Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
213317 conjugacy classes of subgroups.
The subgroup lattice has 212 nodes — too large to display as a diagram.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C24×#121 | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C22×#2656 | C2 | 1 | 112 | yes | maximal, direct product factor |
| 3 | C23×#399 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C23×#392 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C24×#95 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C25×D4 | C22 | 1 | 4 | yes | — |
| 7 | C25×D4 | C22 | 1 | 6 | yes | — |
| 8 | C26×C4 | C22 | 1 | 3 | yes | — |
| 9 | C23×#121 | — | 2 | 192 | no | — |
| 10 | C23×#121 | C22 | 1 | 336 | yes | complemented |
| 11 | C24×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 12 | C24×C22⋊C4 | C22 | 1 | 12 | yes | — |
| 13 | C2×#2656 | C22 | 1 | 1792 | yes | direct product factor |
| 14 | C22×#399 | C22 | 1 | 224 | yes | complemented |
| 15 | C22×#392 | C22 | 1 | 224 | yes | complemented |
| 16 | C23×#95 | C22 | 1 | 4 | yes | — |
| 17 | C23×#95 | C22 | 1 | 56 | yes | complemented |
| 18 | C27 | C23 | 1 | 4 | yes | — |
| 19 | C27 | C23 | 1 | 1 | yes | characteristic |
| 20 | C24×D4 | — | 2 | 192 | no | — |
| 21 | C24×D4 | — | 2 | 96 | no | — |
| 22 | C24×D4 | — | 4 | 96 | no | — |
| 23 | C24×D4 | C23 | 1 | 112 | yes | — |
| 24 | C24×D4 | C23 | 1 | 168 | yes | — |
| 25 | C25×C4 | — | 4 | 48 | no | — |
| 26 | C25×C4 | D4 | 1 | 96 | yes | complemented |
| 27 | C25×C4 | C23 | 1 | 4 | yes | — |
| 28 | C25×C4 | C23 | 1 | 6 | yes | — |
| 29 | C25×C4 | C23 | 1 | 84 | yes | — |
| 30 | C22×#121 | — | 2 | 5376 | no | — |
| 31 | C22×#121 | C23 | 1 | 2688 | yes | complemented |
| 32 | C23×C4⋊C4 | — | 2 | 96 | no | — |
| 33 | C23×C4⋊C4 | C23 | 1 | 168 | yes | — |
| 34 | C23×C22⋊C4 | — | 2 | 192 | no | — |
| 35 | C23×C22⋊C4 | C23 | 1 | 336 | yes | — |
| 36 | #2656 | C23 | 1 | 4096 | yes | direct product factor |
| 37 | C2×#399 | C23 | 1 | 1792 | yes | complemented |
| 38 | C2×#392 | C23 | 1 | 1792 | yes | complemented |
| 39 | C22×#95 | C23 | 1 | 112 | yes | — |
| 40 | C22×#95 | C23 | 1 | 448 | yes | complemented |
| 41 | C26 | — | 2 | 24 | no | — |
| 42 | C26 | — | 2 | 8 | no | — |
| 43 | C26 | — | 2 | 96 | no | — |
| 44 | C26 | — | 2 | 96 | no | — |
| 45 | C26 | — | 2 | 32 | no | — |
| 46 | C26 | C2×D4 | 1 | 48 | yes | — |
| 47 | C26 | C24 | 1 | 56 | yes | — |
| 48 | C26 | C24 | 1 | 14 | yes | — |
| 49 | C26 | C24 | 1 | 1 | yes | characteristic |
| 50 | C23×D4 | — | 2 | 2688 | no | — |
| 51 | C23×D4 | — | 2 | 1344 | no | — |
| 52 | C23×D4 | — | 4 | 1344 | no | — |
| 53 | C23×D4 | — | 4 | 1536 | no | — |
| 54 | C23×D4 | — | 4 | 1024 | no | — |
| 55 | C23×D4 | C24 | 1 | 448 | yes | — |
| 56 | C23×D4 | C24 | 1 | 672 | yes | — |
| 57 | C24×C4 | — | 2 | 96 | no | — |
| 58 | C24×C4 | — | 2 | 96 | no | — |
| 59 | C24×C4 | — | 4 | 768 | no | — |
| 60 | C24×C4 | — | 4 | 672 | no | — |
| 61 | C24×C4 | C2×D4 | 1 | 96 | yes | — |
| 62 | C24×C4 | C2×D4 | 1 | 1344 | yes | complemented |
| 63 | C24×C4 | C24 | 1 | 56 | yes | — |
| 64 | C24×C4 | C24 | 1 | 84 | yes | — |
| 65 | C24×C4 | C24 | 1 | 336 | yes | — |
| 66 | C2×#121 | — | 2 | 21504 | no | — |
| 67 | C2×#121 | C24 | 1 | 3072 | yes | complemented |
| 68 | C22×C4⋊C4 | — | 2 | 1344 | no | — |
| 69 | C22×C4⋊C4 | C24 | 1 | 672 | yes | — |
| 70 | C22×C22⋊C4 | — | 2 | 2688 | no | — |
| 71 | C22×C22⋊C4 | C24 | 1 | 1344 | yes | — |
| 72 | #399 | C24 | 1 | 2048 | yes | complemented |
| 73 | #392 | C24 | 1 | 2048 | yes | complemented |
| 74 | C2×#95 | C24 | 1 | 448 | yes | — |
| 75 | C2×#95 | C24 | 1 | 512 | yes | complemented |
| 76 | C25 | — | 2 | 384 | no | — |
| 77 | C25 | — | 2 | 384 | no | — |
| 78 | C25 | — | 2 | 168 | no | — |
| 79 | C25 | — | 2 | 56 | no | — |
| 80 | C25 | — | 2 | 672 | no | — |
| 81 | C25 | — | 2 | 672 | no | — |
| 82 | C25 | — | 2 | 224 | no | — |
| 83 | C25 | — | 4 | 256 | no | — |
| 84 | C25 | — | 4 | 768 | no | — |
| 85 | C25 | — | 4 | 768 | no | — |
| 86 | C25 | #127 | 1 | 256 | yes | — |
| 87 | C25 | C22×D4 | 1 | 336 | yes | — |
| 88 | C25 | C22×D4 | 1 | 24 | yes | — |
| 89 | C25 | H(5,𝔽2) | 1 | 24 | yes | — |
| 90 | C25 | 2−1+4 | 1 | 8 | yes | — |
| 91 | C25 | C25 | 1 | 112 | yes | — |
| 92 | C25 | C25 | 1 | 28 | yes | — |
| 93 | C25 | C25 | 1 | 7 | yes | — |
| 94 | C22×D4 | — | 2 | 5376 | no | — |
| 95 | C22×D4 | — | 2 | 2688 | no | — |
| 96 | C22×D4 | — | 4 | 2688 | no | — |
| 97 | C22×D4 | — | 4 | 10752 | no | — |
| 98 | C22×D4 | — | 4 | 7168 | no | — |
| 99 | C22×D4 | C25 | 1 | 256 | yes | — |
| 100 | C22×D4 | C25 | 1 | 384 | yes | — |
| 101 | C23×C4 | — | 2 | 768 | no | — |
| 102 | C23×C4 | — | 2 | 672 | no | — |
| 103 | C23×C4 | — | 2 | 672 | no | — |
| 104 | C23×C4 | — | 4 | 5376 | no | — |
| 105 | C23×C4 | — | 4 | 1344 | no | — |
| 106 | C23×C4 | — | 4 | 256 | no | — |
| 107 | C23×C4 | C22×D4 | 1 | 672 | yes | — |
| 108 | C23×C4 | C22×D4 | 1 | 2688 | yes | complemented |
| 109 | C23×C4 | C25 | 1 | 112 | yes | — |
| 110 | C23×C4 | C25 | 1 | 168 | yes | — |
| 111 | C23×C4 | C25 | 1 | 192 | yes | — |
| 112 | #121 | — | 2 | 12288 | no | — |
| 113 | C2×C4⋊C4 | — | 2 | 2688 | no | — |
| 114 | C2×C4⋊C4 | C25 | 1 | 384 | yes | — |
| 115 | C2×C22⋊C4 | — | 2 | 5376 | no | — |
| 116 | C2×C22⋊C4 | C25 | 1 | 768 | yes | — |
| 117 | #95 | C25 | 1 | 256 | yes | — |
| 118 | C24 | — | 2 | 168 | no | — |
| 119 | C24 | — | 2 | 56 | no | — |
| 120 | C24 | — | 2 | 2048 | no | — |
| 121 | C24 | — | 2 | 1344 | no | — |
| 122 | C24 | — | 2 | 1344 | no | — |
| 123 | C24 | — | 2 | 672 | no | — |
| 124 | C24 | — | 2 | 672 | no | — |
| 125 | C24 | — | 2 | 224 | no | — |
| 126 | C24 | — | 4 | 896 | no | — |
| 127 | C24 | — | 4 | 2688 | no | — |
| 128 | C24 | — | 4 | 2688 | no | — |
| 129 | C24 | — | 8 | 2048 | no | — |
| 130 | C24 | C2×#127 | 1 | 896 | yes | — |
| 131 | C24 | C2×#127 | 1 | 64 | yes | — |
| 132 | C24 | #534 | 1 | 192 | yes | — |
| 133 | C24 | #537 | 1 | 192 | yes | — |
| 134 | C24 | C23×D4 | 1 | 336 | yes | — |
| 135 | C24 | C23×D4 | 1 | 84 | yes | — |
| 136 | C24 | C2×H(5,𝔽2) | 1 | 84 | yes | — |
| 137 | C24 | C2×2−1+4 | 1 | 28 | yes | — |
| 138 | C24 | C26 | 1 | 32 | yes | — |
| 139 | C24 | C26 | 1 | 8 | yes | — |
| 140 | C24 | C26 | 1 | 7 | yes | — |
| 141 | C2×D4 | — | 2 | 1536 | no | — |
| 142 | C2×D4 | — | 2 | 768 | no | — |
| 143 | C2×D4 | — | 4 | 768 | no | — |
| 144 | C2×D4 | — | 4 | 10752 | no | — |
| 145 | C2×D4 | — | 4 | 7168 | no | — |
| 146 | C22×C4 | — | 2 | 672 | no | — |
| 147 | C22×C4 | — | 2 | 2688 | no | — |
| 148 | C22×C4 | — | 2 | 672 | no | — |
| 149 | C22×C4 | — | 4 | 5376 | no | — |
| 150 | C22×C4 | — | 4 | 384 | no | — |
| 151 | C22×C4 | — | 4 | 896 | no | — |
| 152 | C22×C4 | C23×D4 | 1 | 672 | yes | — |
| 153 | C22×C4 | C23×D4 | 1 | 768 | yes | complemented |
| 154 | C22×C4 | C26 | 1 | 32 | yes | — |
| 155 | C22×C4 | C26 | 1 | 48 | yes | — |
| 156 | C4⋊C4 | — | 2 | 768 | no | — |
| 157 | C22⋊C4 | — | 2 | 1536 | no | — |
| 158 | C23 | — | 2 | 3584 | no | — |
| 159 | C23 | — | 2 | 24 | no | — |
| 160 | C23 | — | 2 | 8 | no | — |
| 161 | C23 | — | 2 | 672 | no | — |
| 162 | C23 | — | 2 | 672 | no | — |
| 163 | C23 | — | 2 | 96 | no | — |
| 164 | C23 | — | 2 | 96 | no | — |
| 165 | C23 | — | 2 | 32 | no | — |
| 166 | C23 | — | 4 | 448 | no | — |
| 167 | C23 | — | 4 | 1344 | no | — |
| 168 | C23 | — | 4 | 1344 | no | — |
| 169 | C23 | — | 8 | 3584 | no | — |
| 170 | C23 | #2656 | 1 | 512 | yes | direct product factor |
| 171 | C23 | C22×#127 | 1 | 448 | yes | — |
| 172 | C23 | C22×#127 | 1 | 112 | yes | — |
| 173 | C23 | C2×#534 | 1 | 336 | yes | — |
| 174 | C23 | C2×#537 | 1 | 336 | yes | — |
| 175 | C23 | C24×D4 | 1 | 48 | yes | — |
| 176 | C23 | C24×D4 | 1 | 42 | yes | — |
| 177 | C23 | C22×H(5,𝔽2) | 1 | 42 | yes | — |
| 178 | C23 | C22×2−1+4 | 1 | 14 | yes | — |
| 179 | C23 | C27 | 1 | 1 | yes | characteristic |
| 180 | D4 | — | 4 | 1536 | no | — |
| 181 | D4 | — | 4 | 1024 | no | — |
| 182 | C2×C4 | — | 2 | 96 | no | — |
| 183 | C2×C4 | — | 2 | 1344 | no | — |
| 184 | C2×C4 | — | 2 | 96 | no | — |
| 185 | C2×C4 | — | 4 | 768 | no | — |
| 186 | C2×C4 | — | 4 | 448 | no | — |
| 187 | C2×C4 | C24×D4 | 1 | 96 | yes | — |
| 188 | C22 | — | 2 | 48 | no | — |
| 189 | C22 | — | 2 | 48 | no | — |
| 190 | C22 | — | 2 | 896 | no | — |
| 191 | C22 | — | 4 | 32 | no | — |
| 192 | C22 | — | 4 | 96 | no | — |
| 193 | C22 | — | 4 | 96 | no | — |
| 194 | C22 | — | 8 | 896 | no | — |
| 195 | C22 | C2×#2656 | 1 | 448 | yes | direct product factor |
| 196 | C22 | C23×#127 | 1 | 32 | yes | — |
| 197 | C22 | C23×#127 | 1 | 28 | yes | — |
| 198 | C22 | C22×#534 | 1 | 84 | yes | — |
| 199 | C22 | C22×#537 | 1 | 84 | yes | — |
| 200 | C22 | C25×D4 | 1 | 3 | yes | — |
| 201 | C22 | C23×H(5,𝔽2) | 1 | 3 | yes | — |
| 202 | C22 | C23×2−1+4 | 1 | 1 | yes | characteristic |
| 203 | C4 | — | 2 | 96 | no | — |
| 204 | C4 | — | 4 | 32 | no | — |
| 205 | C2 | — | 2 | 32 | no | — |
| 206 | C2 | — | 8 | 32 | no | — |
| 207 | C2 | C22×#2656 | 1 | 56 | yes | direct product factor |
| 208 | C2 | C24×#127 | 1 | 1 | yes | characteristic |
| 209 | C2 | C23×#534 | 1 | 3 | yes | — |
| 210 | C2 | C23×#537 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C24 |
| commutator, Frattini, cosocle | C23 | C27 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×#121 | C2 | semidirect |
| 2. C22×#2656 | C2 | direct (N × H) |
| 3. C23×#399 | C2 | semidirect |
| 4. C23×#392 | C2 | semidirect |
| 5. C24×#95 | C2 | semidirect |
| 10. C23×#121 | C22 | semidirect |
| 13. C2×#2656 | C22 | direct (N × H) |
| 14. C22×#399 | C22 | semidirect |
| 15. C22×#392 | C22 | semidirect |
| 17. C23×#95 | C22 | semidirect |
| 26. C25×C4 | D4 | semidirect |
| 31. C22×#121 | C23 | semidirect |
| 36. #2656 | C23 | direct (N × H) |
| 37. C2×#399 | C23 | semidirect |
| 38. C2×#392 | C23 | semidirect |
| 40. C22×#95 | C23 | semidirect |
| 62. C24×C4 | C2×D4 | semidirect |
| 67. C2×#121 | C24 | semidirect |
| 72. #399 | C24 | semidirect |
| 73. #392 | C24 | semidirect |
| 75. C2×#95 | C24 | semidirect |
| 108. C23×C4 | C22×D4 | semidirect |
| 153. C22×C4 | C23×D4 | semidirect |
| 170. C23 | #2656 | direct (N × H) |
| 195. C22 | C2×#2656 | direct (N × H) |
| 207. C2 | C22×#2656 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×#121 | 2 |
| C22×#2656 | 2 |
| C23×#399 | 2 |
| C23×#392 | 2 |
| C24×#95 | 2 |