← #36469710
Subgroups — #36469710
Order 22784.
422 conjugacy classes of subgroups.
The subgroup lattice has 274 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#19151705 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #26021972 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#19151375 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #26020139 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #26018358 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #26018352 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×#362×C89 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×Q8×Dic89 | — | 2 | 1 | no | — |
| 9 | C2×#15667952 | C22 | 1 | 1 | yes | characteristic |
| 10 | #19151705 | — | 2 | 2 | no | — |
| 11 | #19151705 | C4 | 1 | 4 | yes | complemented |
| 12 | #19151636 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#15667944 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#15667936 | — | 2 | 1 | no | — |
| 15 | #19151556 | — | 2 | 1 | no | — |
| 16 | C2×#15667922 | C22 | 1 | 1 | yes | characteristic |
| 17 | #19151402 | C4 | 1 | 4 | yes | complemented |
| 18 | #19151375 | — | 2 | 2 | no | — |
| 19 | #19151359 | — | 2 | 1 | no | — |
| 20 | C22×Q16×C89 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×M32×C89 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×#108×C89 | C22 | 1 | 1 | yes | characteristic |
| 23 | #362×C89 | C4 | 1 | 4 | yes | complemented |
| 24 | C22×#10623494 | D4 | 1 | 1 | yes | characteristic |
| 25 | Q8×Dic89 | — | 2 | 4 | no | — |
| 26 | Q8×Dic89 | — | 4 | 2 | no | — |
| 27 | #15667971 | — | 2 | 1 | no | — |
| 28 | C2×C4⋊Dic89 | — | 2 | 1 | no | — |
| 29 | C2×C4⋊Dic89 | — | 4 | 1 | no | — |
| 30 | C2×C4×Dic89 | — | 4 | 1 | no | — |
| 31 | C2×C4×Dic89 | D4 | 1 | 1 | yes | characteristic |
| 32 | #15667952 | — | 2 | 1 | no | — |
| 33 | #15667952 | — | 2 | 1 | no | — |
| 34 | C2×#13083549 | C2×C4 | 1 | 2 | yes | — |
| 35 | #15667946 | C2×C4 | 1 | 2 | yes | — |
| 36 | #15667944 | — | 2 | 1 | no | — |
| 37 | #15667944 | — | 2 | 1 | no | — |
| 38 | #15667944 | C2×C4 | 1 | 2 | yes | — |
| 39 | #15667936 | — | 2 | 4 | no | — |
| 40 | #15667922 | — | 4 | 1 | no | — |
| 41 | #15667922 | C2×C4 | 1 | 2 | yes | — |
| 42 | #15667922 | D4 | 1 | 2 | yes | — |
| 43 | C8×Dic89 | — | 4 | 1 | no | — |
| 44 | C22×Q8×C89 | — | 2 | 1 | no | — |
| 45 | C2×Q16×C89 | — | 2 | 1 | no | — |
| 46 | C2×Q16×C89 | — | 2 | 1 | no | — |
| 47 | C2×Q16×C89 | — | 2 | 1 | no | — |
| 48 | C2×Q16×C89 | C2×C4 | 1 | 2 | yes | — |
| 49 | C2×Q16×C89 | C2×C4 | 1 | 2 | yes | — |
| 50 | C2×Q16×C89 | C2×C4 | 1 | 2 | yes | — |
| 51 | C2×M16×C89 | — | 2 | 1 | no | — |
| 52 | C22×C8×C89 | C23 | 1 | 1 | yes | characteristic |
| 53 | M32×C89 | — | 2 | 1 | no | — |
| 54 | M32×C89 | C2×C4 | 1 | 2 | yes | — |
| 55 | C2×C16×C89 | — | 2 | 1 | no | — |
| 56 | #108×C89 | — | 2 | 1 | no | — |
| 57 | #108×C89 | C2×C4 | 1 | 2 | yes | — |
| 58 | C22×Dic89 | — | 2 | 1 | no | — |
| 59 | C22×Dic89 | — | 4 | 1 | no | — |
| 60 | C2×#10623494 | — | 2 | 1 | no | — |
| 61 | C2×#10623494 | — | 2 | 1 | no | — |
| 62 | C2×#10623494 | C22⋊C4 | 1 | 2 | yes | — |
| 63 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 64 | C4⋊Dic89 | — | 2 | 2 | no | — |
| 65 | C4⋊Dic89 | — | 4 | 4 | no | — |
| 66 | C4⋊Dic89 | — | 4 | 1 | no | — |
| 67 | C22⋊Dic89 | — | 4 | 1 | no | — |
| 68 | C4×Dic89 | — | 2 | 1 | no | — |
| 69 | C4×Dic89 | — | 4 | 4 | no | — |
| 70 | C4×Dic89 | — | 4 | 1 | no | — |
| 71 | C4×Dic89 | C22⋊C4 | 1 | 2 | yes | — |
| 72 | #13083549 | — | 2 | 2 | no | — |
| 73 | C2×Q8×C89 | — | 2 | 2 | no | — |
| 74 | C2×Q8×C89 | — | 2 | 2 | no | — |
| 75 | C2×Q8×C89 | — | 2 | 2 | no | — |
| 76 | C2×Q8×C89 | — | 4 | 1 | no | — |
| 77 | C2×Q8×C89 | — | 4 | 1 | no | — |
| 78 | C2×Q8×C89 | — | 4 | 1 | no | — |
| 79 | C22×C4×C89 | — | 4 | 1 | no | — |
| 80 | C22×C4×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 81 | Q16×C89 | — | 2 | 2 | no | — |
| 82 | Q16×C89 | — | 2 | 2 | no | — |
| 83 | Q16×C89 | — | 2 | 2 | no | — |
| 84 | Q16×C89 | C42 | 1 | 4 | yes | — |
| 85 | M16×C89 | — | 2 | 2 | no | — |
| 86 | M16×C89 | — | 4 | 1 | no | — |
| 87 | C2×C8×C89 | — | 4 | 1 | no | — |
| 88 | C2×C8×C89 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C8×C89 | C22×C4 | 1 | 1 | yes | characteristic |
| 90 | C2×C8×C89 | C22×C4 | 1 | 1 | yes | characteristic |
| 91 | C2×C8×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 92 | C2×C8×C89 | #41 | 1 | 1 | yes | characteristic |
| 93 | C2×C8×C89 | #41 | 1 | 1 | yes | characteristic |
| 94 | C16×C89 | — | 2 | 2 | no | — |
| 95 | C23×C89 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 96 | Q8×C89 | — | 2 | 4 | no | — |
| 97 | Q8×C89 | — | 4 | 2 | no | — |
| 98 | Q8×C89 | — | 4 | 2 | no | — |
| 99 | Q8×C89 | — | 4 | 2 | no | — |
| 100 | C2×C4×C89 | — | 4 | 2 | no | — |
| 101 | C2×C4×C89 | — | 4 | 2 | no | — |
| 102 | C2×C4×C89 | — | 4 | 2 | no | — |
| 103 | C2×C4×C89 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C2×C4×C89 | C4×D4 | 1 | 1 | yes | characteristic |
| 105 | C2×C4×C89 | C4×D4 | 1 | 1 | yes | characteristic |
| 106 | C2×C4×C89 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 107 | C2×C4×C89 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 108 | C2×C4×C89 | #137 | 1 | 1 | yes | characteristic |
| 109 | C2×Dic89 | — | 2 | 2 | no | — |
| 110 | C2×Dic89 | — | 4 | 2 | no | — |
| 111 | C2×Dic89 | — | 4 | 2 | no | — |
| 112 | C2×Dic89 | — | 4 | 2 | no | — |
| 113 | C2×Dic89 | — | 8 | 2 | no | — |
| 114 | C8×C89 | — | 4 | 2 | no | — |
| 115 | C8×C89 | C2×C42 | 1 | 1 | yes | characteristic |
| 116 | C8×C89 | #117 | 1 | 1 | yes | characteristic |
| 117 | C8×C89 | C4×D4 | 1 | 1 | yes | characteristic |
| 118 | C8×C89 | C4×D4 | 1 | 1 | yes | characteristic |
| 119 | #10623494 | — | 2 | 2 | no | — |
| 120 | C22×C89 | — | 2 | 2 | no | — |
| 121 | C22×C89 | #418 | 1 | 1 | yes | characteristic |
| 122 | C22×C89 | #440 | 1 | 1 | yes | characteristic |
| 123 | C22×C89 | #442 | 1 | 1 | yes | characteristic |
| 124 | C4×C89 | — | 4 | 4 | no | — |
| 125 | C4×C89 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 126 | C4×C89 | #417 | 1 | 1 | yes | characteristic |
| 127 | C4×C89 | #440 | 1 | 1 | yes | characteristic |
| 128 | C4×C89 | #442 | 1 | 1 | yes | characteristic |
| 129 | Dic89 | — | 4 | 2 | no | — |
| 130 | Dic89 | — | 8 | 2 | no | — |
| 131 | Dic89 | — | 8 | 2 | no | — |
| 132 | #12648 | — | 89 | 1 | no | maximal, Sylow |
| 133 | C2×C89 | — | 2 | 1 | no | — |
| 134 | C2×C89 | — | 2 | 1 | no | — |
| 135 | C2×C89 | #1762 | 1 | 1 | yes | characteristic |
| 136 | C2×C89 | #2180 | 1 | 2 | yes | — |
| 137 | C2×#441 | — | 89 | 1 | no | — |
| 138 | C2×#362 | D89 | 1 | 1 | yes | characteristic |
| 139 | #2141 | — | 89 | 1 | no | — |
| 140 | C2×#347 | — | 89 | 1 | no | — |
| 141 | #1773 | — | 89 | 1 | no | — |
| 142 | #1386 | — | 89 | 1 | no | — |
| 143 | #1383 | — | 89 | 1 | no | — |
| 144 | C89 | #12648 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 145 | C22×Q16 | C2×D89 | 1 | 1 | yes | characteristic |
| 146 | C2×C4×Q8 | — | 178 | 1 | no | — |
| 147 | C2×M32 | — | 89 | 1 | no | — |
| 148 | C2×M32 | C2×D89 | 1 | 1 | yes | characteristic |
| 149 | #441 | — | 89 | 4 | no | — |
| 150 | #441 | — | 178 | 2 | no | — |
| 151 | C2×#108 | C2×D89 | 1 | 1 | yes | characteristic |
| 152 | #427 | — | 89 | 1 | no | — |
| 153 | C2×#106 | — | 89 | 1 | no | — |
| 154 | C2×#103 | — | 178 | 1 | no | — |
| 155 | #405 | — | 178 | 1 | no | — |
| 156 | C2×#97 | — | 89 | 1 | no | — |
| 157 | #362 | Dic89 | 1 | 4 | yes | complemented |
| 158 | #358 | — | 89 | 4 | no | — |
| 159 | #347 | — | 178 | 2 | no | — |
| 160 | #339 | — | 178 | 1 | no | — |
| 161 | C22×Q8 | — | 2 | 1 | no | — |
| 162 | C2×Q16 | — | 2 | 1 | no | — |
| 163 | C2×Q16 | — | 2 | 1 | no | — |
| 164 | C2×Q16 | — | 2 | 1 | no | — |
| 165 | C2×Q16 | C4×D89 | 1 | 2 | yes | — |
| 166 | C2×Q16 | C4×D89 | 1 | 2 | yes | — |
| 167 | C2×Q16 | C2×Dic89 | 1 | 2 | yes | — |
| 168 | C2×M16 | — | 2 | 1 | no | — |
| 169 | C22×C8 | — | 89 | 1 | no | — |
| 170 | C22×C8 | C22×D89 | 1 | 1 | yes | characteristic |
| 171 | C4×Q8 | — | 356 | 2 | no | — |
| 172 | C4×Q8 | — | 178 | 4 | no | — |
| 173 | #117 | — | 178 | 1 | no | — |
| 174 | C2×C4⋊C4 | — | 356 | 1 | no | — |
| 175 | C2×C4⋊C4 | — | 178 | 1 | no | — |
| 176 | C2×C42 | — | 89 | 1 | no | — |
| 177 | C2×C42 | — | 356 | 1 | no | — |
| 178 | M32 | — | 2 | 1 | no | — |
| 179 | M32 | — | 178 | 1 | no | — |
| 180 | M32 | — | 178 | 1 | no | — |
| 181 | M32 | C2×Dic89 | 1 | 2 | yes | — |
| 182 | C2×C16 | — | 2 | 1 | no | — |
| 183 | C2×C16 | — | 89 | 2 | no | — |
| 184 | #108 | — | 2 | 1 | no | — |
| 185 | #108 | C2×Dic89 | 1 | 2 | yes | — |
| 186 | #107 | — | 89 | 2 | no | — |
| 187 | #106 | — | 89 | 2 | no | — |
| 188 | #106 | — | 178 | 1 | no | — |
| 189 | #106 | — | 178 | 1 | no | — |
| 190 | #103 | — | 178 | 4 | no | — |
| 191 | #97 | — | 89 | 2 | no | — |
| 192 | #97 | — | 89 | 2 | no | — |
| 193 | #97 | — | 356 | 1 | no | — |
| 194 | C4×C8 | — | 356 | 1 | no | — |
| 195 | C2×Q8 | — | 2 | 2 | no | — |
| 196 | C2×Q8 | — | 2 | 2 | no | — |
| 197 | C2×Q8 | — | 2 | 2 | no | — |
| 198 | C2×Q8 | — | 4 | 1 | no | — |
| 199 | C2×Q8 | — | 4 | 1 | no | — |
| 200 | C2×Q8 | — | 4 | 1 | no | — |
| 201 | C22×C4 | — | 4 | 1 | no | — |
| 202 | C22×C4 | — | 356 | 1 | no | — |
| 203 | C22×C4 | — | 178 | 1 | no | — |
| 204 | C22×C4 | D4×D89 | 1 | 1 | yes | characteristic |
| 205 | Q16 | — | 2 | 2 | no | — |
| 206 | Q16 | — | 2 | 2 | no | — |
| 207 | Q16 | — | 2 | 2 | no | — |
| 208 | Q16 | C4×Dic89 | 1 | 4 | yes | — |
| 209 | M16 | — | 2 | 2 | no | — |
| 210 | M16 | — | 4 | 1 | no | — |
| 211 | C2×C8 | — | 4 | 1 | no | — |
| 212 | C2×C8 | — | 89 | 2 | no | — |
| 213 | C2×C8 | — | 178 | 1 | no | — |
| 214 | C2×C8 | — | 178 | 1 | no | — |
| 215 | C2×C8 | C2×C4×D89 | 1 | 1 | yes | characteristic |
| 216 | C2×C8 | C2×C4×D89 | 1 | 1 | yes | characteristic |
| 217 | C2×C8 | C22×Dic89 | 1 | 1 | yes | characteristic |
| 218 | C2×C8 | D4×D89 | 1 | 1 | yes | characteristic |
| 219 | C2×C8 | #13083573 | 1 | 1 | yes | characteristic |
| 220 | C2×C8 | #13083573 | 1 | 1 | yes | characteristic |
| 221 | C4⋊C4 | — | 356 | 4 | no | — |
| 222 | C4⋊C4 | — | 356 | 1 | no | — |
| 223 | C4⋊C4 | — | 178 | 2 | no | — |
| 224 | C4⋊C4 | — | 178 | 2 | no | — |
| 225 | C22⋊C4 | — | 356 | 1 | no | — |
| 226 | C42 | — | 89 | 2 | no | — |
| 227 | C42 | — | 356 | 4 | no | — |
| 228 | C42 | — | 356 | 1 | no | — |
| 229 | C42 | — | 178 | 1 | no | — |
| 230 | C16 | — | 2 | 2 | no | — |
| 231 | C16 | — | 178 | 2 | no | — |
| 232 | C23 | #15668039 | 1 | 1 | yes | characteristic |
| 233 | Q8 | — | 2 | 4 | no | — |
| 234 | Q8 | — | 4 | 2 | no | — |
| 235 | Q8 | — | 4 | 2 | no | — |
| 236 | Q8 | — | 4 | 2 | no | — |
| 237 | C2×C4 | — | 4 | 2 | no | — |
| 238 | C2×C4 | — | 4 | 2 | no | — |
| 239 | C2×C4 | — | 4 | 2 | no | — |
| 240 | C2×C4 | — | 356 | 2 | no | — |
| 241 | C2×C4 | — | 356 | 2 | no | — |
| 242 | C2×C4 | — | 356 | 2 | no | — |
| 243 | C2×C4 | — | 178 | 2 | no | — |
| 244 | C2×C4 | — | 712 | 2 | no | — |
| 245 | C2×C4 | C22⋊C4×D89 | 1 | 1 | yes | characteristic |
| 246 | C2×C4 | #15667972 | 1 | 1 | yes | characteristic |
| 247 | C2×C4 | D4×Dic89 | 1 | 1 | yes | characteristic |
| 248 | C2×C4 | #15668036 | 1 | 1 | yes | characteristic |
| 249 | C2×C4 | #15668036 | 1 | 1 | yes | characteristic |
| 250 | C2×C4 | #15668041 | 1 | 1 | yes | characteristic |
| 251 | C8 | — | 4 | 2 | no | — |
| 252 | C8 | — | 178 | 2 | no | — |
| 253 | C8 | C2×C4×Dic89 | 1 | 1 | yes | characteristic |
| 254 | C8 | #15667968 | 1 | 1 | yes | characteristic |
| 255 | C8 | #15667972 | 1 | 1 | yes | characteristic |
| 256 | C8 | D4×Dic89 | 1 | 1 | yes | characteristic |
| 257 | C22 | — | 2 | 2 | no | — |
| 258 | C22 | #19151601 | 1 | 1 | yes | characteristic |
| 259 | C22 | #19151699 | 1 | 1 | yes | characteristic |
| 260 | C22 | #19151707 | 1 | 1 | yes | characteristic |
| 261 | C4 | — | 4 | 4 | no | — |
| 262 | C4 | — | 356 | 2 | no | — |
| 263 | C4 | — | 712 | 2 | no | — |
| 264 | C4 | — | 712 | 2 | no | — |
| 265 | C4 | C22⋊C4×Dic89 | 1 | 1 | yes | characteristic |
| 266 | C4 | #19151596 | 1 | 1 | yes | characteristic |
| 267 | C4 | #19151695 | 1 | 1 | yes | characteristic |
| 268 | C4 | #19151710 | 1 | 1 | yes | characteristic |
| 269 | C2 | — | 2 | 1 | no | — |
| 270 | C2 | — | 2 | 1 | no | — |
| 271 | C2 | #26020079 | 1 | 1 | yes | characteristic |
| 272 | C2 | #26022146 | 1 | 2 | yes | — |
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