← #121415
Subgroups — #121415
Order 512.
636 conjugacy classes of subgroups.
The subgroup lattice has 271 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #21349 | C2 | 1 | 2 | yes | maximal |
| 2 | #17787 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C4×H(3,ℤ/4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #7866 | C2 | 1 | 2 | yes | maximal |
| 5 | #7848 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×#431 | C22 | 1 | 2 | yes | — |
| 7 | C2×#431 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×C4×M16 | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C4×#117 | C22 | 1 | 2 | yes | — |
| 11 | #1939 | — | 2 | 4 | no | — |
| 12 | #1850 | — | 2 | 4 | no | — |
| 13 | C4×C4⋊C8 | — | 2 | 4 | no | — |
| 14 | C2×H(3,ℤ/4) | C4 | 1 | 1 | yes | characteristic |
| 15 | C2×H(3,ℤ/4) | C4 | 1 | 2 | yes | — |
| 16 | C2×H(3,ℤ/4) | C4 | 1 | 1 | yes | characteristic |
| 17 | #1565 | — | 2 | 4 | no | — |
| 18 | #1564 | — | 2 | 2 | no | — |
| 19 | #1276 | — | 2 | 4 | no | — |
| 20 | #1273 | — | 2 | 2 | no | — |
| 21 | C22×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 22 | C22×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C23×C8 | D4 | 1 | 2 | yes | — |
| 24 | C2×#117 | C2×C4 | 1 | 2 | yes | — |
| 25 | C2×#117 | C2×C4 | 1 | 2 | yes | — |
| 26 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 27 | #431 | — | 2 | 2 | no | — |
| 28 | #431 | — | 2 | 2 | no | — |
| 29 | #431 | — | 2 | 2 | no | — |
| 30 | #431 | — | 2 | 2 | no | — |
| 31 | #431 | — | 2 | 2 | no | — |
| 32 | #431 | — | 2 | 2 | no | — |
| 33 | C2×C4⋊C8 | — | 2 | 8 | no | — |
| 34 | C2×C4⋊C8 | D4 | 1 | 4 | yes | — |
| 35 | C2×C4⋊C8 | Q8 | 1 | 2 | yes | — |
| 36 | C2×C22⋊C8 | — | 2 | 2 | no | — |
| 37 | C2×C22⋊C8 | — | 2 | 1 | no | — |
| 38 | C4×M16 | — | 2 | 2 | no | — |
| 39 | C4×M16 | — | 2 | 2 | no | — |
| 40 | C2×#97 | Q8 | 1 | 2 | yes | — |
| 41 | C2×C4×C8 | — | 2 | 2 | no | — |
| 42 | C2×C4×C8 | — | 2 | 2 | no | — |
| 43 | C2×C4×C8 | — | 2 | 1 | no | — |
| 44 | C2×C4×C8 | — | 2 | 2 | no | — |
| 45 | C2×C4×C8 | — | 2 | 1 | no | — |
| 46 | C2×C4×C8 | — | 2 | 1 | no | — |
| 47 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 48 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 49 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 50 | C4×C22⋊C4 | — | 2 | 2 | no | — |
| 51 | C4×C22⋊C4 | — | 2 | 2 | no | — |
| 52 | #376 | Q8 | 1 | 4 | yes | — |
| 53 | C43 | D4 | 1 | 4 | yes | — |
| 54 | H(3,ℤ/4) | — | 2 | 2 | no | — |
| 55 | H(3,ℤ/4) | — | 2 | 4 | no | — |
| 56 | H(3,ℤ/4) | C8 | 1 | 4 | yes | complemented |
| 57 | #336 | — | 2 | 8 | no | — |
| 58 | #335 | — | 2 | 4 | no | — |
| 59 | #334 | — | 2 | 4 | no | — |
| 60 | C23×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 61 | C23×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C23×C4 | #41 | 1 | 1 | yes | characteristic |
| 63 | C2×M16 | — | 2 | 2 | no | — |
| 64 | C2×M16 | — | 2 | 2 | no | — |
| 65 | C2×M16 | — | 2 | 2 | no | — |
| 66 | C2×M16 | — | 2 | 2 | no | — |
| 67 | C2×M16 | — | 2 | 1 | no | — |
| 68 | C2×M16 | — | 2 | 1 | no | — |
| 69 | C2×M16 | C2×C8 | 1 | 2 | yes | — |
| 70 | C2×M16 | M16 | 1 | 2 | yes | — |
| 71 | C22×C8 | — | 2 | 4 | no | — |
| 72 | C22×C8 | — | 2 | 2 | no | — |
| 73 | C22×C8 | — | 2 | 4 | no | — |
| 74 | C22×C8 | — | 2 | 2 | no | — |
| 75 | C22×C8 | — | 4 | 4 | no | — |
| 76 | C22×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 77 | C22×C8 | C4⋊C4 | 1 | 2 | yes | — |
| 78 | #117 | — | 2 | 2 | no | — |
| 79 | #117 | — | 2 | 2 | no | — |
| 80 | #117 | — | 2 | 2 | no | — |
| 81 | #117 | — | 2 | 2 | no | — |
| 82 | #117 | — | 2 | 2 | no | — |
| 83 | #117 | — | 2 | 2 | no | — |
| 84 | #117 | C2×C8 | 1 | 4 | yes | — |
| 85 | #117 | M16 | 1 | 4 | yes | — |
| 86 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 87 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 88 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 89 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 90 | C2×C42 | — | 2 | 4 | no | — |
| 91 | C2×C42 | — | 2 | 4 | no | — |
| 92 | C2×C42 | C4⋊C4 | 1 | 4 | yes | — |
| 93 | C2×C42 | C4⋊C4 | 1 | 4 | yes | — |
| 94 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 95 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 97 | C2×C42 | C2×Q8 | 1 | 1 | yes | characteristic |
| 98 | C2×C42 | C2×Q8 | 1 | 1 | yes | characteristic |
| 99 | C2×C42 | C2×Q8 | 1 | 2 | yes | — |
| 100 | C2×C42 | #41 | 1 | 1 | yes | characteristic |
| 101 | C2×C42 | #41 | 1 | 2 | yes | — |
| 102 | C4⋊C8 | — | 2 | 4 | no | — |
| 103 | C4⋊C8 | — | 2 | 4 | no | — |
| 104 | C4⋊C8 | — | 2 | 4 | no | — |
| 105 | C4⋊C8 | — | 4 | 16 | no | — |
| 106 | C22⋊C8 | — | 4 | 2 | no | — |
| 107 | C22⋊C8 | — | 4 | 2 | no | — |
| 108 | C22⋊C8 | — | 4 | 2 | no | — |
| 109 | #97 | — | 2 | 4 | no | — |
| 110 | C4×C8 | — | 2 | 2 | no | — |
| 111 | C4×C8 | — | 2 | 2 | no | — |
| 112 | C4×C8 | — | 4 | 1 | no | — |
| 113 | C4×C8 | — | 4 | 2 | no | — |
| 114 | C4×C8 | — | 4 | 2 | no | — |
| 115 | C4×C8 | — | 4 | 1 | no | — |
| 116 | C4×C8 | — | 4 | 1 | no | — |
| 117 | C4×C8 | — | 4 | 2 | no | — |
| 118 | C4×C8 | — | 4 | 1 | no | — |
| 119 | #95 | — | 2 | 4 | no | — |
| 120 | #95 | — | 2 | 4 | no | — |
| 121 | C24 | C4×D4 | 1 | 1 | yes | characteristic |
| 122 | C22×C4 | — | 2 | 4 | no | — |
| 123 | C22×C4 | — | 2 | 4 | no | — |
| 124 | C22×C4 | — | 4 | 2 | no | — |
| 125 | C22×C4 | — | 4 | 2 | no | — |
| 126 | C22×C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 127 | C22×C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 128 | C22×C4 | C4×D4 | 1 | 2 | yes | — |
| 129 | C22×C4 | C4×Q8 | 1 | 1 | yes | characteristic |
| 130 | C22×C4 | #120 | 1 | 2 | yes | — |
| 131 | C22×C4 | #121 | 1 | 1 | yes | characteristic |
| 132 | C22×C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 133 | C22×C4 | C22⋊Q8 | 1 | 2 | yes | — |
| 134 | C22×C4 | C22⋊Q8 | 1 | 2 | yes | — |
| 135 | C22×C4 | #123 | 1 | 2 | yes | — |
| 136 | C22×C4 | #125 | 1 | 1 | yes | characteristic |
| 137 | C22×C4 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 138 | C22×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 139 | C22×C4 | C2×M16 | 1 | 1 | yes | characteristic |
| 140 | C22×C4 | C2×M16 | 1 | 2 | yes | — |
| 141 | C22×C4 | #131 | 1 | 2 | yes | — |
| 142 | C22×C4 | #131 | 1 | 1 | yes | characteristic |
| 143 | C22×C4 | #131 | 1 | 1 | yes | characteristic |
| 144 | M16 | — | 2 | 4 | no | — |
| 145 | M16 | — | 2 | 4 | no | — |
| 146 | M16 | — | 4 | 2 | no | — |
| 147 | M16 | — | 4 | 2 | no | — |
| 148 | C2×C8 | — | 2 | 2 | no | — |
| 149 | C2×C8 | — | 2 | 2 | no | — |
| 150 | C2×C8 | — | 2 | 8 | no | — |
| 151 | C2×C8 | — | 2 | 4 | no | — |
| 152 | C2×C8 | — | 2 | 2 | no | — |
| 153 | C2×C8 | — | 2 | 2 | no | — |
| 154 | C2×C8 | — | 2 | 2 | no | — |
| 155 | C2×C8 | — | 4 | 4 | no | — |
| 156 | C2×C8 | — | 4 | 2 | no | — |
| 157 | C2×C8 | — | 4 | 2 | no | — |
| 158 | C2×C8 | — | 4 | 8 | no | — |
| 159 | C2×C8 | — | 4 | 4 | no | — |
| 160 | C2×C8 | — | 4 | 4 | no | — |
| 161 | C2×C8 | — | 4 | 2 | no | — |
| 162 | C2×C8 | C4⋊C8 | 1 | 4 | yes | complemented |
| 163 | C4⋊C4 | — | 2 | 4 | no | — |
| 164 | C4⋊C4 | — | 2 | 4 | no | — |
| 165 | C4⋊C4 | — | 4 | 2 | no | — |
| 166 | C4⋊C4 | — | 4 | 2 | no | — |
| 167 | C22⋊C4 | — | 2 | 4 | no | — |
| 168 | C22⋊C4 | — | 2 | 4 | no | — |
| 169 | C22⋊C4 | — | 4 | 2 | no | — |
| 170 | C22⋊C4 | — | 4 | 2 | no | — |
| 171 | C42 | — | 2 | 4 | no | — |
| 172 | C42 | — | 2 | 4 | no | — |
| 173 | C42 | — | 2 | 4 | no | — |
| 174 | C42 | — | 4 | 8 | no | — |
| 175 | C42 | — | 4 | 8 | no | — |
| 176 | C42 | C4⋊C8 | 1 | 8 | yes | complemented |
| 177 | C42 | #120 | 1 | 1 | yes | characteristic |
| 178 | C42 | #121 | 1 | 2 | yes | — |
| 179 | C42 | C22⋊Q8 | 1 | 2 | yes | — |
| 180 | C42 | C22⋊Q8 | 1 | 2 | yes | — |
| 181 | C42 | #123 | 1 | 1 | yes | characteristic |
| 182 | C42 | #125 | 1 | 2 | yes | — |
| 183 | C42 | #127 | 1 | 1 | yes | characteristic |
| 184 | C42 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 185 | C42 | C4⋊Q8 | 1 | 2 | yes | — |
| 186 | C42 | C4⋊Q8 | 1 | 2 | yes | — |
| 187 | C23 | — | 4 | 1 | no | — |
| 188 | C23 | — | 4 | 1 | no | — |
| 189 | C23 | #385 | 1 | 2 | yes | — |
| 190 | C23 | #389 | 1 | 1 | yes | characteristic |
| 191 | C23 | C8×D4 | 1 | 1 | yes | characteristic |
| 192 | C23 | #435 | 1 | 2 | yes | — |
| 193 | C23 | #436 | 1 | 1 | yes | characteristic |
| 194 | C2×C4 | — | 2 | 4 | no | — |
| 195 | C2×C4 | — | 2 | 2 | no | — |
| 196 | C2×C4 | — | 2 | 4 | no | — |
| 197 | C2×C4 | — | 2 | 2 | no | — |
| 198 | C2×C4 | — | 2 | 8 | no | — |
| 199 | C2×C4 | — | 4 | 2 | no | — |
| 200 | C2×C4 | — | 4 | 2 | no | — |
| 201 | C2×C4 | — | 4 | 1 | no | — |
| 202 | C2×C4 | — | 4 | 1 | no | — |
| 203 | C2×C4 | — | 4 | 4 | no | — |
| 204 | C2×C4 | — | 4 | 4 | no | — |
| 205 | C2×C4 | — | 4 | 4 | no | — |
| 206 | C2×C4 | — | 4 | 4 | no | — |
| 207 | C2×C4 | — | 4 | 4 | no | — |
| 208 | C2×C4 | #384 | 1 | 1 | yes | characteristic |
| 209 | C2×C4 | #385 | 1 | 1 | yes | characteristic |
| 210 | C2×C4 | #389 | 1 | 2 | yes | — |
| 211 | C2×C4 | #392 | 1 | 1 | yes | characteristic |
| 212 | C2×C4 | #395 | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | #395 | 1 | 2 | yes | — |
| 214 | C2×C4 | #398 | 1 | 2 | yes | — |
| 215 | C2×C4 | #398 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | #399 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | C2×C4⋊C8 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | C2×C4⋊C8 | 1 | 2 | yes | — |
| 219 | C2×C4 | #423 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | #423 | 1 | 2 | yes | — |
| 221 | C2×C4 | #424 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #424 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #424 | 1 | 2 | yes | — |
| 224 | C2×C4 | #424 | 1 | 2 | yes | — |
| 225 | C2×C4 | C8×D4 | 1 | 2 | yes | — |
| 226 | C2×C4 | #435 | 1 | 2 | yes | — |
| 227 | C2×C4 | #435 | 1 | 2 | yes | — |
| 228 | C2×C4 | #436 | 1 | 2 | yes | — |
| 229 | C2×C4 | C8×Q8 | 1 | 1 | yes | characteristic |
| 230 | C2×C4 | #446 | 1 | 2 | yes | — |
| 231 | C2×C4 | #446 | 1 | 1 | yes | characteristic |
| 232 | C8 | — | 2 | 4 | no | — |
| 233 | C8 | — | 4 | 2 | no | — |
| 234 | C8 | — | 4 | 4 | no | — |
| 235 | C8 | — | 8 | 2 | no | — |
| 236 | C8 | — | 8 | 4 | no | — |
| 237 | C22 | — | 2 | 2 | no | — |
| 238 | C22 | — | 2 | 1 | no | — |
| 239 | C22 | — | 2 | 2 | no | — |
| 240 | C22 | — | 2 | 1 | no | — |
| 241 | C22 | — | 4 | 1 | no | — |
| 242 | C22 | — | 4 | 2 | no | — |
| 243 | C22 | — | 4 | 1 | no | — |
| 244 | C22 | #1444 | 1 | 1 | yes | characteristic |
| 245 | C22 | #1850 | 1 | 2 | yes | — |
| 246 | C22 | #1851 | 1 | 2 | yes | — |
| 247 | C22 | #1939 | 1 | 1 | yes | characteristic |
| 248 | C22 | #1940 | 1 | 1 | yes | characteristic |
| 249 | C4 | — | 2 | 2 | no | — |
| 250 | C4 | — | 2 | 1 | no | — |
| 251 | C4 | — | 2 | 1 | no | — |
| 252 | C4 | — | 2 | 2 | no | — |
| 253 | C4 | — | 4 | 8 | no | — |
| 254 | C4 | — | 8 | 4 | no | — |
| 255 | C4 | #1842 | 1 | 1 | yes | characteristic |
| 256 | C4 | #1843 | 1 | 1 | yes | characteristic |
| 257 | C4 | #1850 | 1 | 1 | yes | characteristic |
| 258 | C4 | #1851 | 1 | 1 | yes | characteristic |
| 259 | C4 | #1939 | 1 | 2 | yes | — |
| 260 | C4 | #1940 | 1 | 2 | yes | — |
| 261 | C4 | #2010 | 1 | 2 | yes | — |
| 262 | C4 | #2098 | 1 | 2 | yes | — |
| 263 | C2 | — | 2 | 2 | no | — |
| 264 | C2 | — | 2 | 2 | no | — |
| 265 | C2 | — | 2 | 1 | no | — |
| 266 | C2 | — | 2 | 1 | no | — |
| 267 | C2 | #11322 | 1 | 1 | yes | characteristic |
| 268 | C2 | #11335 | 1 | 1 | yes | characteristic |
| 269 | C2 | #11336 | 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