← #72337722
Subgroups — #72337722
Order 2048.
At least 6081 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
The subgroup lattice has 185 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72323548 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #72237780 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72237035 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72236978 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72227421 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #72221174 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×#133822 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C22×#2846 | C22 | 1 | 1 | yes | characteristic |
| 9 | #515326 | C22 | 1 | 1 | yes | characteristic |
| 10 | #391692 | C4 | 1 | 2 | yes | complemented |
| 11 | #382422 | C22 | 1 | 1 | yes | characteristic |
| 12 | #382422 | C22 | 1 | 1 | yes | characteristic |
| 13 | #374022 | C4 | 1 | 2 | yes | complemented |
| 14 | C2×#11069 | C4 | 1 | 2 | yes | complemented |
| 15 | C2×#10965 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#10935 | C4 | 1 | 2 | yes | complemented |
| 17 | C2×#8547 | C22 | 1 | 1 | yes | characteristic |
| 18 | C2×#8541 | C22 | 1 | 1 | yes | characteristic |
| 19 | #133822 | C4 | 1 | 4 | yes | complemented |
| 20 | C28 | D4 | 1 | 1 | yes | characteristic |
| 21 | C23×#120 | D4 | 1 | 2 | yes | — |
| 22 | C23×#120 | C23 | 1 | 1 | yes | characteristic |
| 23 | C2×#2846 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×#2846 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C2×#2846 | C2×C4 | 1 | 2 | yes | — |
| 26 | C2×#2846 | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×#2846 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×#2846 | Q8 | 1 | 2 | yes | — |
| 29 | C2×#2315 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 30 | C22×#379 | C2×C4 | 1 | 2 | yes | — |
| 31 | C22×#379 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×#2273 | C2×C4 | 1 | 1 | yes | characteristic, complemented |
| 33 | C2×#1819 | D4 | 1 | 2 | yes | — |
| 34 | C2×#1781 | D4 | 1 | 2 | yes | — |
| 35 | C22×#342 | C2×C4 | 1 | 2 | yes | complemented |
| 36 | #10965 | C2×C4 | 1 | 2 | yes | — |
| 37 | #10965 | C2×C4 | 1 | 2 | yes | — |
| 38 | #10965 | D4 | 1 | 2 | yes | — |
| 39 | #10965 | D4 | 1 | 2 | yes | — |
| 40 | #8547 | C2×C4 | 1 | 2 | yes | complemented |
| 41 | #8541 | C2×C4 | 1 | 2 | yes | complemented |
| 42 | C27 | C22⋊C4 | 1 | 2 | yes | — |
| 43 | C27 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 44 | C27 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | C27 | C4⋊C4 | 1 | 2 | yes | — |
| 46 | C27 | C2×D4 | 1 | 1 | yes | characteristic |
| 47 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 48 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 49 | C24×D4 | #41 | 1 | 1 | yes | characteristic |
| 50 | C22×#120 | C22⋊C4 | 1 | 2 | yes | — |
| 51 | C22×#120 | C22⋊C4 | 1 | 2 | yes | — |
| 52 | C22×#120 | C4⋊C4 | 1 | 2 | yes | — |
| 53 | C22×#120 | C4⋊C4 | 1 | 2 | yes | — |
| 54 | C22×#120 | C22×C4 | 1 | 1 | yes | characteristic |
| 55 | C22×#120 | C22×C4 | 1 | 1 | yes | characteristic |
| 56 | C22×#120 | C2×D4 | 1 | 1 | yes | characteristic |
| 57 | C22×#120 | C2×D4 | 1 | 1 | yes | characteristic |
| 58 | C22×#120 | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | C22×#120 | C2×Q8 | 1 | 1 | yes | characteristic |
| 60 | C22×#120 | #41 | 1 | 1 | yes | characteristic |
| 61 | C22×#120 | #41 | 1 | 1 | yes | characteristic |
| 62 | C23×C22⋊C4 | C22⋊C4 | 1 | 2 | yes | — |
| 63 | C23×C22⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 64 | C23×C22⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 65 | C23×C22⋊C4 | #41 | 1 | 1 | yes | characteristic |
| 66 | C22×#99 | C22⋊C4 | 1 | 4 | yes | complemented |
| 67 | C2×#379 | C42 | 1 | 2 | yes | — |
| 68 | C2×#379 | C42 | 1 | 2 | yes | — |
| 69 | C2×#379 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C2×#379 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×#379 | C4⋊C4 | 1 | 2 | yes | — |
| 72 | C2×#379 | C4⋊C4 | 1 | 2 | yes | — |
| 73 | C2×#379 | C4⋊C4 | 1 | 2 | yes | — |
| 74 | C2×#342 | C22⋊C4 | 1 | 4 | yes | complemented |
| 75 | C26 | #95 | 1 | 2 | yes | — |
| 76 | C26 | #95 | 1 | 2 | yes | — |
| 77 | C26 | #99 | 1 | 2 | yes | — |
| 78 | C26 | #99 | 1 | 2 | yes | — |
| 79 | C26 | #99 | 1 | 2 | yes | — |
| 80 | C26 | #99 | 1 | 2 | yes | — |
| 81 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 82 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C26 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C26 | #120 | 1 | 1 | yes | characteristic |
| 85 | C26 | #120 | 1 | 1 | yes | characteristic |
| 86 | C26 | #121 | 1 | 1 | yes | characteristic |
| 87 | C26 | #123 | 1 | 1 | yes | characteristic |
| 88 | C26 | #123 | 1 | 1 | yes | characteristic |
| 89 | C23×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 90 | C23×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 91 | C23×D4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 92 | C23×D4 | #117 | 1 | 1 | yes | characteristic |
| 93 | C23×D4 | C4×D4 | 1 | 1 | yes | characteristic |
| 94 | C23×D4 | C4×D4 | 1 | 1 | yes | characteristic |
| 95 | C23×D4 | #120 | 1 | 1 | yes | characteristic |
| 96 | C23×D4 | #121 | 1 | 1 | yes | characteristic |
| 97 | C23×D4 | #121 | 1 | 1 | yes | characteristic |
| 98 | C23×D4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 99 | C23×D4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 100 | C23×D4 | #123 | 1 | 1 | yes | characteristic |
| 101 | C24×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 102 | C24×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 103 | C24×C4 | #121 | 1 | 1 | yes | characteristic |
| 104 | C22×C22⋊C4 | #95 | 1 | 2 | yes | — |
| 105 | C22×C22⋊C4 | #95 | 1 | 2 | yes | — |
| 106 | C22×C22⋊C4 | C2×C42 | 1 | 1 | yes | characteristic |
| 107 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 108 | C22×C22⋊C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 109 | C22×C22⋊C4 | #117 | 1 | 1 | yes | characteristic |
| 110 | C22×C22⋊C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 111 | C22×C22⋊C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 112 | C22×C22⋊C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 113 | C22×C22⋊C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 114 | C22×C22⋊C4 | #120 | 1 | 1 | yes | characteristic |
| 115 | C22×C22⋊C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 116 | C22×C22⋊C4 | C22⋊Q8 | 1 | 1 | yes | characteristic |
| 117 | C22×C22⋊C4 | #123 | 1 | 1 | yes | characteristic |
| 118 | C25 | #342 | 1 | 2 | yes | — |
| 119 | C25 | #342 | 1 | 2 | yes | — |
| 120 | C25 | #342 | 1 | 2 | yes | — |
| 121 | C25 | #342 | 1 | 2 | yes | — |
| 122 | C25 | #353 | 1 | 4 | yes | — |
| 123 | C25 | #354 | 1 | 4 | yes | — |
| 124 | C25 | C2×#95 | 1 | 1 | yes | characteristic |
| 125 | C25 | #379 | 1 | 1 | yes | characteristic |
| 126 | C25 | #380 | 1 | 1 | yes | characteristic |
| 127 | C25 | #381 | 1 | 1 | yes | characteristic |
| 128 | C25 | #385 | 1 | 1 | yes | characteristic |
| 129 | C25 | #385 | 1 | 1 | yes | characteristic |
| 130 | C25 | #386 | 1 | 1 | yes | characteristic |
| 131 | C25 | #386 | 1 | 1 | yes | characteristic |
| 132 | C25 | #386 | 1 | 1 | yes | characteristic |
| 133 | C25 | C2×#99 | 1 | 2 | yes | — |
| 134 | C25 | C2×#99 | 1 | 2 | yes | — |
| 135 | C25 | B(4,𝔽2) | 1 | 2 | yes | — |
| 136 | C25 | #458 | 1 | 2 | yes | — |
| 137 | C23×C4 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 138 | C23×C4 | C4×C22⋊C4 | 1 | 1 | yes | characteristic |
| 139 | C23×C4 | #380 | 1 | 1 | yes | characteristic |
| 140 | C23×C4 | #385 | 1 | 1 | yes | characteristic |
| 141 | C23×C4 | #385 | 1 | 1 | yes | characteristic |
| 142 | C23×C4 | #386 | 1 | 1 | yes | characteristic |
| 143 | C24 | #1393 | 1 | 4 | yes | — |
| 144 | C24 | #1393 | 1 | 4 | yes | — |
| 145 | C24 | C22⋊#95 | 1 | 1 | yes | characteristic |
| 146 | C24 | C2×#342 | 1 | 2 | yes | — |
| 147 | C24 | #1790 | 1 | 2 | yes | — |
| 148 | C24 | #1867 | 1 | 2 | yes | — |
| 149 | C24 | #1889 | 1 | 1 | yes | characteristic |
| 150 | C24 | #1890 | 1 | 1 | yes | characteristic |
| 151 | C24 | #1898 | 1 | 1 | yes | characteristic |
| 152 | C24 | #1898 | 1 | 2 | yes | — |
| 153 | C24 | #1898 | 1 | 1 | yes | characteristic |
| 154 | C24 | #1913 | 1 | 1 | yes | characteristic |
| 155 | C24 | #1913 | 1 | 1 | yes | characteristic |
| 156 | C24 | C2×#353 | 1 | 2 | yes | — |
| 157 | C24 | C2×#354 | 1 | 2 | yes | — |
| 158 | C24 | #2127 | 1 | 2 | yes | — |
| 159 | C24 | #2127 | 1 | 2 | yes | — |
| 160 | C23 | #9495 | 1 | 1 | yes | characteristic |
| 161 | C23 | #9495 | 1 | 1 | yes | characteristic |
| 162 | C23 | #9499 | 1 | 1 | yes | characteristic |
| 163 | C23 | #9499 | 1 | 1 | yes | characteristic |
| 164 | C23 | #10735 | 1 | 1 | yes | characteristic |
| 165 | C23 | #10745 | 1 | 1 | yes | characteristic |
| 166 | C23 | #10754 | 1 | 1 | yes | characteristic |
| 167 | C23 | C2×#1393 | 1 | 2 | yes | — |
| 168 | C23 | #12582 | 1 | 2 | yes | — |
| 169 | C23 | #12753 | 1 | 4 | yes | — |
| 170 | C23 | #13088 | 1 | 2 | yes | — |
| 171 | C23 | #13088 | 1 | 2 | yes | — |
| 172 | C23 | #13090 | 1 | 2 | yes | — |
| 173 | C23 | #13090 | 1 | 2 | yes | — |
| 174 | C22 | #106542 | 1 | 1 | yes | characteristic |
| 175 | C22 | #139718 | 1 | 1 | yes | characteristic |
| 176 | C22 | #139720 | 1 | 1 | yes | characteristic |
| 177 | C22 | #139731 | 1 | 2 | yes | — |
| 178 | C22 | #139830 | 1 | 2 | yes | — |
| 179 | C22 | #146569 | 1 | 2 | yes | — |
| 180 | C22 | #146593 | 1 | 2 | yes | — |
| 181 | C2 | #72218855 | 1 | 1 | yes | characteristic |
| 182 | C2 | #72219297 | 1 | 1 | yes | characteristic |
| 183 | C2 | #72241271 | 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