← #26126205
Subgroups — #26126205
Order 12032.
360 conjugacy classes of subgroups.
The subgroup lattice has 244 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20245191 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20245187 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20241346 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C47×#2931 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C4⋊(C47⋊C16) | — | 2 | 1 | no | — |
| 6 | C4⋊(C47⋊C16) | C22 | 1 | 2 | yes | — |
| 7 | C22⋊(C47⋊C16) | — | 2 | 1 | no | — |
| 8 | C22⋊(C47⋊C16) | — | 2 | 1 | no | — |
| 9 | #15688899 | C22 | 1 | 2 | yes | — |
| 10 | C4×C47⋊C16 | — | 2 | 1 | no | — |
| 11 | C2×#131×C47 | — | 2 | 1 | no | — |
| 12 | C47×#532 | C4 | 1 | 1 | yes | characteristic |
| 13 | C47×#436 | — | 2 | 1 | no | — |
| 14 | C47×#436 | C22 | 1 | 2 | yes | — |
| 15 | C8×D4×C47 | — | 2 | 1 | no | — |
| 16 | C8×D4×C47 | C4 | 1 | 2 | yes | — |
| 17 | C47×#423 | C4 | 1 | 1 | yes | characteristic |
| 18 | C47×#408 | — | 2 | 1 | no | — |
| 19 | C2×C4×C8×C47 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C47⋊C16 | — | 2 | 2 | no | — |
| 21 | C2×C47⋊C16 | — | 2 | 2 | no | — |
| 22 | C2×#41×C47 | — | 2 | 1 | no | — |
| 23 | #131×C47 | — | 4 | 4 | no | — |
| 24 | C2×M16×C47 | — | 2 | 1 | no | — |
| 25 | C2×M16×C47 | — | 2 | 2 | no | — |
| 26 | C22×C8×C47 | — | 2 | 2 | no | — |
| 27 | C22×C8×C47 | D4 | 1 | 1 | yes | characteristic |
| 28 | C22×C8×C47 | D4 | 1 | 1 | yes | characteristic |
| 29 | C4⋊Q8×C47 | C8 | 1 | 1 | yes | characteristic |
| 30 | #127×C47 | C8 | 1 | 1 | yes | characteristic |
| 31 | #124×C47 | — | 2 | 1 | no | — |
| 32 | #121×C47 | — | 2 | 1 | no | — |
| 33 | #121×C47 | C8 | 1 | 2 | yes | — |
| 34 | C4×D4×C47 | — | 2 | 1 | no | — |
| 35 | C4×D4×C47 | C2×C4 | 1 | 2 | yes | — |
| 36 | C2×C42×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C4⋊C8×C47 | — | 2 | 1 | no | — |
| 38 | C4⋊C8×C47 | C2×C4 | 1 | 2 | yes | — |
| 39 | C22⋊C8×C47 | — | 2 | 2 | no | — |
| 40 | C22⋊C8×C47 | — | 2 | 2 | no | — |
| 41 | C4×C8×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C4×C8×C47 | D4 | 1 | 1 | yes | characteristic |
| 43 | C4×C8×C47 | D4 | 1 | 1 | yes | characteristic |
| 44 | C4×C8×C47 | C23 | 1 | 1 | yes | characteristic |
| 45 | #41×C47 | — | 4 | 2 | no | — |
| 46 | #41×C47 | — | 4 | 2 | no | — |
| 47 | C2×Q8×C47 | — | 2 | 1 | no | — |
| 48 | C2×D4×C47 | — | 2 | 2 | no | — |
| 49 | C2×D4×C47 | — | 2 | 1 | no | — |
| 50 | C2×D4×C47 | — | 2 | 1 | no | — |
| 51 | C2×D4×C47 | C2×C8 | 1 | 2 | yes | — |
| 52 | C22×C4×C47 | — | 2 | 2 | no | — |
| 53 | C22×C4×C47 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C22×C4×C47 | C2×C8 | 1 | 1 | yes | characteristic |
| 55 | C22×C4×C47 | M16 | 1 | 1 | yes | characteristic |
| 56 | M16×C47 | — | 4 | 2 | no | — |
| 57 | M16×C47 | — | 4 | 4 | no | — |
| 58 | C2×C8×C47 | — | 2 | 2 | no | — |
| 59 | C2×C8×C47 | — | 2 | 2 | no | — |
| 60 | C2×C8×C47 | — | 4 | 1 | no | — |
| 61 | C2×C8×C47 | — | 4 | 4 | no | — |
| 62 | C2×C8×C47 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 63 | C2×C8×C47 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C2×C8×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C2×C8×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 66 | C4⋊C4×C47 | — | 2 | 1 | no | — |
| 67 | C4⋊C4×C47 | C2×C8 | 1 | 2 | yes | — |
| 68 | C22⋊C4×C47 | — | 2 | 2 | no | — |
| 69 | C22⋊C4×C47 | — | 2 | 2 | no | — |
| 70 | C42×C47 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C42×C47 | C2×C8 | 1 | 1 | yes | characteristic |
| 72 | C42×C47 | M16 | 1 | 1 | yes | characteristic |
| 73 | C42×C47 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C47⋊C16 | — | 4 | 2 | no | — |
| 75 | C47⋊C16 | — | 4 | 2 | no | — |
| 76 | C23×C47 | — | 2 | 2 | no | — |
| 77 | C23×C47 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 78 | Q8×C47 | — | 4 | 2 | no | — |
| 79 | D4×C47 | — | 4 | 2 | no | — |
| 80 | D4×C47 | — | 4 | 2 | no | — |
| 81 | D4×C47 | — | 4 | 4 | no | — |
| 82 | D4×C47 | — | 4 | 2 | no | — |
| 83 | C2×C4×C47 | — | 2 | 2 | no | — |
| 84 | C2×C4×C47 | — | 2 | 1 | no | — |
| 85 | C2×C4×C47 | — | 2 | 1 | no | — |
| 86 | C2×C4×C47 | — | 2 | 2 | no | — |
| 87 | C2×C4×C47 | — | 4 | 1 | no | — |
| 88 | C2×C4×C47 | — | 4 | 2 | no | — |
| 89 | C2×C4×C47 | — | 4 | 2 | no | — |
| 90 | C2×C4×C47 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 91 | C2×C4×C47 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 92 | C2×C4×C47 | C22⋊C8 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C47 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×C47 | C22×C8 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C47 | C2×M16 | 1 | 1 | yes | characteristic |
| 96 | C8×C47 | — | 2 | 1 | no | — |
| 97 | C8×C47 | — | 2 | 1 | no | — |
| 98 | C8×C47 | — | 2 | 1 | no | — |
| 99 | C8×C47 | — | 4 | 2 | no | — |
| 100 | C8×C47 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 101 | #11633 | — | 47 | 1 | no | maximal, Sylow |
| 102 | C22×C47 | — | 2 | 1 | no | — |
| 103 | C22×C47 | — | 2 | 1 | no | — |
| 104 | C22×C47 | — | 2 | 1 | no | — |
| 105 | C22×C47 | — | 4 | 2 | no | — |
| 106 | C22×C47 | — | 4 | 2 | no | — |
| 107 | C22×C47 | — | 4 | 2 | no | — |
| 108 | C22×C47 | C2×C22⋊C8 | 1 | 1 | yes | characteristic |
| 109 | C4×C47 | — | 2 | 1 | no | — |
| 110 | C4×C47 | — | 2 | 1 | no | — |
| 111 | C4×C47 | — | 2 | 1 | no | — |
| 112 | C4×C47 | — | 2 | 1 | no | — |
| 113 | C4×C47 | — | 4 | 2 | no | — |
| 114 | C4×C47 | #350 | 1 | 2 | yes | — |
| 115 | C4×C47 | #418 | 1 | 1 | yes | characteristic |
| 116 | C4×C47 | #421 | 1 | 1 | yes | characteristic |
| 117 | #2931 | D47 | 1 | 1 | yes | characteristic |
| 118 | #2163 | — | 47 | 1 | no | — |
| 119 | #2162 | — | 47 | 1 | no | — |
| 120 | #1330 | — | 47 | 4 | no | — |
| 121 | C2×C47 | — | 4 | 1 | no | — |
| 122 | C2×C47 | — | 8 | 2 | no | — |
| 123 | C2×C47 | #1478 | 1 | 1 | yes | characteristic |
| 124 | C2×C47 | #2115 | 1 | 1 | yes | characteristic |
| 125 | C2×C47 | C2×#350 | 1 | 1 | yes | characteristic |
| 126 | C2×#131 | — | 2 | 1 | no | — |
| 127 | #532 | Dic47 | 1 | 1 | yes | characteristic |
| 128 | #436 | — | 2 | 1 | no | — |
| 129 | #436 | C2×D47 | 1 | 2 | yes | — |
| 130 | C8×D4 | — | 2 | 1 | no | — |
| 131 | C8×D4 | Dic47 | 1 | 2 | yes | — |
| 132 | #423 | Dic47 | 1 | 1 | yes | characteristic |
| 133 | #408 | — | 2 | 1 | no | — |
| 134 | C2×C4×C8 | C2×D47 | 1 | 1 | yes | characteristic |
| 135 | C4⋊C16 | — | 47 | 2 | no | — |
| 136 | C4⋊C16 | — | 94 | 1 | no | — |
| 137 | C22⋊C16 | — | 94 | 1 | no | — |
| 138 | C22⋊C16 | — | 94 | 1 | no | — |
| 139 | #346 | — | 47 | 2 | no | — |
| 140 | C4×C16 | — | 94 | 1 | no | — |
| 141 | C47 | #11633 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 142 | C2×#41 | — | 2 | 1 | no | — |
| 143 | #131 | — | 4 | 4 | no | — |
| 144 | C2×M16 | — | 2 | 1 | no | — |
| 145 | C2×M16 | — | 2 | 2 | no | — |
| 146 | C22×C8 | — | 2 | 2 | no | — |
| 147 | C22×C8 | C22⋊D47 | 1 | 1 | yes | characteristic |
| 148 | C22×C8 | C22⋊D47 | 1 | 1 | yes | characteristic |
| 149 | C4⋊Q8 | C47⋊C8 | 1 | 1 | yes | characteristic |
| 150 | #127 | C47⋊C8 | 1 | 1 | yes | characteristic |
| 151 | #124 | — | 2 | 1 | no | — |
| 152 | #121 | — | 2 | 1 | no | — |
| 153 | #121 | C47⋊C8 | 1 | 2 | yes | — |
| 154 | C4×D4 | — | 2 | 1 | no | — |
| 155 | C4×D4 | C2×Dic47 | 1 | 2 | yes | — |
| 156 | C2×C42 | C2×Dic47 | 1 | 1 | yes | characteristic |
| 157 | C2×C16 | — | 94 | 2 | no | — |
| 158 | C2×C16 | — | 94 | 2 | no | — |
| 159 | C4⋊C8 | — | 2 | 1 | no | — |
| 160 | C4⋊C8 | C2×Dic47 | 1 | 2 | yes | — |
| 161 | C22⋊C8 | — | 2 | 2 | no | — |
| 162 | C22⋊C8 | — | 2 | 2 | no | — |
| 163 | C4×C8 | C2×Dic47 | 1 | 1 | yes | characteristic |
| 164 | C4×C8 | C22⋊D47 | 1 | 1 | yes | characteristic |
| 165 | C4×C8 | C22⋊D47 | 1 | 1 | yes | characteristic |
| 166 | C4×C8 | C22×D47 | 1 | 1 | yes | characteristic |
| 167 | #41 | — | 4 | 2 | no | — |
| 168 | #41 | — | 4 | 2 | no | — |
| 169 | C2×Q8 | — | 2 | 1 | no | — |
| 170 | C2×D4 | — | 2 | 2 | no | — |
| 171 | C2×D4 | — | 2 | 1 | no | — |
| 172 | C2×D4 | — | 2 | 1 | no | — |
| 173 | C2×D4 | C2×C47⋊C8 | 1 | 2 | yes | — |
| 174 | C22×C4 | — | 2 | 2 | no | — |
| 175 | C22×C4 | C2×C47⋊C8 | 1 | 1 | yes | characteristic |
| 176 | C22×C4 | #10625329 | 1 | 1 | yes | characteristic |
| 177 | C22×C4 | C22⋊Dic47 | 1 | 1 | yes | characteristic |
| 178 | M16 | — | 4 | 2 | no | — |
| 179 | M16 | — | 4 | 4 | no | — |
| 180 | C2×C8 | — | 2 | 2 | no | — |
| 181 | C2×C8 | — | 2 | 2 | no | — |
| 182 | C2×C8 | — | 4 | 1 | no | — |
| 183 | C2×C8 | — | 4 | 4 | no | — |
| 184 | C2×C8 | C22⋊Dic47 | 1 | 1 | yes | characteristic |
| 185 | C2×C8 | C22⋊Dic47 | 1 | 1 | yes | characteristic |
| 186 | C2×C8 | C2×C22⋊D47 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | C2×C22⋊D47 | 1 | 1 | yes | characteristic |
| 188 | C4⋊C4 | — | 2 | 1 | no | — |
| 189 | C4⋊C4 | C2×C47⋊C8 | 1 | 2 | yes | — |
| 190 | C22⋊C4 | — | 2 | 2 | no | — |
| 191 | C22⋊C4 | — | 2 | 2 | no | — |
| 192 | C42 | C2×C47⋊C8 | 1 | 1 | yes | characteristic |
| 193 | C42 | #10625329 | 1 | 1 | yes | characteristic |
| 194 | C42 | C22⋊Dic47 | 1 | 1 | yes | characteristic |
| 195 | C42 | C22×Dic47 | 1 | 1 | yes | characteristic |
| 196 | C16 | — | 188 | 2 | no | — |
| 197 | C16 | — | 188 | 2 | no | — |
| 198 | C23 | — | 2 | 2 | no | — |
| 199 | C23 | C22⋊(C47⋊C8) | 1 | 1 | yes | characteristic |
| 200 | Q8 | — | 4 | 2 | no | — |
| 201 | D4 | — | 4 | 2 | no | — |
| 202 | D4 | — | 4 | 2 | no | — |
| 203 | D4 | — | 4 | 4 | no | — |
| 204 | D4 | — | 4 | 2 | no | — |
| 205 | C2×C4 | — | 2 | 1 | no | — |
| 206 | C2×C4 | — | 2 | 1 | no | — |
| 207 | C2×C4 | — | 2 | 2 | no | — |
| 208 | C2×C4 | — | 2 | 2 | no | — |
| 209 | C2×C4 | — | 4 | 1 | no | — |
| 210 | C2×C4 | — | 4 | 2 | no | — |
| 211 | C2×C4 | — | 4 | 2 | no | — |
| 212 | C2×C4 | C22⋊(C47⋊C8) | 1 | 1 | yes | characteristic |
| 213 | C2×C4 | C22⋊(C47⋊C8) | 1 | 1 | yes | characteristic |
| 214 | C2×C4 | C22⋊(C47⋊C8) | 1 | 1 | yes | characteristic |
| 215 | C2×C4 | C2×C22⋊Dic47 | 1 | 1 | yes | characteristic |
| 216 | C2×C4 | C22×C47⋊C8 | 1 | 1 | yes | characteristic |
| 217 | C2×C4 | C2×#10625329 | 1 | 1 | yes | characteristic |
| 218 | C8 | — | 2 | 1 | no | — |
| 219 | C8 | — | 2 | 1 | no | — |
| 220 | C8 | — | 2 | 1 | no | — |
| 221 | C8 | — | 4 | 2 | no | — |
| 222 | C8 | #13094050 | 1 | 2 | yes | — |
| 223 | C22 | — | 2 | 1 | no | — |
| 224 | C22 | — | 2 | 1 | no | — |
| 225 | C22 | — | 2 | 1 | no | — |
| 226 | C22 | — | 4 | 2 | no | — |
| 227 | C22 | — | 4 | 2 | no | — |
| 228 | C22 | — | 4 | 2 | no | — |
| 229 | C22 | C2×C22⋊(C47⋊C8) | 1 | 1 | yes | characteristic |
| 230 | C4 | — | 2 | 1 | no | — |
| 231 | C4 | — | 2 | 1 | no | — |
| 232 | C4 | — | 2 | 1 | no | — |
| 233 | C4 | — | 2 | 1 | no | — |
| 234 | C4 | — | 4 | 2 | no | — |
| 235 | C4 | #15688907 | 1 | 2 | yes | — |
| 236 | C4 | #15689128 | 1 | 1 | yes | characteristic |
| 237 | C4 | #15689140 | 1 | 1 | yes | characteristic |
| 238 | C2 | — | 4 | 1 | no | — |
| 239 | C2 | — | 8 | 2 | no | — |
| 240 | C2 | #20241779 | 1 | 1 | yes | characteristic |
| 241 | C2 | #20245009 | 1 | 1 | yes | characteristic |
| 242 | C2 | C2×#15688907 | 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