← #72314595
Subgroups — #72314595
Order 4096.
At least 13707 conjugacy classes of subgroups; only the 339 normal subgroups are shown.
The subgroup lattice has 220 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72265785 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #72265779 | C4 | 1 | 2 | yes | complemented |
| 3 | #72265678 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #72259108 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #72234688 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #72234620 | C4 | 1 | 2 | yes | complemented |
| 7 | #72225778 | C4 | 1 | 2 | yes | complemented |
| 8 | #72225776 | C4 | 1 | 2 | yes | complemented |
| 9 | #72225629 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #72219227 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#359428 | C22 | 1 | 2 | yes | — |
| 12 | C2×#7670203 | C22 | 1 | 1 | yes | characteristic |
| 13 | #72218744 | C4 | 1 | 2 | yes | complemented |
| 14 | #72218518 | C22 | 1 | 1 | yes | characteristic |
| 15 | #72218518 | C22 | 1 | 2 | yes | — |
| 16 | #72218375 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #72218291 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22×#3484 | D4 | 1 | 1 | yes | characteristic |
| 19 | C22×#3484 | D4 | 1 | 1 | yes | characteristic |
| 20 | C24×#120 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C24×#120 | C23 | 1 | 1 | yes | characteristic |
| 22 | C22×#2846 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C2×#35390 | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×#35348 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C2×#35348 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C2×#35346 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | C2×#34981 | D4 | 1 | 1 | yes | characteristic |
| 28 | C2×#34981 | Q8 | 1 | 1 | yes | characteristic |
| 29 | C2×#34980 | D4 | 1 | 1 | yes | characteristic |
| 30 | C2×#34980 | D4 | 1 | 1 | yes | characteristic |
| 31 | C2×#34198 | D4 | 1 | 1 | yes | characteristic |
| 32 | C2×#34198 | Q8 | 1 | 1 | yes | characteristic |
| 33 | C23×#379 | C2×C4 | 1 | 2 | yes | complemented |
| 34 | C2×#14679 | C2×C4 | 1 | 2 | yes | complemented |
| 35 | #392109 | D4 | 1 | 4 | yes | — |
| 36 | C2×#8547 | C2×C4 | 1 | 2 | yes | complemented |
| 37 | C2×#8541 | C2×C4 | 1 | 4 | yes | — |
| 38 | C2×#8541 | C2×C4 | 1 | 2 | yes | complemented |
| 39 | C2×#8541 | D4 | 1 | 4 | yes | — |
| 40 | C28 | C22×C4 | 1 | 1 | yes | characteristic |
| 41 | C23×H(5,𝔽2) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 42 | C25×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 43 | C25×D4 | C2×D4 | 1 | 2 | yes | — |
| 44 | C25×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 45 | C22×#534 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 46 | C22×#534 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 47 | C22×#534 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 48 | C22×#534 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 49 | C22×#534 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 50 | C23×#120 | C42 | 1 | 1 | yes | characteristic |
| 51 | C23×#120 | C42 | 1 | 1 | yes | characteristic |
| 52 | C23×#120 | C42 | 1 | 1 | yes | characteristic |
| 53 | C23×#120 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C23×#120 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 55 | C23×#120 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 56 | C23×#120 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 57 | C23×#120 | C22×C4 | 1 | 1 | yes | characteristic |
| 58 | C23×#120 | C22×C4 | 1 | 1 | yes | characteristic |
| 59 | C23×#120 | C2×D4 | 1 | 1 | yes | characteristic |
| 60 | C23×#120 | C2×D4 | 1 | 2 | yes | — |
| 61 | C23×#120 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C23×#120 | #41 | 1 | 2 | yes | — |
| 63 | C22×#518 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 64 | C22×#518 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 65 | C24×C22⋊C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C24×C22⋊C4 | C2×Q8 | 1 | 1 | yes | characteristic |
| 67 | C24×C22⋊C4 | #41 | 1 | 2 | yes | — |
| 68 | C2×#2315 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C2×#2315 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C2×#2308 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C2×#2308 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 72 | C2×#2308 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C2×#2308 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 74 | C2×#2308 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 75 | C22×#379 | C42 | 1 | 1 | yes | characteristic |
| 76 | C22×#379 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 77 | C22×#379 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 78 | C2×#2273 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 79 | C2×#1781 | C22⋊C4 | 1 | 8 | yes | complemented |
| 80 | C27 | #99 | 1 | 1 | yes | characteristic |
| 81 | C27 | C2×C42 | 1 | 1 | yes | characteristic |
| 82 | C27 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C27 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C27 | C4×D4 | 1 | 2 | yes | — |
| 85 | C27 | C4×D4 | 1 | 2 | yes | — |
| 86 | C27 | #120 | 1 | 1 | yes | characteristic |
| 87 | C22×H(5,𝔽2) | #99 | 1 | 2 | yes | — |
| 88 | C22×H(5,𝔽2) | #99 | 1 | 2 | yes | — |
| 89 | C24×D4 | #95 | 1 | 1 | yes | characteristic |
| 90 | C24×D4 | #95 | 1 | 1 | yes | characteristic |
| 91 | C24×D4 | #95 | 1 | 1 | yes | characteristic |
| 92 | C24×D4 | #99 | 1 | 1 | yes | characteristic |
| 93 | C24×D4 | #99 | 1 | 1 | yes | characteristic |
| 94 | C24×D4 | #99 | 1 | 2 | yes | — |
| 95 | C24×D4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 96 | C24×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C24×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C24×D4 | C4×D4 | 1 | 2 | yes | — |
| 99 | C24×D4 | #120 | 1 | 2 | yes | — |
| 100 | C24×D4 | #120 | 1 | 1 | yes | characteristic |
| 101 | C24×D4 | #121 | 1 | 2 | yes | — |
| 102 | C24×D4 | #123 | 1 | 1 | yes | characteristic |
| 103 | C25×C4 | #99 | 1 | 1 | yes | characteristic |
| 104 | C25×C4 | C22⋊Q8 | 1 | 2 | yes | — |
| 105 | C25×C4 | #123 | 1 | 1 | yes | characteristic |
| 106 | C2×#534 | #99 | 1 | 2 | yes | — |
| 107 | C2×#534 | #99 | 1 | 2 | yes | — |
| 108 | C2×#534 | #99 | 1 | 4 | yes | — |
| 109 | C23×C22⋊C4 | #95 | 1 | 1 | yes | characteristic |
| 110 | C23×C22⋊C4 | #95 | 1 | 1 | yes | characteristic |
| 111 | C23×C22⋊C4 | #95 | 1 | 1 | yes | characteristic |
| 112 | C23×C22⋊C4 | #95 | 1 | 1 | yes | characteristic |
| 113 | C23×C22⋊C4 | #95 | 1 | 1 | yes | characteristic |
| 114 | C23×C22⋊C4 | #99 | 1 | 2 | yes | — |
| 115 | C23×C22⋊C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C23×C22⋊C4 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 117 | C23×C22⋊C4 | #117 | 1 | 2 | yes | — |
| 118 | C23×C22⋊C4 | C4×D4 | 1 | 2 | yes | — |
| 119 | C23×C22⋊C4 | #121 | 1 | 2 | yes | — |
| 120 | C23×C22⋊C4 | C22⋊Q8 | 1 | 2 | yes | — |
| 121 | C23×C22⋊C4 | #123 | 1 | 2 | yes | — |
| 122 | #2308 | #99 | 1 | 4 | yes | — |
| 123 | C26 | #342 | 1 | 1 | yes | characteristic |
| 124 | C26 | #342 | 1 | 1 | yes | characteristic |
| 125 | C26 | #342 | 1 | 1 | yes | characteristic |
| 126 | C26 | C2×#95 | 1 | 1 | yes | characteristic |
| 127 | C26 | C4×C22⋊C4 | 1 | 2 | yes | — |
| 128 | C26 | #379 | 1 | 1 | yes | characteristic |
| 129 | C26 | #385 | 1 | 1 | yes | characteristic |
| 130 | C26 | #385 | 1 | 1 | yes | characteristic |
| 131 | C26 | #386 | 1 | 1 | yes | characteristic |
| 132 | C26 | #386 | 1 | 2 | yes | — |
| 133 | C26 | C2×#99 | 1 | 1 | yes | characteristic |
| 134 | C26 | C2×#99 | 1 | 2 | yes | — |
| 135 | C26 | C2×#99 | 1 | 1 | yes | characteristic |
| 136 | C23×D4 | #342 | 1 | 1 | yes | characteristic |
| 137 | C23×D4 | #342 | 1 | 1 | yes | characteristic |
| 138 | C23×D4 | #342 | 1 | 1 | yes | characteristic |
| 139 | C23×D4 | #342 | 1 | 1 | yes | characteristic |
| 140 | C23×D4 | #342 | 1 | 2 | yes | — |
| 141 | C23×D4 | #342 | 1 | 2 | yes | — |
| 142 | C23×D4 | C2×#99 | 1 | 2 | yes | — |
| 143 | C23×D4 | C2×#99 | 1 | 2 | yes | — |
| 144 | C23×D4 | C2×#99 | 1 | 2 | yes | — |
| 145 | C23×D4 | C2×#99 | 1 | 2 | yes | — |
| 146 | C23×D4 | B(4,𝔽2) | 1 | 4 | yes | — |
| 147 | C23×D4 | #458 | 1 | 4 | yes | — |
| 148 | C24×C4 | #342 | 1 | 1 | yes | characteristic |
| 149 | C24×C4 | #342 | 1 | 1 | yes | characteristic |
| 150 | C24×C4 | #342 | 1 | 2 | yes | — |
| 151 | C24×C4 | #342 | 1 | 1 | yes | characteristic |
| 152 | C24×C4 | #380 | 1 | 2 | yes | — |
| 153 | C24×C4 | #381 | 1 | 1 | yes | characteristic |
| 154 | C24×C4 | #385 | 1 | 2 | yes | — |
| 155 | C24×C4 | #386 | 1 | 1 | yes | characteristic |
| 156 | C22×C22⋊C4 | #342 | 1 | 2 | yes | — |
| 157 | C22×C22⋊C4 | #342 | 1 | 2 | yes | — |
| 158 | C22×C22⋊C4 | #342 | 1 | 2 | yes | — |
| 159 | C22×C22⋊C4 | #342 | 1 | 2 | yes | — |
| 160 | C25 | #1304 | 1 | 1 | yes | characteristic |
| 161 | C25 | #1304 | 1 | 1 | yes | characteristic |
| 162 | C25 | #1304 | 1 | 2 | yes | — |
| 163 | C25 | C22⋊#95 | 1 | 1 | yes | characteristic |
| 164 | C25 | C2×#342 | 1 | 1 | yes | characteristic |
| 165 | C25 | C2×#342 | 1 | 1 | yes | characteristic |
| 166 | C25 | C2×#342 | 1 | 1 | yes | characteristic |
| 167 | C25 | C2×#342 | 1 | 1 | yes | characteristic |
| 168 | C25 | C2×#342 | 1 | 1 | yes | characteristic |
| 169 | C25 | C2×#342 | 1 | 1 | yes | characteristic |
| 170 | C25 | #1781 | 1 | 2 | yes | — |
| 171 | C25 | #1790 | 1 | 4 | yes | — |
| 172 | C25 | #1889 | 1 | 2 | yes | — |
| 173 | C25 | #1890 | 1 | 2 | yes | — |
| 174 | C25 | #1898 | 1 | 2 | yes | — |
| 175 | C25 | #1898 | 1 | 2 | yes | — |
| 176 | C25 | #1898 | 1 | 2 | yes | — |
| 177 | C25 | #1913 | 1 | 2 | yes | — |
| 178 | C25 | #1913 | 1 | 2 | yes | — |
| 179 | C23×C4 | #1304 | 1 | 1 | yes | characteristic |
| 180 | C23×C4 | #1304 | 1 | 1 | yes | characteristic |
| 181 | C23×C4 | #1304 | 1 | 2 | yes | — |
| 182 | C23×C4 | #1819 | 1 | 2 | yes | — |
| 183 | C23×C4 | #1867 | 1 | 4 | yes | — |
| 184 | C23×C4 | #1898 | 1 | 2 | yes | — |
| 185 | C24 | #7310 | 1 | 2 | yes | complemented |
| 186 | C24 | C2×#1304 | 1 | 1 | yes | characteristic |
| 187 | C24 | C2×#1304 | 1 | 1 | yes | characteristic |
| 188 | C24 | #8101 | 1 | 1 | yes | characteristic |
| 189 | C24 | #8101 | 1 | 1 | yes | characteristic |
| 190 | C24 | #8534 | 1 | 4 | yes | — |
| 191 | C24 | #9495 | 1 | 2 | yes | — |
| 192 | C24 | #9495 | 1 | 2 | yes | — |
| 193 | C24 | #9499 | 1 | 2 | yes | — |
| 194 | C24 | #9499 | 1 | 2 | yes | — |
| 195 | C24 | #10725 | 1 | 1 | yes | characteristic |
| 196 | C24 | #10735 | 1 | 1 | yes | characteristic |
| 197 | C24 | #10735 | 1 | 1 | yes | characteristic |
| 198 | C24 | #10745 | 1 | 2 | yes | — |
| 199 | C24 | #10754 | 1 | 2 | yes | — |
| 200 | C22×C4 | #7310 | 1 | 2 | yes | complemented |
| 201 | C23 | C22⋊#1304 | 1 | 1 | yes | characteristic |
| 202 | C23 | #106542 | 1 | 2 | yes | — |
| 203 | C23 | #106766 | 1 | 2 | yes | — |
| 204 | C23 | #107114 | 1 | 1 | yes | characteristic |
| 205 | C23 | #108222 | 1 | 1 | yes | characteristic |
| 206 | C23 | C2×#7310 | 1 | 1 | yes | characteristic |
| 207 | C23 | C2×#7310 | 1 | 1 | yes | characteristic |
| 208 | C23 | #128062 | 1 | 2 | yes | — |
| 209 | C23 | #139749 | 1 | 2 | yes | — |
| 210 | C23 | #139750 | 1 | 2 | yes | — |
| 211 | C23 | #139843 | 1 | 4 | yes | — |
| 212 | C22 | #72218381 | 1 | 1 | yes | characteristic |
| 213 | C22 | #72218854 | 1 | 1 | yes | characteristic |
| 214 | C22 | #72246453 | 1 | 2 | yes | — |
| 215 | C22 | #72258381 | 1 | 1 | yes | characteristic |
| 216 | C22 | #72265902 | 1 | 2 | yes | — |
| 217 | C2 | #72266198 | 1 | 1 | yes | characteristic |
| 218 | C2 | #72266373 | 1 | 2 | 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