← C23×#3549 #72289451
Subgroups — C23×#3549 #72289451
Order 1024.
557373 conjugacy classes of subgroups.
The subgroup lattice has 244 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×#3549 | C2 | 1 | 224 | yes | maximal, direct product factor |
| 2 | C23×H(3,𝔽4) | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | C23×#555 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C23×D42 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C23×#535 | C2 | 1 | 12 | yes | maximal, complemented |
| 6 | C23×#532 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C24×#41 | C22 | 1 | 6 | yes | complemented |
| 8 | C25×D4 | C22 | 1 | 12 | yes | complemented |
| 9 | C2×#3549 | C22 | 1 | 7168 | yes | direct product factor |
| 10 | C22×#555 | C22 | 1 | 672 | yes | complemented |
| 11 | C22×D42 | C22 | 1 | 672 | yes | complemented |
| 12 | C22×H(3,𝔽4) | C22 | 1 | 448 | yes | complemented |
| 13 | C22×#535 | C22 | 1 | 1344 | yes | complemented |
| 14 | C22×#532 | C22 | 1 | 336 | yes | complemented |
| 15 | C23×C4⋊Q8 | C22 | 1 | 3 | yes | complemented |
| 16 | C23×#127 | C22 | 1 | 3 | yes | — |
| 17 | C23×#127 | C22 | 1 | 8 | yes | complemented |
| 18 | C23×#124 | C22 | 1 | 6 | yes | complemented |
| 19 | C23×#124 | C22 | 1 | 24 | yes | complemented |
| 20 | C23×#121 | C22 | 1 | 12 | yes | complemented |
| 21 | C23×#121 | C22 | 1 | 24 | yes | complemented |
| 22 | C23×#120 | C22 | 1 | 8 | yes | complemented |
| 23 | C23×#120 | C22 | 1 | 24 | yes | complemented |
| 24 | C23×C4×D4 | C22 | 1 | 12 | yes | complemented |
| 25 | C23×#117 | C22 | 1 | 12 | yes | complemented |
| 26 | C24×C42 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | C27 | C23 | 1 | 8 | yes | — |
| 28 | C23×#41 | — | 4 | 96 | no | — |
| 29 | C23×#41 | C23 | 1 | 336 | yes | complemented |
| 30 | C24×Q8 | C23 | 1 | 6 | yes | — |
| 31 | C24×D4 | — | 4 | 192 | no | — |
| 32 | C24×D4 | C23 | 1 | 12 | yes | — |
| 33 | C24×D4 | C23 | 1 | 24 | yes | — |
| 34 | C24×D4 | C23 | 1 | 6 | yes | — |
| 35 | C24×D4 | C23 | 1 | 12 | yes | — |
| 36 | C24×D4 | C23 | 1 | 672 | yes | complemented |
| 37 | C25×C4 | C23 | 1 | 12 | yes | — |
| 38 | C25×C4 | C23 | 1 | 3 | yes | — |
| 39 | #3549 | C23 | 1 | 32768 | yes | direct product factor |
| 40 | C2×H(3,𝔽4) | C23 | 1 | 7168 | yes | complemented |
| 41 | C2×#555 | C23 | 1 | 10752 | yes | complemented |
| 42 | C2×D42 | C23 | 1 | 10752 | yes | complemented |
| 43 | C2×#535 | C23 | 1 | 21504 | yes | complemented |
| 44 | C2×#532 | C23 | 1 | 5376 | yes | complemented |
| 45 | C22×C4⋊Q8 | C23 | 1 | 168 | yes | complemented |
| 46 | C22×#127 | C23 | 1 | 168 | yes | — |
| 47 | C22×#127 | C23 | 1 | 448 | yes | complemented |
| 48 | C22×#124 | C23 | 1 | 336 | yes | complemented |
| 49 | C22×#124 | C23 | 1 | 1344 | yes | complemented |
| 50 | C22×#121 | C23 | 1 | 672 | yes | complemented |
| 51 | C22×#121 | C23 | 1 | 1344 | yes | complemented |
| 52 | C22×#120 | C23 | 1 | 448 | yes | complemented |
| 53 | C22×#120 | C23 | 1 | 1344 | yes | complemented |
| 54 | C22×C4×D4 | C23 | 1 | 672 | yes | complemented |
| 55 | C22×#117 | C23 | 1 | 672 | yes | complemented |
| 56 | C23×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 57 | C23×C22⋊C4 | C23 | 1 | 24 | yes | — |
| 58 | C23×C22⋊C4 | C23 | 1 | 24 | yes | — |
| 59 | C23×C42 | C23 | 1 | 3 | yes | — |
| 60 | C23×C42 | C23 | 1 | 1 | yes | characteristic |
| 61 | C23×C42 | C23 | 1 | 8 | yes | — |
| 62 | C23×C42 | C23 | 1 | 56 | yes | complemented |
| 63 | C26 | — | 4 | 192 | no | — |
| 64 | C26 | C24 | 1 | 1 | yes | characteristic |
| 65 | C26 | C24 | 1 | 12 | yes | — |
| 66 | C26 | C24 | 1 | 224 | yes | — |
| 67 | C22×#41 | — | 4 | 2688 | no | — |
| 68 | C22×#41 | C24 | 1 | 2688 | yes | complemented |
| 69 | C23×Q8 | — | 4 | 48 | no | — |
| 70 | C23×Q8 | C24 | 1 | 168 | yes | — |
| 71 | C23×D4 | — | 2 | 192 | no | — |
| 72 | C23×D4 | — | 4 | 5376 | no | — |
| 73 | C23×D4 | — | 4 | 192 | no | — |
| 74 | C23×D4 | — | 4 | 96 | no | — |
| 75 | C23×D4 | — | 4 | 48 | no | — |
| 76 | C23×D4 | C24 | 1 | 336 | yes | — |
| 77 | C23×D4 | C24 | 1 | 672 | yes | — |
| 78 | C23×D4 | C24 | 1 | 168 | yes | — |
| 79 | C23×D4 | C24 | 1 | 336 | yes | — |
| 80 | C23×D4 | C24 | 1 | 5376 | yes | complemented |
| 81 | C24×C4 | — | 2 | 192 | no | — |
| 82 | C24×C4 | — | 2 | 48 | no | — |
| 83 | C24×C4 | C24 | 1 | 336 | yes | — |
| 84 | C24×C4 | C24 | 1 | 84 | yes | — |
| 85 | C24×C4 | C24 | 1 | 12 | yes | — |
| 86 | C24×C4 | C24 | 1 | 3 | yes | — |
| 87 | C24×C4 | C24 | 1 | 3 | yes | — |
| 88 | H(3,𝔽4) | C24 | 1 | 16384 | yes | complemented |
| 89 | #555 | C24 | 1 | 24576 | yes | complemented |
| 90 | D42 | C24 | 1 | 24576 | yes | complemented |
| 91 | #535 | C24 | 1 | 49152 | yes | complemented |
| 92 | #532 | C24 | 1 | 12288 | yes | complemented |
| 93 | C2×C4⋊Q8 | C24 | 1 | 1344 | yes | complemented |
| 94 | C2×#127 | C24 | 1 | 1344 | yes | — |
| 95 | C2×#127 | C24 | 1 | 3584 | yes | complemented |
| 96 | C2×#124 | C24 | 1 | 2688 | yes | complemented |
| 97 | C2×#124 | C24 | 1 | 10752 | yes | complemented |
| 98 | C2×#121 | C24 | 1 | 5376 | yes | complemented |
| 99 | C2×#121 | C24 | 1 | 10752 | yes | complemented |
| 100 | C2×#120 | C24 | 1 | 3584 | yes | complemented |
| 101 | C2×#120 | C24 | 1 | 10752 | yes | complemented |
| 102 | C2×C4×D4 | C24 | 1 | 5376 | yes | complemented |
| 103 | C2×#117 | C24 | 1 | 5376 | yes | complemented |
| 104 | C22×C4⋊C4 | C24 | 1 | 336 | yes | — |
| 105 | C22×C22⋊C4 | C24 | 1 | 672 | yes | — |
| 106 | C22×C22⋊C4 | C24 | 1 | 672 | yes | — |
| 107 | C22×C42 | C24 | 1 | 84 | yes | — |
| 108 | C22×C42 | C24 | 1 | 28 | yes | — |
| 109 | C22×C42 | C24 | 1 | 224 | yes | — |
| 110 | C22×C42 | C24 | 1 | 448 | yes | complemented |
| 111 | C25 | — | 2 | 24 | no | — |
| 112 | C25 | — | 2 | 96 | no | — |
| 113 | C25 | — | 2 | 192 | no | — |
| 114 | C25 | — | 4 | 2688 | no | — |
| 115 | C25 | — | 8 | 1024 | no | — |
| 116 | C25 | C25 | 1 | 168 | yes | — |
| 117 | C25 | C25 | 1 | 14 | yes | — |
| 118 | C25 | C25 | 1 | 896 | yes | — |
| 119 | C25 | C25 | 1 | 1 | yes | characteristic |
| 120 | C2×#41 | — | 4 | 10752 | no | — |
| 121 | C2×#41 | C25 | 1 | 3072 | yes | complemented |
| 122 | C22×Q8 | — | 4 | 672 | no | — |
| 123 | C22×Q8 | C25 | 1 | 672 | yes | — |
| 124 | C22×D4 | — | 2 | 2688 | no | — |
| 125 | C22×D4 | — | 4 | 21504 | no | — |
| 126 | C22×D4 | — | 4 | 2688 | no | — |
| 127 | C22×D4 | — | 4 | 1344 | no | — |
| 128 | C22×D4 | — | 4 | 672 | no | — |
| 129 | C22×D4 | C25 | 1 | 1344 | yes | — |
| 130 | C22×D4 | C25 | 1 | 672 | yes | — |
| 131 | C22×D4 | C25 | 1 | 2688 | yes | — |
| 132 | C22×D4 | C25 | 1 | 1344 | yes | — |
| 133 | C22×D4 | C25 | 1 | 6144 | yes | complemented |
| 134 | C23×C4 | — | 2 | 96 | no | — |
| 135 | C23×C4 | — | 2 | 24 | no | — |
| 136 | C23×C4 | — | 2 | 2688 | no | — |
| 137 | C23×C4 | — | 2 | 672 | no | — |
| 138 | C23×C4 | H(5,𝔽2) | 1 | 48 | yes | — |
| 139 | C23×C4 | C25 | 1 | 42 | yes | — |
| 140 | C23×C4 | C25 | 1 | 168 | yes | — |
| 141 | C23×C4 | C25 | 1 | 1344 | yes | — |
| 142 | C23×C4 | C25 | 1 | 336 | yes | — |
| 143 | C23×C4 | C25 | 1 | 42 | yes | — |
| 144 | C4⋊Q8 | C25 | 1 | 1536 | yes | complemented |
| 145 | #127 | C25 | 1 | 1536 | yes | — |
| 146 | #127 | C25 | 1 | 4096 | yes | complemented |
| 147 | #124 | C25 | 1 | 3072 | yes | complemented |
| 148 | #124 | C25 | 1 | 12288 | yes | complemented |
| 149 | #121 | C25 | 1 | 6144 | yes | complemented |
| 150 | #121 | C25 | 1 | 12288 | yes | complemented |
| 151 | #120 | C25 | 1 | 4096 | yes | complemented |
| 152 | #120 | C25 | 1 | 12288 | yes | complemented |
| 153 | C4×D4 | C25 | 1 | 6144 | yes | complemented |
| 154 | #117 | C25 | 1 | 6144 | yes | complemented |
| 155 | C2×C4⋊C4 | C25 | 1 | 1344 | yes | — |
| 156 | C2×C22⋊C4 | C25 | 1 | 2688 | yes | — |
| 157 | C2×C22⋊C4 | C25 | 1 | 2688 | yes | — |
| 158 | C2×C42 | C25 | 1 | 896 | yes | — |
| 159 | C2×C42 | C25 | 1 | 336 | yes | — |
| 160 | C2×C42 | C25 | 1 | 112 | yes | — |
| 161 | C2×C42 | C25 | 1 | 512 | yes | complemented |
| 162 | C24 | — | 2 | 168 | no | — |
| 163 | C24 | — | 2 | 672 | no | — |
| 164 | C24 | — | 2 | 1344 | no | — |
| 165 | C24 | — | 4 | 5376 | no | — |
| 166 | C24 | — | 4 | 64 | no | — |
| 167 | C24 | — | 4 | 768 | no | — |
| 168 | C24 | — | 8 | 7168 | no | — |
| 169 | C24 | C2×H(5,𝔽2) | 1 | 24 | yes | — |
| 170 | C24 | C26 | 1 | 336 | yes | — |
| 171 | C24 | C26 | 1 | 28 | yes | — |
| 172 | C24 | C26 | 1 | 512 | yes | — |
| 173 | C24 | C26 | 1 | 7 | yes | — |
| 174 | #41 | — | 4 | 6144 | no | — |
| 175 | C2×Q8 | — | 4 | 1344 | no | — |
| 176 | C2×Q8 | C26 | 1 | 384 | yes | — |
| 177 | C2×D4 | — | 2 | 5376 | no | — |
| 178 | C2×D4 | — | 4 | 12288 | no | — |
| 179 | C2×D4 | — | 4 | 5376 | no | — |
| 180 | C2×D4 | — | 4 | 2688 | no | — |
| 181 | C2×D4 | — | 4 | 1344 | no | — |
| 182 | C2×D4 | C26 | 1 | 768 | yes | — |
| 183 | C2×D4 | C26 | 1 | 384 | yes | — |
| 184 | C2×D4 | C26 | 1 | 1536 | yes | — |
| 185 | C2×D4 | C26 | 1 | 768 | yes | — |
| 186 | C22×C4 | — | 2 | 672 | no | — |
| 187 | C22×C4 | — | 2 | 168 | no | — |
| 188 | C22×C4 | — | 2 | 5376 | no | — |
| 189 | C22×C4 | — | 2 | 1344 | no | — |
| 190 | C22×C4 | C2×H(5,𝔽2) | 1 | 336 | yes | — |
| 191 | C22×C4 | C26 | 1 | 84 | yes | — |
| 192 | C22×C4 | C26 | 1 | 336 | yes | — |
| 193 | C22×C4 | C26 | 1 | 192 | yes | — |
| 194 | C22×C4 | C26 | 1 | 768 | yes | — |
| 195 | C22×C4 | C26 | 1 | 84 | yes | — |
| 196 | C4⋊C4 | C26 | 1 | 768 | yes | — |
| 197 | C22⋊C4 | C26 | 1 | 1536 | yes | — |
| 198 | C22⋊C4 | C26 | 1 | 1536 | yes | — |
| 199 | C42 | C26 | 1 | 192 | yes | — |
| 200 | C42 | C26 | 1 | 64 | yes | — |
| 201 | C42 | C26 | 1 | 512 | yes | — |
| 202 | C23 | — | 2 | 168 | no | — |
| 203 | C23 | — | 2 | 672 | no | — |
| 204 | C23 | — | 2 | 1344 | no | — |
| 205 | C23 | — | 4 | 1536 | no | — |
| 206 | C23 | — | 4 | 224 | no | — |
| 207 | C23 | — | 4 | 2688 | no | — |
| 208 | C23 | — | 8 | 7168 | no | — |
| 209 | C23 | #3549 | 1 | 64 | yes | direct product factor |
| 210 | C23 | C22×H(5,𝔽2) | 1 | 84 | yes | — |
| 211 | C23 | C27 | 1 | 96 | yes | — |
| 212 | C23 | C27 | 1 | 8 | yes | — |
| 213 | C23 | C27 | 1 | 7 | yes | — |
| 214 | Q8 | — | 4 | 384 | no | — |
| 215 | D4 | — | 2 | 1536 | no | — |
| 216 | D4 | — | 4 | 1536 | no | — |
| 217 | D4 | — | 4 | 768 | no | — |
| 218 | D4 | — | 4 | 384 | no | — |
| 219 | C2×C4 | — | 2 | 672 | no | — |
| 220 | C2×C4 | — | 2 | 168 | no | — |
| 221 | C2×C4 | — | 2 | 1536 | no | — |
| 222 | C2×C4 | — | 2 | 384 | no | — |
| 223 | C2×C4 | C22×H(5,𝔽2) | 1 | 336 | yes | — |
| 224 | C2×C4 | C27 | 1 | 24 | yes | — |
| 225 | C2×C4 | C27 | 1 | 96 | yes | — |
| 226 | C2×C4 | C27 | 1 | 24 | yes | — |
| 227 | C22 | — | 2 | 24 | no | — |
| 228 | C22 | — | 2 | 96 | no | — |
| 229 | C22 | — | 2 | 192 | no | — |
| 230 | C22 | — | 4 | 112 | no | — |
| 231 | C22 | — | 4 | 1344 | no | — |
| 232 | C22 | — | 8 | 1024 | no | — |
| 233 | C22 | C2×#3549 | 1 | 112 | yes | direct product factor |
| 234 | C22 | C23×H(5,𝔽2) | 1 | 42 | yes | — |
| 235 | C22 | C28 | 1 | 1 | yes | characteristic |
| 236 | C4 | — | 2 | 96 | no | — |
| 237 | C4 | — | 2 | 24 | no | — |
| 238 | C4 | C23×H(5,𝔽2) | 1 | 48 | yes | — |
| 239 | C2 | — | 4 | 8 | no | — |
| 240 | C2 | — | 4 | 96 | no | — |
| 241 | C2 | C22×#3549 | 1 | 28 | yes | direct product factor |
| 242 | C2 | C24×H(5,𝔽2) | 1 | 3 | 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