← #135921
Subgroups — #135921
Order 512.
354 conjugacy classes of subgroups.
The subgroup lattice has 280 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#1589 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×#357 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#1372 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#1331 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #9594 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7171 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7136 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×C4×D8 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#436 | — | 2 | 1 | no | — |
| 10 | C2×#357 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#357 | C4 | 1 | 1 | yes | characteristic |
| 12 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4×C16 | C22 | 1 | 1 | yes | characteristic |
| 14 | #1939 | — | 2 | 1 | no | — |
| 15 | C4×#107 | C22 | 1 | 1 | yes | characteristic |
| 16 | #1589 | — | 2 | 2 | no | — |
| 17 | #1589 | — | 2 | 2 | no | — |
| 18 | C2×#335 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#335 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×D4⋊C8 | — | 2 | 1 | no | — |
| 21 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1380 | C4 | 1 | 1 | yes | characteristic |
| 23 | #1380 | C4 | 1 | 1 | yes | characteristic |
| 24 | #1372 | — | 2 | 2 | no | — |
| 25 | #1331 | — | 2 | 2 | no | — |
| 26 | #1279 | — | 2 | 1 | no | — |
| 27 | C22×D8 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C22×M16 | — | 4 | 1 | no | — |
| 29 | C2×C4×D4 | — | 2 | 1 | no | — |
| 30 | C22×C16 | — | 2 | 1 | no | — |
| 31 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C22×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C4×D8 | — | 2 | 1 | no | — |
| 34 | C4×D8 | — | 2 | 1 | no | — |
| 35 | C4×D8 | D4 | 1 | 2 | yes | — |
| 36 | C4×D8 | Q8 | 1 | 2 | yes | — |
| 37 | #436 | — | 2 | 4 | no | — |
| 38 | #436 | — | 4 | 2 | no | — |
| 39 | C2×#107 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C2×#107 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C2×#107 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 43 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 44 | C2×C4⋊C8 | — | 4 | 1 | no | — |
| 45 | C2×#102 | — | 2 | 1 | no | — |
| 46 | C2×C22⋊C8 | — | 4 | 1 | no | — |
| 47 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 48 | C2×C4×C8 | D4 | 1 | 1 | yes | characteristic |
| 49 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 50 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 51 | #357 | — | 2 | 2 | no | — |
| 52 | #357 | — | 2 | 2 | no | — |
| 53 | C4×C16 | — | 2 | 1 | no | — |
| 54 | C4×C16 | — | 2 | 1 | no | — |
| 55 | C4×C16 | — | 2 | 1 | no | — |
| 56 | C4×C16 | — | 2 | 1 | no | — |
| 57 | #340 | — | 2 | 1 | no | — |
| 58 | #336 | — | 4 | 1 | no | — |
| 59 | #335 | — | 2 | 1 | no | — |
| 60 | #335 | — | 2 | 1 | no | — |
| 61 | #335 | — | 2 | 1 | no | — |
| 62 | #335 | D4 | 1 | 2 | yes | — |
| 63 | D4⋊C8 | — | 2 | 4 | no | — |
| 64 | C82 | — | 2 | 1 | no | — |
| 65 | C82 | — | 2 | 1 | no | — |
| 66 | C22×D4 | — | 2 | 1 | no | — |
| 67 | C23×C4 | — | 4 | 1 | no | — |
| 68 | C2×D8 | — | 2 | 1 | no | — |
| 69 | C2×D8 | — | 2 | 1 | no | — |
| 70 | C2×D8 | — | 2 | 1 | no | — |
| 71 | C2×D8 | — | 2 | 1 | no | — |
| 72 | C2×D8 | — | 2 | 1 | no | — |
| 73 | C2×D8 | C4⋊C4 | 1 | 2 | yes | — |
| 74 | C2×M16 | — | 4 | 4 | no | — |
| 75 | C2×M16 | — | 8 | 2 | no | — |
| 76 | C2×M16 | — | 8 | 2 | no | — |
| 77 | C22×C8 | — | 2 | 1 | no | — |
| 78 | C22×C8 | — | 4 | 1 | no | — |
| 79 | C22×C8 | — | 4 | 1 | no | — |
| 80 | C22×C8 | D8 | 1 | 1 | yes | characteristic |
| 81 | C22×C8 | D8 | 1 | 1 | yes | characteristic |
| 82 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 84 | C4×D4 | — | 2 | 2 | no | — |
| 85 | C4×D4 | — | 2 | 2 | no | — |
| 86 | C4×D4 | — | 4 | 1 | no | — |
| 87 | C4×D4 | — | 4 | 1 | no | — |
| 88 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 89 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 90 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 91 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 92 | C2×C42 | — | 4 | 1 | no | — |
| 93 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C16 | — | 2 | 1 | no | — |
| 95 | C2×C16 | — | 2 | 1 | no | — |
| 96 | C2×C16 | — | 2 | 2 | no | — |
| 97 | C2×C16 | — | 2 | 1 | no | — |
| 98 | C2×C16 | — | 2 | 1 | no | — |
| 99 | C2×C16 | — | 2 | 1 | no | — |
| 100 | C2×C16 | — | 4 | 1 | no | — |
| 101 | C2×C16 | — | 4 | 1 | no | — |
| 102 | C2×C16 | M16 | 1 | 2 | yes | complemented |
| 103 | #107 | — | 2 | 1 | no | — |
| 104 | #107 | — | 2 | 1 | no | — |
| 105 | #107 | — | 2 | 1 | no | — |
| 106 | #107 | — | 2 | 1 | no | — |
| 107 | #107 | C4⋊C4 | 1 | 2 | yes | — |
| 108 | #107 | M16 | 1 | 2 | yes | complemented |
| 109 | C4⋊C8 | — | 2 | 2 | no | — |
| 110 | C4⋊C8 | — | 4 | 1 | no | — |
| 111 | C4⋊C8 | — | 4 | 1 | no | — |
| 112 | C4⋊C8 | — | 4 | 1 | no | — |
| 113 | C4⋊C8 | — | 8 | 2 | no | — |
| 114 | #102 | — | 2 | 2 | no | — |
| 115 | #102 | — | 2 | 2 | no | — |
| 116 | C22⋊C8 | — | 4 | 4 | no | — |
| 117 | C4×C8 | — | 2 | 1 | no | — |
| 118 | C4×C8 | — | 2 | 1 | no | — |
| 119 | C4×C8 | — | 2 | 1 | no | — |
| 120 | C4×C8 | — | 2 | 1 | no | — |
| 121 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 122 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 123 | C4×C8 | C2×Q8 | 1 | 1 | yes | characteristic |
| 124 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 125 | #95 | — | 4 | 1 | no | — |
| 126 | C24 | — | 4 | 1 | no | — |
| 127 | C2×D4 | — | 2 | 2 | no | — |
| 128 | C2×D4 | — | 4 | 1 | no | — |
| 129 | C2×D4 | — | 4 | 1 | no | — |
| 130 | C2×D4 | — | 4 | 1 | no | — |
| 131 | C2×D4 | — | 4 | 1 | no | — |
| 132 | C2×D4 | — | 4 | 1 | no | — |
| 133 | C22×C4 | — | 4 | 1 | no | — |
| 134 | C22×C4 | — | 4 | 1 | no | — |
| 135 | C22×C4 | — | 4 | 2 | no | — |
| 136 | C22×C4 | — | 4 | 2 | no | — |
| 137 | C22×C4 | — | 4 | 1 | no | — |
| 138 | C22×C4 | — | 8 | 1 | no | — |
| 139 | C22×C4 | — | 8 | 1 | no | — |
| 140 | C22×C4 | — | 8 | 1 | no | — |
| 141 | C22×C4 | — | 8 | 1 | no | — |
| 142 | C22×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 143 | C22×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 144 | C22×C4 | #135 | 1 | 1 | yes | characteristic |
| 145 | D8 | — | 2 | 2 | no | — |
| 146 | D8 | — | 2 | 2 | no | — |
| 147 | D8 | — | 4 | 1 | no | — |
| 148 | D8 | — | 4 | 1 | no | — |
| 149 | M16 | — | 8 | 8 | no | — |
| 150 | C2×C8 | — | 2 | 1 | no | — |
| 151 | C2×C8 | — | 2 | 1 | no | — |
| 152 | C2×C8 | — | 2 | 1 | no | — |
| 153 | C2×C8 | — | 2 | 1 | no | — |
| 154 | C2×C8 | — | 2 | 2 | no | — |
| 155 | C2×C8 | — | 2 | 1 | no | — |
| 156 | C2×C8 | — | 2 | 1 | no | — |
| 157 | C2×C8 | — | 2 | 1 | no | — |
| 158 | C2×C8 | — | 4 | 1 | no | — |
| 159 | C2×C8 | — | 4 | 1 | no | — |
| 160 | C2×C8 | — | 4 | 2 | no | — |
| 161 | C2×C8 | — | 8 | 1 | no | — |
| 162 | C2×C8 | — | 8 | 1 | no | — |
| 163 | C2×C8 | — | 8 | 1 | no | — |
| 164 | C2×C8 | — | 8 | 1 | no | — |
| 165 | C2×C8 | — | 8 | 1 | no | — |
| 166 | C2×C8 | SD32 | 1 | 2 | yes | — |
| 167 | C2×C8 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 168 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 169 | C2×C8 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 170 | C2×C8 | #123 | 1 | 1 | yes | characteristic |
| 171 | C2×C8 | C2×M16 | 1 | 1 | yes | characteristic |
| 172 | C2×C8 | C2×M16 | 1 | 1 | yes | characteristic |
| 173 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 174 | C2×C8 | #131 | 1 | 1 | yes | characteristic |
| 175 | C4⋊C4 | — | 2 | 2 | no | — |
| 176 | C4⋊C4 | — | 2 | 2 | no | — |
| 177 | C4⋊C4 | — | 4 | 1 | no | — |
| 178 | C4⋊C4 | — | 4 | 1 | no | — |
| 179 | C4⋊C4 | — | 4 | 1 | no | — |
| 180 | C4⋊C4 | — | 4 | 1 | no | — |
| 181 | C22⋊C4 | — | 4 | 2 | no | — |
| 182 | C22⋊C4 | — | 4 | 2 | no | — |
| 183 | C42 | — | 8 | 1 | no | — |
| 184 | C42 | — | 8 | 1 | no | — |
| 185 | C42 | #120 | 1 | 1 | yes | characteristic |
| 186 | C42 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 187 | C42 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 188 | C42 | #137 | 1 | 1 | yes | characteristic |
| 189 | C16 | — | 2 | 2 | no | — |
| 190 | C16 | — | 4 | 2 | no | — |
| 191 | C16 | — | 4 | 1 | no | — |
| 192 | C23 | — | 4 | 2 | no | — |
| 193 | C23 | — | 8 | 1 | no | — |
| 194 | C23 | — | 8 | 1 | no | — |
| 195 | C23 | — | 8 | 1 | no | — |
| 196 | C23 | — | 8 | 1 | no | — |
| 197 | C23 | — | 8 | 1 | no | — |
| 198 | C23 | — | 8 | 1 | no | — |
| 199 | C23 | C4×D8 | 1 | 1 | yes | characteristic |
| 200 | D4 | — | 4 | 2 | no | — |
| 201 | D4 | — | 4 | 2 | no | — |
| 202 | D4 | — | 8 | 1 | no | — |
| 203 | D4 | — | 8 | 1 | no | — |
| 204 | C2×C4 | — | 2 | 1 | no | — |
| 205 | C2×C4 | — | 2 | 1 | no | — |
| 206 | C2×C4 | — | 4 | 2 | no | — |
| 207 | C2×C4 | — | 4 | 2 | no | — |
| 208 | C2×C4 | — | 8 | 1 | no | — |
| 209 | C2×C4 | — | 8 | 1 | no | — |
| 210 | C2×C4 | — | 8 | 1 | no | — |
| 211 | C2×C4 | — | 8 | 1 | no | — |
| 212 | C2×C4 | — | 8 | 2 | no | — |
| 213 | C2×C4 | — | 8 | 2 | no | — |
| 214 | C2×C4 | — | 8 | 2 | no | — |
| 215 | C2×C4 | — | 8 | 2 | no | — |
| 216 | C2×C4 | — | 8 | 1 | no | — |
| 217 | C2×C4 | — | 8 | 1 | no | — |
| 218 | C2×C4 | — | 8 | 1 | no | — |
| 219 | C2×C4 | #385 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | #435 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #436 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #440 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #443 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | #444 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | #452 | 1 | 1 | yes | characteristic |
| 227 | C2×C4 | #474 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | #478 | 1 | 1 | yes | characteristic |
| 229 | C2×C4 | C2×SD32 | 1 | 1 | yes | characteristic |
| 230 | C2×C4 | #508 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | #509 | 1 | 1 | yes | characteristic |
| 232 | C2×C4 | #509 | 1 | 1 | yes | characteristic |
| 233 | C8 | — | 2 | 1 | no | — |
| 234 | C8 | — | 2 | 1 | no | — |
| 235 | C8 | — | 2 | 2 | no | — |
| 236 | C8 | — | 4 | 2 | no | — |
| 237 | C8 | — | 4 | 1 | no | — |
| 238 | C8 | — | 8 | 2 | no | — |
| 239 | C8 | — | 16 | 1 | no | — |
| 240 | C8 | #423 | 1 | 1 | yes | characteristic |
| 241 | C8 | #424 | 1 | 1 | yes | characteristic |
| 242 | C8 | #435 | 1 | 1 | yes | characteristic |
| 243 | C8 | #436 | 1 | 1 | yes | characteristic |
| 244 | C22 | — | 8 | 2 | no | — |
| 245 | C22 | — | 8 | 2 | no | — |
| 246 | C22 | — | 8 | 2 | no | — |
| 247 | C22 | — | 16 | 1 | no | — |
| 248 | C22 | — | 16 | 1 | no | — |
| 249 | C22 | — | 16 | 1 | no | — |
| 250 | C22 | — | 16 | 1 | no | — |
| 251 | C22 | #1581 | 1 | 1 | yes | characteristic |
| 252 | C22 | #1589 | 1 | 1 | yes | characteristic |
| 253 | C22 | #1862 | 1 | 1 | yes | characteristic |
| 254 | C22 | C4×SD32 | 1 | 1 | yes | characteristic |
| 255 | C22 | #2177 | 1 | 1 | yes | characteristic |
| 256 | C22 | #2178 | 1 | 1 | yes | characteristic |
| 257 | C22 | #2179 | 1 | 1 | yes | characteristic |
| 258 | C4 | — | 2 | 1 | no | — |
| 259 | C4 | — | 2 | 1 | no | — |
| 260 | C4 | — | 8 | 2 | no | — |
| 261 | C4 | — | 8 | 2 | no | — |
| 262 | C4 | — | 16 | 1 | no | — |
| 263 | C4 | #1588 | 1 | 1 | yes | characteristic |
| 264 | C4 | #1592 | 1 | 1 | yes | characteristic |
| 265 | C4 | #1851 | 1 | 1 | yes | characteristic |
| 266 | C4 | #1866 | 1 | 1 | yes | characteristic |
| 267 | C4 | #2186 | 1 | 1 | yes | characteristic |
| 268 | C4 | #2187 | 1 | 1 | yes | characteristic |
| 269 | C4 | #2226 | 1 | 1 | yes | characteristic |
| 270 | C4 | #2228 | 1 | 1 | yes | characteristic |
| 271 | C2 | — | 16 | 2 | no | — |
| 272 | C2 | #8847 | 1 | 1 | yes | characteristic |
| 273 | C2 | #12071 | 1 | 1 | yes | characteristic |
| 274 | C2 | #12073 | 1 | 1 | yes | characteristic |
| 275 | C2 | #12082 | 1 | 1 | yes | characteristic |
| 276 | C2 | #12086 | 1 | 1 | yes | characteristic |
| 277 | C2 | #12937 | 1 | 1 | yes | characteristic |
| 278 | C2 | #12939 | 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