← #26107593
Subgroups — #26107593
Order 12032.
342 conjugacy classes of subgroups.
The subgroup lattice has 232 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20242136 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #20241675 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | M16×C47⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#15688845 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #20241226 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #20241226 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C4×#108×C47 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #15689176 | C22 | 1 | 1 | yes | characteristic |
| 9 | #15689176 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C4×C47⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C4×C47⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 12 | #15688876 | C4 | 1 | 2 | yes | — |
| 13 | #15688874 | C4 | 1 | 2 | yes | — |
| 14 | #15688845 | — | 2 | 1 | no | — |
| 15 | #15688845 | — | 2 | 1 | no | — |
| 16 | #15688845 | C4 | 1 | 2 | yes | — |
| 17 | #15688845 | C4 | 1 | 2 | yes | — |
| 18 | #15688844 | — | 2 | 1 | no | — |
| 19 | C8×C47⋊C8 | — | 2 | 1 | no | — |
| 20 | C2×#108×C47 | C4 | 1 | 2 | yes | — |
| 21 | C4×M16×C47 | C22 | 1 | 1 | yes | characteristic |
| 22 | C4×M16×C47 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×C4×C8×C47 | C22 | 1 | 1 | yes | characteristic |
| 24 | C47×#341 | C4 | 1 | 2 | yes | — |
| 25 | C22×C47⋊C8 | — | 2 | 1 | no | — |
| 26 | C22×C47⋊C8 | Q8 | 1 | 2 | yes | — |
| 27 | C4⋊(C47⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 28 | C4⋊(C47⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 29 | C22⋊(C47⋊C8) | — | 2 | 1 | no | — |
| 30 | C22⋊(C47⋊C8) | — | 2 | 1 | no | — |
| 31 | C4×C47⋊C8 | — | 2 | 1 | no | — |
| 32 | C4×C47⋊C8 | — | 2 | 1 | no | — |
| 33 | C4×C47⋊C8 | — | 2 | 1 | no | — |
| 34 | C4×C47⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 35 | C4×C47⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 36 | C4×C47⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 37 | C4×C47⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×M16×C47 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×M16×C47 | C2×C4 | 1 | 2 | yes | — |
| 40 | C22×C8×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C22×C8×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 42 | C2×C42×C47 | C23 | 1 | 1 | yes | characteristic |
| 43 | #108×C47 | — | 2 | 4 | no | — |
| 44 | #97×C47 | — | 2 | 1 | no | — |
| 45 | #97×C47 | — | 2 | 1 | no | — |
| 46 | C4×C8×C47 | — | 2 | 1 | no | — |
| 47 | C4×C8×C47 | — | 2 | 1 | no | — |
| 48 | C4×C8×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 49 | C4×C8×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | C4×C8×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 51 | C4×C8×C47 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C22×C4×C47 | C22×C4 | 1 | 1 | yes | characteristic |
| 53 | C22×C4×C47 | C2×Q8 | 1 | 1 | yes | characteristic |
| 54 | C22×C4×C47 | #41 | 1 | 1 | yes | characteristic |
| 55 | M16×C47 | — | 2 | 2 | no | — |
| 56 | M16×C47 | — | 2 | 2 | no | — |
| 57 | M16×C47 | — | 2 | 2 | no | — |
| 58 | M16×C47 | M16 | 1 | 4 | yes | — |
| 59 | C2×C8×C47 | — | 2 | 2 | no | — |
| 60 | C2×C8×C47 | — | 2 | 2 | no | — |
| 61 | C2×C8×C47 | — | 2 | 2 | no | — |
| 62 | C2×C8×C47 | — | 2 | 2 | no | — |
| 63 | C2×C8×C47 | C42 | 1 | 1 | yes | characteristic |
| 64 | C2×C8×C47 | C42 | 1 | 1 | yes | characteristic |
| 65 | C2×C8×C47 | C42 | 1 | 1 | yes | characteristic |
| 66 | C2×C8×C47 | C42 | 1 | 1 | yes | characteristic |
| 67 | C42×C47 | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | C42×C47 | C22×C4 | 1 | 1 | yes | characteristic |
| 69 | C42×C47 | C2×D4 | 1 | 1 | yes | characteristic |
| 70 | C42×C47 | #41 | 1 | 1 | yes | characteristic |
| 71 | C2×C47⋊C8 | — | 2 | 4 | no | — |
| 72 | C2×C47⋊C8 | — | 2 | 2 | no | — |
| 73 | C2×C47⋊C8 | — | 2 | 2 | no | — |
| 74 | C2×C47⋊C8 | — | 2 | 2 | no | — |
| 75 | C2×C47⋊C8 | — | 4 | 2 | no | — |
| 76 | C2×C47⋊C8 | C4⋊C4 | 1 | 2 | yes | — |
| 77 | C2×C47⋊C8 | C4⋊C4 | 1 | 2 | yes | — |
| 78 | C23×C47 | C4×Q8 | 1 | 1 | yes | characteristic |
| 79 | C2×C4×C47 | — | 2 | 2 | no | — |
| 80 | C2×C4×C47 | — | 2 | 2 | no | — |
| 81 | C2×C4×C47 | C2×C42 | 1 | 1 | yes | characteristic |
| 82 | C2×C4×C47 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C2×C4×C47 | #117 | 1 | 1 | yes | characteristic |
| 84 | C2×C4×C47 | C4×D4 | 1 | 1 | yes | characteristic |
| 85 | C2×C4×C47 | C4×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×C4×C47 | C4×Q8 | 1 | 1 | yes | characteristic |
| 87 | C2×C4×C47 | C2×M16 | 1 | 2 | yes | — |
| 88 | C2×C4×C47 | #131 | 1 | 2 | yes | — |
| 89 | C8×C47 | — | 2 | 1 | no | — |
| 90 | C8×C47 | — | 2 | 1 | no | — |
| 91 | C8×C47 | — | 2 | 1 | no | — |
| 92 | C8×C47 | — | 2 | 1 | no | — |
| 93 | C8×C47 | — | 2 | 4 | no | — |
| 94 | C8×C47 | — | 4 | 2 | no | — |
| 95 | C47⋊C8 | — | 2 | 4 | no | — |
| 96 | C47⋊C8 | — | 2 | 2 | no | — |
| 97 | C47⋊C8 | — | 2 | 2 | no | — |
| 98 | C47⋊C8 | — | 4 | 2 | no | — |
| 99 | C47⋊C8 | — | 4 | 2 | no | — |
| 100 | #8296 | — | 47 | 1 | no | maximal, Sylow |
| 101 | C22×C47 | — | 2 | 1 | no | — |
| 102 | C22×C47 | — | 2 | 1 | no | — |
| 103 | C22×C47 | C4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 104 | C22×C47 | #446 | 1 | 2 | yes | — |
| 105 | C4×C47 | — | 2 | 1 | no | — |
| 106 | C4×C47 | — | 2 | 1 | no | — |
| 107 | C4×C47 | C4×M16 | 1 | 1 | yes | characteristic |
| 108 | C4×C47 | #405 | 1 | 1 | yes | characteristic |
| 109 | C4×C47 | #430 | 1 | 2 | yes | — |
| 110 | C4×C47 | #435 | 1 | 2 | yes | — |
| 111 | C4×C47 | #443 | 1 | 2 | yes | — |
| 112 | C4×#108 | D47 | 1 | 1 | yes | characteristic |
| 113 | #1564 | — | 47 | 1 | no | — |
| 114 | #1451 | — | 47 | 1 | no | — |
| 115 | C8×M16 | — | 47 | 1 | no | — |
| 116 | C2×#322 | — | 47 | 1 | no | — |
| 117 | #1278 | — | 47 | 1 | no | — |
| 118 | #1278 | — | 47 | 1 | no | — |
| 119 | C2×C47 | — | 2 | 2 | no | — |
| 120 | C2×C47 | #1771 | 1 | 1 | yes | characteristic |
| 121 | C2×C47 | #1772 | 1 | 1 | yes | characteristic |
| 122 | C2×C47 | #1778 | 1 | 1 | yes | characteristic |
| 123 | #431 | — | 47 | 1 | no | — |
| 124 | #431 | — | 47 | 1 | no | — |
| 125 | C2×#108 | Dic47 | 1 | 2 | yes | — |
| 126 | C4×M16 | C2×D47 | 1 | 1 | yes | characteristic |
| 127 | C4×M16 | C2×D47 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C8 | — | 47 | 1 | no | — |
| 129 | C2×C4×C8 | — | 47 | 1 | no | — |
| 130 | C2×C4×C8 | C2×D47 | 1 | 1 | yes | characteristic |
| 131 | #341 | Dic47 | 1 | 2 | yes | — |
| 132 | #335 | — | 47 | 2 | no | — |
| 133 | #334 | — | 47 | 2 | no | — |
| 134 | #322 | — | 47 | 2 | no | — |
| 135 | #322 | — | 47 | 2 | no | — |
| 136 | #322 | — | 94 | 1 | no | — |
| 137 | #322 | — | 94 | 1 | no | — |
| 138 | #322 | — | 94 | 1 | no | — |
| 139 | C82 | — | 94 | 1 | no | — |
| 140 | C47 | #8296 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 141 | C2×M16 | C2×Dic47 | 1 | 2 | yes | — |
| 142 | C2×M16 | C2×Dic47 | 1 | 2 | yes | — |
| 143 | C22×C8 | — | 47 | 2 | no | — |
| 144 | C22×C8 | — | 94 | 1 | no | — |
| 145 | C22×C8 | C2×Dic47 | 1 | 1 | yes | characteristic |
| 146 | C22×C8 | C2×Dic47 | 1 | 1 | yes | characteristic |
| 147 | C2×C42 | C22×D47 | 1 | 1 | yes | characteristic |
| 148 | #108 | — | 2 | 4 | no | — |
| 149 | C4⋊C8 | — | 47 | 2 | no | — |
| 150 | C4⋊C8 | — | 47 | 2 | no | — |
| 151 | C22⋊C8 | — | 94 | 1 | no | — |
| 152 | C22⋊C8 | — | 94 | 1 | no | — |
| 153 | #97 | — | 2 | 1 | no | — |
| 154 | #97 | — | 2 | 1 | no | — |
| 155 | C4×C8 | — | 2 | 1 | no | — |
| 156 | C4×C8 | — | 2 | 1 | no | — |
| 157 | C4×C8 | — | 47 | 2 | no | — |
| 158 | C4×C8 | — | 47 | 2 | no | — |
| 159 | C4×C8 | — | 47 | 1 | no | — |
| 160 | C4×C8 | — | 47 | 1 | no | — |
| 161 | C4×C8 | — | 94 | 1 | no | — |
| 162 | C4×C8 | — | 94 | 1 | no | — |
| 163 | C4×C8 | — | 94 | 1 | no | — |
| 164 | C4×C8 | C4×D47 | 1 | 1 | yes | characteristic |
| 165 | C4×C8 | C4×D47 | 1 | 1 | yes | characteristic |
| 166 | C4×C8 | C4×D47 | 1 | 1 | yes | characteristic |
| 167 | C4×C8 | C4×D47 | 1 | 1 | yes | characteristic |
| 168 | C22×C4 | Q8×D47 | 1 | 1 | yes | characteristic |
| 169 | C22×C4 | #10625354 | 1 | 1 | yes | characteristic |
| 170 | C22×C4 | C22×Dic47 | 1 | 1 | yes | characteristic |
| 171 | M16 | — | 2 | 2 | no | — |
| 172 | M16 | — | 2 | 2 | no | — |
| 173 | M16 | — | 2 | 2 | no | — |
| 174 | M16 | #10625329 | 1 | 4 | yes | — |
| 175 | C2×C8 | — | 2 | 2 | no | — |
| 176 | C2×C8 | — | 2 | 2 | no | — |
| 177 | C2×C8 | — | 2 | 2 | no | — |
| 178 | C2×C8 | — | 2 | 2 | no | — |
| 179 | C2×C8 | — | 47 | 2 | no | — |
| 180 | C2×C8 | — | 47 | 2 | no | — |
| 181 | C2×C8 | — | 94 | 4 | no | — |
| 182 | C2×C8 | — | 94 | 2 | no | — |
| 183 | C2×C8 | — | 94 | 2 | no | — |
| 184 | C2×C8 | — | 94 | 2 | no | — |
| 185 | C2×C8 | — | 188 | 2 | no | — |
| 186 | C2×C8 | C4×Dic47 | 1 | 1 | yes | characteristic |
| 187 | C2×C8 | C4×Dic47 | 1 | 1 | yes | characteristic |
| 188 | C2×C8 | C4×Dic47 | 1 | 1 | yes | characteristic |
| 189 | C2×C8 | C4×Dic47 | 1 | 1 | yes | characteristic |
| 190 | C42 | C2×C4×D47 | 1 | 1 | yes | characteristic |
| 191 | C42 | C2×C4×D47 | 1 | 1 | yes | characteristic |
| 192 | C42 | D4×D47 | 1 | 1 | yes | characteristic |
| 193 | C42 | #10625352 | 1 | 1 | yes | characteristic |
| 194 | C23 | Q8×Dic47 | 1 | 1 | yes | characteristic |
| 195 | C2×C4 | — | 2 | 2 | no | — |
| 196 | C2×C4 | — | 2 | 2 | no | — |
| 197 | C2×C4 | C2×C4×Dic47 | 1 | 1 | yes | characteristic |
| 198 | C2×C4 | C4⋊C4×D47 | 1 | 1 | yes | characteristic |
| 199 | C2×C4 | #13093982 | 1 | 1 | yes | characteristic |
| 200 | C2×C4 | #13093986 | 1 | 1 | yes | characteristic |
| 201 | C2×C4 | D4×Dic47 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | #13093989 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | C2×#10625329 | 1 | 2 | yes | — |
| 204 | C2×C4 | #13094031 | 1 | 2 | yes | — |
| 205 | C8 | — | 2 | 1 | no | — |
| 206 | C8 | — | 2 | 1 | no | — |
| 207 | C8 | — | 2 | 1 | no | — |
| 208 | C8 | — | 2 | 1 | no | — |
| 209 | C8 | — | 2 | 4 | no | — |
| 210 | C8 | — | 4 | 2 | no | — |
| 211 | C8 | — | 94 | 4 | no | — |
| 212 | C8 | — | 94 | 2 | no | — |
| 213 | C8 | — | 94 | 2 | no | — |
| 214 | C8 | — | 188 | 2 | no | — |
| 215 | C8 | — | 188 | 2 | no | — |
| 216 | C22 | — | 2 | 1 | no | — |
| 217 | C22 | — | 2 | 1 | no | — |
| 218 | C22 | C4⋊C4×Dic47 | 1 | 1 | yes | characteristic |
| 219 | C22 | #15689254 | 1 | 2 | yes | — |
| 220 | C4 | — | 2 | 1 | no | — |
| 221 | C4 | — | 2 | 1 | no | — |
| 222 | C4 | C4×#10625329 | 1 | 1 | yes | characteristic |
| 223 | C4 | #15689079 | 1 | 1 | yes | characteristic |
| 224 | C4 | #15689170 | 1 | 2 | yes | — |
| 225 | C4 | #15689195 | 1 | 2 | yes | — |
| 226 | C4 | #15689239 | 1 | 2 | yes | — |
| 227 | C2 | — | 2 | 2 | no | — |
| 228 | C2 | #20243263 | 1 | 1 | yes | characteristic |
| 229 | C2 | #20243270 | 1 | 1 | yes | characteristic |
| 230 | C2 | #20243296 | 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