← #3227×C251 #44547814
Subgroups — #3227×C251 #44547814
Order 32128.
404 conjugacy classes of subgroups.
The subgroup lattice has 200 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #532×C251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C4⋊Q8×C251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #498×C251 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #486×C251 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #481×C251 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #476×C251 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #431×C251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×#41×C251 | — | 2 | 1 | no | — |
| 9 | C22×Q8×C251 | — | 2 | 1 | no | — |
| 10 | C4⋊Q8×C251 | — | 2 | 1 | no | — |
| 11 | C4⋊Q8×C251 | — | 2 | 1 | no | — |
| 12 | C4⋊Q8×C251 | C22 | 1 | 2 | yes | — |
| 13 | C4⋊Q8×C251 | C22 | 1 | 2 | yes | — |
| 14 | C4⋊Q8×C251 | C22 | 1 | 1 | yes | characteristic |
| 15 | #127×C251 | C22 | 1 | 1 | yes | characteristic |
| 16 | #124×C251 | — | 2 | 1 | no | — |
| 17 | #121×C251 | — | 2 | 1 | no | — |
| 18 | #121×C251 | C22 | 1 | 2 | yes | — |
| 19 | C4×D4×C251 | — | 2 | 1 | no | — |
| 20 | C4×D4×C251 | C22 | 1 | 2 | yes | — |
| 21 | C2×C4⋊C4×C251 | — | 2 | 1 | no | — |
| 22 | C2×C4⋊C4×C251 | C22 | 1 | 2 | yes | — |
| 23 | C2×C42×C251 | C22 | 1 | 1 | yes | characteristic |
| 24 | #106×C251 | C22 | 1 | 8 | yes | — |
| 25 | C4⋊C8×C251 | C22 | 1 | 2 | yes | — |
| 26 | #102×C251 | C22 | 1 | 8 | yes | — |
| 27 | C22⋊C8×C251 | C22 | 1 | 2 | yes | — |
| 28 | C4×C8×C251 | C22 | 1 | 2 | yes | — |
| 29 | #41×C251 | — | 4 | 4 | no | — |
| 30 | C2×Q8×C251 | — | 2 | 1 | no | — |
| 31 | C2×Q8×C251 | — | 2 | 2 | no | — |
| 32 | C2×Q8×C251 | — | 2 | 2 | no | — |
| 33 | C2×Q8×C251 | — | 4 | 4 | no | — |
| 34 | C2×D4×C251 | — | 2 | 2 | no | — |
| 35 | C2×D4×C251 | — | 2 | 1 | no | — |
| 36 | C2×D4×C251 | — | 2 | 1 | no | — |
| 37 | C2×D4×C251 | C23 | 1 | 2 | yes | — |
| 38 | C22×C4×C251 | — | 2 | 2 | no | — |
| 39 | C22×C4×C251 | — | 2 | 2 | no | — |
| 40 | C22×C4×C251 | D4 | 1 | 1 | yes | characteristic |
| 41 | C22×C4×C251 | D4 | 1 | 1 | yes | characteristic |
| 42 | C22×C4×C251 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×C8×C251 | C23 | 1 | 4 | yes | — |
| 44 | C4⋊C4×C251 | — | 2 | 2 | no | — |
| 45 | C4⋊C4×C251 | — | 2 | 2 | no | — |
| 46 | C4⋊C4×C251 | — | 2 | 2 | no | — |
| 47 | C4⋊C4×C251 | — | 2 | 1 | no | — |
| 48 | C4⋊C4×C251 | C23 | 1 | 4 | yes | — |
| 49 | C4⋊C4×C251 | C23 | 1 | 2 | yes | — |
| 50 | C22⋊C4×C251 | — | 2 | 2 | no | — |
| 51 | C22⋊C4×C251 | — | 2 | 2 | no | — |
| 52 | C42×C251 | D4 | 1 | 1 | yes | characteristic |
| 53 | C42×C251 | D4 | 1 | 1 | yes | characteristic |
| 54 | C42×C251 | C23 | 1 | 1 | yes | characteristic |
| 55 | C42×C251 | C23 | 1 | 1 | yes | characteristic |
| 56 | C23×C251 | — | 2 | 2 | no | — |
| 57 | C23×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | Q8×C251 | — | 4 | 4 | no | — |
| 59 | Q8×C251 | — | 4 | 2 | no | — |
| 60 | Q8×C251 | — | 4 | 4 | no | — |
| 61 | D4×C251 | — | 2 | 4 | no | — |
| 62 | D4×C251 | — | 4 | 4 | no | — |
| 63 | D4×C251 | — | 4 | 2 | no | — |
| 64 | D4×C251 | — | 4 | 2 | no | — |
| 65 | C2×C4×C251 | — | 2 | 2 | no | — |
| 66 | C2×C4×C251 | — | 2 | 2 | no | — |
| 67 | C2×C4×C251 | — | 2 | 2 | no | — |
| 68 | C2×C4×C251 | — | 2 | 4 | no | — |
| 69 | C2×C4×C251 | — | 4 | 4 | no | — |
| 70 | C2×C4×C251 | — | 4 | 4 | no | — |
| 71 | C2×C4×C251 | SD16 | 1 | 4 | yes | — |
| 72 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 73 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 76 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C2×C4×C251 | C24 | 1 | 1 | yes | characteristic |
| 78 | C8×C251 | — | 2 | 4 | no | — |
| 79 | C22×C251 | — | 2 | 1 | no | — |
| 80 | C22×C251 | — | 2 | 1 | no | — |
| 81 | C22×C251 | — | 4 | 2 | no | — |
| 82 | C22×C251 | — | 4 | 2 | no | — |
| 83 | C22×C251 | — | 4 | 2 | no | — |
| 84 | C22×C251 | C2×SD16 | 1 | 2 | yes | — |
| 85 | C22×C251 | C22×D4 | 1 | 1 | yes | characteristic |
| 86 | C4×C251 | — | 2 | 1 | no | — |
| 87 | C4×C251 | — | 2 | 1 | no | — |
| 88 | C4×C251 | — | 4 | 2 | no | — |
| 89 | C4×C251 | — | 4 | 4 | no | — |
| 90 | C4×C251 | C2×SD16 | 1 | 2 | yes | — |
| 91 | C4×C251 | C2×SD16 | 1 | 2 | yes | — |
| 92 | C4×C251 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 93 | C4×C251 | 2−1+4 | 1 | 2 | yes | — |
| 94 | C2×C251 | — | 2 | 2 | no | — |
| 95 | C2×C251 | — | 8 | 2 | no | — |
| 96 | C2×C251 | #536 | 1 | 1 | yes | characteristic |
| 97 | C2×C251 | C22×SD16 | 1 | 1 | yes | characteristic |
| 98 | C2×C251 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 99 | C251 | #3227 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 100 | #3227 | C251 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 101 | #532 | C2×C251 | 1 | 1 | yes | characteristic |
| 102 | C2×C4⋊Q8 | C2×C251 | 1 | 1 | yes | characteristic, complemented |
| 103 | #498 | C2×C251 | 1 | 2 | yes | complemented |
| 104 | #486 | C2×C251 | 1 | 2 | yes | complemented |
| 105 | #481 | C2×C251 | 1 | 4 | yes | complemented |
| 106 | #476 | C2×C251 | 1 | 4 | yes | complemented |
| 107 | #431 | C2×C251 | 1 | 1 | yes | characteristic, complemented |
| 108 | C2×#41 | — | 2 | 1 | no | — |
| 109 | C22×Q8 | — | 2 | 1 | no | — |
| 110 | C4⋊Q8 | — | 2 | 1 | no | — |
| 111 | C4⋊Q8 | — | 2 | 1 | no | — |
| 112 | C4⋊Q8 | C22×C251 | 1 | 2 | yes | — |
| 113 | C4⋊Q8 | C22×C251 | 1 | 2 | yes | — |
| 114 | C4⋊Q8 | C22×C251 | 1 | 1 | yes | characteristic |
| 115 | #127 | C22×C251 | 1 | 1 | yes | characteristic |
| 116 | #124 | — | 2 | 1 | no | — |
| 117 | #121 | — | 2 | 1 | no | — |
| 118 | #121 | C22×C251 | 1 | 2 | yes | — |
| 119 | C4×D4 | — | 2 | 1 | no | — |
| 120 | C4×D4 | C22×C251 | 1 | 2 | yes | — |
| 121 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 122 | C2×C4⋊C4 | C22×C251 | 1 | 2 | yes | — |
| 123 | C2×C42 | C22×C251 | 1 | 1 | yes | characteristic |
| 124 | #106 | C22×C251 | 1 | 8 | yes | — |
| 125 | C4⋊C8 | C22×C251 | 1 | 2 | yes | — |
| 126 | #102 | C22×C251 | 1 | 8 | yes | — |
| 127 | C22⋊C8 | C22×C251 | 1 | 2 | yes | — |
| 128 | C4×C8 | C22×C251 | 1 | 2 | yes | — |
| 129 | #41 | — | 4 | 4 | no | — |
| 130 | C2×Q8 | — | 2 | 1 | no | — |
| 131 | C2×Q8 | — | 2 | 2 | no | — |
| 132 | C2×Q8 | — | 2 | 2 | no | — |
| 133 | C2×Q8 | — | 4 | 4 | no | — |
| 134 | C2×D4 | — | 2 | 2 | no | — |
| 135 | C2×D4 | — | 2 | 1 | no | — |
| 136 | C2×D4 | — | 2 | 1 | no | — |
| 137 | C2×D4 | C23×C251 | 1 | 2 | yes | — |
| 138 | C22×C4 | — | 2 | 2 | no | — |
| 139 | C22×C4 | — | 2 | 2 | no | — |
| 140 | C22×C4 | D4×C251 | 1 | 1 | yes | characteristic |
| 141 | C22×C4 | D4×C251 | 1 | 1 | yes | characteristic |
| 142 | C22×C4 | C23×C251 | 1 | 1 | yes | characteristic |
| 143 | C2×C8 | C23×C251 | 1 | 4 | yes | — |
| 144 | C4⋊C4 | — | 2 | 2 | no | — |
| 145 | C4⋊C4 | — | 2 | 2 | no | — |
| 146 | C4⋊C4 | — | 2 | 2 | no | — |
| 147 | C4⋊C4 | — | 2 | 1 | no | — |
| 148 | C4⋊C4 | C23×C251 | 1 | 4 | yes | — |
| 149 | C4⋊C4 | C23×C251 | 1 | 2 | yes | — |
| 150 | C22⋊C4 | — | 2 | 2 | no | — |
| 151 | C22⋊C4 | — | 2 | 2 | no | — |
| 152 | C42 | D4×C251 | 1 | 1 | yes | characteristic |
| 153 | C42 | D4×C251 | 1 | 1 | yes | characteristic |
| 154 | C42 | C23×C251 | 1 | 1 | yes | characteristic |
| 155 | C42 | C23×C251 | 1 | 1 | yes | characteristic |
| 156 | C23 | — | 2 | 2 | no | — |
| 157 | C23 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 158 | Q8 | — | 4 | 4 | no | — |
| 159 | Q8 | — | 4 | 2 | no | — |
| 160 | Q8 | — | 4 | 4 | no | — |
| 161 | D4 | — | 2 | 4 | no | — |
| 162 | D4 | — | 4 | 4 | no | — |
| 163 | D4 | — | 4 | 2 | no | — |
| 164 | D4 | — | 4 | 2 | no | — |
| 165 | C2×C4 | — | 2 | 2 | no | — |
| 166 | C2×C4 | — | 2 | 2 | no | — |
| 167 | C2×C4 | — | 2 | 2 | no | — |
| 168 | C2×C4 | — | 2 | 4 | no | — |
| 169 | C2×C4 | — | 4 | 4 | no | — |
| 170 | C2×C4 | — | 4 | 4 | no | — |
| 171 | C2×C4 | SD16×C251 | 1 | 4 | yes | — |
| 172 | C2×C4 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 173 | C2×C4 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 174 | C2×C4 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 175 | C2×C4 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 176 | C2×C4 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 177 | C2×C4 | C24×C251 | 1 | 1 | yes | characteristic |
| 178 | C8 | — | 2 | 4 | no | — |
| 179 | C22 | — | 2 | 1 | no | — |
| 180 | C22 | — | 2 | 1 | no | — |
| 181 | C22 | — | 4 | 2 | no | — |
| 182 | C22 | — | 4 | 2 | no | — |
| 183 | C22 | — | 4 | 2 | no | — |
| 184 | C22 | C2×SD16×C251 | 1 | 2 | yes | — |
| 185 | C22 | C22×D4×C251 | 1 | 1 | yes | characteristic |
| 186 | C4 | — | 2 | 1 | no | — |
| 187 | C4 | — | 2 | 1 | no | — |
| 188 | C4 | — | 4 | 2 | no | — |
| 189 | C4 | — | 4 | 4 | no | — |
| 190 | C4 | C2×SD16×C251 | 1 | 2 | yes | — |
| 191 | C4 | C2×SD16×C251 | 1 | 2 | yes | — |
| 192 | C4 | AGL(1,ℤ/8)×C251 | 1 | 2 | yes | — |
| 193 | C4 | 2−1+4×C251 | 1 | 2 | yes | — |
| 194 | C2 | — | 2 | 2 | no | — |
| 195 | C2 | — | 8 | 2 | no | — |
| 196 | C2 | #536×C251 | 1 | 1 | yes | characteristic |
| 197 | C2 | C22×SD16×C251 | 1 | 1 | yes | characteristic |
| 198 | C2 | C2×AGL(1,ℤ/8)×C251 | 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