← #27122811
Subgroups — #27122811
Order 12032.
2038 conjugacy classes of subgroups.
The subgroup lattice has 248 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #20258162 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #20256541 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #20256536 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #20256529 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #20256524 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | C2×#15689555 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | C2×C47×#504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #15689959 | — | 2 | 2 | no | — |
| 9 | #15689940 | C22 | 1 | 8 | yes | complemented |
| 10 | #15689935 | — | 2 | 8 | no | — |
| 11 | #15689928 | C22 | 1 | 8 | yes | complemented |
| 12 | #15689923 | — | 2 | 6 | no | — |
| 13 | C2×#13094046 | C22 | 1 | 12 | yes | complemented |
| 14 | #15689566 | C22 | 1 | 24 | yes | complemented |
| 15 | #15689561 | C22 | 1 | 24 | yes | complemented |
| 16 | #15689555 | C22 | 1 | 12 | yes | complemented |
| 17 | #15689555 | C22 | 1 | 12 | yes | complemented |
| 18 | #15689555 | C22 | 1 | 12 | yes | complemented |
| 19 | #15689555 | C22 | 1 | 4 | yes | complemented |
| 20 | C2×Dic376 | C22 | 1 | 6 | yes | complemented |
| 21 | C2×#13093969 | C22 | 1 | 12 | yes | complemented |
| 22 | C2×D752 | C22 | 1 | 6 | yes | complemented |
| 23 | C2×#131×C47 | C22 | 1 | 1 | yes | characteristic |
| 24 | C47×#504 | C22 | 1 | 4 | yes | complemented |
| 25 | C47×#504 | C22 | 1 | 4 | yes | complemented |
| 26 | C2×M32×C47 | C22 | 1 | 3 | yes | complemented |
| 27 | C22×C16×C47 | C22 | 1 | 3 | yes | complemented |
| 28 | #13094072 | — | 4 | 2 | no | — |
| 29 | #13094071 | — | 2 | 8 | no | — |
| 30 | #13094070 | — | 2 | 8 | no | — |
| 31 | #13094069 | — | 4 | 6 | no | — |
| 32 | C2×#10625350 | — | 2 | 12 | no | — |
| 33 | #41×D47 | — | 4 | 8 | no | — |
| 34 | #13094057 | — | 2 | 24 | no | — |
| 35 | #13094052 | — | 2 | 24 | no | — |
| 36 | #13094046 | — | 2 | 12 | no | — |
| 37 | #13094046 | — | 2 | 12 | no | — |
| 38 | #13094046 | C23 | 1 | 8 | yes | — |
| 39 | #13094046 | C23 | 1 | 24 | yes | — |
| 40 | C2×Dic188 | C23 | 1 | 12 | yes | — |
| 41 | C2×#10625325 | — | 2 | 12 | no | — |
| 42 | C2×D376 | C23 | 1 | 12 | yes | — |
| 43 | Dic376 | C23 | 1 | 4 | yes | — |
| 44 | Dic376 | C23 | 1 | 12 | yes | — |
| 45 | #13093969 | C23 | 1 | 8 | yes | — |
| 46 | #13093969 | C23 | 1 | 24 | yes | — |
| 47 | D752 | C23 | 1 | 4 | yes | — |
| 48 | D752 | C23 | 1 | 12 | yes | — |
| 49 | C2×#41×C47 | D4 | 1 | 1 | yes | characteristic |
| 50 | #131×C47 | D4 | 1 | 4 | yes | — |
| 51 | #131×C47 | C23 | 1 | 4 | yes | — |
| 52 | C2×M16×C47 | D4 | 1 | 3 | yes | — |
| 53 | C22×C8×C47 | C23 | 1 | 3 | yes | — |
| 54 | M32×C47 | C23 | 1 | 12 | yes | — |
| 55 | C2×C16×C47 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C16×C47 | C23 | 1 | 3 | yes | — |
| 57 | C2×C16×C47 | C23 | 1 | 6 | yes | — |
| 58 | C2×C16×C47 | C23 | 1 | 6 | yes | — |
| 59 | #41×C47 | D8 | 1 | 4 | yes | — |
| 60 | #41×C47 | C2×D4 | 1 | 4 | yes | — |
| 61 | C2×Q8×C47 | D8 | 1 | 1 | yes | characteristic |
| 62 | C2×D4×C47 | D8 | 1 | 3 | yes | — |
| 63 | C22×C4×C47 | C2×D4 | 1 | 3 | yes | — |
| 64 | C2×C22⋊D47 | — | 4 | 12 | no | — |
| 65 | #10625354 | — | 4 | 8 | no | — |
| 66 | Q8×D47 | — | 4 | 8 | no | — |
| 67 | #10625352 | — | 4 | 24 | no | — |
| 68 | D4×D47 | — | 4 | 24 | no | — |
| 69 | #10625350 | — | 2 | 8 | no | — |
| 70 | #10625350 | — | 2 | 24 | no | — |
| 71 | #10625350 | — | 4 | 12 | no | — |
| 72 | #10625350 | — | 4 | 12 | no | — |
| 73 | C2×D188 | — | 2 | 12 | no | — |
| 74 | C2×C4×D47 | — | 4 | 12 | no | — |
| 75 | C2×Dic94 | — | 2 | 12 | no | — |
| 76 | M16×C47 | C2×D4 | 1 | 12 | yes | — |
| 77 | C2×C8×C47 | C2×D4 | 1 | 3 | yes | — |
| 78 | C2×C8×C47 | C2×D4 | 1 | 6 | yes | — |
| 79 | C2×C8×C47 | C24 | 1 | 1 | yes | characteristic |
| 80 | C2×C8×C47 | C24 | 1 | 6 | yes | — |
| 81 | Dic188 | — | 2 | 12 | no | — |
| 82 | Dic188 | C24 | 1 | 8 | yes | — |
| 83 | D376 | — | 2 | 12 | no | — |
| 84 | D376 | C24 | 1 | 8 | yes | — |
| 85 | #10625325 | — | 2 | 8 | no | — |
| 86 | #10625325 | — | 2 | 24 | no | — |
| 87 | C16×C47 | C24 | 1 | 1 | yes | characteristic |
| 88 | C16×C47 | C24 | 1 | 1 | yes | characteristic |
| 89 | C16×C47 | C24 | 1 | 6 | yes | — |
| 90 | C23×C47 | C2×D8 | 1 | 3 | yes | — |
| 91 | C22×D47 | — | 4 | 12 | no | — |
| 92 | Q8×C47 | C2×D8 | 1 | 4 | yes | — |
| 93 | D4×C47 | C2×D8 | 1 | 12 | yes | — |
| 94 | C2×C4×C47 | C2×D8 | 1 | 6 | yes | — |
| 95 | C2×C4×C47 | C2×D8 | 1 | 3 | yes | — |
| 96 | C2×C4×C47 | C22×D4 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C47 | C22×D4 | 1 | 6 | yes | — |
| 98 | C22⋊D47 | — | 4 | 8 | no | — |
| 99 | C22⋊D47 | — | 4 | 24 | no | — |
| 100 | C2×Dic47 | — | 4 | 12 | no | — |
| 101 | D188 | — | 2 | 8 | no | — |
| 102 | D188 | — | 4 | 12 | no | — |
| 103 | C4×D47 | — | 4 | 8 | no | — |
| 104 | C4×D47 | — | 4 | 24 | no | — |
| 105 | Dic94 | — | 2 | 8 | no | — |
| 106 | Dic94 | — | 4 | 12 | no | — |
| 107 | C8×C47 | C22×D4 | 1 | 1 | yes | characteristic |
| 108 | C8×C47 | C22×D4 | 1 | 6 | yes | — |
| 109 | C8×C47 | C25 | 1 | 1 | yes | characteristic |
| 110 | #62635 | — | 47 | 1 | no | maximal, Sylow |
| 111 | C22×C47 | — | 4 | 3 | no | — |
| 112 | C22×C47 | C22×D8 | 1 | 6 | yes | — |
| 113 | C22×C47 | C22×D8 | 1 | 1 | yes | characteristic |
| 114 | C2×D47 | — | 4 | 8 | no | — |
| 115 | C2×D47 | — | 16 | 12 | no | — |
| 116 | C4×C47 | C22×D8 | 1 | 6 | yes | — |
| 117 | C4×C47 | C22×D8 | 1 | 1 | yes | characteristic |
| 118 | C4×C47 | C23×D4 | 1 | 1 | yes | characteristic |
| 119 | Dic47 | — | 4 | 8 | no | — |
| 120 | #3584 | — | 47 | 2 | no | — |
| 121 | #3417 | — | 47 | 4 | no | — |
| 122 | #3416 | — | 47 | 8 | no | — |
| 123 | #3415 | — | 47 | 4 | no | — |
| 124 | #3414 | — | 47 | 6 | no | — |
| 125 | C2×#508 | — | 47 | 6 | no | — |
| 126 | C2×#504 | D47 | 1 | 1 | yes | characteristic, complemented |
| 127 | C2×C47 | — | 2 | 6 | no | — |
| 128 | C2×C47 | — | 2 | 1 | no | — |
| 129 | C2×C47 | C23×D8 | 1 | 1 | yes | characteristic |
| 130 | D47 | — | 8 | 8 | no | — |
| 131 | #585 | — | 94 | 2 | no | — |
| 132 | #578 | — | 47 | 8 | no | — |
| 133 | #577 | — | 94 | 8 | no | — |
| 134 | #576 | — | 47 | 8 | no | — |
| 135 | #575 | — | 94 | 6 | no | — |
| 136 | C2×#135 | — | 47 | 12 | no | — |
| 137 | C2×#131 | C2×D47 | 1 | 1 | yes | characteristic |
| 138 | #510 | — | 47 | 24 | no | — |
| 139 | #509 | — | 47 | 24 | no | — |
| 140 | #508 | — | 47 | 4 | no | — |
| 141 | #508 | — | 47 | 12 | no | — |
| 142 | #508 | — | 47 | 12 | no | — |
| 143 | #508 | — | 47 | 12 | no | — |
| 144 | C2×Q32 | — | 47 | 6 | no | — |
| 145 | C2×SD32 | — | 47 | 12 | no | — |
| 146 | C2×D16 | — | 47 | 6 | no | — |
| 147 | #504 | C2×D47 | 1 | 4 | yes | complemented |
| 148 | #504 | C2×D47 | 1 | 4 | yes | complemented |
| 149 | C2×M32 | C2×D47 | 1 | 3 | yes | complemented |
| 150 | C22×C16 | C2×D47 | 1 | 3 | yes | complemented |
| 151 | C47 | #62635 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 152 | 2−1+4 | — | 94 | 8 | no | — |
| 153 | 2−1+4 | — | 188 | 2 | no | — |
| 154 | H(5,𝔽2) | — | 94 | 8 | no | — |
| 155 | H(5,𝔽2) | — | 188 | 6 | no | — |
| 156 | C2×#41 | — | 94 | 12 | no | — |
| 157 | C2×#41 | — | 188 | 8 | no | — |
| 158 | C2×#41 | D188 | 1 | 1 | yes | characteristic |
| 159 | #137 | — | 94 | 24 | no | — |
| 160 | AGL(1,ℤ/8) | — | 94 | 24 | no | — |
| 161 | #135 | — | 47 | 8 | no | — |
| 162 | #135 | — | 47 | 24 | no | — |
| 163 | #135 | — | 94 | 12 | no | — |
| 164 | #135 | — | 94 | 12 | no | — |
| 165 | C2×Q16 | — | 47 | 12 | no | — |
| 166 | C2×SD16 | — | 94 | 12 | no | — |
| 167 | C2×D8 | — | 47 | 12 | no | — |
| 168 | #131 | D188 | 1 | 4 | yes | — |
| 169 | #131 | C22×D47 | 1 | 4 | yes | — |
| 170 | C2×M16 | D188 | 1 | 3 | yes | — |
| 171 | C22×C8 | C22×D47 | 1 | 3 | yes | — |
| 172 | Q32 | — | 47 | 4 | no | — |
| 173 | Q32 | — | 47 | 12 | no | — |
| 174 | SD32 | — | 47 | 8 | no | — |
| 175 | SD32 | — | 47 | 24 | no | — |
| 176 | D16 | — | 47 | 4 | no | — |
| 177 | D16 | — | 47 | 12 | no | — |
| 178 | M32 | C22×D47 | 1 | 12 | yes | — |
| 179 | C2×C16 | C22×D47 | 1 | 1 | yes | characteristic |
| 180 | C2×C16 | C22×D47 | 1 | 3 | yes | — |
| 181 | C2×C16 | C22×D47 | 1 | 6 | yes | — |
| 182 | C2×C16 | C22×D47 | 1 | 6 | yes | — |
| 183 | #41 | — | 94 | 8 | no | — |
| 184 | #41 | — | 94 | 24 | no | — |
| 185 | #41 | — | 188 | 8 | no | — |
| 186 | #41 | — | 188 | 24 | no | — |
| 187 | #41 | — | 188 | 12 | no | — |
| 188 | #41 | — | 188 | 12 | no | — |
| 189 | #41 | D376 | 1 | 4 | yes | — |
| 190 | #41 | C2×D188 | 1 | 4 | yes | — |
| 191 | C2×Q8 | — | 94 | 12 | no | — |
| 192 | C2×Q8 | — | 188 | 8 | no | — |
| 193 | C2×Q8 | D376 | 1 | 1 | yes | characteristic |
| 194 | C2×D4 | — | 94 | 12 | no | — |
| 195 | C2×D4 | — | 188 | 12 | no | — |
| 196 | C2×D4 | — | 188 | 24 | no | — |
| 197 | C2×D4 | D376 | 1 | 3 | yes | — |
| 198 | C22×C4 | — | 188 | 12 | no | — |
| 199 | C22×C4 | C2×D188 | 1 | 3 | yes | — |
| 200 | Q16 | — | 47 | 8 | no | — |
| 201 | Q16 | — | 94 | 12 | no | — |
| 202 | SD16 | — | 94 | 8 | no | — |
| 203 | SD16 | — | 94 | 24 | no | — |
| 204 | D8 | — | 47 | 8 | no | — |
| 205 | D8 | — | 94 | 12 | no | — |
| 206 | M16 | C2×D188 | 1 | 12 | yes | — |
| 207 | C2×C8 | C2×D188 | 1 | 3 | yes | — |
| 208 | C2×C8 | C2×D188 | 1 | 6 | yes | — |
| 209 | C2×C8 | C23×D47 | 1 | 1 | yes | characteristic |
| 210 | C2×C8 | C23×D47 | 1 | 6 | yes | — |
| 211 | C16 | C23×D47 | 1 | 1 | yes | characteristic |
| 212 | C16 | C23×D47 | 1 | 1 | yes | characteristic |
| 213 | C16 | C23×D47 | 1 | 6 | yes | — |
| 214 | C23 | — | 188 | 12 | no | — |
| 215 | C23 | C2×D376 | 1 | 3 | yes | — |
| 216 | Q8 | — | 94 | 8 | no | — |
| 217 | Q8 | — | 188 | 12 | no | — |
| 218 | Q8 | C2×D376 | 1 | 4 | yes | — |
| 219 | D4 | — | 94 | 8 | no | — |
| 220 | D4 | — | 188 | 24 | no | — |
| 221 | D4 | — | 188 | 8 | no | — |
| 222 | D4 | — | 188 | 12 | no | — |
| 223 | D4 | C2×D376 | 1 | 12 | yes | — |
| 224 | C2×C4 | — | 188 | 12 | no | — |
| 225 | C2×C4 | — | 188 | 24 | no | — |
| 226 | C2×C4 | — | 188 | 8 | no | — |
| 227 | C2×C4 | C2×D376 | 1 | 3 | yes | — |
| 228 | C2×C4 | C2×D376 | 1 | 6 | yes | — |
| 229 | C2×C4 | C22×D188 | 1 | 6 | yes | — |
| 230 | C2×C4 | C22×D188 | 1 | 1 | yes | characteristic |
| 231 | C8 | C22×D188 | 1 | 1 | yes | characteristic |
| 232 | C8 | C22×D188 | 1 | 6 | yes | — |
| 233 | C8 | C24×D47 | 1 | 1 | yes | characteristic |
| 234 | C22 | — | 4 | 3 | no | — |
| 235 | C22 | — | 188 | 8 | no | — |
| 236 | C22 | — | 752 | 12 | no | — |
| 237 | C22 | C22×D376 | 1 | 6 | yes | — |
| 238 | C22 | C22×D376 | 1 | 1 | yes | characteristic |
| 239 | C4 | — | 188 | 8 | no | — |
| 240 | C4 | C22×D376 | 1 | 6 | yes | — |
| 241 | C4 | C22×D376 | 1 | 1 | yes | characteristic |
| 242 | C4 | C23×D188 | 1 | 1 | yes | characteristic |
| 243 | C2 | — | 2 | 1 | no | — |
| 244 | C2 | — | 2 | 6 | no | — |
| 245 | C2 | — | 376 | 8 | no | — |
| 246 | C2 | C23×D376 | 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