← #28315874
Subgroups — #28315874
Order 13568.
964 conjugacy classes of subgroups.
The subgroup lattice has 344 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20291633 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20291363 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #20290232 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #20290229 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #20290216 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | D53⋊#333 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #20288532 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#13133305 | — | 2 | 1 | no | — |
| 9 | Q8×#88324 | — | 2 | 1 | no | — |
| 10 | #16811651 | — | 2 | 3 | no | — |
| 11 | #16811641 | C22 | 1 | 3 | yes | — |
| 12 | #16811630 | — | 2 | 3 | no | — |
| 13 | #16811628 | C22 | 1 | 3 | yes | — |
| 14 | D53⋊(C4⋊C8) | C22 | 1 | 3 | yes | — |
| 15 | D53⋊(C4⋊C8) | C22 | 1 | 3 | yes | — |
| 16 | #16811483 | C22 | 1 | 3 | yes | — |
| 17 | #16811483 | C22 | 1 | 6 | yes | — |
| 18 | #16811483 | C22 | 1 | 3 | yes | — |
| 19 | #16811482 | C22 | 1 | 1 | yes | characteristic |
| 20 | #16811482 | C22 | 1 | 3 | yes | — |
| 21 | Q8×Dic106 | — | 2 | 1 | no | — |
| 22 | #16811352 | C4 | 1 | 1 | yes | characteristic |
| 23 | #16811329 | — | 2 | 3 | no | — |
| 24 | Q8×D212 | C4 | 1 | 3 | yes | — |
| 25 | #16811288 | C4 | 1 | 3 | yes | — |
| 26 | #16811253 | — | 2 | 3 | no | — |
| 27 | #16811177 | C22 | 1 | 3 | yes | — |
| 28 | C4⋊Q8×D53 | C22 | 1 | 3 | yes | — |
| 29 | #16811105 | — | 2 | 1 | no | — |
| 30 | C4×#11719949 | C22 | 1 | 1 | yes | characteristic |
| 31 | C53×#557 | C4 | 1 | 1 | yes | characteristic |
| 32 | #13133305 | — | 4 | 4 | no | — |
| 33 | C2×#11719936 | — | 2 | 3 | no | — |
| 34 | C2×D53⋊C8 | — | 2 | 3 | no | — |
| 35 | C2×D53⋊C8 | — | 2 | 3 | no | — |
| 36 | #13133284 | — | 2 | 3 | no | — |
| 37 | #13133284 | C23 | 1 | 1 | yes | characteristic |
| 38 | #13133284 | C23 | 1 | 3 | yes | — |
| 39 | #13133284 | C23 | 1 | 3 | yes | — |
| 40 | #13133284 | C23 | 1 | 1 | yes | characteristic |
| 41 | C4⋊#88324 | — | 2 | 3 | no | — |
| 42 | C4⋊#88324 | — | 2 | 3 | no | — |
| 43 | #13133274 | — | 2 | 3 | no | — |
| 44 | C4×#88324 | — | 2 | 3 | no | — |
| 45 | C2×#11719949 | D4 | 1 | 1 | yes | characteristic |
| 46 | C2×Q8×D53 | C2×C4 | 1 | 6 | yes | — |
| 47 | #13133217 | — | 2 | 3 | no | — |
| 48 | #13133217 | C2×C4 | 1 | 3 | yes | — |
| 49 | #13133216 | — | 2 | 3 | no | — |
| 50 | #13133216 | — | 2 | 3 | no | — |
| 51 | #13133216 | — | 2 | 3 | no | — |
| 52 | #13133216 | C23 | 1 | 1 | yes | characteristic |
| 53 | #13133215 | C23 | 1 | 3 | yes | — |
| 54 | #13133208 | — | 2 | 3 | no | — |
| 55 | #13133207 | — | 2 | 3 | no | — |
| 56 | #13133197 | — | 2 | 6 | no | — |
| 57 | #13133197 | C2×C4 | 1 | 3 | yes | — |
| 58 | #13133195 | — | 2 | 6 | no | — |
| 59 | #13133195 | — | 2 | 3 | no | — |
| 60 | #13133193 | C2×C4 | 1 | 3 | yes | — |
| 61 | Q8×Dic53 | — | 2 | 1 | no | — |
| 62 | Q8×Dic53 | D4 | 1 | 1 | yes | characteristic |
| 63 | C4×Dic106 | — | 2 | 3 | no | — |
| 64 | C4×Dic106 | C2×C4 | 1 | 3 | yes | — |
| 65 | C4×D212 | — | 2 | 3 | no | — |
| 66 | C4×D212 | C2×C4 | 1 | 3 | yes | — |
| 67 | #13133179 | — | 2 | 3 | no | — |
| 68 | #13133179 | D4 | 1 | 3 | yes | — |
| 69 | #13133175 | D4 | 1 | 3 | yes | — |
| 70 | #13133174 | — | 2 | 3 | no | — |
| 71 | C4⋊C4×D53 | — | 2 | 3 | no | — |
| 72 | C4⋊C4×D53 | — | 2 | 3 | no | — |
| 73 | C42×D53 | C23 | 1 | 3 | yes | — |
| 74 | C4⋊Q8×C53 | C2×C4 | 1 | 3 | yes | — |
| 75 | #125×C53 | — | 2 | 3 | no | — |
| 76 | C4×Q8×C53 | — | 2 | 1 | no | — |
| 77 | C4×Q8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 78 | C4×Q8×C53 | C2×C4 | 1 | 3 | yes | — |
| 79 | C2×Q8×C53 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 80 | C2×Q8×C53 | C22×C4 | 1 | 3 | yes | — |
| 81 | #11719949 | SD16 | 1 | 4 | yes | — |
| 82 | #11719949 | Q16 | 1 | 4 | yes | — |
| 83 | Q8×D53 | — | 8 | 6 | no | — |
| 84 | C2×D212 | C22⋊C4 | 1 | 3 | yes | — |
| 85 | C2×C4×D53 | — | 2 | 6 | no | — |
| 86 | C2×C4×D53 | C22×C4 | 1 | 3 | yes | — |
| 87 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 88 | C2×C4×D53 | C2×D4 | 1 | 3 | yes | — |
| 89 | C2×Dic106 | — | 2 | 1 | no | — |
| 90 | C2×Dic106 | — | 2 | 6 | no | — |
| 91 | C2×Dic106 | C22×C4 | 1 | 3 | yes | — |
| 92 | C2×#88324 | — | 2 | 1 | no | — |
| 93 | C2×#88324 | — | 2 | 3 | no | — |
| 94 | C2×#88324 | — | 2 | 1 | no | — |
| 95 | C2×#88324 | — | 2 | 3 | no | — |
| 96 | #11719936 | — | 4 | 6 | no | — |
| 97 | D53⋊C8 | — | 4 | 6 | no | — |
| 98 | D53⋊C8 | — | 8 | 3 | no | — |
| 99 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 100 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 101 | C4⋊C4×C53 | — | 2 | 3 | no | — |
| 102 | C4⋊C4×C53 | C22⋊C4 | 1 | 3 | yes | — |
| 103 | C4⋊C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 104 | C42×C53 | — | 2 | 3 | no | — |
| 105 | C42×C53 | C22×C4 | 1 | 3 | yes | — |
| 106 | #11719921 | — | 2 | 3 | no | — |
| 107 | #11719921 | — | 2 | 3 | no | — |
| 108 | #11719921 | C22⋊C4 | 1 | 6 | yes | — |
| 109 | C4⋊Dic53 | — | 2 | 3 | no | — |
| 110 | C4⋊Dic53 | — | 2 | 3 | no | — |
| 111 | C4⋊Dic53 | — | 2 | 3 | no | — |
| 112 | C4⋊Dic53 | C22⋊C4 | 1 | 3 | yes | — |
| 113 | C4⋊Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 114 | #11719919 | — | 2 | 3 | no | — |
| 115 | #11719919 | — | 2 | 3 | no | — |
| 116 | #11719919 | — | 2 | 3 | no | — |
| 117 | #11719919 | — | 2 | 3 | no | — |
| 118 | #11719919 | — | 2 | 1 | no | — |
| 119 | C4×Dic53 | — | 2 | 3 | no | — |
| 120 | C4×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 121 | C4×Dic53 | C2×D4 | 1 | 3 | yes | — |
| 122 | C4×Dic53 | C24 | 1 | 1 | yes | characteristic |
| 123 | C22×D53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 124 | Q8×C53 | — | 4 | 3 | no | — |
| 125 | Q8×C53 | #103 | 1 | 4 | yes | — |
| 126 | C2×C4×C53 | — | 2 | 3 | no | — |
| 127 | C2×C4×C53 | — | 2 | 1 | no | — |
| 128 | C2×C4×C53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 129 | C2×C4×C53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 130 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 131 | C2×Dic53 | — | 2 | 3 | no | — |
| 132 | C2×Dic53 | — | 2 | 1 | no | — |
| 133 | C2×Dic53 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 134 | C2×Dic53 | C22×D4 | 1 | 1 | yes | characteristic |
| 135 | C2×Dic53 | C22×D4 | 1 | 1 | yes | characteristic |
| 136 | D212 | #103 | 1 | 12 | yes | — |
| 137 | C4×D53 | — | 2 | 3 | no | — |
| 138 | C4×D53 | — | 2 | 3 | no | — |
| 139 | C4×D53 | — | 4 | 3 | no | — |
| 140 | C4×D53 | — | 8 | 6 | no | — |
| 141 | C4×D53 | C2×SD16 | 1 | 6 | yes | — |
| 142 | C4×D53 | C2×Q16 | 1 | 6 | yes | — |
| 143 | Dic106 | — | 4 | 1 | no | — |
| 144 | Dic106 | — | 4 | 1 | no | — |
| 145 | Dic106 | — | 4 | 3 | no | — |
| 146 | Dic106 | — | 8 | 6 | no | — |
| 147 | #88324 | — | 4 | 3 | no | — |
| 148 | #88324 | — | 4 | 1 | no | — |
| 149 | #88324 | — | 4 | 3 | no | — |
| 150 | #88324 | — | 4 | 1 | no | — |
| 151 | #17722 | — | 53 | 1 | no | maximal, Sylow |
| 152 | C22×C53 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 153 | C2×D53 | — | 2 | 3 | no | — |
| 154 | C2×D53 | — | 2 | 3 | no | — |
| 155 | C2×D53 | C2×#103 | 1 | 6 | yes | — |
| 156 | C4×C53 | — | 2 | 1 | no | — |
| 157 | C4×C53 | — | 2 | 3 | no | — |
| 158 | C4×C53 | — | 4 | 3 | no | — |
| 159 | C4×C53 | — | 4 | 1 | no | — |
| 160 | C4×C53 | C2×#103 | 1 | 6 | yes | — |
| 161 | Dic53 | — | 2 | 3 | no | — |
| 162 | Dic53 | — | 4 | 3 | no | — |
| 163 | Dic53 | — | 4 | 1 | no | — |
| 164 | Dic53 | C22×SD16 | 1 | 1 | yes | characteristic |
| 165 | Dic53 | C22×Q16 | 1 | 1 | yes | characteristic |
| 166 | Dic53 | #577 | 1 | 1 | yes | characteristic |
| 167 | Dic53 | #578 | 1 | 1 | yes | characteristic |
| 168 | #3503 | — | 53 | 1 | no | — |
| 169 | #2988 | — | 53 | 1 | no | — |
| 170 | #2906 | — | 53 | 1 | no | — |
| 171 | #1550 | — | 53 | 3 | no | — |
| 172 | #1548 | — | 53 | 3 | no | — |
| 173 | #1542 | — | 53 | 3 | no | — |
| 174 | C2×#333 | — | 53 | 3 | no | — |
| 175 | C2×C53 | #2888 | 1 | 1 | yes | characteristic |
| 176 | C2×C53 | C22×#103 | 1 | 1 | yes | characteristic |
| 177 | C2×C53 | #2898 | 1 | 1 | yes | characteristic |
| 178 | D53 | — | 2 | 6 | no | — |
| 179 | C2×#131 | — | 106 | 1 | no | — |
| 180 | Q82 | — | 106 | 1 | no | — |
| 181 | #557 | — | 53 | 1 | no | — |
| 182 | #557 | #6157 | 1 | 1 | yes | characteristic |
| 183 | #554 | — | 106 | 3 | no | — |
| 184 | D4×Q8 | — | 53 | 3 | no | — |
| 185 | D4×Q8 | — | 53 | 3 | no | — |
| 186 | #544 | — | 106 | 3 | no | — |
| 187 | #533 | — | 53 | 3 | no | — |
| 188 | C2×C4⋊Q8 | — | 53 | 3 | no | — |
| 189 | #520 | — | 106 | 1 | no | — |
| 190 | C4×#41 | — | 53 | 1 | no | — |
| 191 | C8×Q8 | — | 106 | 1 | no | — |
| 192 | C8×D4 | — | 106 | 3 | no | — |
| 193 | #431 | — | 53 | 3 | no | — |
| 194 | #424 | — | 106 | 3 | no | — |
| 195 | #423 | — | 53 | 3 | no | — |
| 196 | C2×C4⋊C8 | — | 53 | 3 | no | — |
| 197 | C2×C4⋊C8 | — | 53 | 3 | no | — |
| 198 | #333 | — | 53 | 3 | no | — |
| 199 | #333 | — | 53 | 6 | no | — |
| 200 | #333 | — | 53 | 3 | no | — |
| 201 | #332 | — | 53 | 1 | no | — |
| 202 | #332 | — | 53 | 3 | no | — |
| 203 | C53 | #17722 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 204 | C2×#41 | — | 53 | 1 | no | — |
| 205 | C22×Q8 | — | 53 | 6 | no | — |
| 206 | #131 | — | 212 | 4 | no | — |
| 207 | C2×M16 | — | 106 | 3 | no | — |
| 208 | C22×C8 | — | 106 | 3 | no | — |
| 209 | C22×C8 | — | 106 | 3 | no | — |
| 210 | C4⋊Q8 | — | 53 | 1 | no | — |
| 211 | C4⋊Q8 | — | 53 | 3 | no | — |
| 212 | C4⋊Q8 | — | 53 | 3 | no | — |
| 213 | C4⋊Q8 | — | 106 | 3 | no | — |
| 214 | C4⋊Q8 | — | 106 | 3 | no | — |
| 215 | C4⋊Q8 | — | 106 | 3 | no | — |
| 216 | C4⋊Q8 | — | 106 | 3 | no | — |
| 217 | C4⋊Q8 | C2×#6157 | 1 | 3 | yes | — |
| 218 | #125 | — | 2 | 3 | no | — |
| 219 | #125 | — | 106 | 3 | no | — |
| 220 | #125 | — | 106 | 3 | no | — |
| 221 | C22⋊Q8 | — | 53 | 3 | no | — |
| 222 | C22⋊Q8 | — | 53 | 3 | no | — |
| 223 | C22⋊Q8 | — | 106 | 6 | no | — |
| 224 | C22⋊Q8 | — | 106 | 6 | no | — |
| 225 | C22⋊Q8 | — | 106 | 3 | no | — |
| 226 | C4×Q8 | — | 2 | 1 | no | — |
| 227 | C4×Q8 | — | 53 | 1 | no | — |
| 228 | C4×Q8 | — | 53 | 3 | no | — |
| 229 | C4×Q8 | — | 106 | 3 | no | — |
| 230 | C4×Q8 | — | 106 | 1 | no | — |
| 231 | C4×Q8 | C2×#6157 | 1 | 3 | yes | — |
| 232 | C4×Q8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 233 | C4×D4 | — | 53 | 3 | no | — |
| 234 | C4×D4 | — | 53 | 3 | no | — |
| 235 | C4×D4 | — | 106 | 3 | no | — |
| 236 | C4×D4 | — | 106 | 3 | no | — |
| 237 | #117 | — | 53 | 3 | no | — |
| 238 | #117 | — | 106 | 3 | no | — |
| 239 | C2×C4⋊C4 | — | 106 | 3 | no | — |
| 240 | C2×C4⋊C4 | — | 106 | 3 | no | — |
| 241 | C2×C42 | — | 53 | 3 | no | — |
| 242 | C4⋊C8 | — | 53 | 1 | no | — |
| 243 | C4⋊C8 | — | 53 | 3 | no | — |
| 244 | C4⋊C8 | — | 53 | 3 | no | — |
| 245 | C4⋊C8 | — | 53 | 1 | no | — |
| 246 | C4⋊C8 | — | 106 | 3 | no | — |
| 247 | C4⋊C8 | — | 106 | 3 | no | — |
| 248 | C4⋊C8 | — | 106 | 3 | no | — |
| 249 | C22⋊C8 | — | 106 | 3 | no | — |
| 250 | C4×C8 | — | 106 | 3 | no | — |
| 251 | #41 | — | 53 | 4 | no | — |
| 252 | #41 | — | 53 | 4 | no | — |
| 253 | C2×Q8 | — | 53 | 3 | no | — |
| 254 | C2×Q8 | — | 106 | 6 | no | — |
| 255 | C2×Q8 | — | 106 | 1 | no | — |
| 256 | C2×Q8 | — | 424 | 6 | no | — |
| 257 | C2×Q8 | C22⋊#6157 | 1 | 1 | yes | characteristic |
| 258 | C2×Q8 | C22×#6157 | 1 | 3 | yes | — |
| 259 | C2×D4 | — | 53 | 3 | no | — |
| 260 | C22×C4 | — | 53 | 3 | no | — |
| 261 | C22×C4 | — | 53 | 3 | no | — |
| 262 | C22×C4 | — | 53 | 3 | no | — |
| 263 | C22×C4 | — | 106 | 6 | no | — |
| 264 | M16 | — | 212 | 6 | no | — |
| 265 | C2×C8 | — | 106 | 1 | no | — |
| 266 | C2×C8 | — | 106 | 3 | no | — |
| 267 | C2×C8 | — | 106 | 1 | no | — |
| 268 | C2×C8 | — | 106 | 3 | no | — |
| 269 | C2×C8 | — | 424 | 3 | no | — |
| 270 | C2×C8 | — | 212 | 6 | no | — |
| 271 | C4⋊C4 | — | 2 | 3 | no | — |
| 272 | C4⋊C4 | — | 2 | 3 | no | — |
| 273 | C4⋊C4 | — | 2 | 3 | no | — |
| 274 | C4⋊C4 | — | 53 | 3 | no | — |
| 275 | C4⋊C4 | — | 53 | 1 | no | — |
| 276 | C4⋊C4 | — | 106 | 3 | no | — |
| 277 | C4⋊C4 | — | 106 | 3 | no | — |
| 278 | C4⋊C4 | — | 106 | 1 | no | — |
| 279 | C4⋊C4 | — | 106 | 3 | no | — |
| 280 | C4⋊C4 | — | 106 | 3 | no | — |
| 281 | C4⋊C4 | — | 106 | 3 | no | — |
| 282 | C4⋊C4 | — | 106 | 3 | no | — |
| 283 | C4⋊C4 | — | 106 | 3 | no | — |
| 284 | C4⋊C4 | C22⋊#6157 | 1 | 3 | yes | — |
| 285 | C4⋊C4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 286 | C22⋊C4 | — | 53 | 6 | no | — |
| 287 | C22⋊C4 | — | 106 | 3 | no | — |
| 288 | C22⋊C4 | — | 106 | 3 | no | — |
| 289 | C42 | — | 2 | 3 | no | — |
| 290 | C42 | — | 53 | 3 | no | — |
| 291 | C42 | — | 53 | 3 | no | — |
| 292 | C42 | — | 53 | 1 | no | — |
| 293 | C42 | — | 106 | 3 | no | — |
| 294 | C42 | C22×#6157 | 1 | 3 | yes | — |
| 295 | C23 | — | 53 | 3 | no | — |
| 296 | Q8 | — | 4 | 3 | no | — |
| 297 | Q8 | — | 424 | 6 | no | — |
| 298 | Q8 | — | 212 | 3 | no | — |
| 299 | Q8 | — | 212 | 1 | no | — |
| 300 | Q8 | — | 212 | 1 | no | — |
| 301 | Q8 | #13133280 | 1 | 4 | yes | — |
| 302 | D4 | — | 53 | 12 | no | — |
| 303 | C2×C4 | — | 2 | 1 | no | — |
| 304 | C2×C4 | — | 2 | 3 | no | — |
| 305 | C2×C4 | — | 53 | 1 | no | — |
| 306 | C2×C4 | — | 53 | 1 | no | — |
| 307 | C2×C4 | — | 53 | 6 | no | — |
| 308 | C2×C4 | — | 53 | 6 | no | — |
| 309 | C2×C4 | — | 53 | 3 | no | — |
| 310 | C2×C4 | — | 106 | 3 | no | — |
| 311 | C2×C4 | — | 106 | 3 | no | — |
| 312 | C2×C4 | — | 106 | 1 | no | — |
| 313 | C2×C4 | — | 106 | 3 | no | — |
| 314 | C2×C4 | — | 424 | 6 | no | — |
| 315 | C2×C4 | — | 212 | 3 | no | — |
| 316 | C2×C4 | C2×C22⋊#6157 | 1 | 3 | yes | — |
| 317 | C2×C4 | C2×C22⋊#6157 | 1 | 3 | yes | — |
| 318 | C2×C4 | C23×#6157 | 1 | 1 | yes | characteristic |
| 319 | C8 | — | 212 | 1 | no | — |
| 320 | C8 | — | 212 | 3 | no | — |
| 321 | C8 | — | 212 | 1 | no | — |
| 322 | C8 | — | 212 | 3 | no | — |
| 323 | C22 | — | 53 | 6 | no | — |
| 324 | C22 | — | 106 | 3 | no | — |
| 325 | C22 | — | 106 | 3 | no | — |
| 326 | C22 | C22×C22⋊#6157 | 1 | 1 | yes | characteristic |
| 327 | C4 | — | 2 | 3 | no | — |
| 328 | C4 | — | 2 | 1 | no | — |
| 329 | C4 | — | 4 | 1 | no | — |
| 330 | C4 | — | 4 | 3 | no | — |
| 331 | C4 | — | 53 | 1 | no | — |
| 332 | C4 | — | 53 | 1 | no | — |
| 333 | C4 | — | 53 | 1 | no | — |
| 334 | C4 | — | 53 | 1 | no | — |
| 335 | C4 | — | 106 | 3 | no | — |
| 336 | C4 | — | 212 | 1 | no | — |
| 337 | C4 | — | 212 | 3 | no | — |
| 338 | C4 | C2×#13133280 | 1 | 6 | yes | — |
| 339 | C2 | — | 106 | 6 | no | — |
| 340 | C2 | #20291326 | 1 | 1 | yes | characteristic |
| 341 | C2 | C22×#13133280 | 1 | 1 | yes | characteristic |
| 342 | C2 | #20291337 | 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