← #36453166
Subgroups — #36453166
Order 22784.
310 conjugacy classes of subgroups.
The subgroup lattice has 252 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #26020423 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #26020122 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #26019022 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #26018998 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #26018154 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #26018153 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C89×#1570 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #19151651 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#15667947 | — | 2 | 1 | no | — |
| 10 | C2×#15667947 | — | 2 | 1 | no | — |
| 11 | #19151622 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#15667922 | — | 2 | 1 | no | — |
| 13 | C2×C4×#10623494 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×C4⋊Dic89 | C22 | 1 | 1 | yes | characteristic |
| 15 | #19151352 | — | 2 | 1 | no | — |
| 16 | #19151350 | C4 | 1 | 2 | yes | complemented |
| 17 | #19151348 | — | 2 | 2 | no | — |
| 18 | #19151348 | C4 | 1 | 2 | yes | complemented |
| 19 | #431×C89 | C22 | 1 | 1 | yes | characteristic |
| 20 | #423×C89 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C4⋊C8×C89 | C22 | 1 | 1 | yes | characteristic |
| 22 | #334×C89 | — | 2 | 2 | no | — |
| 23 | C2×#13083558 | — | 2 | 1 | no | — |
| 24 | C22×#10623494 | D4 | 1 | 1 | yes | characteristic |
| 25 | C22×#10623494 | D4 | 1 | 1 | yes | characteristic |
| 26 | C2×C4⋊Dic89 | — | 2 | 1 | no | — |
| 27 | C2×C4×Dic89 | — | 2 | 1 | no | — |
| 28 | #15667947 | — | 4 | 2 | no | — |
| 29 | #15667947 | — | 4 | 2 | no | — |
| 30 | C4⋊#10623494 | — | 2 | 1 | no | — |
| 31 | C4⋊#10623494 | — | 2 | 1 | no | — |
| 32 | C4⋊#10623494 | C2×C4 | 1 | 2 | yes | — |
| 33 | C22⋊#10623494 | — | 2 | 1 | no | — |
| 34 | #15667922 | — | 4 | 2 | no | — |
| 35 | C4×#10623494 | — | 2 | 1 | no | — |
| 36 | C4×#10623494 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C4×#10623494 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C4×#10623494 | Q8 | 1 | 1 | yes | characteristic |
| 39 | C4×#10623494 | Q8 | 1 | 1 | yes | characteristic |
| 40 | #15667919 | Q8 | 1 | 2 | yes | — |
| 41 | C2×M16×C89 | — | 2 | 1 | no | — |
| 42 | C22×C8×C89 | — | 2 | 1 | no | — |
| 43 | C2×C42×C89 | C23 | 1 | 1 | yes | characteristic |
| 44 | C4⋊C8×C89 | — | 2 | 1 | no | — |
| 45 | C4⋊C8×C89 | — | 2 | 1 | no | — |
| 46 | C4⋊C8×C89 | — | 2 | 1 | no | — |
| 47 | C4⋊C8×C89 | C2×C4 | 1 | 2 | yes | — |
| 48 | C4⋊C8×C89 | D4 | 1 | 2 | yes | — |
| 49 | C22⋊C8×C89 | — | 2 | 1 | no | — |
| 50 | C4×C8×C89 | — | 2 | 1 | no | — |
| 51 | C22×Dic89 | — | 2 | 2 | no | — |
| 52 | #13083558 | — | 4 | 1 | no | — |
| 53 | #13083558 | — | 4 | 1 | no | — |
| 54 | C2×#10623494 | — | 2 | 1 | no | — |
| 55 | C2×#10623494 | — | 2 | 1 | no | — |
| 56 | C2×#10623494 | — | 2 | 2 | no | — |
| 57 | C2×#10623494 | — | 4 | 1 | no | — |
| 58 | C2×#10623494 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C2×#10623494 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C2×#10623494 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 61 | C2×#10623494 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 62 | C2×#10623494 | SD16 | 1 | 2 | yes | — |
| 63 | C2×#10623494 | SD16 | 1 | 2 | yes | — |
| 64 | C4⋊Dic89 | — | 4 | 2 | no | — |
| 65 | C4×Dic89 | — | 4 | 1 | no | — |
| 66 | C4×Dic89 | — | 4 | 1 | no | — |
| 67 | C22×C4×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 68 | C22×C4×C89 | C2×Q8 | 1 | 1 | yes | characteristic |
| 69 | C22×C4×C89 | #41 | 1 | 1 | yes | characteristic |
| 70 | M16×C89 | — | 4 | 1 | no | — |
| 71 | M16×C89 | — | 4 | 1 | no | — |
| 72 | C2×C8×C89 | — | 2 | 2 | no | — |
| 73 | C2×C8×C89 | — | 2 | 2 | no | — |
| 74 | C2×C8×C89 | — | 4 | 1 | no | — |
| 75 | C2×C8×C89 | — | 4 | 1 | no | — |
| 76 | C2×C8×C89 | — | 4 | 1 | no | — |
| 77 | C42×C89 | C22×C4 | 1 | 1 | yes | characteristic |
| 78 | C42×C89 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C42×C89 | C2×Q8 | 1 | 1 | yes | characteristic |
| 80 | C42×C89 | #41 | 1 | 1 | yes | characteristic |
| 81 | C23×C89 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×C89 | — | 2 | 2 | no | — |
| 83 | C2×C4×C89 | — | 4 | 1 | no | — |
| 84 | C2×C4×C89 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C89 | C4×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C89 | C4×Q8 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C89 | #121 | 1 | 1 | yes | characteristic |
| 88 | C2×C4×C89 | #125 | 1 | 1 | yes | characteristic |
| 89 | C2×C4×C89 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C89 | C2×SD16 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C89 | C2×SD16 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C89 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C89 | #137 | 1 | 1 | yes | characteristic |
| 94 | C2×Dic89 | — | 4 | 2 | no | — |
| 95 | C2×Dic89 | — | 4 | 2 | no | — |
| 96 | C2×Dic89 | — | 4 | 2 | no | — |
| 97 | C8×C89 | — | 4 | 2 | no | — |
| 98 | C8×C89 | — | 4 | 2 | no | — |
| 99 | C8×C89 | — | 8 | 1 | no | — |
| 100 | #10623494 | — | 2 | 1 | no | — |
| 101 | #10623494 | — | 2 | 1 | no | — |
| 102 | #10623494 | — | 4 | 2 | no | — |
| 103 | #10623494 | — | 8 | 1 | no | — |
| 104 | #10623494 | #106 | 1 | 2 | yes | — |
| 105 | #10623494 | #106 | 1 | 2 | yes | — |
| 106 | C22×C89 | — | 2 | 1 | no | — |
| 107 | C22×C89 | — | 2 | 1 | no | — |
| 108 | C22×C89 | #389 | 1 | 1 | yes | characteristic |
| 109 | C22×C89 | #475 | 1 | 1 | yes | characteristic |
| 110 | C22×C89 | #476 | 1 | 1 | yes | characteristic |
| 111 | C4×C89 | — | 2 | 1 | no | — |
| 112 | C4×C89 | — | 2 | 1 | no | — |
| 113 | C4×C89 | — | 2 | 1 | no | — |
| 114 | C4×C89 | C2×#106 | 1 | 1 | yes | characteristic |
| 115 | C4×C89 | #428 | 1 | 1 | yes | characteristic |
| 116 | C4×C89 | #444 | 1 | 2 | yes | — |
| 117 | C4×C89 | #460 | 1 | 1 | yes | characteristic |
| 118 | C4×C89 | #461 | 1 | 1 | yes | characteristic |
| 119 | Dic89 | — | 8 | 2 | no | — |
| 120 | #9631 | — | 89 | 1 | no | maximal, Sylow |
| 121 | C2×C89 | — | 2 | 1 | no | — |
| 122 | C2×C89 | — | 2 | 1 | no | — |
| 123 | C2×C89 | #1946 | 1 | 1 | yes | characteristic |
| 124 | C2×C89 | #1948 | 1 | 1 | yes | characteristic |
| 125 | C2×C89 | #1953 | 1 | 1 | yes | characteristic |
| 126 | #1848 | — | 89 | 1 | no | — |
| 127 | #1770 | — | 89 | 1 | no | — |
| 128 | #1570 | — | 89 | 1 | no | — |
| 129 | #1570 | D89 | 1 | 1 | yes | characteristic |
| 130 | #1564 | — | 89 | 1 | no | — |
| 131 | #1303 | — | 89 | 1 | no | — |
| 132 | #1303 | — | 89 | 1 | no | — |
| 133 | C89 | #9631 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 134 | #431 | — | 89 | 1 | no | — |
| 135 | #431 | C2×D89 | 1 | 1 | yes | characteristic |
| 136 | C2×#108 | — | 178 | 1 | no | — |
| 137 | C2×#108 | — | 178 | 1 | no | — |
| 138 | #423 | — | 89 | 1 | no | — |
| 139 | #423 | C2×D89 | 1 | 1 | yes | characteristic |
| 140 | C2×C4⋊C8 | C2×D89 | 1 | 1 | yes | characteristic |
| 141 | C2×#97 | — | 178 | 1 | no | — |
| 142 | C2×C4×C8 | — | 89 | 1 | no | — |
| 143 | C4×C4⋊C4 | — | 89 | 1 | no | — |
| 144 | #336 | — | 178 | 1 | no | — |
| 145 | #335 | — | 89 | 2 | no | — |
| 146 | #334 | — | 2 | 2 | no | — |
| 147 | #334 | — | 89 | 2 | no | — |
| 148 | #334 | — | 178 | 2 | no | — |
| 149 | C2×M16 | — | 2 | 1 | no | — |
| 150 | C2×M16 | — | 178 | 1 | no | — |
| 151 | C22×C8 | — | 2 | 1 | no | — |
| 152 | C22×C8 | — | 89 | 1 | no | — |
| 153 | C22×C8 | — | 89 | 1 | no | — |
| 154 | C2×C4⋊C4 | — | 178 | 1 | no | — |
| 155 | C2×C42 | — | 178 | 1 | no | — |
| 156 | C2×C42 | C22×D89 | 1 | 1 | yes | characteristic |
| 157 | #108 | — | 356 | 2 | no | — |
| 158 | #108 | — | 356 | 2 | no | — |
| 159 | C4⋊C8 | — | 2 | 1 | no | — |
| 160 | C4⋊C8 | — | 2 | 1 | no | — |
| 161 | C4⋊C8 | — | 2 | 1 | no | — |
| 162 | C4⋊C8 | — | 89 | 2 | no | — |
| 163 | C4⋊C8 | — | 178 | 1 | no | — |
| 164 | C4⋊C8 | — | 178 | 1 | no | — |
| 165 | C4⋊C8 | C4×D89 | 1 | 2 | yes | — |
| 166 | C4⋊C8 | C22⋊D89 | 1 | 2 | yes | — |
| 167 | C22⋊C8 | — | 2 | 1 | no | — |
| 168 | C22⋊C8 | — | 178 | 1 | no | — |
| 169 | #97 | — | 356 | 2 | no | — |
| 170 | C4×C8 | — | 2 | 1 | no | — |
| 171 | C4×C8 | — | 89 | 1 | no | — |
| 172 | C4×C8 | — | 89 | 1 | no | — |
| 173 | C4×C8 | — | 89 | 1 | no | — |
| 174 | C4×C8 | — | 89 | 1 | no | — |
| 175 | C4×C8 | — | 178 | 1 | no | — |
| 176 | #95 | — | 89 | 2 | no | — |
| 177 | C22×C4 | — | 178 | 2 | no | — |
| 178 | C22×C4 | D4×D89 | 1 | 1 | yes | characteristic |
| 179 | C22×C4 | Q8×D89 | 1 | 1 | yes | characteristic |
| 180 | C22×C4 | #13083574 | 1 | 1 | yes | characteristic |
| 181 | M16 | — | 4 | 1 | no | — |
| 182 | M16 | — | 4 | 1 | no | — |
| 183 | M16 | — | 356 | 1 | no | — |
| 184 | M16 | — | 356 | 1 | no | — |
| 185 | C2×C8 | — | 2 | 2 | no | — |
| 186 | C2×C8 | — | 2 | 2 | no | — |
| 187 | C2×C8 | — | 4 | 1 | no | — |
| 188 | C2×C8 | — | 4 | 1 | no | — |
| 189 | C2×C8 | — | 4 | 1 | no | — |
| 190 | C2×C8 | — | 89 | 2 | no | — |
| 191 | C2×C8 | — | 89 | 2 | no | — |
| 192 | C2×C8 | — | 89 | 1 | no | — |
| 193 | C2×C8 | — | 89 | 1 | no | — |
| 194 | C2×C8 | — | 89 | 1 | no | — |
| 195 | C2×C8 | — | 89 | 1 | no | — |
| 196 | C2×C8 | — | 356 | 1 | no | — |
| 197 | C2×C8 | — | 178 | 2 | no | — |
| 198 | C2×C8 | — | 178 | 1 | no | — |
| 199 | C2×C8 | — | 178 | 1 | no | — |
| 200 | C4⋊C4 | — | 356 | 2 | no | — |
| 201 | C42 | — | 356 | 1 | no | — |
| 202 | C42 | — | 356 | 1 | no | — |
| 203 | C42 | C2×C4×D89 | 1 | 1 | yes | characteristic |
| 204 | C42 | C2×C22⋊D89 | 1 | 1 | yes | characteristic |
| 205 | C42 | Q8×D89 | 1 | 1 | yes | characteristic |
| 206 | C42 | #13083573 | 1 | 1 | yes | characteristic |
| 207 | C23 | #15667990 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | — | 2 | 2 | no | — |
| 209 | C2×C4 | — | 4 | 1 | no | — |
| 210 | C2×C4 | — | 356 | 2 | no | — |
| 211 | C2×C4 | — | 356 | 2 | no | — |
| 212 | C2×C4 | — | 356 | 2 | no | — |
| 213 | C2×C4 | C4⋊C4×D89 | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | C4×C22⋊D89 | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | #15667977 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | #15667984 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | #15668002 | 1 | 1 | yes | characteristic |
| 218 | C2×C4 | #15668010 | 1 | 1 | yes | characteristic |
| 219 | C2×C4 | SD16×D89 | 1 | 1 | yes | characteristic |
| 220 | C2×C4 | SD16×D89 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | #15668039 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #15668041 | 1 | 1 | yes | characteristic |
| 223 | C8 | — | 4 | 2 | no | — |
| 224 | C8 | — | 4 | 2 | no | — |
| 225 | C8 | — | 8 | 1 | no | — |
| 226 | C8 | — | 89 | 2 | no | — |
| 227 | C8 | — | 89 | 2 | no | — |
| 228 | C8 | — | 356 | 2 | no | — |
| 229 | C8 | — | 178 | 1 | no | — |
| 230 | C8 | — | 178 | 1 | no | — |
| 231 | C8 | — | 712 | 1 | no | — |
| 232 | C22 | — | 2 | 1 | no | — |
| 233 | C22 | — | 2 | 1 | no | — |
| 234 | C22 | #19151493 | 1 | 1 | yes | characteristic |
| 235 | C22 | #19151882 | 1 | 1 | yes | characteristic |
| 236 | C22 | #19151887 | 1 | 1 | yes | characteristic |
| 237 | C4 | — | 2 | 1 | no | — |
| 238 | C4 | — | 2 | 1 | no | — |
| 239 | C4 | — | 2 | 1 | no | — |
| 240 | C4 | — | 712 | 2 | no | — |
| 241 | C4 | #106×D89 | 1 | 1 | yes | characteristic |
| 242 | C4 | #19151637 | 1 | 1 | yes | characteristic |
| 243 | C4 | #19151718 | 1 | 2 | yes | — |
| 244 | C4 | #19151796 | 1 | 1 | yes | characteristic |
| 245 | C4 | #19151801 | 1 | 1 | yes | characteristic |
| 246 | C2 | — | 2 | 1 | no | — |
| 247 | C2 | — | 2 | 1 | no | — |
| 248 | C2 | #26021002 | 1 | 1 | yes | characteristic |
| 249 | C2 | #26021012 | 1 | 1 | yes | characteristic |
| 250 | C2 | #26021039 | 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