← #72234200
Subgroups — #72234200
Order 2048.
At least 6187 conjugacy classes of subgroups; only the 860 normal subgroups are shown.
The subgroup lattice has 244 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72234187 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #72234164 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72233910 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #120×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #72232755 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #72232738 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #72232733 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C24×H(5,𝔽2) | C4 | 1 | 1 | yes | characteristic, complemented |
| 9 | C2×D4×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 10 | #7631287 | C4 | 1 | 3 | yes | complemented |
| 11 | #7631286 | C22 | 1 | 3 | yes | — |
| 12 | #7630939 | C4 | 1 | 3 | yes | complemented |
| 13 | C2×D4×#120 | C22 | 1 | 3 | yes | — |
| 14 | C22⋊C4×H(5,𝔽2) | C4 | 1 | 1 | yes | characteristic, complemented |
| 15 | C23×AΣL(1,𝔽16) | C22 | 1 | 1 | yes | characteristic, complemented |
| 16 | #496225 | C22 | 1 | 6 | yes | complemented |
| 17 | C2×#13086 | C22 | 1 | 3 | yes | complemented |
| 18 | C2×#13082 | C22 | 1 | 3 | yes | complemented |
| 19 | C2×#12687 | C22 | 1 | 3 | yes | complemented |
| 20 | #359583 | C22 | 1 | 3 | yes | complemented |
| 21 | #359579 | C22 | 1 | 3 | yes | complemented |
| 22 | #188087 | C22 | 1 | 3 | yes | complemented |
| 23 | #188035 | C22 | 1 | 3 | yes | complemented |
| 24 | C23×H(5,𝔽2) | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C23×H(5,𝔽2) | C2×C4 | 1 | 3 | yes | — |
| 26 | C23×H(5,𝔽2) | D4 | 1 | 1 | yes | characteristic |
| 27 | C25×D4 | C2×C4 | 1 | 3 | yes | — |
| 28 | C2×#3496 | D4 | 1 | 3 | yes | — |
| 29 | C2×#3484 | C2×C4 | 1 | 3 | yes | — |
| 30 | C2×#3484 | D4 | 1 | 3 | yes | — |
| 31 | C22×D42 | C23 | 1 | 3 | yes | — |
| 32 | C23×#120 | D4 | 1 | 3 | yes | — |
| 33 | C23×#120 | C23 | 1 | 1 | yes | characteristic |
| 34 | C2×C4×H(5,𝔽2) | D4 | 1 | 1 | yes | characteristic |
| 35 | C23×#100 | C23 | 1 | 1 | yes | characteristic, complemented |
| 36 | C23×#99 | C23 | 1 | 1 | yes | characteristic, complemented |
| 37 | C2×#2847 | D4 | 1 | 3 | yes | — |
| 38 | C2×#2846 | C2×C4 | 1 | 3 | yes | — |
| 39 | C2×#2846 | D4 | 1 | 3 | yes | — |
| 40 | #35527 | D4 | 1 | 6 | yes | — |
| 41 | #35512 | D4 | 1 | 6 | yes | — |
| 42 | #35347 | D4 | 1 | 6 | yes | — |
| 43 | #35346 | D4 | 1 | 6 | yes | — |
| 44 | #34980 | C2×C4 | 1 | 6 | yes | — |
| 45 | C2×#2433 | D4 | 1 | 3 | yes | — |
| 46 | C2×#2408 | D4 | 1 | 3 | yes | — |
| 47 | C2×#2410 | C2×C4 | 1 | 3 | yes | — |
| 48 | C2×#2410 | D4 | 1 | 3 | yes | — |
| 49 | C2×#2403 | D4 | 1 | 3 | yes | — |
| 50 | C2×#2403 | C23 | 1 | 3 | yes | — |
| 51 | #34171 | C2×C4 | 1 | 3 | yes | — |
| 52 | C2×D4×C22⋊C4 | C2×C4 | 1 | 3 | yes | — |
| 53 | C2×C4×#120 | D4 | 1 | 3 | yes | — |
| 54 | C2×#1783 | C23 | 1 | 3 | yes | complemented |
| 55 | C2×#1781 | C23 | 1 | 3 | yes | complemented |
| 56 | C27 | C22⋊C4 | 1 | 3 | yes | complemented |
| 57 | C27 | C22×C4 | 1 | 1 | yes | characteristic |
| 58 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 59 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 3 | yes | — |
| 60 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 3 | yes | — |
| 61 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 6 | yes | complemented |
| 62 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 6 | yes | complemented |
| 63 | C24×D4 | C22⋊C4 | 1 | 3 | yes | complemented |
| 64 | C24×D4 | C22⋊C4 | 1 | 3 | yes | complemented |
| 65 | C24×D4 | C22×C4 | 1 | 3 | yes | — |
| 66 | C24×D4 | C22×C4 | 1 | 3 | yes | — |
| 67 | C24×D4 | C2×D4 | 1 | 3 | yes | — |
| 68 | C24×D4 | C2×D4 | 1 | 3 | yes | — |
| 69 | C24×D4 | C24 | 1 | 1 | yes | characteristic |
| 70 | C25×C4 | C22⋊C4 | 1 | 3 | yes | complemented |
| 71 | C2×#534 | C22⋊C4 | 1 | 6 | yes | — |
| 72 | C22×#127 | C2×D4 | 1 | 3 | yes | — |
| 73 | C22×#121 | C2×D4 | 1 | 3 | yes | — |
| 74 | C22×#121 | C2×D4 | 1 | 3 | yes | — |
| 75 | C22×#121 | C2×D4 | 1 | 3 | yes | — |
| 76 | C22×#121 | C2×D4 | 1 | 3 | yes | — |
| 77 | C22×#120 | C22⋊C4 | 1 | 3 | yes | complemented |
| 78 | C22×#120 | C22⋊C4 | 1 | 3 | yes | complemented |
| 79 | C22×#120 | C22⋊C4 | 1 | 3 | yes | complemented |
| 80 | C22×#120 | C22×C4 | 1 | 3 | yes | — |
| 81 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 82 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 83 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 84 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 85 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 86 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 87 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 88 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 89 | C22×#120 | C2×D4 | 1 | 3 | yes | — |
| 90 | C2×#518 | C22⋊C4 | 1 | 3 | yes | — |
| 91 | C22×C4×D4 | C2×D4 | 1 | 3 | yes | — |
| 92 | C22×C4×D4 | C2×D4 | 1 | 3 | yes | — |
| 93 | C23×C22⋊C4 | C22⋊C4 | 1 | 3 | yes | complemented |
| 94 | C23×C22⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 95 | #2405 | C22⋊C4 | 1 | 12 | yes | complemented |
| 96 | #2403 | C22⋊C4 | 1 | 12 | yes | complemented |
| 97 | #2403 | C2×D4 | 1 | 12 | yes | — |
| 98 | #2403 | C2×D4 | 1 | 12 | yes | — |
| 99 | #2309 | C22⋊C4 | 1 | 6 | yes | complemented |
| 100 | #2308 | C22⋊C4 | 1 | 6 | yes | complemented |
| 101 | C2×#386 | C22⋊C4 | 1 | 3 | yes | complemented |
| 102 | C2×#386 | C2×D4 | 1 | 3 | yes | — |
| 103 | C2×#386 | C2×D4 | 1 | 3 | yes | — |
| 104 | C2×#385 | C22⋊C4 | 1 | 3 | yes | complemented |
| 105 | C2×#380 | C22⋊C4 | 1 | 3 | yes | complemented |
| 106 | C2×#379 | C22⋊C4 | 1 | 3 | yes | complemented |
| 107 | C2×#379 | C22⋊C4 | 1 | 3 | yes | complemented |
| 108 | C2×#379 | C22⋊C4 | 1 | 3 | yes | complemented |
| 109 | C2×#379 | C22⋊C4 | 1 | 3 | yes | complemented |
| 110 | C2×#379 | C22×C4 | 1 | 3 | yes | — |
| 111 | C2×C4×C22⋊C4 | C22⋊C4 | 1 | 3 | yes | complemented |
| 112 | C26 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 113 | C26 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 114 | C26 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 115 | C26 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 116 | C26 | #120 | 1 | 3 | yes | — |
| 117 | C26 | C23×C4 | 1 | 1 | yes | characteristic |
| 118 | C26 | C22×D4 | 1 | 1 | yes | characteristic |
| 119 | C2×H(5,𝔽2) | #99 | 1 | 4 | yes | — |
| 120 | C2×H(5,𝔽2) | #99 | 1 | 12 | yes | — |
| 121 | C23×D4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 122 | C23×D4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 123 | C23×D4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 124 | C23×D4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 125 | C23×D4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 126 | C23×D4 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 127 | C23×D4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 128 | C23×D4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 129 | C23×D4 | #120 | 1 | 3 | yes | — |
| 130 | C23×D4 | #120 | 1 | 3 | yes | — |
| 131 | C23×D4 | #120 | 1 | 3 | yes | — |
| 132 | C23×D4 | #120 | 1 | 3 | yes | — |
| 133 | C23×D4 | #120 | 1 | 3 | yes | — |
| 134 | C23×D4 | C22×D4 | 1 | 3 | yes | — |
| 135 | C23×D4 | C22×D4 | 1 | 3 | yes | — |
| 136 | C23×D4 | C22×D4 | 1 | 3 | yes | — |
| 137 | C23×D4 | C22×D4 | 1 | 3 | yes | — |
| 138 | C24×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 139 | C24×C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 140 | C24×C4 | #120 | 1 | 3 | yes | — |
| 141 | C24×C4 | #120 | 1 | 3 | yes | — |
| 142 | C24×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 143 | C2×#120 | #120 | 1 | 3 | yes | — |
| 144 | C2×#120 | #120 | 1 | 3 | yes | — |
| 145 | C2×#120 | #120 | 1 | 3 | yes | — |
| 146 | C2×#120 | #120 | 1 | 1 | yes | characteristic |
| 147 | C2×#120 | #120 | 1 | 6 | yes | — |
| 148 | C2×#120 | #120 | 1 | 6 | yes | — |
| 149 | C2×#120 | #120 | 1 | 6 | yes | — |
| 150 | C2×#120 | #120 | 1 | 12 | yes | — |
| 151 | C22×C4⋊C4 | #120 | 1 | 3 | yes | — |
| 152 | C22×C4⋊C4 | #120 | 1 | 3 | yes | — |
| 153 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 154 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 155 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 156 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 157 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 158 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 159 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 3 | yes | — |
| 160 | C22×C22⋊C4 | #120 | 1 | 3 | yes | — |
| 161 | C22×C22⋊C4 | #120 | 1 | 3 | yes | — |
| 162 | C22×C22⋊C4 | #120 | 1 | 3 | yes | — |
| 163 | C22×C22⋊C4 | #120 | 1 | 3 | yes | — |
| 164 | C22×C22⋊C4 | #120 | 1 | 3 | yes | — |
| 165 | C22×C42 | #120 | 1 | 3 | yes | — |
| 166 | #386 | #120 | 1 | 12 | yes | — |
| 167 | #386 | #120 | 1 | 6 | yes | — |
| 168 | #386 | #120 | 1 | 6 | yes | — |
| 169 | #379 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 170 | C25 | #379 | 1 | 3 | yes | — |
| 171 | C25 | #379 | 1 | 3 | yes | — |
| 172 | C25 | #379 | 1 | 3 | yes | — |
| 173 | C25 | #379 | 1 | 1 | yes | characteristic |
| 174 | C25 | #379 | 1 | 3 | yes | — |
| 175 | C25 | #379 | 1 | 6 | yes | — |
| 176 | C25 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 177 | C25 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 178 | C25 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 179 | C25 | C2×#120 | 1 | 3 | yes | — |
| 180 | C25 | C2×#120 | 1 | 3 | yes | — |
| 181 | H(5,𝔽2) | AΣL(1,𝔽16) | 1 | 16 | yes | complemented |
| 182 | C22×D4 | C2×#99 | 1 | 6 | yes | — |
| 183 | C22×D4 | C2×#99 | 1 | 6 | yes | — |
| 184 | C22×D4 | C2×#99 | 1 | 12 | yes | — |
| 185 | C22×D4 | C2×#120 | 1 | 3 | yes | — |
| 186 | C22×D4 | C2×#120 | 1 | 3 | yes | — |
| 187 | C22×D4 | C2×#120 | 1 | 3 | yes | — |
| 188 | C22×D4 | C2×#120 | 1 | 6 | yes | — |
| 189 | C22×D4 | C2×#120 | 1 | 1 | yes | characteristic |
| 190 | C23×C4 | #379 | 1 | 3 | yes | — |
| 191 | C23×C4 | #379 | 1 | 3 | yes | — |
| 192 | C23×C4 | #379 | 1 | 3 | yes | — |
| 193 | C23×C4 | #379 | 1 | 3 | yes | — |
| 194 | C23×C4 | #379 | 1 | 3 | yes | — |
| 195 | C23×C4 | #379 | 1 | 6 | yes | — |
| 196 | C23×C4 | C2×#120 | 1 | 3 | yes | — |
| 197 | C23×C4 | C2×#120 | 1 | 3 | yes | — |
| 198 | #120 | B(4,𝔽2) | 1 | 2 | yes | — |
| 199 | #120 | #458 | 1 | 2 | yes | — |
| 200 | C2×C22⋊C4 | #379 | 1 | 3 | yes | — |
| 201 | C2×C22⋊C4 | #379 | 1 | 1 | yes | characteristic |
| 202 | C2×C22⋊C4 | #379 | 1 | 6 | yes | — |
| 203 | C2×C22⋊C4 | #379 | 1 | 6 | yes | — |
| 204 | C24 | #1781 | 1 | 6 | yes | — |
| 205 | C24 | #1781 | 1 | 6 | yes | — |
| 206 | C24 | #1790 | 1 | 2 | yes | complemented |
| 207 | C24 | C2×#379 | 1 | 3 | yes | — |
| 208 | C24 | C2×#379 | 1 | 3 | yes | — |
| 209 | C24 | C2×#379 | 1 | 1 | yes | characteristic |
| 210 | C24 | #2846 | 1 | 3 | yes | — |
| 211 | C24 | #2846 | 1 | 1 | yes | characteristic |
| 212 | C24 | C22×#99 | 1 | 3 | yes | — |
| 213 | C24 | C22×#99 | 1 | 1 | yes | characteristic |
| 214 | C2×D4 | C2×AΣL(1,𝔽16) | 1 | 24 | yes | — |
| 215 | C2×D4 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 216 | C2×D4 | C2×#458 | 1 | 1 | yes | characteristic |
| 217 | C2×D4 | #3026 | 1 | 1 | yes | characteristic |
| 218 | C2×D4 | #3026 | 1 | 1 | yes | characteristic |
| 219 | C22×C4 | #1781 | 1 | 6 | yes | — |
| 220 | C22×C4 | #1781 | 1 | 6 | yes | — |
| 221 | C22×C4 | #2846 | 1 | 3 | yes | — |
| 222 | C22×C4 | #2846 | 1 | 1 | yes | characteristic |
| 223 | C22⋊C4 | #1790 | 1 | 2 | yes | complemented |
| 224 | C23 | #12687 | 1 | 12 | yes | complemented |
| 225 | C23 | #14679 | 1 | 1 | yes | characteristic |
| 226 | C23 | C2×#1781 | 1 | 3 | yes | — |
| 227 | C23 | C2×#1781 | 1 | 3 | yes | — |
| 228 | C23 | C2×#1790 | 1 | 1 | yes | characteristic |
| 229 | C23 | #20609 | 1 | 1 | yes | characteristic |
| 230 | C23 | #32898 | 1 | 1 | yes | characteristic |
| 231 | C23 | #32900 | 1 | 1 | yes | characteristic |
| 232 | C23 | C22×AΣL(1,𝔽16) | 1 | 4 | yes | — |
| 233 | C2×C4 | #12687 | 1 | 12 | yes | complemented |
| 234 | C2×C4 | #32898 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | #32900 | 1 | 1 | yes | characteristic |
| 236 | C22 | #392109 | 1 | 1 | yes | characteristic |
| 237 | C22 | #392125 | 1 | 1 | yes | characteristic |
| 238 | C22 | #392135 | 1 | 1 | yes | characteristic |
| 239 | C22 | C2×#12687 | 1 | 6 | yes | — |
| 240 | C2 | #72215179 | 1 | 1 | yes | characteristic |
| 241 | C2 | #72234164 | 1 | 1 | yes | characteristic |
| 242 | C2 | #72234199 | 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