1 | yes | characteristic |
| 254 | C8 | #553 | 1 | 1 | yes | characteristic |
| 255 | C22 | — | 4 | 1 | no | — |
| 256 | C22 | — | 4 | 2 | no | — |
| 257 | C22 | — | 16 | 1 | no | — |
| 258 | C22 | #3083 | 1 | 1 | yes | characteristic |
| 259 | C22 | #3302 | 1 | 1 | yes | characteristic |
| 260 | C22 | #3367 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 2 | 2 | no | — |
| 262 | C4 | — | 4 | 1 | no | — |
| 263 | C4 | — | 4 | 2 | no | — |
| 264 | C4 | — | 8 | 2 | no | — |
| 265 | C4 | — | 16 | 1 | no | — |
| 266 | C4 | #2442 | 1 | 1 | yes | characteristic |
| 267 | C4 | C2×#478 | 1 | 1 | yes | characteristic |
| 268 | C4 | #3325 | 1 | 1 | yes | characteristic |
| 269 | C4 | #3361 | 1 | 1 | yes | characteristic |
| 270 | C2 | — | 2 | 1 | no | — |
| 271 | C2 | — | 2 | 1 | no | — |
| 272 | C2 | — | 2 | 1 | no | — |
| 273 | C2 | — | 8 | 2 | no | — |
| 274 | C2 | #26111 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta