← #72219247
Subgroups — #72219247
Order 2048.
At least 6224 conjugacy classes of subgroups; only the 924 normal subgroups are shown.
The subgroup lattice has 222 nodes (normal subgroups only) — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #72219244 | C2 | 1 | 8 | yes | maximal, complemented |
| 2 | #72219243 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #72219227 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72218291 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #72217511 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×#62817 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C24×#120 | C22 | 1 | 1 | yes | characteristic |
| 8 | #7642273 | C4 | 1 | 4 | yes | complemented |
| 9 | #7642244 | C4 | 1 | 2 | yes | complemented |
| 10 | #7630944 | C22 | 1 | 4 | yes | — |
| 11 | #7630939 | C22 | 1 | 2 | yes | — |
| 12 | #7626202 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C22×#1438 | C22 | 1 | 2 | yes | complemented |
| 14 | #392113 | C22 | 1 | 2 | yes | complemented |
| 15 | #196858 | C22 | 1 | 16 | yes | complemented |
| 16 | #184773 | C22 | 1 | 4 | yes | complemented |
| 17 | C2×#7555 | C22 | 1 | 4 | yes | complemented |
| 18 | C28 | D4 | 1 | 1 | yes | characteristic |
| 19 | C25×D4 | D4 | 1 | 2 | yes | — |
| 20 | C25×D4 | C23 | 1 | 1 | yes | characteristic |
| 21 | C2×#3490 | C2×C4 | 1 | 4 | yes | — |
| 22 | C2×#3484 | C2×C4 | 1 | 2 | yes | — |
| 23 | C2×#3484 | D4 | 1 | 4 | yes | — |
| 24 | C23×#120 | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C23×#120 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | C23×#120 | C2×C4 | 1 | 2 | yes | — |
| 27 | C23×#120 | D4 | 1 | 2 | yes | — |
| 28 | C23×#120 | D4 | 1 | 1 | yes | characteristic |
| 29 | C23×#120 | D4 | 1 | 1 | yes | characteristic |
| 30 | C24×C22⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 31 | C24×C22⋊C4 | C23 | 1 | 2 | yes | — |
| 32 | #36614 | C23 | 1 | 4 | yes | — |
| 33 | #35222 | C2×C4 | 1 | 4 | yes | — |
| 34 | #35222 | D4 | 1 | 8 | yes | — |
| 35 | #35050 | C2×C4 | 1 | 8 | yes | — |
| 36 | #34988 | D4 | 1 | 16 | yes | — |
| 37 | #34982 | D4 | 1 | 8 | yes | — |
| 38 | #34981 | D4 | 1 | 8 | yes | — |
| 39 | #34176 | C2×C4 | 1 | 4 | yes | — |
| 40 | #34171 | C2×C4 | 1 | 2 | yes | — |
| 41 | #34171 | D4 | 1 | 4 | yes | — |
| 42 | C23×#95 | C23 | 1 | 4 | yes | complemented |
| 43 | #20579 | C23 | 1 | 4 | yes | complemented |
| 44 | C27 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | C27 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 46 | C27 | C22⋊C4 | 1 | 2 | yes | — |
| 47 | C27 | C2×D4 | 1 | 2 | yes | — |
| 48 | C27 | C2×D4 | 1 | 1 | yes | characteristic |
| 49 | C27 | C2×D4 | 1 | 1 | yes | characteristic |
| 50 | C22×H(5,𝔽2) | C22⋊C4 | 1 | 2 | yes | complemented |
| 51 | C24×D4 | C22⋊C4 | 1 | 2 | yes | — |
| 52 | C24×D4 | C22⋊C4 | 1 | 2 | yes | — |
| 53 | C24×D4 | C22⋊C4 | 1 | 2 | yes | — |
| 54 | C24×D4 | C22⋊C4 | 1 | 2 | yes | — |
| 55 | C24×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 56 | C24×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 57 | C24×D4 | C2×D4 | 1 | 4 | yes | — |
| 58 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 59 | C24×D4 | C2×D4 | 1 | 1 | yes | characteristic |
| 60 | C24×D4 | C2×D4 | 1 | 4 | yes | — |
| 61 | C25×C4 | C2×D4 | 1 | 2 | yes | — |
| 62 | C25×C4 | C24 | 1 | 1 | yes | characteristic |
| 63 | C2×#543 | C22⋊C4 | 1 | 4 | yes | complemented |
| 64 | C2×#538 | C22⋊C4 | 1 | 8 | yes | complemented |
| 65 | C2×#535 | C22⋊C4 | 1 | 4 | yes | complemented |
| 66 | C2×#535 | C22⋊C4 | 1 | 8 | yes | complemented |
| 67 | C2×#534 | C22⋊C4 | 1 | 4 | yes | complemented |
| 68 | C2×#525 | C22×C4 | 1 | 4 | yes | — |
| 69 | C2×#525 | C2×D4 | 1 | 8 | yes | — |
| 70 | C2×#518 | C22⋊C4 | 1 | 2 | yes | complemented |
| 71 | C2×#518 | C22⋊C4 | 1 | 4 | yes | complemented |
| 72 | C23×C22⋊C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 73 | C23×C22⋊C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 74 | C23×C22⋊C4 | C22⋊C4 | 1 | 2 | yes | — |
| 75 | C23×C22⋊C4 | C22×C4 | 1 | 2 | yes | — |
| 76 | C23×C22⋊C4 | C22×C4 | 1 | 2 | yes | — |
| 77 | C23×C22⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 78 | C23×C22⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 79 | C23×C22⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 80 | C23×C22⋊C4 | C2×D4 | 1 | 8 | yes | — |
| 81 | #2848 | C2×D4 | 1 | 16 | yes | — |
| 82 | #2847 | C2×D4 | 1 | 16 | yes | — |
| 83 | #2626 | C22⋊C4 | 1 | 16 | yes | complemented |
| 84 | #2617 | C22⋊C4 | 1 | 16 | yes | complemented |
| 85 | #2315 | C22⋊C4 | 1 | 8 | yes | complemented |
| 86 | #2312 | C22⋊C4 | 1 | 8 | yes | complemented |
| 87 | #2309 | C22⋊C4 | 1 | 4 | yes | complemented |
| 88 | #2309 | C22⋊C4 | 1 | 8 | yes | complemented |
| 89 | #2280 | C22×C4 | 1 | 4 | yes | — |
| 90 | #2280 | C2×D4 | 1 | 8 | yes | — |
| 91 | C26 | #99 | 1 | 4 | yes | — |
| 92 | C26 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 93 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 94 | C26 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 95 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 96 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 97 | C26 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 98 | C26 | #120 | 1 | 4 | yes | — |
| 99 | C26 | #120 | 1 | 2 | yes | — |
| 100 | C26 | #120 | 1 | 4 | yes | — |
| 101 | C26 | #120 | 1 | 4 | yes | — |
| 102 | C26 | #120 | 1 | 2 | yes | — |
| 103 | C26 | C22×D4 | 1 | 1 | yes | characteristic |
| 104 | C22×#41 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 105 | C23×D4 | #99 | 1 | 8 | yes | — |
| 106 | C23×D4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 107 | C23×D4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 108 | C23×D4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 109 | C23×D4 | #120 | 1 | 2 | yes | — |
| 110 | C23×D4 | #120 | 1 | 4 | yes | — |
| 111 | C23×D4 | #120 | 1 | 2 | yes | — |
| 112 | C24×C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 113 | C24×C4 | C2×C22⋊C4 | 1 | 2 | yes | — |
| 114 | C24×C4 | #120 | 1 | 8 | yes | — |
| 115 | C24×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 116 | C24×C4 | C22×D4 | 1 | 4 | yes | — |
| 117 | C24×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 118 | C24×C4 | C22×D4 | 1 | 8 | yes | — |
| 119 | C2×C22⋊Q8 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 120 | C2×C22⋊Q8 | #120 | 1 | 16 | yes | — |
| 121 | C2×#121 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 122 | C2×#121 | #120 | 1 | 16 | yes | — |
| 123 | C2×#117 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 124 | C2×#117 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 125 | C22×C22⋊C4 | #99 | 1 | 4 | yes | — |
| 126 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 127 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 4 | yes | — |
| 128 | C22×C22⋊C4 | C2×C22⋊C4 | 1 | 8 | yes | — |
| 129 | C22×C22⋊C4 | #120 | 1 | 4 | yes | — |
| 130 | C22×C22⋊C4 | #120 | 1 | 8 | yes | — |
| 131 | C22×C22⋊C4 | #120 | 1 | 4 | yes | — |
| 132 | #380 | C2×C22⋊C4 | 1 | 16 | yes | — |
| 133 | #380 | #120 | 1 | 32 | yes | — |
| 134 | C25 | #379 | 1 | 2 | yes | — |
| 135 | C25 | #379 | 1 | 4 | yes | — |
| 136 | C25 | #379 | 1 | 2 | yes | — |
| 137 | C25 | #379 | 1 | 1 | yes | characteristic |
| 138 | C25 | #379 | 1 | 2 | yes | — |
| 139 | C25 | #379 | 1 | 2 | yes | — |
| 140 | C25 | #379 | 1 | 2 | yes | — |
| 141 | C25 | #379 | 1 | 1 | yes | characteristic |
| 142 | C25 | C2×#99 | 1 | 4 | yes | — |
| 143 | C25 | C2×#99 | 1 | 2 | yes | — |
| 144 | C25 | C2×#99 | 1 | 4 | yes | — |
| 145 | C25 | C2×#99 | 1 | 2 | yes | — |
| 146 | C25 | C2×#99 | 1 | 4 | yes | — |
| 147 | C25 | #453 | 1 | 8 | yes | — |
| 148 | C25 | #455 | 1 | 8 | yes | — |
| 149 | C25 | B(4,𝔽2) | 1 | 4 | yes | — |
| 150 | C25 | #458 | 1 | 4 | yes | — |
| 151 | C25 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 152 | C25 | C2×#120 | 1 | 4 | yes | — |
| 153 | C25 | C2×#120 | 1 | 2 | yes | — |
| 154 | C22×Q8 | #379 | 1 | 4 | yes | complemented |
| 155 | C22×D4 | #379 | 1 | 4 | yes | complemented |
| 156 | C23×C4 | #379 | 1 | 4 | yes | — |
| 157 | C23×C4 | #379 | 1 | 4 | yes | — |
| 158 | C23×C4 | C2×#99 | 1 | 4 | yes | — |
| 159 | C23×C4 | C2×#99 | 1 | 4 | yes | — |
| 160 | C23×C4 | C22×C22⋊C4 | 1 | 2 | yes | — |
| 161 | C23×C4 | C22×C22⋊C4 | 1 | 4 | yes | — |
| 162 | C23×C4 | C2×#120 | 1 | 2 | yes | — |
| 163 | C23×C4 | C2×#120 | 1 | 4 | yes | — |
| 164 | C23×C4 | C2×#120 | 1 | 16 | yes | — |
| 165 | C2×C4⋊C4 | #379 | 1 | 8 | yes | complemented |
| 166 | C2×C4⋊C4 | #379 | 1 | 16 | yes | complemented |
| 167 | C24 | #1781 | 1 | 4 | yes | — |
| 168 | C24 | #1781 | 1 | 2 | yes | — |
| 169 | C24 | #1781 | 1 | 4 | yes | — |
| 170 | C24 | #1781 | 1 | 4 | yes | — |
| 171 | C24 | #1781 | 1 | 2 | yes | — |
| 172 | C24 | #1790 | 1 | 4 | yes | — |
| 173 | C24 | #1790 | 1 | 4 | yes | — |
| 174 | C24 | #1792 | 1 | 8 | yes | — |
| 175 | C24 | #1792 | 1 | 8 | yes | — |
| 176 | C24 | C2×#379 | 1 | 1 | yes | characteristic |
| 177 | C24 | C2×#379 | 1 | 2 | yes | — |
| 178 | C24 | #2846 | 1 | 4 | yes | — |
| 179 | C24 | C22×#99 | 1 | 1 | yes | characteristic |
| 180 | C24 | C22×#99 | 1 | 2 | yes | — |
| 181 | C24 | C22×#99 | 1 | 1 | yes | characteristic |
| 182 | C24 | C2×#453 | 1 | 4 | yes | — |
| 183 | C24 | C2×#455 | 1 | 4 | yes | — |
| 184 | C24 | #3019 | 1 | 8 | yes | — |
| 185 | C24 | C2×B(4,𝔽2) | 1 | 2 | yes | — |
| 186 | C24 | C2×#458 | 1 | 2 | yes | — |
| 187 | C24 | #3026 | 1 | 4 | yes | — |
| 188 | C22×C4 | #1781 | 1 | 8 | yes | — |
| 189 | C22×C4 | C2×#379 | 1 | 4 | yes | — |
| 190 | C22×C4 | #2846 | 1 | 4 | yes | — |
| 191 | C23 | #8453 | 1 | 8 | yes | complemented |
| 192 | C23 | #8534 | 1 | 4 | yes | — |
| 193 | C23 | #14679 | 1 | 1 | yes | characteristic |
| 194 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 195 | C23 | C2×#1781 | 1 | 1 | yes | characteristic |
| 196 | C23 | C2×#1781 | 1 | 2 | yes | — |
| 197 | C23 | C2×#1781 | 1 | 2 | yes | — |
| 198 | C23 | C2×#1790 | 1 | 2 | yes | — |
| 199 | C23 | #20609 | 1 | 2 | yes | — |
| 200 | C23 | C2×#1792 | 1 | 4 | yes | — |
| 201 | C23 | #20616 | 1 | 4 | yes | — |
| 202 | C23 | #32888 | 1 | 8 | yes | — |
| 203 | C23 | #32891 | 1 | 8 | yes | — |
| 204 | C23 | #32898 | 1 | 4 | yes | — |
| 205 | C23 | #32900 | 1 | 4 | yes | — |
| 206 | C22 | C2×#8453 | 1 | 2 | yes | — |
| 207 | C22 | #186808 | 1 | 4 | yes | — |
| 208 | C22 | #186835 | 1 | 2 | yes | — |
| 209 | C22 | C2×#8534 | 1 | 1 | yes | characteristic |
| 210 | C22 | #187928 | 1 | 2 | yes | — |
| 211 | C22 | #187944 | 1 | 1 | yes | characteristic |
| 212 | C22 | #346446 | 1 | 4 | yes | — |
| 213 | C22 | #346456 | 1 | 4 | yes | — |
| 214 | C22 | #346466 | 1 | 8 | yes | — |
| 215 | C22 | #392109 | 1 | 1 | yes | characteristic |
| 216 | C22 | #392125 | 1 | 2 | yes | — |
| 217 | C22 | #392128 | 1 | 4 | yes | — |
| 218 | C2 | #72218467 | 1 | 1 | yes | characteristic |
| 219 | C2 | #72219245 | 1 | 2 | yes | — |
| 220 | C2 | #72219246 | 1 | 4 | yes | — |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Finite Group Databasebeta