← #3342×C251 #44547929
Subgroups — #3342×C251 #44547929
Order 32128.
340 conjugacy classes of subgroups.
The subgroup lattice has 166 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #556×C251 | C2 | 1 | 2 | yes | maximal |
| 2 | #500×C251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #485×C251 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #475×C251 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #467×C251 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C4×Q16×C251 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C8×D4×C251 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #427×C251 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | C2×Q16×C251 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×C8×C251 | C22 | 1 | 2 | yes | — |
| 11 | C4⋊Q8×C251 | C22 | 1 | 2 | yes | — |
| 12 | #126×C251 | — | 2 | 4 | no | — |
| 13 | #124×C251 | — | 2 | 2 | no | — |
| 14 | C22⋊Q8×C251 | C22 | 1 | 4 | yes | — |
| 15 | C4×Q8×C251 | — | 2 | 2 | no | — |
| 16 | C4×Q8×C251 | C22 | 1 | 2 | yes | — |
| 17 | C4×D4×C251 | C22 | 1 | 1 | yes | characteristic |
| 18 | #117×C251 | C22 | 1 | 4 | yes | — |
| 19 | #107×C251 | C22 | 1 | 1 | yes | characteristic |
| 20 | #107×C251 | C22 | 1 | 2 | yes | — |
| 21 | #107×C251 | C22 | 1 | 2 | yes | — |
| 22 | #106×C251 | C22 | 1 | 4 | yes | — |
| 23 | C4⋊C8×C251 | C22 | 1 | 1 | yes | characteristic |
| 24 | #103×C251 | C22 | 1 | 2 | yes | — |
| 25 | #103×C251 | C22 | 1 | 4 | yes | — |
| 26 | C22⋊C8×C251 | C22 | 1 | 2 | yes | — |
| 27 | C4×C8×C251 | C22 | 1 | 1 | yes | characteristic |
| 28 | C2×Q8×C251 | — | 2 | 2 | no | — |
| 29 | C2×Q8×C251 | C23 | 1 | 2 | yes | — |
| 30 | C2×D4×C251 | D4 | 1 | 1 | yes | characteristic |
| 31 | C22×C4×C251 | C23 | 1 | 2 | yes | — |
| 32 | Q16×C251 | — | 2 | 2 | no | — |
| 33 | C2×C8×C251 | — | 2 | 4 | no | — |
| 34 | C2×C8×C251 | C23 | 1 | 1 | yes | characteristic |
| 35 | C2×C8×C251 | C23 | 1 | 2 | yes | — |
| 36 | C2×C8×C251 | C23 | 1 | 1 | yes | characteristic |
| 37 | C4⋊C4×C251 | — | 2 | 4 | no | — |
| 38 | C4⋊C4×C251 | — | 2 | 2 | no | — |
| 39 | C4⋊C4×C251 | — | 2 | 2 | no | — |
| 40 | C4⋊C4×C251 | D4 | 1 | 1 | yes | characteristic |
| 41 | C4⋊C4×C251 | C23 | 1 | 2 | yes | — |
| 42 | C4⋊C4×C251 | C23 | 1 | 4 | yes | — |
| 43 | C22⋊C4×C251 | — | 2 | 4 | no | — |
| 44 | C22⋊C4×C251 | — | 2 | 4 | no | — |
| 45 | C22⋊C4×C251 | D4 | 1 | 2 | yes | — |
| 46 | C42×C251 | — | 2 | 4 | no | — |
| 47 | C42×C251 | — | 2 | 2 | no | — |
| 48 | C42×C251 | C23 | 1 | 1 | yes | characteristic |
| 49 | C23×C251 | C2×D4 | 1 | 2 | yes | — |
| 50 | Q8×C251 | — | 2 | 2 | no | — |
| 51 | Q8×C251 | — | 2 | 2 | no | — |
| 52 | Q8×C251 | — | 4 | 4 | no | — |
| 53 | D4×C251 | — | 2 | 2 | no | — |
| 54 | C2×C4×C251 | — | 2 | 2 | no | — |
| 55 | C2×C4×C251 | — | 2 | 2 | no | — |
| 56 | C2×C4×C251 | — | 2 | 2 | no | — |
| 57 | C2×C4×C251 | — | 2 | 2 | no | — |
| 58 | C2×C4×C251 | — | 2 | 4 | no | — |
| 59 | C2×C4×C251 | C2×D4 | 1 | 2 | yes | — |
| 60 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C2×C4×C251 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C2×C4×C251 | C24 | 1 | 1 | yes | characteristic |
| 63 | C8×C251 | — | 2 | 2 | no | — |
| 64 | C8×C251 | — | 2 | 1 | no | — |
| 65 | C8×C251 | #41 | 1 | 2 | yes | — |
| 66 | C22×C251 | — | 2 | 2 | no | — |
| 67 | C22×C251 | — | 2 | 2 | no | — |
| 68 | C22×C251 | — | 2 | 2 | no | — |
| 69 | C22×C251 | C22×D4 | 1 | 1 | yes | characteristic |
| 70 | C4×C251 | — | 2 | 2 | no | — |
| 71 | C4×C251 | — | 2 | 1 | no | — |
| 72 | C4×C251 | — | 4 | 2 | no | — |
| 73 | C4×C251 | — | 4 | 4 | no | — |
| 74 | C4×C251 | — | 4 | 2 | no | — |
| 75 | C4×C251 | #135 | 1 | 2 | yes | — |
| 76 | C4×C251 | C2×#41 | 1 | 1 | yes | characteristic |
| 77 | C4×C251 | 2−1+4 | 1 | 1 | yes | characteristic |
| 78 | C2×C251 | — | 4 | 2 | no | — |
| 79 | C2×C251 | #547 | 1 | 1 | yes | characteristic |
| 80 | C2×C251 | C2×#135 | 1 | 1 | yes | characteristic |
| 81 | C2×C251 | #578 | 1 | 1 | yes | characteristic |
| 82 | C251 | #3342 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 83 | #3342 | C251 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 84 | #556 | C2×C251 | 1 | 2 | yes | — |
| 85 | #500 | C2×C251 | 1 | 1 | yes | characteristic, complemented |
| 86 | #485 | C2×C251 | 1 | 4 | yes | complemented |
| 87 | #475 | C2×C251 | 1 | 2 | yes | complemented |
| 88 | #467 | C2×C251 | 1 | 2 | yes | complemented |
| 89 | C4×Q16 | C2×C251 | 1 | 1 | yes | characteristic, complemented |
| 90 | C8×D4 | C2×C251 | 1 | 1 | yes | characteristic |
| 91 | #427 | C2×C251 | 1 | 2 | yes | complemented |
| 92 | C2×Q16 | C22×C251 | 1 | 1 | yes | characteristic |
| 93 | C22×C8 | C22×C251 | 1 | 2 | yes | — |
| 94 | C4⋊Q8 | C22×C251 | 1 | 2 | yes | — |
| 95 | #126 | — | 2 | 4 | no | — |
| 96 | #124 | — | 2 | 2 | no | — |
| 97 | C22⋊Q8 | C22×C251 | 1 | 4 | yes | — |
| 98 | C4×Q8 | — | 2 | 2 | no | — |
| 99 | C4×Q8 | C22×C251 | 1 | 2 | yes | — |
| 100 | C4×D4 | C22×C251 | 1 | 1 | yes | characteristic |
| 101 | #117 | C22×C251 | 1 | 4 | yes | — |
| 102 | #107 | C22×C251 | 1 | 1 | yes | characteristic |
| 103 | #107 | C22×C251 | 1 | 2 | yes | — |
| 104 | #107 | C22×C251 | 1 | 2 | yes | — |
| 105 | #106 | C22×C251 | 1 | 4 | yes | — |
| 106 | C4⋊C8 | C22×C251 | 1 | 1 | yes | characteristic |
| 107 | #103 | C22×C251 | 1 | 2 | yes | — |
| 108 | #103 | C22×C251 | 1 | 4 | yes | — |
| 109 | C22⋊C8 | C22×C251 | 1 | 2 | yes | — |
| 110 | C4×C8 | C22×C251 | 1 | 1 | yes | characteristic |
| 111 | C2×Q8 | — | 2 | 2 | no | — |
| 112 | C2×Q8 | C23×C251 | 1 | 2 | yes | — |
| 113 | C2×D4 | D4×C251 | 1 | 1 | yes | characteristic |
| 114 | C22×C4 | C23×C251 | 1 | 2 | yes | — |
| 115 | Q16 | — | 2 | 2 | no | — |
| 116 | C2×C8 | — | 2 | 4 | no | — |
| 117 | C2×C8 | C23×C251 | 1 | 1 | yes | characteristic |
| 118 | C2×C8 | C23×C251 | 1 | 2 | yes | — |
| 119 | C2×C8 | C23×C251 | 1 | 1 | yes | characteristic |
| 120 | C4⋊C4 | — | 2 | 4 | no | — |
| 121 | C4⋊C4 | — | 2 | 2 | no | — |
| 122 | C4⋊C4 | — | 2 | 2 | no | — |
| 123 | C4⋊C4 | D4×C251 | 1 | 1 | yes | characteristic |
| 124 | C4⋊C4 | C23×C251 | 1 | 2 | yes | — |
| 125 | C4⋊C4 | C23×C251 | 1 | 4 | yes | — |
| 126 | C22⋊C4 | — | 2 | 4 | no | — |
| 127 | C22⋊C4 | — | 2 | 4 | no | — |
| 128 | C22⋊C4 | D4×C251 | 1 | 2 | yes | — |
| 129 | C42 | — | 2 | 4 | no | — |
| 130 | C42 | — | 2 | 2 | no | — |
| 131 | C42 | C23×C251 | 1 | 1 | yes | characteristic |
| 132 | C23 | C2×D4×C251 | 1 | 2 | yes | — |
| 133 | Q8 | — | 2 | 2 | no | — |
| 134 | Q8 | — | 2 | 2 | no | — |
| 135 | Q8 | — | 4 | 4 | no | — |
| 136 | D4 | — | 2 | 2 | no | — |
| 137 | C2×C4 | — | 2 | 2 | no | — |
| 138 | C2×C4 | — | 2 | 2 | no | — |
| 139 | C2×C4 | — | 2 | 2 | no | — |
| 140 | C2×C4 | — | 2 | 2 | no | — |
| 141 | C2×C4 | — | 2 | 4 | no | — |
| 142 | C2×C4 | C2×D4×C251 | 1 | 2 | yes | — |
| 143 | C2×C4 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 144 | C2×C4 | C2×D4×C251 | 1 | 1 | yes | characteristic |
| 145 | C2×C4 | C24×C251 | 1 | 1 | yes | characteristic |
| 146 | C8 | — | 2 | 2 | no | — |
| 147 | C8 | — | 2 | 1 | no | — |
| 148 | C8 | #41×C251 | 1 | 2 | yes | — |
| 149 | C22 | — | 2 | 2 | no | — |
| 150 | C22 | — | 2 | 2 | no | — |
| 151 | C22 | — | 2 | 2 | no | — |
| 152 | C22 | C22×D4×C251 | 1 | 1 | yes | characteristic |
| 153 | C4 | — | 2 | 2 | no | — |
| 154 | C4 | — | 2 | 1 | no | — |
| 155 | C4 | — | 4 | 2 | no | — |
| 156 | C4 | — | 4 | 4 | no | — |
| 157 | C4 | — | 4 | 2 | no | — |
| 158 | C4 | #135×C251 | 1 | 2 | yes | — |
| 159 | C4 | C2×#41×C251 | 1 | 1 | yes | characteristic |
| 160 | C4 | 2−1+4×C251 | 1 | 1 | yes | characteristic |
| 161 | C2 | — | 4 | 2 | no | — |
| 162 | C2 | #547×C251 | 1 | 1 | yes | characteristic |
| 163 | C2 | C2×#135×C251 | 1 | 1 | yes | characteristic |
| 164 | C2 | #578×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