← #28317494
Subgroups — #28317494
Order 13568.
656 conjugacy classes of subgroups.
The subgroup lattice has 340 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20291552 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20290355 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #20290320 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | D8×#88324 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #20290277 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #20290046 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C8×#11719940 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #135×Dic53 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #16811674 | C22 | 1 | 4 | yes | — |
| 10 | #16811654 | — | 2 | 4 | no | — |
| 11 | D4×#88324 | C22 | 1 | 4 | yes | — |
| 12 | #16811644 | C22 | 1 | 2 | yes | — |
| 13 | #16811643 | — | 2 | 2 | no | — |
| 14 | #16811640 | C22 | 1 | 1 | yes | characteristic |
| 15 | C4×#11719940 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#13133272 | — | 2 | 2 | no | — |
| 17 | #16811486 | C22 | 1 | 2 | yes | — |
| 18 | #16811485 | C22 | 1 | 2 | yes | — |
| 19 | Q8⋊#88324 | C22 | 1 | 4 | yes | — |
| 20 | D4⋊#88324 | C22 | 1 | 4 | yes | — |
| 21 | #88324⋊C8 | C22 | 1 | 2 | yes | — |
| 22 | C8×#88324 | C22 | 1 | 2 | yes | — |
| 23 | #41×Dic53 | C22 | 1 | 2 | yes | — |
| 24 | Q16×Dic53 | C4 | 1 | 1 | yes | characteristic |
| 25 | Q16×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 26 | SD16×Dic53 | C4 | 1 | 2 | yes | — |
| 27 | SD16×Dic53 | C22 | 1 | 2 | yes | — |
| 28 | D8×Dic53 | C4 | 1 | 1 | yes | characteristic |
| 29 | D8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 30 | #16810661 | C4 | 1 | 1 | yes | characteristic |
| 31 | #16810619 | C4 | 1 | 2 | yes | — |
| 32 | C2×C8×Dic53 | C22 | 1 | 1 | yes | characteristic |
| 33 | C2×#135×C53 | C4 | 1 | 1 | yes | characteristic |
| 34 | C2×#11719940 | D4 | 1 | 2 | yes | — |
| 35 | C22×#88324 | — | 2 | 4 | no | — |
| 36 | #13133284 | — | 2 | 1 | no | — |
| 37 | #13133284 | — | 2 | 4 | no | — |
| 38 | C4⋊#88324 | C23 | 1 | 4 | yes | — |
| 39 | C22⋊#88324 | — | 2 | 2 | no | — |
| 40 | C22⋊#88324 | — | 2 | 1 | no | — |
| 41 | C22⋊#88324 | — | 2 | 4 | no | — |
| 42 | #13133272 | — | 2 | 2 | no | — |
| 43 | #13133272 | — | 2 | 4 | no | — |
| 44 | #13133272 | D4 | 1 | 1 | yes | characteristic |
| 45 | #13133272 | D4 | 1 | 1 | yes | characteristic |
| 46 | C4×#88324 | C23 | 1 | 2 | yes | — |
| 47 | C4×#88324 | C23 | 1 | 2 | yes | — |
| 48 | C22×#88322 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | Q8×Dic53 | C2×C4 | 1 | 2 | yes | — |
| 50 | Q8×Dic53 | C23 | 1 | 2 | yes | — |
| 51 | D4×Dic53 | — | 2 | 2 | no | — |
| 52 | D4×Dic53 | — | 2 | 2 | no | — |
| 53 | D4×Dic53 | C2×C4 | 1 | 2 | yes | — |
| 54 | D4×Dic53 | C23 | 1 | 2 | yes | — |
| 55 | #13133176 | — | 2 | 2 | no | — |
| 56 | #13133176 | C2×C4 | 1 | 2 | yes | — |
| 57 | C2×C4×Dic53 | — | 2 | 2 | no | — |
| 58 | C2×C4×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 59 | #13133151 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | #13133151 | C2×C4 | 1 | 1 | yes | characteristic |
| 61 | #13133149 | C2×C4 | 1 | 2 | yes | — |
| 62 | #13133141 | C2×C4 | 1 | 2 | yes | — |
| 63 | #13133141 | C2×C4 | 1 | 2 | yes | — |
| 64 | #13133138 | C2×C4 | 1 | 2 | yes | — |
| 65 | #13133138 | C2×C4 | 1 | 2 | yes | — |
| 66 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 67 | C8×Dic53 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | C8×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic53 | C23 | 1 | 1 | yes | characteristic |
| 70 | C2×#41×C53 | C2×C4 | 1 | 2 | yes | — |
| 71 | #135×C53 | — | 2 | 4 | no | — |
| 72 | C2×Q16×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 73 | C2×SD16×C53 | C2×C4 | 1 | 2 | yes | — |
| 74 | C2×D8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 75 | C22×C8×C53 | C2×C4 | 1 | 1 | yes | characteristic |
| 76 | #41×C53 | — | 2 | 4 | no | — |
| 77 | #41×C53 | — | 4 | 2 | no | — |
| 78 | C2×Q8×C53 | C22×C4 | 1 | 2 | yes | — |
| 79 | C2×D4×C53 | — | 2 | 2 | no | — |
| 80 | C2×D4×C53 | C22×C4 | 1 | 2 | yes | — |
| 81 | C22×C4×C53 | — | 2 | 2 | no | — |
| 82 | C22×C4×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C22×Dic53 | — | 2 | 2 | no | — |
| 84 | C22×Dic53 | — | 2 | 2 | no | — |
| 85 | C22×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C22×Dic53 | #41 | 1 | 1 | yes | characteristic |
| 87 | #11719940 | — | 2 | 4 | no | — |
| 88 | C2×#88324 | — | 2 | 4 | no | — |
| 89 | C2×#88324 | — | 2 | 2 | no | — |
| 90 | C2×#88324 | — | 4 | 8 | no | — |
| 91 | C2×#88324 | C2×D4 | 1 | 2 | yes | — |
| 92 | C2×#88324 | C2×D4 | 1 | 2 | yes | — |
| 93 | Q16×C53 | — | 2 | 2 | no | — |
| 94 | SD16×C53 | — | 2 | 4 | no | — |
| 95 | D8×C53 | M16 | 1 | 4 | yes | complemented |
| 96 | C2×C8×C53 | — | 2 | 2 | no | — |
| 97 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 98 | C2×C8×C53 | C22×C4 | 1 | 1 | yes | characteristic |
| 99 | C22⋊Dic53 | — | 2 | 2 | no | — |
| 100 | C22⋊Dic53 | — | 2 | 2 | no | — |
| 101 | C22⋊Dic53 | — | 2 | 2 | no | — |
| 102 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 103 | C4⋊Dic53 | — | 2 | 2 | no | — |
| 104 | C4⋊Dic53 | C22×C4 | 1 | 2 | yes | — |
| 105 | C4⋊Dic53 | C22×C4 | 1 | 2 | yes | — |
| 106 | C4×Dic53 | — | 2 | 2 | no | — |
| 107 | C4×Dic53 | — | 2 | 2 | no | — |
| 108 | C4×Dic53 | — | 2 | 2 | no | — |
| 109 | C4×Dic53 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C4×Dic53 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C4×Dic53 | #41 | 1 | 1 | yes | characteristic |
| 112 | C4×Dic53 | C24 | 1 | 1 | yes | characteristic |
| 113 | C2×#88322 | — | 4 | 1 | no | — |
| 114 | C2×#88322 | C22×C4 | 1 | 1 | yes | characteristic |
| 115 | C2×#88322 | C22×C4 | 1 | 1 | yes | characteristic |
| 116 | C23×C53 | — | 2 | 2 | no | — |
| 117 | C23×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 118 | Q8×C53 | — | 2 | 2 | no | — |
| 119 | Q8×C53 | — | 2 | 2 | no | — |
| 120 | D4×C53 | — | 2 | 2 | no | — |
| 121 | D4×C53 | — | 4 | 4 | no | — |
| 122 | D4×C53 | C2×M16 | 1 | 4 | yes | — |
| 123 | C2×C4×C53 | — | 2 | 1 | no | — |
| 124 | C2×C4×C53 | — | 2 | 1 | no | — |
| 125 | C2×C4×C53 | — | 2 | 2 | no | — |
| 126 | C2×C4×C53 | — | 4 | 4 | no | — |
| 127 | C2×C4×C53 | C4×D4 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C53 | C23×C4 | 1 | 1 | yes | characteristic |
| 129 | C2×Dic53 | — | 2 | 2 | no | — |
| 130 | C2×Dic53 | — | 2 | 2 | no | — |
| 131 | C2×Dic53 | — | 2 | 2 | no | — |
| 132 | C2×Dic53 | — | 4 | 1 | no | — |
| 133 | C2×Dic53 | — | 4 | 4 | no | — |
| 134 | C2×Dic53 | — | 8 | 2 | no | — |
| 135 | C2×Dic53 | C4×D4 | 1 | 1 | yes | characteristic |
| 136 | C2×Dic53 | C4×D4 | 1 | 1 | yes | characteristic |
| 137 | C2×Dic53 | C22×D4 | 1 | 1 | yes | characteristic |
| 138 | C2×Dic53 | C2×#41 | 1 | 1 | yes | characteristic |
| 139 | #88324 | — | 2 | 2 | no | — |
| 140 | #88324 | — | 4 | 2 | no | — |
| 141 | #88324 | — | 4 | 4 | no | — |
| 142 | #88324 | #135 | 1 | 4 | yes | complemented |
| 143 | C8×C53 | — | 2 | 1 | no | — |
| 144 | C8×C53 | C2×M16 | 1 | 2 | yes | — |
| 145 | #88322 | — | 2 | 1 | no | — |
| 146 | #88322 | — | 2 | 1 | no | — |
| 147 | #18035 | — | 53 | 1 | no | maximal, Sylow |
| 148 | C22×C53 | — | 2 | 1 | no | — |
| 149 | C22×C53 | — | 2 | 4 | no | — |
| 150 | C22×C53 | — | 2 | 1 | no | — |
| 151 | C22×C53 | — | 2 | 1 | no | — |
| 152 | C22×C53 | — | 4 | 2 | no | — |
| 153 | C22×C53 | — | 4 | 2 | no | — |
| 154 | C22×C53 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 155 | C4×C53 | — | 2 | 1 | no | — |
| 156 | C4×C53 | — | 4 | 2 | no | — |
| 157 | C4×C53 | C22×M16 | 1 | 1 | yes | characteristic |
| 158 | C4×C53 | #568 | 1 | 1 | yes | characteristic |
| 159 | Dic53 | — | 2 | 1 | no | — |
| 160 | Dic53 | — | 2 | 1 | no | — |
| 161 | Dic53 | — | 2 | 1 | no | — |
| 162 | Dic53 | — | 4 | 2 | no | — |
| 163 | Dic53 | — | 4 | 2 | no | — |
| 164 | Dic53 | C2×#135 | 1 | 2 | yes | — |
| 165 | #2970 | — | 53 | 2 | no | — |
| 166 | C4×#135 | — | 53 | 1 | no | — |
| 167 | #1591 | — | 53 | 2 | no | — |
| 168 | #1582 | — | 53 | 4 | no | — |
| 169 | C8×D8 | — | 53 | 2 | no | — |
| 170 | #1568 | — | 53 | 1 | no | — |
| 171 | #1488 | — | 53 | 2 | no | — |
| 172 | C8×M16 | — | 53 | 1 | no | — |
| 173 | C2×C53 | — | 4 | 1 | no | — |
| 174 | C2×C53 | — | 4 | 4 | no | — |
| 175 | C2×C53 | D4×M16 | 1 | 1 | yes | characteristic |
| 176 | C2×C53 | C4×#135 | 1 | 1 | yes | characteristic |
| 177 | C2×C53 | #2957 | 1 | 1 | yes | characteristic |
| 178 | C2×#135 | #6157 | 1 | 1 | yes | characteristic |
| 179 | C4×#41 | — | 53 | 2 | no | — |
| 180 | #446 | — | 53 | 4 | no | — |
| 181 | C4×Q16 | — | 53 | 1 | no | — |
| 182 | C4×Q16 | — | 53 | 1 | no | — |
| 183 | C4×SD16 | — | 53 | 2 | no | — |
| 184 | C4×SD16 | — | 53 | 2 | no | — |
| 185 | C4×D8 | — | 53 | 1 | no | — |
| 186 | C4×D8 | — | 53 | 1 | no | — |
| 187 | #435 | — | 106 | 4 | no | — |
| 188 | C8×D4 | — | 53 | 4 | no | — |
| 189 | #432 | — | 53 | 2 | no | — |
| 190 | #432 | — | 106 | 2 | no | — |
| 191 | #431 | — | 53 | 1 | no | — |
| 192 | #427 | — | 53 | 1 | no | — |
| 193 | #416 | — | 53 | 2 | no | — |
| 194 | C4×M16 | — | 53 | 1 | no | — |
| 195 | C2×#97 | — | 106 | 2 | no | — |
| 196 | C2×C4×C8 | — | 53 | 1 | no | — |
| 197 | #335 | — | 53 | 2 | no | — |
| 198 | #334 | — | 53 | 2 | no | — |
| 199 | Q8⋊C8 | — | 53 | 4 | no | — |
| 200 | D4⋊C8 | — | 53 | 4 | no | — |
| 201 | #322 | — | 53 | 2 | no | — |
| 202 | C82 | — | 53 | 2 | no | — |
| 203 | C53 | #18035 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 204 | C2×#41 | C2×#6157 | 1 | 2 | yes | — |
| 205 | #135 | — | 2 | 4 | no | — |
| 206 | C2×Q16 | C2×#6157 | 1 | 1 | yes | characteristic |
| 207 | C2×SD16 | C2×#6157 | 1 | 2 | yes | — |
| 208 | C2×D8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 209 | C2×M16 | — | 53 | 2 | no | — |
| 210 | C22×C8 | — | 53 | 1 | no | — |
| 211 | C22×C8 | — | 106 | 4 | no | — |
| 212 | C22×C8 | C2×#6157 | 1 | 1 | yes | characteristic |
| 213 | C4×Q8 | — | 53 | 2 | no | — |
| 214 | C4×Q8 | — | 53 | 2 | no | — |
| 215 | C4×D4 | — | 53 | 2 | no | — |
| 216 | C4×D4 | — | 53 | 2 | no | — |
| 217 | C4×D4 | — | 106 | 2 | no | — |
| 218 | C4×D4 | — | 106 | 2 | no | — |
| 219 | #117 | — | 53 | 2 | no | — |
| 220 | #117 | — | 106 | 2 | no | — |
| 221 | C2×C42 | — | 53 | 1 | no | — |
| 222 | C2×C42 | — | 106 | 2 | no | — |
| 223 | #107 | — | 53 | 1 | no | — |
| 224 | #107 | — | 53 | 1 | no | — |
| 225 | #106 | — | 53 | 2 | no | — |
| 226 | C4⋊C8 | — | 53 | 4 | no | — |
| 227 | C4⋊C8 | — | 106 | 1 | no | — |
| 228 | C4⋊C8 | — | 106 | 4 | no | — |
| 229 | #103 | — | 53 | 2 | no | — |
| 230 | #103 | — | 53 | 2 | no | — |
| 231 | #102 | — | 53 | 2 | no | — |
| 232 | #102 | — | 53 | 2 | no | — |
| 233 | C22⋊C8 | — | 106 | 4 | no | — |
| 234 | C22⋊C8 | — | 106 | 1 | no | — |
| 235 | C22⋊C8 | — | 106 | 2 | no | — |
| 236 | #97 | — | 53 | 1 | no | — |
| 237 | #97 | — | 53 | 1 | no | — |
| 238 | #97 | — | 106 | 2 | no | — |
| 239 | #97 | — | 106 | 4 | no | — |
| 240 | C4×C8 | — | 53 | 2 | no | — |
| 241 | C4×C8 | — | 53 | 1 | no | — |
| 242 | C4×C8 | — | 53 | 1 | no | — |
| 243 | C4×C8 | — | 53 | 1 | no | — |
| 244 | C4×C8 | — | 53 | 1 | no | — |
| 245 | C4×C8 | — | 53 | 2 | no | — |
| 246 | #41 | — | 2 | 4 | no | — |
| 247 | #41 | — | 4 | 2 | no | — |
| 248 | C2×Q8 | C22×#6157 | 1 | 2 | yes | — |
| 249 | C2×D4 | — | 2 | 2 | no | — |
| 250 | C2×D4 | C22×#6157 | 1 | 2 | yes | — |
| 251 | C22×C4 | — | 2 | 2 | no | — |
| 252 | C22×C4 | — | 53 | 1 | no | — |
| 253 | C22×C4 | — | 53 | 1 | no | — |
| 254 | C22×C4 | — | 106 | 2 | no | — |
| 255 | C22×C4 | — | 106 | 2 | no | — |
| 256 | C22×C4 | C22×#6157 | 1 | 1 | yes | characteristic |
| 257 | Q16 | — | 2 | 2 | no | — |
| 258 | SD16 | — | 2 | 4 | no | — |
| 259 | D8 | #11719940 | 1 | 4 | yes | complemented |
| 260 | M16 | — | 106 | 4 | no | — |
| 261 | C2×C8 | — | 2 | 2 | no | — |
| 262 | C2×C8 | — | 53 | 2 | no | — |
| 263 | C2×C8 | — | 53 | 2 | no | — |
| 264 | C2×C8 | — | 53 | 1 | no | — |
| 265 | C2×C8 | — | 53 | 1 | no | — |
| 266 | C2×C8 | — | 106 | 2 | no | — |
| 267 | C2×C8 | — | 106 | 4 | no | — |
| 268 | C2×C8 | — | 212 | 8 | no | — |
| 269 | C2×C8 | — | 212 | 1 | no | — |
| 270 | C2×C8 | C22×#6157 | 1 | 1 | yes | characteristic |
| 271 | C2×C8 | C22×#6157 | 1 | 1 | yes | characteristic |
| 272 | C4⋊C4 | — | 53 | 2 | no | — |
| 273 | C4⋊C4 | — | 53 | 2 | no | — |
| 274 | C4⋊C4 | — | 106 | 2 | no | — |
| 275 | C4⋊C4 | — | 106 | 2 | no | — |
| 276 | C22⋊C4 | — | 106 | 2 | no | — |
| 277 | C22⋊C4 | — | 106 | 2 | no | — |
| 278 | C22⋊C4 | — | 106 | 2 | no | — |
| 279 | C42 | — | 53 | 1 | no | — |
| 280 | C42 | — | 53 | 1 | no | — |
| 281 | C42 | — | 53 | 1 | no | — |
| 282 | C42 | — | 53 | 1 | no | — |
| 283 | C42 | — | 106 | 2 | no | — |
| 284 | C42 | — | 106 | 2 | no | — |
| 285 | C42 | — | 106 | 2 | no | — |
| 286 | C23 | — | 2 | 2 | no | — |
| 287 | C23 | D4×#6157 | 1 | 1 | yes | characteristic |
| 288 | Q8 | — | 2 | 2 | no | — |
| 289 | Q8 | — | 2 | 2 | no | — |
| 290 | D4 | — | 2 | 2 | no | — |
| 291 | D4 | — | 4 | 4 | no | — |
| 292 | D4 | C2×#11719940 | 1 | 4 | yes | — |
| 293 | C2×C4 | — | 2 | 1 | no | — |
| 294 | C2×C4 | — | 2 | 1 | no | — |
| 295 | C2×C4 | — | 2 | 2 | no | — |
| 296 | C2×C4 | — | 4 | 4 | no | — |
| 297 | C2×C4 | — | 53 | 1 | no | — |
| 298 | C2×C4 | — | 53 | 1 | no | — |
| 299 | C2×C4 | — | 53 | 1 | no | — |
| 300 | C2×C4 | — | 53 | 1 | no | — |
| 301 | C2×C4 | — | 106 | 2 | no | — |
| 302 | C2×C4 | — | 106 | 2 | no | — |
| 303 | C2×C4 | — | 106 | 2 | no | — |
| 304 | C2×C4 | — | 424 | 2 | no | — |
| 305 | C2×C4 | — | 212 | 4 | no | — |
| 306 | C2×C4 | — | 212 | 1 | no | — |
| 307 | C2×C4 | D4×#6157 | 1 | 1 | yes | characteristic |
| 308 | C2×C4 | C23×#6157 | 1 | 1 | yes | characteristic |
| 309 | C8 | — | 2 | 1 | no | — |
| 310 | C8 | — | 53 | 4 | no | — |
| 311 | C8 | — | 106 | 2 | no | — |
| 312 | C8 | — | 106 | 1 | no | — |
| 313 | C8 | — | 106 | 1 | no | — |
| 314 | C8 | — | 212 | 4 | no | — |
| 315 | C8 | — | 212 | 2 | no | — |
| 316 | C8 | C2×#11719940 | 1 | 2 | yes | — |
| 317 | C22 | — | 2 | 1 | no | — |
| 318 | C22 | — | 2 | 4 | no | — |
| 319 | C22 | — | 2 | 1 | no | — |
| 320 | C22 | — | 2 | 1 | no | — |
| 321 | C22 | — | 4 | 2 | no | — |
| 322 | C22 | — | 4 | 2 | no | — |
| 323 | C22 | C2×D4×#6157 | 1 | 1 | yes | characteristic |
| 324 | C4 | — | 2 | 1 | no | — |
| 325 | C4 | — | 4 | 2 | no | — |
| 326 | C4 | — | 53 | 2 | no | — |
| 327 | C4 | — | 106 | 1 | no | — |
| 328 | C4 | — | 106 | 1 | no | — |
| 329 | C4 | — | 106 | 1 | no | — |
| 330 | C4 | — | 212 | 2 | no | — |
| 331 | C4 | — | 212 | 2 | no | — |
| 332 | C4 | C22×#11719940 | 1 | 1 | yes | characteristic |
| 333 | C4 | #16811701 | 1 | 1 | yes | characteristic |
| 334 | C2 | — | 4 | 1 | no | — |
| 335 | C2 | — | 4 | 4 | no | — |
| 336 | C2 | D4×#11719940 | 1 | 1 | yes | characteristic |
| 337 | C2 | #135×#6157 | 1 | 1 | yes | characteristic |
| 338 | C2 | #20291492 | 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