← #72266242
Subgroups — #72266242
Order 2048.
At least 7060 conjugacy classes of subgroups; only the 820 normal subgroups are shown.
The subgroup lattice has 284 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72266241 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #72234675 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72225621 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72219377 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72219022 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #72218057 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C2×#62817 | C4 | 1 | 1 | yes | characteristic, complemented |
| 8 | C23×D42 | C22 | 1 | 1 | yes | characteristic |
| 9 | #7642350 | C4 | 1 | 2 | yes | complemented |
| 10 | #7642350 | C22 | 1 | 2 | yes | — |
| 11 | #7642273 | C22 | 1 | 2 | yes | — |
| 12 | #7642257 | C4 | 1 | 2 | yes | complemented |
| 13 | #7630945 | C4 | 1 | 2 | yes | complemented |
| 14 | #7630939 | C22 | 1 | 2 | yes | — |
| 15 | #7626202 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | C22×#1781 | C22 | 1 | 2 | yes | complemented |
| 17 | #494713 | C22 | 1 | 16 | yes | complemented |
| 18 | #392109 | C22 | 1 | 2 | yes | complemented |
| 19 | C2×#10980 | C22 | 1 | 2 | yes | complemented |
| 20 | C2×#10965 | C22 | 1 | 2 | yes | complemented |
| 21 | C2×#8542 | C22 | 1 | 2 | yes | complemented |
| 22 | C2×#8541 | C22 | 1 | 2 | yes | complemented |
| 23 | C25×D4 | D4 | 1 | 1 | yes | characteristic |
| 24 | C25×D4 | D4 | 1 | 1 | yes | characteristic |
| 25 | C25×D4 | D4 | 1 | 1 | yes | characteristic |
| 26 | C25×D4 | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×#3490 | C2×C4 | 1 | 2 | yes | — |
| 28 | C2×#3490 | C2×C4 | 1 | 2 | yes | — |
| 29 | C2×#3484 | C2×C4 | 1 | 2 | yes | — |
| 30 | C2×#3484 | D4 | 1 | 4 | yes | — |
| 31 | C23×#127 | D4 | 1 | 1 | yes | characteristic |
| 32 | C23×#121 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C23×#121 | C2×C4 | 1 | 1 | yes | characteristic |
| 34 | C23×#121 | D4 | 1 | 1 | yes | characteristic |
| 35 | C23×#121 | C23 | 1 | 1 | yes | characteristic |
| 36 | C23×#120 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C23×#120 | D4 | 1 | 1 | yes | characteristic |
| 38 | C23×#120 | D4 | 1 | 1 | yes | characteristic |
| 39 | C23×#120 | C23 | 1 | 1 | yes | characteristic |
| 40 | C23×C4×D4 | C2×C4 | 1 | 1 | yes | characteristic |
| 41 | C23×C4×D4 | D4 | 1 | 1 | yes | characteristic |
| 42 | C23×#99 | C23 | 1 | 4 | yes | complemented |
| 43 | #36718 | D4 | 1 | 4 | yes | — |
| 44 | #36717 | D4 | 1 | 4 | yes | — |
| 45 | #36703 | D4 | 1 | 8 | yes | — |
| 46 | #36614 | C2×C4 | 1 | 4 | yes | — |
| 47 | #35395 | D4 | 1 | 8 | yes | — |
| 48 | #35390 | D4 | 1 | 4 | yes | — |
| 49 | #35367 | D4 | 1 | 4 | yes | — |
| 50 | #35222 | D4 | 1 | 8 | yes | — |
| 51 | #35222 | C23 | 1 | 4 | yes | — |
| 52 | #35057 | C2×C4 | 1 | 4 | yes | — |
| 53 | #35050 | C2×C4 | 1 | 4 | yes | — |
| 54 | #34176 | C2×C4 | 1 | 2 | yes | — |
| 55 | #34176 | C2×C4 | 1 | 2 | yes | — |
| 56 | #34171 | C2×C4 | 1 | 2 | yes | — |
| 57 | #34171 | D4 | 1 | 4 | yes | — |
| 58 | C2×#1781 | C23 | 1 | 4 | yes | complemented |
| 59 | C27 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 60 | C27 | C2×D4 | 1 | 1 | yes | characteristic |
| 61 | C27 | C2×D4 | 1 | 1 | yes | characteristic |
| 62 | C27 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 2 | yes | complemented |
| 64 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 65 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 66 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 67 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 68 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 69 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 70 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 71 | C24×D4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 72 | C24×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 73 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 74 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 75 | C24×D4 | C2×D4 | 1 | 4 | yes | — |
| 76 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 78 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 79 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 80 | C24×D4 | C2×D4 | 1 | 4 | yes | — |
| 81 | C24×D4 | C24 | 1 | 1 | yes | characteristic |
| 82 | C25×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 83 | C25×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 84 | C25×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 85 | C25×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 86 | C2×#543 | C22⋊C4 | 1 | 2 | yes | complemented |
| 87 | C2×#543 | C22⋊C4 | 1 | 2 | yes | complemented |
| 88 | C2×#538 | C22⋊C4 | 1 | 4 | yes | complemented |
| 89 | C2×#538 | C22⋊C4 | 1 | 4 | yes | complemented |
| 90 | C2×#535 | C22⋊C4 | 1 | 4 | yes | complemented |
| 91 | C2×#535 | C2×D4 | 1 | 8 | yes | — |
| 92 | C2×#534 | C22⋊C4 | 1 | 2 | yes | complemented |
| 93 | C2×#534 | C22⋊C4 | 1 | 2 | yes | complemented |
| 94 | C2×#525 | C22⋊C4 | 1 | 8 | yes | complemented |
| 95 | C2×#525 | C22×C4 | 1 | 4 | yes | — |
| 96 | C22×#121 | C2×D4 | 1 | 4 | yes | — |
| 97 | C22×#121 | C2×D4 | 1 | 4 | yes | — |
| 98 | C22×#120 | C2×D4 | 1 | 4 | yes | — |
| 99 | C22×#120 | C2×D4 | 1 | 4 | yes | — |
| 100 | C2×#518 | C22⋊C4 | 1 | 2 | yes | complemented |
| 101 | C2×#518 | C22⋊C4 | 1 | 4 | yes | complemented |
| 102 | C23×C4⋊C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 103 | C23×C4⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 104 | C23×C22⋊C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 105 | C23×C22⋊C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 106 | C23×C22⋊C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 107 | C23×C22⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 108 | C23×C22⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C23×C22⋊C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C23×C22⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C23×C22⋊C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C23×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 113 | #2591 | C22⋊C4 | 1 | 8 | yes | complemented |
| 114 | #2583 | C22⋊C4 | 1 | 8 | yes | complemented |
| 115 | #2561 | C22⋊C4 | 1 | 8 | yes | complemented |
| 116 | #2539 | C22⋊C4 | 1 | 8 | yes | complemented |
| 117 | #2435 | C2×D4 | 1 | 8 | yes | — |
| 118 | #2435 | C2×D4 | 1 | 8 | yes | — |
| 119 | #2433 | C2×D4 | 1 | 8 | yes | — |
| 120 | #2433 | C2×D4 | 1 | 8 | yes | — |
| 121 | #2312 | C22⋊C4 | 1 | 4 | yes | complemented |
| 122 | #2312 | C22⋊C4 | 1 | 4 | yes | complemented |
| 123 | #2309 | C22⋊C4 | 1 | 4 | yes | complemented |
| 124 | #2309 | C2×D4 | 1 | 8 | yes | — |
| 125 | #2280 | C22⋊C4 | 1 | 8 | yes | complemented |
| 126 | #2280 | C22×C4 | 1 | 4 | yes | — |
| 127 | #2273 | C22⋊C4 | 1 | 4 | yes | complemented |
| 128 | #2273 | C22⋊C4 | 1 | 4 | yes | complemented |
| 129 | C26 | #99 | 1 | 4 | yes | — |
| 130 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 131 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 132 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 133 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 134 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 135 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 136 | C26 | #120 | 1 | 4 | yes | — |
| 137 | C26 | #120 | 1 | 2 | yes | — |
| 138 | C26 | #120 | 1 | 2 | yes | — |
| 139 | C26 | #120 | 1 | 2 | yes | — |
| 140 | C26 | #120 | 1 | 2 | yes | — |
| 141 | C26 | C22×D4 | 1 | 1 | yes | characteristic |
| 142 | C26 | C22×D4 | 1 | 1 | yes | characteristic |
| 143 | C22×#41 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 144 | C23×D4 | #99 | 1 | 4 | yes | — |
| 145 | C23×D4 | #99 | 1 | 4 | yes | — |
| 146 | C23×D4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 147 | C23×D4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 148 | C23×D4 | #120 | 1 | 2 | yes | — |
| 149 | C23×D4 | #120 | 1 | 2 | yes | — |
| 150 | C23×D4 | #120 | 1 | 4 | yes | — |
| 151 | C23×D4 | #120 | 1 | 2 | yes | — |
| 152 | C23×D4 | #120 | 1 | 2 | yes | — |
| 153 | C23×D4 | #120 | 1 | 2 | yes | — |
| 154 | C23×D4 | #120 | 1 | 2 | yes | — |
| 155 | C23×D4 | #120 | 1 | 2 | yes | — |
| 156 | C23×D4 | #120 | 1 | 2 | yes | — |
| 157 | C23×D4 | #120 | 1 | 4 | yes | — |
| 158 | C23×D4 | C22×D4 | 1 | 4 | yes | — |
| 159 | C23×D4 | C22×D4 | 1 | 4 | yes | — |
| 160 | C23×D4 | C22×D4 | 1 | 4 | yes | — |
| 161 | C24×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 162 | C24×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 163 | C24×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 164 | C24×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 165 | C24×C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 166 | C24×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 167 | C24×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 168 | C24×C4 | #120 | 1 | 4 | yes | — |
| 169 | C24×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 170 | C2×#127 | #120 | 1 | 16 | yes | — |
| 171 | C2×#124 | #120 | 1 | 16 | yes | — |
| 172 | C2×C22⋊Q8 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 173 | C2×C22⋊Q8 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 174 | C2×#121 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 175 | C2×#121 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 176 | C2×#117 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 177 | C2×#117 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 178 | C22×C4⋊C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 179 | C22×C22⋊C4 | #99 | 1 | 4 | yes | — |
| 180 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 181 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 182 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 183 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 184 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 185 | C22×C22⋊C4 | #120 | 1 | 4 | yes | — |
| 186 | C22×C22⋊C4 | #120 | 1 | 4 | yes | — |
| 187 | #390 | #120 | 1 | 16 | yes | — |
| 188 | #390 | #120 | 1 | 16 | yes | — |
| 189 | C4×C22⋊C4 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 190 | C4×C22⋊C4 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 191 | C25 | #379 | 1 | 2 | yes | — |
| 192 | C25 | #379 | 1 | 1 | yes | characteristic |
| 193 | C25 | #379 | 1 | 2 | yes | — |
| 194 | C25 | #379 | 1 | 1 | yes | characteristic |
| 195 | C25 | #379 | 1 | 1 | yes | characteristic |
| 196 | C25 | #379 | 1 | 2 | yes | — |
| 197 | C25 | #379 | 1 | 1 | yes | characteristic |
| 198 | C25 | #379 | 1 | 1 | yes | characteristic |
| 199 | C25 | #379 | 1 | 1 | yes | characteristic |
| 200 | C25 | C2×#99 | 1 | 2 | yes | — |
| 201 | C25 | C2×#99 | 1 | 2 | yes | — |
| 202 | C25 | C2×#99 | 1 | 2 | yes | — |
| 203 | C25 | C2×#99 | 1 | 2 | yes | — |
| 204 | C25 | C2×#99 | 1 | 2 | yes | — |
| 205 | C25 | C2×#99 | 1 | 2 | yes | — |
| 206 | C25 | C2×#99 | 1 | 2 | yes | — |
| 207 | C25 | C2×#99 | 1 | 2 | yes | — |
| 208 | C25 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 209 | C25 | C2×#120 | 1 | 2 | yes | — |
| 210 | C25 | C2×#120 | 1 | 2 | yes | — |
| 211 | C25 | C2×#120 | 1 | 2 | yes | — |
| 212 | C25 | C2×#120 | 1 | 2 | yes | — |
| 213 | C25 | C2×#120 | 1 | 2 | yes | — |
| 214 | C25 | C2×#120 | 1 | 2 | yes | — |
| 215 | C22×Q8 | #379 | 1 | 4 | yes | — |
| 216 | C22×D4 | #379 | 1 | 4 | yes | — |
| 217 | C22×D4 | B(4,𝔽2) | 1 | 4 | yes | — |
| 218 | C22×D4 | #458 | 1 | 4 | yes | — |
| 219 | C22×D4 | C2×#120 | 1 | 16 | yes | — |
| 220 | C23×C4 | #379 | 1 | 2 | yes | — |
| 221 | C23×C4 | #379 | 1 | 2 | yes | — |
| 222 | C23×C4 | #379 | 1 | 2 | yes | — |
| 223 | C23×C4 | #379 | 1 | 2 | yes | — |
| 224 | C23×C4 | #379 | 1 | 2 | yes | — |
| 225 | C23×C4 | #379 | 1 | 2 | yes | — |
| 226 | C23×C4 | C2×#99 | 1 | 2 | yes | — |
| 227 | C23×C4 | C2×#99 | 1 | 2 | yes | — |
| 228 | C23×C4 | C2×#99 | 1 | 2 | yes | — |
| 229 | C23×C4 | C2×#99 | 1 | 2 | yes | — |
| 230 | C23×C4 | C22×C22⋊C4 | 1 | 2 | yes | — |
| 231 | C23×C4 | C22×C22⋊C4 | 1 | 2 | yes | — |
| 232 | C23×C4 | C22×C22⋊C4 | 1 | 2 | yes | — |
| 233 | C2×C4⋊C4 | #379 | 1 | 4 | yes | — |
| 234 | C2×C4⋊C4 | #379 | 1 | 4 | yes | — |
| 235 | C2×C42 | #379 | 1 | 16 | yes | — |
| 236 | C24 | #1781 | 1 | 2 | yes | — |
| 237 | C24 | #1781 | 1 | 2 | yes | — |
| 238 | C24 | #1781 | 1 | 4 | yes | — |
| 239 | C24 | #1781 | 1 | 2 | yes | — |
| 240 | C24 | #1781 | 1 | 2 | yes | — |
| 241 | C24 | #1781 | 1 | 2 | yes | — |
| 242 | C24 | #1781 | 1 | 2 | yes | — |
| 243 | C24 | #1790 | 1 | 4 | yes | — |
| 244 | C24 | C2×#379 | 1 | 1 | yes | characteristic |
| 245 | C24 | C2×#379 | 1 | 1 | yes | characteristic |
| 246 | C24 | C2×#379 | 1 | 1 | yes | characteristic |
| 247 | C24 | #2846 | 1 | 4 | yes | — |
| 248 | C24 | #2846 | 1 | 4 | yes | — |
| 249 | C24 | C22×#99 | 1 | 1 | yes | characteristic |
| 250 | C24 | C22×#99 | 1 | 1 | yes | characteristic |
| 251 | C24 | C22×#99 | 1 | 1 | yes | characteristic |
| 252 | C24 | C22×#99 | 1 | 1 | yes | characteristic |
| 253 | C24 | C2×B(4,𝔽2) | 1 | 2 | yes | — |
| 254 | C24 | C2×#458 | 1 | 2 | yes | — |
| 255 | C24 | #3026 | 1 | 4 | yes | — |
| 256 | C22×C4 | #1781 | 1 | 4 | yes | — |
| 257 | C22×C4 | #1781 | 1 | 4 | yes | — |
| 258 | C22×C4 | #1790 | 1 | 4 | yes | — |
| 259 | C22×C4 | C2×#379 | 1 | 4 | yes | — |
| 260 | C23 | #8534 | 1 | 4 | yes | — |
| 261 | C23 | #12707 | 1 | 4 | yes | complemented |
| 262 | C23 | #14679 | 1 | 1 | yes | characteristic |
| 263 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 264 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 265 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 266 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 267 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 268 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 269 | C23 | C2×#1790 | 1 | 2 | yes | — |
| 270 | C23 | #20609 | 1 | 2 | yes | — |
| 271 | C23 | #32898 | 1 | 4 | yes | — |
| 272 | C23 | #32900 | 1 | 4 | yes | — |
| 273 | C22 | C2×#8534 | 1 | 1 | yes | characteristic |
| 274 | C22 | #187928 | 1 | 2 | yes | — |
| 275 | C22 | #187944 | 1 | 1 | yes | characteristic |
| 276 | C22 | #392109 | 1 | 1 | yes | characteristic |
| 277 | C22 | #392125 | 1 | 2 | yes | — |
| 278 | C22 | C2×#12707 | 1 | 2 | yes | — |
| 279 | C22 | #493398 | 1 | 2 | yes | — |
| 280 | C2 | #72210555 | 1 | 1 | yes | characteristic |
| 281 | C2 | #72218467 | 1 | 1 | yes | characteristic |
| 282 | C2 | #72258026 | 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