← #72340451
Subgroups — #72340451
Order 2048.
Only the 830 normal subgroups are shown; non-normal subgroups are omitted.
The subgroup lattice has 256 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72340431 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #72311883 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C2×#128062 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #72310930 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#392135 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #72213462 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C22×H(7,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 8 | #7630502 | C4 | 1 | 2 | yes | complemented |
| 9 | #7630502 | C22 | 1 | 1 | yes | characteristic |
| 10 | #6628902 | C4 | 1 | 2 | yes | complemented |
| 11 | #6628902 | C22 | 1 | 2 | yes | — |
| 12 | #6628901 | C4 | 1 | 2 | yes | complemented |
| 13 | #6628901 | C22 | 1 | 2 | yes | — |
| 14 | C2×#20586 | C4 | 1 | 2 | yes | complemented |
| 15 | C2×#20586 | C22 | 1 | 1 | yes | characteristic |
| 16 | C2×#20585 | C22 | 1 | 4 | yes | — |
| 17 | #496205 | C22 | 1 | 4 | yes | complemented |
| 18 | #496203 | C22 | 1 | 4 | yes | complemented |
| 19 | #392135 | C22 | 1 | 8 | yes | — |
| 20 | #128062 | C22 | 1 | 8 | yes | complemented |
| 21 | C2×H(7,𝔽2) | C2×C4 | 1 | 2 | yes | — |
| 22 | C2×H(7,𝔽2) | D4 | 1 | 4 | yes | — |
| 23 | C2×H(7,𝔽2) | C23 | 1 | 2 | yes | — |
| 24 | C23×H(5,𝔽2) | C2×C4 | 1 | 2 | yes | — |
| 25 | C23×H(5,𝔽2) | D4 | 1 | 4 | yes | — |
| 26 | C23×H(5,𝔽2) | C23 | 1 | 1 | yes | characteristic |
| 27 | C2×#3449 | D4 | 1 | 2 | yes | — |
| 28 | C22×#535 | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C23×#99 | D4 | 1 | 8 | yes | complemented |
| 30 | #35616 | C2×C4 | 1 | 1 | yes | characteristic |
| 31 | #35616 | C2×C4 | 1 | 4 | yes | complemented |
| 32 | #35616 | D4 | 1 | 2 | yes | — |
| 33 | #35616 | C23 | 1 | 1 | yes | characteristic |
| 34 | #35527 | C2×C4 | 1 | 1 | yes | characteristic |
| 35 | #35527 | C2×C4 | 1 | 4 | yes | complemented |
| 36 | #35527 | D4 | 1 | 2 | yes | — |
| 37 | #35527 | C23 | 1 | 1 | yes | characteristic |
| 38 | #34044 | D4 | 1 | 2 | yes | — |
| 39 | C2×#2309 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C2×#2309 | C2×C4 | 1 | 2 | yes | — |
| 41 | #20586 | C2×C4 | 1 | 2 | yes | — |
| 42 | #20586 | C2×C4 | 1 | 4 | yes | complemented |
| 43 | #20586 | C2×C4 | 1 | 4 | yes | complemented |
| 44 | #20586 | D4 | 1 | 4 | yes | — |
| 45 | #20586 | C23 | 1 | 2 | yes | — |
| 46 | #20585 | D4 | 1 | 8 | yes | complemented |
| 47 | #20585 | C23 | 1 | 8 | yes | — |
| 48 | C2×#1781 | D4 | 1 | 4 | yes | — |
| 49 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 4 | yes | — |
| 50 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 8 | yes | — |
| 51 | C22×H(5,𝔽2) | C22×C4 | 1 | 1 | yes | characteristic |
| 52 | C22×H(5,𝔽2) | C22×C4 | 1 | 4 | yes | — |
| 53 | C22×H(5,𝔽2) | C2×D4 | 1 | 2 | yes | — |
| 54 | C22×H(5,𝔽2) | C2×D4 | 1 | 2 | yes | — |
| 55 | C22×H(5,𝔽2) | C2×D4 | 1 | 8 | yes | — |
| 56 | C22×H(5,𝔽2) | C2×D4 | 1 | 2 | yes | — |
| 57 | C22×H(5,𝔽2) | C2×D4 | 1 | 4 | yes | — |
| 58 | C22×H(5,𝔽2) | #41 | 1 | 4 | yes | — |
| 59 | C22×H(5,𝔽2) | C24 | 1 | 1 | yes | characteristic |
| 60 | C23×#41 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 61 | C23×#41 | C2×D4 | 1 | 2 | yes | — |
| 62 | C23×#41 | #41 | 1 | 4 | yes | — |
| 63 | C24×D4 | C22⋊C4 | 1 | 4 | yes | — |
| 64 | C24×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 65 | C24×D4 | C2×D4 | 1 | 4 | yes | — |
| 66 | C24×D4 | C2×D4 | 1 | 2 | yes | — |
| 67 | C24×D4 | C2×D4 | 1 | 2 | yes | — |
| 68 | C2×#536 | C22⋊C4 | 1 | 2 | yes | complemented |
| 69 | C2×#536 | C2×D4 | 1 | 2 | yes | — |
| 70 | C2×#535 | C22⋊C4 | 1 | 2 | yes | complemented |
| 71 | C2×#535 | C2×D4 | 1 | 2 | yes | — |
| 72 | C22×#127 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 73 | C22×#127 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C22×#124 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 75 | C22×#124 | C22×C4 | 1 | 1 | yes | characteristic |
| 76 | C22×#117 | C22⋊C4 | 1 | 2 | yes | complemented |
| 77 | C22×#117 | C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 78 | C22×#99 | C2×D4 | 1 | 16 | yes | — |
| 79 | #2309 | C22⋊C4 | 1 | 2 | yes | complemented |
| 80 | #2309 | C22⋊C4 | 1 | 4 | yes | complemented |
| 81 | #2309 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | #2309 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | #2309 | C22×C4 | 1 | 4 | yes | — |
| 84 | #2297 | C22⋊C4 | 1 | 2 | yes | complemented |
| 85 | #2297 | C22⋊C4 | 1 | 2 | yes | complemented |
| 86 | #2297 | C2×D4 | 1 | 2 | yes | — |
| 87 | #2297 | C2×D4 | 1 | 2 | yes | — |
| 88 | C2×#379 | C22⋊C4 | 1 | 4 | yes | complemented |
| 89 | #1781 | C22⋊C4 | 1 | 8 | yes | complemented |
| 90 | #1781 | C2×D4 | 1 | 8 | yes | — |
| 91 | C26 | #99 | 1 | 4 | yes | — |
| 92 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 93 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 94 | C26 | #120 | 1 | 2 | yes | — |
| 95 | C26 | #127 | 1 | 2 | yes | — |
| 96 | C22×#41 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 97 | C22×#41 | C4×D4 | 1 | 4 | yes | — |
| 98 | C22×#41 | #120 | 1 | 4 | yes | — |
| 99 | C22×#41 | #120 | 1 | 2 | yes | — |
| 100 | C22×#41 | #121 | 1 | 4 | yes | — |
| 101 | C22×#41 | #121 | 1 | 4 | yes | — |
| 102 | C22×#41 | #121 | 1 | 4 | yes | — |
| 103 | C22×#41 | #121 | 1 | 4 | yes | — |
| 104 | C22×#41 | #123 | 1 | 4 | yes | — |
| 105 | C22×#41 | #123 | 1 | 2 | yes | — |
| 106 | C22×#41 | #123 | 1 | 4 | yes | — |
| 107 | C22×#41 | #124 | 1 | 4 | yes | — |
| 108 | C22×#41 | C22×D4 | 1 | 2 | yes | — |
| 109 | C22×#41 | C2×#41 | 1 | 4 | yes | — |
| 110 | C23×Q8 | C2×C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 111 | C23×Q8 | #123 | 1 | 2 | yes | — |
| 112 | C23×D4 | #99 | 1 | 8 | yes | — |
| 113 | C23×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 114 | C23×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 115 | C23×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 116 | C23×D4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 117 | C23×D4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 118 | C23×D4 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 119 | C23×D4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 120 | C23×D4 | C4×D4 | 1 | 4 | yes | — |
| 121 | C23×D4 | C4×D4 | 1 | 4 | yes | — |
| 122 | C23×D4 | C4×D4 | 1 | 4 | yes | — |
| 123 | C23×D4 | #120 | 1 | 2 | yes | — |
| 124 | C23×D4 | #120 | 1 | 4 | yes | — |
| 125 | C23×D4 | #120 | 1 | 2 | yes | — |
| 126 | C23×D4 | #120 | 1 | 4 | yes | — |
| 127 | C23×D4 | #120 | 1 | 4 | yes | — |
| 128 | C23×D4 | #120 | 1 | 4 | yes | — |
| 129 | C23×D4 | #120 | 1 | 4 | yes | — |
| 130 | C23×D4 | #121 | 1 | 4 | yes | — |
| 131 | C23×D4 | #121 | 1 | 4 | yes | — |
| 132 | C23×D4 | #121 | 1 | 4 | yes | — |
| 133 | C23×D4 | #121 | 1 | 4 | yes | — |
| 134 | C23×D4 | #121 | 1 | 4 | yes | — |
| 135 | C23×D4 | #121 | 1 | 4 | yes | — |
| 136 | C23×D4 | #121 | 1 | 4 | yes | — |
| 137 | C23×D4 | #123 | 1 | 2 | yes | — |
| 138 | C23×D4 | #124 | 1 | 2 | yes | — |
| 139 | C23×D4 | #127 | 1 | 2 | yes | — |
| 140 | C23×D4 | #127 | 1 | 4 | yes | — |
| 141 | C23×D4 | C23×C4 | 1 | 1 | yes | characteristic |
| 142 | C23×D4 | C22×D4 | 1 | 4 | yes | — |
| 143 | C23×D4 | C22×D4 | 1 | 2 | yes | — |
| 144 | C23×D4 | C22×D4 | 1 | 2 | yes | — |
| 145 | C24×C4 | #99 | 1 | 4 | yes | — |
| 146 | C24×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 147 | C24×C4 | #121 | 1 | 4 | yes | — |
| 148 | C24×C4 | #123 | 1 | 2 | yes | — |
| 149 | C24×C4 | #124 | 1 | 2 | yes | — |
| 150 | C2×#117 | C2×C22⋊C4 | 1 | 2 | yes | complemented |
| 151 | C2×#117 | C2×C22⋊C4 | 1 | 2 | yes | complemented |
| 152 | C22×C42 | C2×C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 153 | C22×C42 | C2×C22⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 154 | #379 | C2×C22⋊C4 | 1 | 8 | yes | complemented |
| 155 | C25 | #379 | 1 | 1 | yes | characteristic |
| 156 | C25 | #379 | 1 | 1 | yes | characteristic |
| 157 | C25 | #379 | 1 | 1 | yes | characteristic |
| 158 | C25 | #386 | 1 | 2 | yes | — |
| 159 | C25 | #386 | 1 | 2 | yes | — |
| 160 | C25 | #386 | 1 | 2 | yes | — |
| 161 | C25 | #390 | 1 | 2 | yes | — |
| 162 | C25 | #390 | 1 | 2 | yes | — |
| 163 | C25 | #390 | 1 | 2 | yes | — |
| 164 | C25 | #392 | 1 | 4 | yes | — |
| 165 | C25 | #392 | 1 | 4 | yes | — |
| 166 | C25 | #392 | 1 | 4 | yes | — |
| 167 | C25 | C2×#99 | 1 | 8 | yes | — |
| 168 | C25 | C2×#99 | 1 | 4 | yes | — |
| 169 | C25 | C2×#99 | 1 | 4 | yes | — |
| 170 | C25 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 171 | C25 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 172 | C25 | C2×#120 | 1 | 2 | yes | — |
| 173 | C25 | C2×#127 | 1 | 2 | yes | — |
| 174 | C22×Q8 | #386 | 1 | 2 | yes | — |
| 175 | C22×Q8 | #394 | 1 | 4 | yes | — |
| 176 | C22×Q8 | C2×#123 | 1 | 2 | yes | — |
| 177 | C22×D4 | #379 | 1 | 4 | yes | — |
| 178 | C22×D4 | #386 | 1 | 4 | yes | — |
| 179 | C22×D4 | #386 | 1 | 4 | yes | — |
| 180 | C22×D4 | #386 | 1 | 2 | yes | — |
| 181 | C22×D4 | #386 | 1 | 4 | yes | — |
| 182 | C22×D4 | #390 | 1 | 4 | yes | — |
| 183 | C22×D4 | #390 | 1 | 4 | yes | — |
| 184 | C22×D4 | #392 | 1 | 4 | yes | — |
| 185 | C22×D4 | #392 | 1 | 4 | yes | — |
| 186 | C22×D4 | #392 | 1 | 4 | yes | — |
| 187 | C22×D4 | #392 | 1 | 4 | yes | — |
| 188 | C22×D4 | #394 | 1 | 4 | yes | — |
| 189 | C22×D4 | #394 | 1 | 4 | yes | — |
| 190 | C22×D4 | #394 | 1 | 4 | yes | — |
| 191 | C22×D4 | B(4,𝔽2) | 1 | 16 | yes | — |
| 192 | C22×D4 | C2×C4×D4 | 1 | 4 | yes | — |
| 193 | C22×D4 | C2×#120 | 1 | 4 | yes | — |
| 194 | C22×D4 | C2×#120 | 1 | 2 | yes | — |
| 195 | C22×D4 | C2×#121 | 1 | 4 | yes | — |
| 196 | C22×D4 | C2×#121 | 1 | 4 | yes | — |
| 197 | C23×C4 | #379 | 1 | 1 | yes | characteristic |
| 198 | C23×C4 | #386 | 1 | 2 | yes | — |
| 199 | C23×C4 | #386 | 1 | 2 | yes | — |
| 200 | C23×C4 | #386 | 1 | 2 | yes | — |
| 201 | C23×C4 | #386 | 1 | 4 | yes | — |
| 202 | C23×C4 | #390 | 1 | 2 | yes | — |
| 203 | C23×C4 | #392 | 1 | 4 | yes | — |
| 204 | C23×C4 | #394 | 1 | 4 | yes | — |
| 205 | C23×C4 | #394 | 1 | 4 | yes | — |
| 206 | C23×C4 | #394 | 1 | 4 | yes | — |
| 207 | C23×C4 | #394 | 1 | 4 | yes | — |
| 208 | C23×C4 | C2×#99 | 1 | 8 | yes | — |
| 209 | C23×C4 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 210 | C23×C4 | C2×#121 | 1 | 4 | yes | — |
| 211 | C23×C4 | C2×#123 | 1 | 2 | yes | — |
| 212 | C23×C4 | C2×#124 | 1 | 2 | yes | — |
| 213 | C24 | #1438 | 1 | 2 | yes | — |
| 214 | C24 | #1438 | 1 | 1 | yes | characteristic |
| 215 | C24 | #1438 | 1 | 2 | yes | — |
| 216 | C24 | #1438 | 1 | 4 | yes | — |
| 217 | C24 | #1438 | 1 | 1 | yes | characteristic |
| 218 | C24 | #1781 | 1 | 4 | yes | — |
| 219 | C24 | #1781 | 1 | 2 | yes | — |
| 220 | C24 | #1781 | 1 | 2 | yes | — |
| 221 | C24 | #1898 | 1 | 8 | yes | — |
| 222 | C24 | #1898 | 1 | 4 | yes | — |
| 223 | C24 | #1898 | 1 | 4 | yes | — |
| 224 | C24 | #2021 | 1 | 16 | yes | — |
| 225 | C24 | C2×#379 | 1 | 1 | yes | characteristic |
| 226 | C24 | C2×#386 | 1 | 2 | yes | — |
| 227 | C24 | C2×#390 | 1 | 2 | yes | — |
| 228 | C24 | C2×#392 | 1 | 4 | yes | — |
| 229 | C24 | C22×#99 | 1 | 4 | yes | — |
| 230 | C2×D4 | #1889 | 1 | 8 | yes | — |
| 231 | C2×D4 | #2021 | 1 | 16 | yes | — |
| 232 | C2×D4 | C2×B(4,𝔽2) | 1 | 8 | yes | — |
| 233 | C22×C4 | #1438 | 1 | 2 | yes | — |
| 234 | C22×C4 | #1438 | 1 | 4 | yes | — |
| 235 | C22×C4 | #1889 | 1 | 8 | yes | — |
| 236 | C22×C4 | C2×#386 | 1 | 2 | yes | — |
| 237 | C22×C4 | C2×#394 | 1 | 4 | yes | — |
| 238 | C23 | #10965 | 1 | 2 | yes | — |
| 239 | C23 | #10965 | 1 | 4 | yes | — |
| 240 | C23 | #10965 | 1 | 1 | yes | characteristic |
| 241 | C23 | #10965 | 1 | 1 | yes | characteristic |
| 242 | C23 | #10991 | 1 | 8 | yes | — |
| 243 | C23 | #10991 | 1 | 4 | yes | — |
| 244 | C23 | C2×#1438 | 1 | 1 | yes | characteristic |
| 245 | C23 | C2×#1781 | 1 | 2 | yes | — |
| 246 | C23 | C2×#1898 | 1 | 4 | yes | — |
| 247 | C23 | C2×#2021 | 1 | 8 | yes | — |
| 248 | C2×C4 | #10991 | 1 | 4 | yes | — |
| 249 | C2×C4 | C2×#1889 | 1 | 4 | yes | — |
| 250 | C22 | #115690 | 1 | 2 | yes | — |
| 251 | C22 | #115690 | 1 | 2 | yes | — |
| 252 | C22 | C2×#10965 | 1 | 1 | yes | characteristic |
| 253 | C22 | C2×#10991 | 1 | 2 | yes | — |
| 254 | C2 | C2×#115690 | 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