← #138132
Subgroups — #138132
Order 512.
355 conjugacy classes of subgroups.
The subgroup lattice has 315 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #33221 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #17788 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #12787 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11935 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#1363 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #7280 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #7276 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#499 | C22 | 1 | 1 | yes | characteristic |
| 9 | #3080 | — | 2 | 1 | no | — |
| 10 | #2988 | — | 2 | 1 | no | — |
| 11 | C4×#131 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2150 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×C4⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#350 | — | 2 | 1 | no | — |
| 15 | #1998 | — | 2 | 1 | no | — |
| 16 | #1952 | C4 | 1 | 1 | yes | characteristic |
| 17 | #1952 | C4 | 1 | 1 | yes | characteristic |
| 18 | #1951 | — | 2 | 1 | no | — |
| 19 | #1848 | C22 | 1 | 1 | yes | characteristic |
| 20 | #1816 | C4 | 1 | 1 | yes | characteristic |
| 21 | #1816 | C4 | 1 | 1 | yes | characteristic |
| 22 | #1815 | — | 2 | 1 | no | — |
| 23 | #1565 | — | 2 | 1 | no | — |
| 24 | #1565 | — | 2 | 2 | no | — |
| 25 | #1564 | — | 2 | 1 | no | — |
| 26 | #1564 | C22 | 1 | 1 | yes | characteristic |
| 27 | C2×#334 | — | 2 | 1 | no | — |
| 28 | C2×#334 | C22 | 1 | 1 | yes | characteristic |
| 29 | #1379 | — | 2 | 1 | no | — |
| 30 | #1363 | — | 2 | 2 | no | — |
| 31 | #1363 | — | 2 | 1 | no | — |
| 32 | #1363 | — | 2 | 1 | no | — |
| 33 | C2×#131 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C2×#131 | D4 | 1 | 1 | yes | characteristic |
| 35 | C2×#125 | — | 2 | 1 | no | — |
| 36 | C4×#41 | D4 | 1 | 1 | yes | characteristic |
| 37 | C2×M32 | — | 2 | 1 | no | — |
| 38 | C22×C16 | — | 2 | 1 | no | — |
| 39 | #499 | — | 2 | 1 | no | — |
| 40 | #499 | — | 2 | 1 | no | — |
| 41 | #499 | — | 2 | 1 | no | — |
| 42 | #499 | — | 2 | 1 | no | — |
| 43 | #479 | — | 4 | 1 | no | — |
| 44 | #479 | — | 4 | 1 | no | — |
| 45 | #478 | — | 4 | 1 | no | — |
| 46 | #478 | — | 4 | 1 | no | — |
| 47 | C8×Q8 | — | 4 | 1 | no | — |
| 48 | C8×D4 | — | 4 | 2 | no | — |
| 49 | C8×D4 | — | 4 | 1 | no | — |
| 50 | #431 | — | 2 | 2 | no | — |
| 51 | #431 | — | 2 | 1 | no | — |
| 52 | C2×#108 | — | 2 | 1 | no | — |
| 53 | C2×#108 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C2×#108 | C2×C4 | 1 | 1 | yes | characteristic |
| 55 | #428 | — | 2 | 1 | no | — |
| 56 | #428 | — | 2 | 1 | no | — |
| 57 | C2×#107 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | C2×#107 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C2×#106 | — | 2 | 1 | no | — |
| 60 | #423 | — | 2 | 1 | no | — |
| 61 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 62 | C2×C4⋊C8 | — | 2 | 1 | no | — |
| 63 | #417 | — | 2 | 1 | no | — |
| 64 | #417 | — | 2 | 1 | no | — |
| 65 | #405 | — | 2 | 1 | no | — |
| 66 | #405 | — | 2 | 1 | no | — |
| 67 | #405 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | #405 | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×M16 | — | 2 | 1 | no | — |
| 70 | C4×M16 | D4 | 1 | 1 | yes | characteristic |
| 71 | C2×C4×C8 | — | 2 | 1 | no | — |
| 72 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 73 | C4⋊C16 | — | 2 | 1 | no | — |
| 74 | C4⋊C16 | — | 2 | 1 | no | — |
| 75 | C4⋊C16 | — | 2 | 1 | no | — |
| 76 | C4⋊C16 | — | 2 | 1 | no | — |
| 77 | #350 | — | 4 | 2 | no | — |
| 78 | #343 | — | 2 | 1 | no | — |
| 79 | #343 | — | 2 | 1 | no | — |
| 80 | #343 | — | 2 | 1 | no | — |
| 81 | #343 | — | 2 | 1 | no | — |
| 82 | #335 | — | 2 | 2 | no | — |
| 83 | #335 | — | 2 | 1 | no | — |
| 84 | #334 | — | 2 | 2 | no | — |
| 85 | #334 | — | 2 | 1 | no | — |
| 86 | #334 | D4 | 1 | 1 | yes | characteristic |
| 87 | #334 | D4 | 1 | 2 | yes | — |
| 88 | #334 | D4 | 1 | 1 | yes | characteristic |
| 89 | C2×#41 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 90 | #131 | — | 2 | 1 | no | — |
| 91 | #131 | — | 2 | 1 | no | — |
| 92 | #131 | — | 2 | 2 | no | — |
| 93 | #131 | — | 4 | 1 | no | — |
| 94 | #131 | SD16 | 1 | 2 | yes | — |
| 95 | #131 | Q16 | 1 | 2 | yes | — |
| 96 | C2×M16 | — | 2 | 1 | no | — |
| 97 | C2×M16 | — | 2 | 1 | no | — |
| 98 | C2×M16 | — | 2 | 1 | no | — |
| 99 | C2×M16 | — | 2 | 1 | no | — |
| 100 | C2×M16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 101 | C2×M16 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 102 | C22×C8 | — | 2 | 1 | no | — |
| 103 | C22×C8 | — | 2 | 1 | no | — |
| 104 | C22×C8 | — | 4 | 2 | no | — |
| 105 | C22×C8 | — | 4 | 1 | no | — |
| 106 | C22×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 107 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | #125 | — | 4 | 1 | no | — |
| 109 | #125 | — | 4 | 1 | no | — |
| 110 | #125 | — | 4 | 1 | no | — |
| 111 | #125 | — | 4 | 1 | no | — |
| 112 | C4×Q8 | D8 | 1 | 1 | yes | characteristic |
| 113 | C4×Q8 | SD16 | 1 | 1 | yes | characteristic |
| 114 | C4×D4 | — | 2 | 1 | no | — |
| 115 | C4×D4 | — | 2 | 1 | no | — |
| 116 | C4×D4 | D8 | 1 | 1 | yes | characteristic |
| 117 | C4×D4 | SD16 | 1 | 1 | yes | characteristic |
| 118 | #117 | — | 2 | 1 | no | — |
| 119 | #117 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 120 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 121 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 122 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 123 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 124 | C2×C42 | — | 2 | 1 | no | — |
| 125 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 126 | M32 | — | 4 | 1 | no | — |
| 127 | M32 | — | 4 | 1 | no | — |
| 128 | C2×C16 | — | 4 | 1 | no | — |
| 129 | C2×C16 | — | 4 | 1 | no | — |
| 130 | C2×C16 | — | 4 | 1 | no | — |
| 131 | C2×C16 | — | 4 | 1 | no | — |
| 132 | #108 | — | 4 | 1 | no | — |
| 133 | #108 | — | 4 | 1 | no | — |
| 134 | #108 | — | 4 | 1 | no | — |
| 135 | #108 | — | 4 | 1 | no | — |
| 136 | #107 | — | 2 | 1 | no | — |
| 137 | #107 | — | 2 | 1 | no | — |
| 138 | #107 | — | 2 | 1 | no | — |
| 139 | #107 | — | 2 | 1 | no | — |
| 140 | #107 | — | 4 | 1 | no | — |
| 141 | #107 | — | 4 | 1 | no | — |
| 142 | #106 | — | 4 | 1 | no | — |
| 143 | #106 | — | 4 | 1 | no | — |
| 144 | #106 | — | 4 | 1 | no | — |
| 145 | #106 | — | 4 | 1 | no | — |
| 146 | C4⋊C8 | — | 2 | 1 | no | — |
| 147 | C4⋊C8 | — | 2 | 2 | no | — |
| 148 | C4⋊C8 | — | 2 | 1 | no | — |
| 149 | C4⋊C8 | — | 4 | 2 | no | — |
| 150 | C4⋊C8 | — | 4 | 1 | no | — |
| 151 | C4⋊C8 | — | 4 | 1 | no | — |
| 152 | C4⋊C8 | — | 4 | 1 | no | — |
| 153 | #103 | — | 4 | 1 | no | — |
| 154 | #103 | — | 4 | 1 | no | — |
| 155 | #102 | — | 4 | 1 | no | — |
| 156 | #102 | — | 4 | 1 | no | — |
| 157 | C22⋊C8 | — | 4 | 2 | no | — |
| 158 | C22⋊C8 | — | 4 | 1 | no | — |
| 159 | #97 | — | 2 | 1 | no | — |
| 160 | #97 | — | 2 | 1 | no | — |
| 161 | #97 | — | 2 | 1 | no | — |
| 162 | C4×C8 | — | 2 | 1 | no | — |
| 163 | C4×C8 | — | 2 | 1 | no | — |
| 164 | C4×C8 | — | 2 | 1 | no | — |
| 165 | C4×C8 | — | 4 | 2 | no | — |
| 166 | C4×C8 | — | 4 | 1 | no | — |
| 167 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 168 | C4×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 169 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 170 | C4×C8 | #41 | 1 | 1 | yes | characteristic |
| 171 | #41 | — | 2 | 1 | no | — |
| 172 | #41 | — | 2 | 1 | no | — |
| 173 | #41 | #101 | 1 | 2 | yes | — |
| 174 | #41 | #103 | 1 | 2 | yes | — |
| 175 | C2×Q8 | #102 | 1 | 1 | yes | characteristic |
| 176 | C2×D4 | — | 2 | 1 | no | — |
| 177 | C2×D4 | #102 | 1 | 1 | yes | characteristic |
| 178 | C22×C4 | — | 2 | 1 | no | — |
| 179 | C22×C4 | — | 2 | 1 | no | — |
| 180 | C22×C4 | — | 2 | 1 | no | — |
| 181 | C22×C4 | — | 4 | 1 | no | — |
| 182 | C22×C4 | — | 4 | 1 | no | — |
| 183 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 184 | C22×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 185 | C22×C4 | #137 | 1 | 1 | yes | characteristic |
| 186 | M16 | — | 2 | 1 | no | — |
| 187 | M16 | — | 2 | 2 | no | — |
| 188 | M16 | — | 2 | 2 | no | — |
| 189 | M16 | — | 2 | 1 | no | — |
| 190 | M16 | — | 2 | 1 | no | — |
| 191 | M16 | — | 4 | 2 | no | — |
| 192 | M16 | — | 4 | 1 | no | — |
| 193 | M16 | — | 4 | 1 | no | — |
| 194 | M16 | — | 4 | 1 | no | — |
| 195 | M16 | — | 4 | 1 | no | — |
| 196 | M16 | #103 | 1 | 2 | yes | — |
| 197 | C2×C8 | — | 2 | 1 | no | — |
| 198 | C2×C8 | — | 2 | 1 | no | — |
| 199 | C2×C8 | — | 2 | 1 | no | — |
| 200 | C2×C8 | — | 2 | 2 | no | — |
| 201 | C2×C8 | — | 2 | 2 | no | — |
| 202 | C2×C8 | — | 2 | 1 | no | — |
| 203 | C2×C8 | — | 4 | 2 | no | — |
| 204 | C2×C8 | — | 4 | 1 | no | — |
| 205 | C2×C8 | — | 4 | 1 | no | — |
| 206 | C2×C8 | — | 4 | 1 | no | — |
| 207 | C2×C8 | — | 4 | 2 | no | — |
| 208 | C2×C8 | — | 4 | 1 | no | — |
| 209 | C2×C8 | — | 16 | 1 | no | — |
| 210 | C2×C8 | — | 16 | 2 | no | — |
| 211 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 212 | C2×C8 | C4×D4 | 1 | 1 | yes | characteristic |
| 213 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 214 | C2×C8 | #121 | 1 | 1 | yes | characteristic |
| 215 | C2×C8 | C2×SD16 | 1 | 1 | yes | characteristic |
| 216 | C2×C8 | C2×Q16 | 1 | 1 | yes | characteristic |
| 217 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 218 | C2×C8 | #135 | 1 | 1 | yes | characteristic |
| 219 | C4⋊C4 | — | 2 | 1 | no | — |
| 220 | C4⋊C4 | — | 2 | 1 | no | — |
| 221 | C4⋊C4 | — | 4 | 1 | no | — |
| 222 | C4⋊C4 | — | 4 | 1 | no | — |
| 223 | C4⋊C4 | — | 4 | 1 | no | — |
| 224 | C4⋊C4 | — | 4 | 1 | no | — |
| 225 | C4⋊C4 | — | 8 | 1 | no | — |
| 226 | C4⋊C4 | — | 8 | 1 | no | — |
| 227 | C4⋊C4 | — | 8 | 1 | no | — |
| 228 | C4⋊C4 | — | 8 | 1 | no | — |
| 229 | C4⋊C4 | #102 | 1 | 1 | yes | characteristic |
| 230 | C4⋊C4 | #102 | 1 | 1 | yes | characteristic |
| 231 | C22⋊C4 | — | 2 | 1 | no | — |
| 232 | C22⋊C4 | — | 2 | 1 | no | — |
| 233 | C22⋊C4 | — | 2 | 1 | no | — |
| 234 | C42 | — | 2 | 1 | no | — |
| 235 | C42 | — | 2 | 1 | no | — |
| 236 | C42 | — | 2 | 1 | no | — |
| 237 | C42 | #120 | 1 | 1 | yes | characteristic |
| 238 | C42 | #123 | 1 | 1 | yes | characteristic |
| 239 | C42 | C2×D8 | 1 | 1 | yes | characteristic |
| 240 | C42 | C2×SD16 | 1 | 1 | yes | characteristic |
| 241 | C16 | — | 8 | 1 | no | — |
| 242 | C16 | — | 8 | 1 | no | — |
| 243 | C23 | — | 2 | 1 | no | — |
| 244 | C23 | #417 | 1 | 1 | yes | characteristic |
| 245 | Q8 | — | 2 | 1 | no | — |
| 246 | Q8 | #332 | 1 | 2 | yes | — |
| 247 | D4 | — | 2 | 2 | no | — |
| 248 | D4 | — | 2 | 2 | no | — |
| 249 | D4 | — | 2 | 1 | no | — |
| 250 | D4 | #332 | 1 | 2 | yes | — |
| 251 | C2×C4 | — | 2 | 1 | no | — |
| 252 | C2×C4 | — | 2 | 2 | no | — |
| 253 | C2×C4 | — | 2 | 2 | no | — |
| 254 | C2×C4 | — | 2 | 1 | no | — |
| 255 | C2×C4 | — | 2 | 1 | no | — |
| 256 | C2×C4 | — | 4 | 1 | no | — |
| 257 | C2×C4 | — | 4 | 1 | no | — |
| 258 | C2×C4 | — | 4 | 2 | no | — |
| 259 | C2×C4 | — | 8 | 1 | no | — |
| 260 | C2×C4 | — | 8 | 1 | no | — |
| 261 | C2×C4 | — | 8 | 1 | no | — |
| 262 | C2×C4 | — | 16 | 1 | no | — |
| 263 | C2×C4 | — | 16 | 1 | no | — |
| 264 | C2×C4 | #386 | 1 | 1 | yes | characteristic |
| 265 | C2×C4 | C2×#101 | 1 | 1 | yes | characteristic |
| 266 | C2×C4 | #413 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | C2×#102 | 1 | 1 | yes | characteristic |
| 268 | C2×C4 | C2×#103 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | #416 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #447 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | #448 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | #480 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | #481 | 1 | 1 | yes | characteristic |
| 274 | C8 | — | 2 | 2 | no | — |
| 275 | C8 | — | 2 | 1 | no | — |
| 276 | C8 | — | 2 | 1 | no | — |
| 277 | C8 | — | 4 | 1 | no | — |
| 278 | C8 | — | 8 | 1 | no | — |
| 279 | C8 | — | 8 | 2 | no | — |
| 280 | C8 | — | 8 | 1 | no | — |
| 281 | C8 | — | 8 | 1 | no | — |
| 282 | C8 | — | 8 | 1 | no | — |
| 283 | C8 | #465 | 1 | 1 | yes | characteristic |
| 284 | C8 | #465 | 1 | 1 | yes | characteristic |
| 285 | C8 | #467 | 1 | 1 | yes | characteristic |
| 286 | C8 | #467 | 1 | 1 | yes | characteristic |
| 287 | C22 | — | 2 | 1 | no | — |
| 288 | C22 | — | 2 | 1 | no | — |
| 289 | C22 | — | 2 | 2 | no | — |
| 290 | C22 | — | 4 | 1 | no | — |
| 291 | C22 | — | 4 | 1 | no | — |
| 292 | C22 | #1550 | 1 | 1 | yes | characteristic |
| 293 | C22 | #1552 | 1 | 1 | yes | characteristic |
| 294 | C22 | #1874 | 1 | 1 | yes | characteristic |
| 295 | C4 | — | 2 | 2 | no | — |
| 296 | C4 | — | 2 | 1 | no | — |
| 297 | C4 | — | 2 | 1 | no | — |
| 298 | C4 | — | 2 | 1 | no | — |
| 299 | C4 | — | 2 | 1 | no | — |
| 300 | C4 | — | 4 | 1 | no | — |
| 301 | C4 | — | 4 | 1 | no | — |
| 302 | C4 | — | 16 | 1 | no | — |
| 303 | C4 | — | 16 | 1 | no | — |
| 304 | C4 | C2×#332 | 1 | 1 | yes | characteristic |
| 305 | C4 | #1540 | 1 | 1 | yes | characteristic |
| 306 | C4 | #1892 | 1 | 1 | yes | characteristic |
| 307 | C4 | #1901 | 1 | 1 | yes | characteristic |
| 308 | C2 | — | 2 | 1 | no | — |
| 309 | C2 | — | 2 | 1 | no | — |
| 310 | C2 | — | 4 | 2 | no | — |
| 311 | C2 | #9131 | 1 | 1 | yes | characteristic |
| 312 | C2 | #12995 | 1 | 1 | yes | characteristic |
| 313 | C2 | #13005 | 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