Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
413477 conjugacy classes of subgroups; only the 5957 normal subgroups are shown.
This group has 413477 conjugacy classes of subgroups — too many to draw. Only its 5957 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C27×C4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#14693 | C2 | 1 | 14 | yes | maximal, complemented |
| 3 | C2×#14679 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#14679 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #154077 | C2 | 1 | 14 | yes | maximal, complemented |
| 6 | C28 | C22 | 1 | 1 | yes | characteristic |
| 7 | C26×C4 | — | 2 | 8 | no | — |
| 8 | C26×C4 | — | 2 | 112 | no | — |
| 9 | C26×C4 | C22 | 1 | 14 | yes | — |
| 10 | C22×#385 | C22 | 1 | 28 | yes | complemented |
| 11 | C22×#379 | C22 | 1 | 7 | yes | — |
| 12 | C22×#379 | C22 | 1 | 7 | yes | — |
| 13 | C22×#379 | C22 | 1 | 14 | yes | — |
| 14 | #14693 | — | 2 | 112 | no | — |
| 15 | #14679 | — | 2 | 8 | no | — |
| 16 | #14679 | C4 | 1 | 16 | yes | complemented |
| 17 | C2×C22⋊#95 | C22 | 1 | 84 | yes | complemented |
| 18 | C27 | — | 2 | 56 | no | — |
| 19 | C27 | — | 2 | 56 | no | — |
| 20 | C27 | C2×C4 | 1 | 8 | yes | — |
| 21 | C27 | D4 | 1 | 8 | yes | — |
| 22 | C27 | Q8 | 1 | 8 | yes | — |
| 23 | C27 | C23 | 1 | 7 | yes | — |
| 24 | C25×C4 | — | 2 | 896 | no | — |
| 25 | C25×C4 | — | 2 | 3584 | no | — |
| 26 | C25×C4 | — | 2 | 56 | no | — |
| 27 | C25×C4 | — | 2 | 56 | no | — |
| 28 | C25×C4 | — | 2 | 672 | no | — |
| 29 | C25×C4 | D4 | 1 | 112 | yes | — |
| 30 | C25×C4 | C23 | 1 | 28 | yes | — |
| 31 | C23×C4⋊C4 | C23 | 1 | 8 | yes | complemented |
| 32 | C23×C22⋊C4 | C23 | 1 | 21 | yes | — |
| 33 | C23×C22⋊C4 | C23 | 1 | 7 | yes | — |
| 34 | C23×C22⋊C4 | C23 | 1 | 28 | yes | — |
| 35 | C2×#385 | — | 2 | 336 | no | — |
| 36 | C2×#379 | — | 2 | 56 | no | — |
| 37 | C2×#379 | — | 2 | 28 | no | — |
| 38 | C2×#379 | — | 2 | 28 | no | — |
| 39 | C2×#379 | — | 2 | 56 | no | — |
| 40 | C2×#379 | — | 2 | 56 | no | — |
| 41 | C2×#379 | — | 2 | 56 | no | — |
| 42 | C2×#379 | C2×C4 | 1 | 112 | yes | complemented |
| 43 | C22×#95 | C23 | 1 | 56 | yes | complemented |
| 44 | C22⋊#95 | — | 2 | 336 | no | — |
| 45 | C26 | — | 2 | 896 | no | — |
| 46 | C26 | — | 2 | 2688 | no | — |
| 47 | C26 | — | 2 | 448 | no | — |
| 48 | C26 | — | 2 | 224 | no | — |
| 49 | C26 | — | 2 | 448 | no | — |
| 50 | C26 | — | 2 | 224 | no | — |
| 51 | C26 | — | 2 | 168 | no | — |
| 52 | C26 | — | 2 | 168 | no | — |
| 53 | C26 | C4⋊C4 | 1 | 64 | yes | complemented |
| 54 | C26 | C22×C4 | 1 | 28 | yes | — |
| 55 | C26 | C2×D4 | 1 | 28 | yes | — |
| 56 | C26 | C2×D4 | 1 | 56 | yes | — |
| 57 | C26 | C2×Q8 | 1 | 28 | yes | — |
| 58 | C26 | #41 | 1 | 56 | yes | — |
| 59 | C26 | C24 | 1 | 7 | yes | — |
| 60 | C24×C4 | — | 2 | 14336 | no | — |
| 61 | C24×C4 | — | 2 | 16384 | no | — |
| 62 | C24×C4 | — | 2 | 2688 | no | — |
| 63 | C24×C4 | — | 2 | 2688 | no | — |
| 64 | C24×C4 | — | 2 | 7168 | no | — |
| 65 | C24×C4 | — | 2 | 448 | no | — |
| 66 | C24×C4 | — | 2 | 56 | no | — |
| 67 | C24×C4 | — | 2 | 168 | no | — |
| 68 | C24×C4 | — | 2 | 2688 | no | — |
| 69 | C24×C4 | — | 2 | 448 | no | — |
| 70 | C24×C4 | C2×D4 | 1 | 336 | yes | — |
| 71 | C24×C4 | C24 | 1 | 1 | yes | characteristic |
| 72 | C24×C4 | C24 | 1 | 7 | yes | — |
| 73 | C24×C4 | C24 | 1 | 8 | yes | — |
| 74 | C24×C4 | C24 | 1 | 8 | yes | — |
| 75 | C22×C4⋊C4 | — | 2 | 112 | no | — |
| 76 | C22×C22⋊C4 | — | 2 | 168 | no | — |
| 77 | C22×C22⋊C4 | — | 2 | 168 | no | — |
| 78 | C22×C22⋊C4 | — | 2 | 84 | no | — |
| 79 | C22×C22⋊C4 | — | 2 | 14 | no | — |
| 80 | C22×C22⋊C4 | — | 2 | 42 | no | — |
| 81 | C22×C22⋊C4 | — | 2 | 84 | no | — |
| 82 | C22×C22⋊C4 | — | 2 | 168 | no | — |
| 83 | C22×C22⋊C4 | C22×C4 | 1 | 112 | yes | complemented |
| 84 | #385 | — | 4 | 448 | no | — |
| 85 | #379 | — | 2 | 448 | no | — |
| 86 | #379 | — | 4 | 112 | no | — |
| 87 | #379 | — | 4 | 112 | no | — |
| 88 | C2×#95 | — | 2 | 336 | no | — |
| 89 | C25 | — | 2 | 2048 | no | — |
| 90 | C25 | — | 2 | 1792 | no | — |
| 91 | C25 | — | 2 | 896 | no | — |
| 92 | C25 | — | 2 | 14336 | no | — |
| 93 | C25 | — | 2 | 7168 | no | — |
| 94 | C25 | — | 2 | 5376 | no | — |
| 95 | C25 | — | 2 | 2688 | no | — |
| 96 | C25 | — | 2 | 7168 | no | — |
| 97 | C25 | — | 2 | 672 | no | — |
| 98 | C25 | — | 2 | 672 | no | — |
| 99 | C25 | — | 2 | 336 | no | — |
| 100 | C25 | — | 2 | 672 | no | — |
| 101 | C25 | — | 2 | 336 | no | — |
| 102 | C25 | — | 2 | 672 | no | — |
| 103 | C25 | — | 2 | 1344 | no | — |
| 104 | C25 | — | 2 | 56 | no | — |
| 105 | C25 | — | 2 | 56 | no | — |
| 106 | C25 | — | 2 | 1792 | no | — |
| 107 | C25 | C2×C4⋊C4 | 1 | 112 | yes | — |
| 108 | C25 | C4×D4 | 1 | 224 | yes | — |
| 109 | C25 | #120 | 1 | 112 | yes | — |
| 110 | C25 | C22⋊Q8 | 1 | 224 | yes | — |
| 111 | C25 | #123 | 1 | 112 | yes | — |
| 112 | C25 | C23×C4 | 1 | 14 | yes | — |
| 113 | C25 | C22×D4 | 1 | 14 | yes | — |
| 114 | C25 | C22×D4 | 1 | 84 | yes | — |
| 115 | C25 | C22×Q8 | 1 | 14 | yes | — |
| 116 | C25 | C2×#41 | 1 | 84 | yes | — |
| 117 | C25 | C25 | 1 | 1 | yes | characteristic |
| 118 | C23×C4 | — | 2 | 32768 | no | — |
| 119 | C23×C4 | — | 2 | 14336 | no | — |
| 120 | C23×C4 | — | 2 | 14336 | no | — |
| 121 | C23×C4 | — | 2 | 10752 | no | — |
| 122 | C23×C4 | — | 2 | 2688 | no | — |
| 123 | C23×C4 | — | 2 | 896 | no | — |
| 124 | C23×C4 | — | 2 | 14336 | no | — |
| 125 | C23×C4 | — | 2 | 672 | no | — |
| 126 | C23×C4 | — | 2 | 672 | no | — |
| 127 | C23×C4 | — | 2 | 8 | no | — |
| 128 | C23×C4 | — | 2 | 56 | no | — |
| 129 | C23×C4 | — | 2 | 2688 | no | — |
| 130 | C23×C4 | — | 2 | 56 | no | — |
| 131 | C23×C4 | — | 2 | 56 | no | — |
| 132 | C23×C4 | — | 2 | 1 | no | — |
| 133 | C23×C4 | — | 2 | 7 | no | — |
| 134 | C23×C4 | — | 2 | 14 | no | — |
| 135 | C23×C4 | — | 2 | 42 | no | — |
| 136 | C23×C4 | — | 2 | 56 | no | — |
| 137 | C23×C4 | #120 | 1 | 448 | yes | — |
| 138 | C23×C4 | C23×C4 | 1 | 16 | yes | complemented |
| 139 | C23×C4 | C22×D4 | 1 | 112 | yes | — |
| 140 | C2×C4⋊C4 | — | 4 | 224 | no | — |
| 141 | C2×C22⋊C4 | — | 2 | 672 | no | — |
| 142 | C2×C22⋊C4 | — | 4 | 112 | no | — |
| 143 | C2×C22⋊C4 | — | 4 | 336 | no | — |
| 144 | C2×C22⋊C4 | — | 4 | 84 | no | — |
| 145 | C2×C22⋊C4 | — | 4 | 84 | no | — |
| 146 | C2×C22⋊C4 | — | 4 | 112 | no | — |
| 147 | C2×C22⋊C4 | — | 4 | 168 | no | — |
| 148 | C2×C22⋊C4 | — | 4 | 336 | no | — |
| 149 | #95 | — | 4 | 224 | no | — |
| 150 | C24 | — | 2 | 2048 | no | — |
| 151 | C24 | — | 2 | 896 | no | — |
| 152 | C24 | — | 2 | 1792 | no | — |
| 153 | C24 | — | 2 | 16384 | no | — |
| 154 | C24 | — | 2 | 16384 | no | — |
| 155 | C24 | — | 2 | 14336 | no | — |
| 156 | C24 | — | 2 | 3584 | no | — |
| 157 | C24 | — | 2 | 2688 | no | — |
| 158 | C24 | — | 2 | 5376 | no | — |
| 159 | C24 | — | 2 | 7168 | no | — |
| 160 | C24 | — | 2 | 14336 | no | — |
| 161 | C24 | — | 2 | 3584 | no | — |
| 162 | C24 | — | 2 | 672 | no | — |
| 163 | C24 | — | 2 | 2688 | no | — |
| 164 | C24 | — | 2 | 672 | no | — |
| 165 | C24 | — | 2 | 1344 | no | — |
| 166 | C24 | — | 2 | 112 | no | — |
| 167 | C24 | — | 2 | 336 | no | — |
| 168 | C24 | — | 2 | 56 | no | — |
| 169 | C24 | — | 2 | 112 | no | — |
| 170 | C24 | — | 2 | 56 | no | — |
| 171 | C24 | — | 2 | 336 | no | — |
| 172 | C24 | — | 2 | 672 | no | — |
| 173 | C24 | — | 2 | 1344 | no | — |
| 174 | C24 | — | 2 | 2688 | no | — |
| 175 | C24 | #385 | 1 | 448 | yes | complemented |
| 176 | C24 | C22×C4⋊C4 | 1 | 28 | yes | — |
| 177 | C24 | C2×C4×D4 | 1 | 168 | yes | — |
| 178 | C24 | C2×#120 | 1 | 112 | yes | — |
| 179 | C24 | C2×#120 | 1 | 84 | yes | — |
| 180 | C24 | C2×C22⋊Q8 | 1 | 168 | yes | — |
| 181 | C24 | C2×#123 | 1 | 84 | yes | — |
| 182 | C24 | #525 | 1 | 336 | yes | — |
| 183 | C24 | C24×C4 | 1 | 1 | yes | characteristic |
| 184 | C24 | C23×D4 | 1 | 1 | yes | characteristic |
| 185 | C24 | C23×D4 | 1 | 14 | yes | — |
| 186 | C24 | C23×Q8 | 1 | 1 | yes | characteristic |
| 187 | C24 | C22×#41 | 1 | 14 | yes | — |
| 188 | C22×C4 | — | 2 | 28672 | no | — |
| 189 | C22×C4 | — | 2 | 5376 | no | — |
| 190 | C22×C4 | — | 2 | 5376 | no | — |
| 191 | C22×C4 | — | 2 | 896 | no | — |
| 192 | C22×C4 | — | 2 | 336 | no | — |
| 193 | C22×C4 | — | 2 | 112 | no | — |
| 194 | C22×C4 | — | 2 | 1792 | no | — |
| 195 | C22×C4 | — | 2 | 112 | no | — |
| 196 | C22×C4 | — | 4 | 56 | no | — |
| 197 | C22×C4 | — | 4 | 168 | no | — |
| 198 | C22×C4 | — | 4 | 28 | no | — |
| 199 | C22×C4 | — | 4 | 7 | no | — |
| 200 | C22×C4 | — | 4 | 21 | no | — |
| 201 | C22×C4 | — | 4 | 28 | no | — |
| 202 | C22×C4 | — | 4 | 28 | no | — |
| 203 | C22×C4 | — | 4 | 168 | no | — |
| 204 | C22×C4 | C2×#120 | 1 | 224 | yes | — |
| 205 | C4⋊C4 | — | 8 | 64 | no | — |
| 206 | C22⋊C4 | — | 4 | 448 | no | — |
| 207 | C22⋊C4 | — | 8 | 56 | no | — |
| 208 | C22⋊C4 | — | 8 | 168 | no | — |
| 209 | C23 | — | 2 | 896 | no | — |
| 210 | C23 | — | 2 | 16384 | no | — |
| 211 | C23 | — | 2 | 3584 | no | — |
| 212 | C23 | — | 2 | 7168 | no | — |
| 213 | C23 | — | 2 | 7168 | no | — |
| 214 | C23 | — | 2 | 2688 | no | — |
| 215 | C23 | — | 2 | 3584 | no | — |
| 216 | C23 | — | 2 | 168 | no | — |
| 217 | C23 | — | 2 | 224 | no | — |
| 218 | C23 | — | 2 | 672 | no | — |
| 219 | C23 | — | 2 | 168 | no | — |
| 220 | C23 | — | 2 | 336 | no | — |
| 221 | C23 | — | 2 | 224 | no | — |
| 222 | C23 | — | 2 | 1344 | no | — |
| 223 | C23 | — | 2 | 2688 | no | — |
| 224 | C23 | — | 2 | 112 | no | — |
| 225 | C23 | — | 2 | 672 | no | — |
| 226 | C23 | C2×#385 | 1 | 168 | yes | — |
| 227 | C23 | C4×#120 | 1 | 224 | yes | — |
| 228 | C23 | #2846 | 1 | 56 | yes | — |
| 229 | C23 | #2848 | 1 | 224 | yes | — |
| 230 | C23 | #2849 | 1 | 168 | yes | — |
| 231 | C23 | C23×C4⋊C4 | 1 | 1 | yes | characteristic |
| 232 | C23 | C22×C4×D4 | 1 | 14 | yes | — |
| 233 | C23 | C22×#120 | 1 | 28 | yes | — |
| 234 | C23 | C22×#120 | 1 | 7 | yes | — |
| 235 | C23 | C22×C22⋊Q8 | 1 | 14 | yes | — |
| 236 | C23 | C22×#123 | 1 | 7 | yes | — |
| 237 | C23 | C2×#525 | 1 | 84 | yes | — |
| 238 | C2×C4 | — | 2 | 3584 | no | — |
| 239 | C2×C4 | — | 2 | 224 | no | — |
| 240 | C2×C4 | — | 2 | 224 | no | — |
| 241 | C2×C4 | — | 4 | 112 | no | — |
| 242 | C2×C4 | — | 8 | 8 | no | — |
| 243 | C2×C4 | — | 8 | 56 | no | — |
| 244 | C2×C4 | — | 8 | 7 | no | — |
| 245 | C2×C4 | — | 8 | 7 | no | — |
| 246 | C2×C4 | — | 8 | 8 | no | — |
| 247 | C2×C4 | — | 8 | 42 | no | — |
| 248 | C2×C4 | — | 8 | 56 | no | — |
| 249 | C2×C4 | #2846 | 1 | 64 | yes | — |
| 250 | C22 | — | 2 | 3584 | no | — |
| 251 | C22 | — | 2 | 336 | no | — |
| 252 | C22 | — | 2 | 448 | no | — |
| 253 | C22 | — | 2 | 56 | no | — |
| 254 | C22 | — | 2 | 448 | no | — |
| 255 | C22 | — | 2 | 56 | no | — |
| 256 | C22 | — | 2 | 336 | no | — |
| 257 | C22 | #14693 | 1 | 112 | yes | complemented |
| 258 | C22 | C22×#385 | 1 | 7 | yes | — |
| 259 | C22 | C2×C4×#120 | 1 | 28 | yes | — |
| 260 | C22 | C2×#2846 | 1 | 7 | yes | — |
| 261 | C22 | C2×#2846 | 1 | 8 | yes | — |
| 262 | C22 | C2×#2848 | 1 | 28 | yes | — |
| 263 | C22 | C2×#2849 | 1 | 21 | yes | — |
| 264 | C22 | #36614 | 1 | 56 | yes | — |
| 265 | C4 | — | 2 | 64 | no | — |
| 266 | C4 | — | 8 | 16 | no | — |
| 267 | C4 | — | 16 | 1 | no | — |
| 268 | C4 | — | 16 | 7 | no | — |
| 269 | C2 | — | 2 | 112 | no | — |
| 270 | C2 | C2×#14693 | 1 | 7 | yes | — |
| 271 | C2 | C4×#2846 | 1 | 8 | yes | — |
| 272 | C2 | #6342371 | 1 | 1 | yes | characteristic |
| 273 | C2 | #6342373 | 1 | 8 | yes | — |
| 274 | C2 | #6342374 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C25 |
| commutator | C24 | C24×C4 |
| prime radical | C28 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#14693 | C2 | semidirect |
| 5. #154077 | C2 | semidirect |
| 10. C22×#385 | C22 | semidirect |
| 16. #14679 | C4 | semidirect |
| 17. C2×C22⋊#95 | C22 | semidirect |
| 31. C23×C4⋊C4 | C23 | semidirect |
| 42. C2×#379 | C2×C4 | semidirect |
| 43. C22×#95 | C23 | semidirect |
| 53. C26 | C4⋊C4 | semidirect |
| 83. C22×C22⋊C4 | C22×C4 | semidirect |
| 138. C23×C4 | C23×C4 | semidirect |
| 175. C24 | #385 | semidirect |
| 257. C22 | #14693 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C27×C4 | 2 |
| C2×#14693 | 2 |
| C2×#14679 | 2 |
| #154077 | 2 |