← #72267568
Subgroups — #72267568
Order 4096.
Only the 1771 normal subgroups are shown; non-normal subgroups are omitted.
The subgroup lattice has 182 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72267567 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #72267566 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72267555 | C2 | 1 | 18 | yes | maximal, complemented |
| 4 | #72267554 | C22 | 1 | 9 | yes | complemented |
| 5 | C22×#13379 | C22 | 1 | 12 | yes | complemented |
| 6 | #72267493 | C2 | 1 | 3 | yes | maximal |
| 7 | #72267485 | C22 | 1 | 3 | yes | — |
| 8 | #72267473 | C2 | 1 | 6 | yes | maximal, complemented |
| 9 | #72267472 | C22 | 1 | 3 | yes | complemented |
| 10 | C2×#392114 | C22 | 1 | 3 | yes | — |
| 11 | C25×2−1+4 | C22 | 1 | 1 | yes | characteristic |
| 12 | #72266781 | C22 | 1 | 12 | yes | complemented |
| 13 | #72266781 | C22 | 1 | 36 | yes | complemented |
| 14 | #72266775 | C22 | 1 | 18 | yes | complemented |
| 15 | #72266775 | C22 | 1 | 18 | yes | complemented |
| 16 | C2×#392129 | C22 | 1 | 18 | yes | — |
| 17 | C2×#392129 | C22 | 1 | 18 | yes | — |
| 18 | C22×#13378 | C22 | 1 | 4 | yes | complemented |
| 19 | C24×2−1+4 | Q8 | 1 | 8 | yes | complemented |
| 20 | C24×2−1+4 | C23 | 1 | 6 | yes | — |
| 21 | C26×Q8 | C23 | 1 | 1 | yes | characteristic |
| 22 | C22×#2811 | C23 | 1 | 1 | yes | characteristic, complemented |
| 23 | C22×#2809 | C23 | 1 | 3 | yes | complemented |
| 24 | C22×#1793 | C23 | 1 | 36 | yes | — |
| 25 | C22×#1793 | C23 | 1 | 12 | yes | — |
| 26 | C22×#1784 | C23 | 1 | 18 | yes | — |
| 27 | C2×#14680 | D4 | 1 | 3 | yes | — |
| 28 | C2×#13922 | D4 | 1 | 9 | yes | — |
| 29 | C2×#13922 | D4 | 1 | 9 | yes | — |
| 30 | C2×#13922 | C23 | 1 | 9 | yes | — |
| 31 | C2×#13922 | C23 | 1 | 9 | yes | — |
| 32 | C2×C4×#379 | D4 | 1 | 3 | yes | — |
| 33 | C22×#1394 | C23 | 1 | 36 | yes | complemented |
| 34 | C22×#1394 | C23 | 1 | 12 | yes | complemented |
| 35 | #154039 | C23 | 1 | 3 | yes | complemented |
| 36 | #154039 | C23 | 1 | 9 | yes | complemented |
| 37 | C23×2−1+4 | C2×Q8 | 1 | 24 | yes | — |
| 38 | C23×2−1+4 | C24 | 1 | 4 | yes | — |
| 39 | C25×Q8 | C2×D4 | 1 | 9 | yes | — |
| 40 | C25×Q8 | C2×Q8 | 1 | 4 | yes | — |
| 41 | C25×Q8 | #41 | 1 | 12 | yes | — |
| 42 | C25×Q8 | C24 | 1 | 3 | yes | — |
| 43 | C26×C4 | C2×D4 | 1 | 3 | yes | — |
| 44 | C23×#101 | C24 | 1 | 12 | yes | — |
| 45 | C22×#391 | C2×D4 | 1 | 18 | yes | — |
| 46 | C22×#391 | C2×D4 | 1 | 9 | yes | — |
| 47 | C22×#391 | C2×D4 | 1 | 9 | yes | — |
| 48 | C22×#391 | C24 | 1 | 9 | yes | — |
| 49 | C22×#391 | C24 | 1 | 3 | yes | — |
| 50 | C22×#380 | C2×D4 | 1 | 9 | yes | — |
| 51 | C22×#380 | C2×D4 | 1 | 9 | yes | — |
| 52 | C22×C4×C22⋊C4 | C2×D4 | 1 | 9 | yes | — |
| 53 | C22×C4×C22⋊C4 | C2×D4 | 1 | 9 | yes | — |
| 54 | C27 | #120 | 1 | 1 | yes | characteristic |
| 55 | C22×2−1+4 | C22×Q8 | 1 | 8 | yes | — |
| 56 | C22×2−1+4 | H(5,𝔽2) | 1 | 24 | yes | — |
| 57 | C22×2−1+4 | H(5,𝔽2) | 1 | 24 | yes | — |
| 58 | C24×Q8 | #120 | 1 | 3 | yes | — |
| 59 | C24×Q8 | #120 | 1 | 3 | yes | — |
| 60 | C24×Q8 | C22⋊Q8 | 1 | 36 | yes | — |
| 61 | C24×Q8 | #124 | 1 | 36 | yes | — |
| 62 | C24×Q8 | C22×D4 | 1 | 9 | yes | — |
| 63 | C24×Q8 | C22×Q8 | 1 | 6 | yes | — |
| 64 | C24×Q8 | C2×#41 | 1 | 18 | yes | — |
| 65 | C24×Q8 | C25 | 1 | 1 | yes | characteristic |
| 66 | C25×C4 | #120 | 1 | 9 | yes | — |
| 67 | C25×C4 | C22⋊Q8 | 1 | 12 | yes | — |
| 68 | C25×C4 | #124 | 1 | 12 | yes | — |
| 69 | C25×C4 | C22×D4 | 1 | 9 | yes | — |
| 70 | C23×C4⋊C4 | C22×D4 | 1 | 9 | yes | — |
| 71 | C23×C4⋊C4 | C22×D4 | 1 | 3 | yes | — |
| 72 | C23×C42 | C22×D4 | 1 | 9 | yes | — |
| 73 | C23×C42 | C22×D4 | 1 | 3 | yes | — |
| 74 | C26 | #393 | 1 | 4 | yes | — |
| 75 | C26 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 76 | C26 | B(4,𝔽2) | 1 | 3 | yes | — |
| 77 | C26 | B(4,𝔽2) | 1 | 4 | yes | — |
| 78 | C26 | C2×#120 | 1 | 3 | yes | — |
| 79 | C2×2−1+4 | #543 | 1 | 48 | yes | — |
| 80 | C2×2−1+4 | H(3,𝔽4) | 1 | 48 | yes | — |
| 81 | C2×2−1+4 | H(3,𝔽4) | 1 | 16 | yes | — |
| 82 | C23×Q8 | #393 | 1 | 12 | yes | — |
| 83 | C23×Q8 | #393 | 1 | 12 | yes | — |
| 84 | C23×Q8 | B(4,𝔽2) | 1 | 12 | yes | — |
| 85 | C23×Q8 | B(4,𝔽2) | 1 | 12 | yes | — |
| 86 | C23×Q8 | C2×C22⋊Q8 | 1 | 18 | yes | — |
| 87 | C23×Q8 | C2×#124 | 1 | 18 | yes | — |
| 88 | C23×Q8 | C23×Q8 | 1 | 1 | yes | characteristic |
| 89 | C23×Q8 | C22×#41 | 1 | 3 | yes | — |
| 90 | C23×Q8 | C2×H(5,𝔽2) | 1 | 12 | yes | — |
| 91 | C23×Q8 | C2×H(5,𝔽2) | 1 | 12 | yes | — |
| 92 | C24×C4 | #393 | 1 | 36 | yes | — |
| 93 | C24×C4 | C2×#120 | 1 | 9 | yes | — |
| 94 | C24×C4 | C2×C22⋊Q8 | 1 | 18 | yes | — |
| 95 | C24×C4 | C2×#124 | 1 | 18 | yes | — |
| 96 | C24×C4 | C23×D4 | 1 | 3 | yes | — |
| 97 | C25 | #2029 | 1 | 4 | yes | — |
| 98 | C25 | #2029 | 1 | 12 | yes | — |
| 99 | C25 | C2×#393 | 1 | 6 | yes | — |
| 100 | C25 | C2×B(4,𝔽2) | 1 | 3 | yes | — |
| 101 | C25 | C2×B(4,𝔽2) | 1 | 6 | yes | — |
| 102 | C25 | C2×B(4,𝔽2) | 1 | 3 | yes | — |
| 103 | C25 | C22×#120 | 1 | 1 | yes | characteristic |
| 104 | 2−1+4 | #2840 | 1 | 32 | yes | — |
| 105 | C22×Q8 | #2029 | 1 | 48 | yes | — |
| 106 | C22×Q8 | C2×#543 | 1 | 12 | yes | — |
| 107 | C22×Q8 | #3490 | 1 | 36 | yes | — |
| 108 | C22×Q8 | C2×H(3,𝔽4) | 1 | 12 | yes | — |
| 109 | C22×Q8 | C2×H(3,𝔽4) | 1 | 4 | yes | — |
| 110 | C22×Q8 | #3532 | 1 | 36 | yes | — |
| 111 | C22×Q8 | #3532 | 1 | 12 | yes | — |
| 112 | C23×C4 | C2×#393 | 1 | 18 | yes | — |
| 113 | C23×C4 | C2×B(4,𝔽2) | 1 | 18 | yes | — |
| 114 | C23×C4 | C2×B(4,𝔽2) | 1 | 18 | yes | — |
| 115 | C23×C4 | C22×C22⋊Q8 | 1 | 3 | yes | — |
| 116 | C23×C4 | C22×#124 | 1 | 3 | yes | — |
| 117 | C23×C4 | #3484 | 1 | 18 | yes | — |
| 118 | C23×C4 | #3484 | 1 | 18 | yes | — |
| 119 | C24 | C2×#2029 | 1 | 6 | yes | — |
| 120 | C24 | C2×#2029 | 1 | 6 | yes | — |
| 121 | C24 | #33568 | 1 | 6 | yes | — |
| 122 | C24 | #33569 | 1 | 6 | yes | — |
| 123 | C24 | C22×#393 | 1 | 1 | yes | characteristic |
| 124 | C24 | #35346 | 1 | 3 | yes | — |
| 125 | C24 | #35346 | 1 | 3 | yes | — |
| 126 | C24 | C22×B(4,𝔽2) | 1 | 3 | yes | — |
| 127 | C24 | C22×B(4,𝔽2) | 1 | 3 | yes | — |
| 128 | C24 | C22×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 129 | C24 | C22×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 130 | C2×Q8 | C2×#2840 | 1 | 4 | yes | — |
| 131 | C2×Q8 | #37475 | 1 | 12 | yes | — |
| 132 | C2×Q8 | #45458 | 1 | 36 | yes | — |
| 133 | C2×Q8 | #45458 | 1 | 12 | yes | — |
| 134 | C22×C4 | C2×#2029 | 1 | 36 | yes | — |
| 135 | C22×C4 | #34982 | 1 | 18 | yes | — |
| 136 | C22×C4 | #35367 | 1 | 18 | yes | — |
| 137 | C22×C4 | #45439 | 1 | 36 | yes | — |
| 138 | C22×C4 | #45439 | 1 | 12 | yes | — |
| 139 | C23 | #498990 | 1 | 12 | yes | — |
| 140 | C23 | #5021138 | 1 | 3 | yes | — |
| 141 | C23 | #5021138 | 1 | 1 | yes | characteristic |
| 142 | C23 | #5955304 | 1 | 3 | yes | — |
| 143 | C23 | C22×#2029 | 1 | 1 | yes | characteristic |
| 144 | C23 | C22×#2029 | 1 | 3 | yes | — |
| 145 | C23 | #6499867 | 1 | 3 | yes | — |
| 146 | C23 | #6499867 | 1 | 3 | yes | — |
| 147 | C23 | #6499867 | 1 | 6 | yes | — |
| 148 | C23 | #6499867 | 1 | 3 | yes | — |
| 149 | C23 | #6499867 | 1 | 3 | yes | — |
| 150 | C23 | #6499867 | 1 | 3 | yes | — |
| 151 | C23 | #6499867 | 1 | 3 | yes | — |
| 152 | C23 | C2×#33568 | 1 | 3 | yes | — |
| 153 | C23 | C2×#33569 | 1 | 3 | yes | — |
| 154 | C23 | #7622904 | 1 | 1 | yes | characteristic |
| 155 | C23 | #7622904 | 1 | 1 | yes | characteristic |
| 156 | C2×C4 | #4414129 | 1 | 12 | yes | — |
| 157 | C2×C4 | #5020751 | 1 | 12 | yes | — |
| 158 | C22 | #72209710 | 1 | 3 | yes | — |
| 159 | C22 | #72209710 | 1 | 1 | yes | characteristic |
| 160 | C22 | #72209710 | 1 | 3 | yes | — |
| 161 | C22 | #72209710 | 1 | 1 | yes | characteristic |
| 162 | C22 | #72210404 | 1 | 3 | yes | — |
| 163 | C22 | #72210417 | 1 | 3 | yes | — |
| 164 | C22 | #72212257 | 1 | 3 | yes | — |
| 165 | C22 | #72212272 | 1 | 3 | yes | — |
| 166 | C22 | #72219195 | 1 | 1 | yes | characteristic |
| 167 | C22 | #72226612 | 1 | 3 | yes | — |
| 168 | C22 | #72226612 | 1 | 3 | yes | — |
| 169 | C22 | #72267434 | 1 | 3 | yes | — |
| 170 | C22 | #72267531 | 1 | 1 | yes | characteristic |
| 171 | C22 | C2×#498990 | 1 | 3 | yes | — |
| 172 | C22 | #72267535 | 1 | 6 | yes | — |
| 173 | C22 | #72267544 | 1 | 3 | yes | — |
| 174 | C2 | #72210423 | 1 | 1 | yes | characteristic |
| 175 | C2 | #72212279 | 1 | 1 | yes | characteristic |
| 176 | C2 | #72226613 | 1 | 1 | yes | characteristic |
| 177 | C2 | #72267543 | 1 | 3 | yes | — |
| 178 | C2 | #72267546 | 1 | 3 | yes | — |
| 179 | C2 | #72267549 | 1 | 3 | yes | — |
| 180 | C2 | #72267552 | 1 | 3 | yes | — |
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