← #62845
Subgroups — #62845
Order 256.
3551 conjugacy classes of subgroups.
The subgroup lattice has 199 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #3507 | C2 | 1 | 12 | yes | maximal, complemented |
| 2 | #3506 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | #3493 | C2 | 1 | 12 | yes | maximal, complemented |
| 4 | #3492 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #3459 | C2 | 1 | 8 | yes | maximal, complemented |
| 6 | #3458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #3449 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C2×#520 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | C22×#41 | C22 | 1 | 3 | yes | — |
| 10 | #559 | C22 | 1 | 24 | yes | complemented |
| 11 | #557 | C22 | 1 | 8 | yes | complemented |
| 12 | #556 | C22 | 1 | 24 | yes | complemented |
| 13 | #556 | C22 | 1 | 24 | yes | complemented |
| 14 | #555 | C22 | 1 | 24 | yes | complemented |
| 15 | #554 | C22 | 1 | 24 | yes | complemented |
| 16 | #553 | C22 | 1 | 48 | yes | complemented |
| 17 | #552 | C22 | 1 | 48 | yes | complemented |
| 18 | #550 | C22 | 1 | 8 | yes | complemented |
| 19 | #547 | C22 | 1 | 24 | yes | complemented |
| 20 | #544 | C22 | 1 | 6 | yes | complemented |
| 21 | #544 | C22 | 1 | 6 | yes | complemented |
| 22 | #543 | C22 | 1 | 4 | yes | — |
| 23 | #539 | C22 | 1 | 48 | yes | complemented |
| 24 | #538 | C22 | 1 | 48 | yes | complemented |
| 25 | #537 | C22 | 1 | 6 | yes | — |
| 26 | #537 | C22 | 1 | 6 | yes | — |
| 27 | #536 | C22 | 1 | 12 | yes | complemented |
| 28 | #535 | C22 | 1 | 12 | yes | complemented |
| 29 | #534 | C22 | 1 | 12 | yes | — |
| 30 | #533 | C22 | 1 | 12 | yes | complemented |
| 31 | #532 | C22 | 1 | 12 | yes | — |
| 32 | #529 | C22 | 1 | 48 | yes | complemented |
| 33 | #525 | C22 | 1 | 24 | yes | complemented |
| 34 | #525 | C22 | 1 | 48 | yes | complemented |
| 35 | #520 | C22 | 1 | 24 | yes | complemented |
| 36 | #520 | C22 | 1 | 24 | yes | complemented |
| 37 | C2×C4×Q8 | C22 | 1 | 6 | yes | complemented |
| 38 | C2×C4×D4 | C22 | 1 | 12 | yes | — |
| 39 | C2×C4×D4 | C22 | 1 | 6 | yes | complemented |
| 40 | C2×#117 | C22 | 1 | 3 | yes | — |
| 41 | C2×#117 | C22 | 1 | 6 | yes | complemented |
| 42 | C22×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 43 | C22×C4⋊C4 | C22 | 1 | 6 | yes | complemented |
| 44 | C2×#41 | — | 2 | 24 | no | — |
| 45 | C2×#41 | C23 | 1 | 12 | yes | — |
| 46 | C2×#41 | C23 | 1 | 12 | yes | — |
| 47 | C2×#41 | C23 | 1 | 16 | yes | — |
| 48 | C22×Q8 | C23 | 1 | 3 | yes | — |
| 49 | C22×D4 | C23 | 1 | 6 | yes | — |
| 50 | C22×D4 | C23 | 1 | 3 | yes | — |
| 51 | C23×C4 | C23 | 1 | 6 | yes | — |
| 52 | C23×C4 | C23 | 1 | 2 | yes | — |
| 53 | C23×C4 | C23 | 1 | 12 | yes | — |
| 54 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 55 | C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 56 | C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 57 | C4⋊Q8 | C23 | 1 | 12 | yes | — |
| 58 | #127 | C23 | 1 | 12 | yes | — |
| 59 | #126 | C23 | 1 | 96 | yes | — |
| 60 | #125 | C23 | 1 | 48 | yes | — |
| 61 | #125 | C23 | 1 | 24 | yes | — |
| 62 | #124 | C23 | 1 | 48 | yes | — |
| 63 | #124 | C23 | 1 | 24 | yes | — |
| 64 | #123 | C23 | 1 | 48 | yes | — |
| 65 | #123 | C23 | 1 | 96 | yes | — |
| 66 | C22⋊Q8 | C23 | 1 | 24 | yes | — |
| 67 | C22⋊Q8 | C23 | 1 | 24 | yes | — |
| 68 | C22⋊Q8 | C23 | 1 | 24 | yes | — |
| 69 | C22⋊Q8 | C23 | 1 | 48 | yes | — |
| 70 | C22⋊Q8 | C23 | 1 | 24 | yes | — |
| 71 | #121 | C23 | 1 | 24 | yes | — |
| 72 | #121 | C23 | 1 | 24 | yes | — |
| 73 | #121 | C23 | 1 | 24 | yes | — |
| 74 | #121 | C23 | 1 | 48 | yes | — |
| 75 | #121 | C23 | 1 | 24 | yes | — |
| 76 | #120 | C23 | 1 | 48 | yes | — |
| 77 | C4×Q8 | C23 | 1 | 24 | yes | — |
| 78 | C4×Q8 | C23 | 1 | 24 | yes | — |
| 79 | C4×Q8 | C23 | 1 | 16 | yes | — |
| 80 | C4×Q8 | C23 | 1 | 8 | yes | — |
| 81 | C4×Q8 | C23 | 1 | 8 | yes | — |
| 82 | C4×D4 | C23 | 1 | 96 | yes | — |
| 83 | C4×D4 | C23 | 1 | 24 | yes | — |
| 84 | C4×D4 | C23 | 1 | 24 | yes | — |
| 85 | C4×D4 | C23 | 1 | 48 | yes | — |
| 86 | C4×D4 | C23 | 1 | 24 | yes | — |
| 87 | C4×D4 | C23 | 1 | 24 | yes | — |
| 88 | #117 | C23 | 1 | 12 | yes | — |
| 89 | #117 | C23 | 1 | 12 | yes | — |
| 90 | #117 | C23 | 1 | 48 | yes | — |
| 91 | #117 | C23 | 1 | 48 | yes | — |
| 92 | C2×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 93 | C2×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 94 | C2×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 95 | C2×C4⋊C4 | C23 | 1 | 12 | yes | — |
| 96 | C2×C4⋊C4 | C23 | 1 | 48 | yes | — |
| 97 | C2×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 98 | C2×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 99 | C2×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 100 | C2×C42 | C23 | 1 | 6 | yes | — |
| 101 | C2×C42 | C23 | 1 | 12 | yes | — |
| 102 | C24 | C24 | 1 | 6 | yes | — |
| 103 | C24 | C24 | 1 | 1 | yes | characteristic |
| 104 | #41 | — | 2 | 48 | no | — |
| 105 | #41 | — | 2 | 48 | no | — |
| 106 | #41 | — | 8 | 16 | no | — |
| 107 | C2×Q8 | — | 2 | 12 | no | — |
| 108 | C2×Q8 | C24 | 1 | 6 | yes | — |
| 109 | C2×Q8 | C24 | 1 | 6 | yes | — |
| 110 | C2×Q8 | C24 | 1 | 8 | yes | — |
| 111 | C2×Q8 | C24 | 1 | 8 | yes | — |
| 112 | C2×D4 | — | 2 | 24 | no | — |
| 113 | C2×D4 | — | 2 | 12 | no | — |
| 114 | C2×D4 | C24 | 1 | 24 | yes | — |
| 115 | C2×D4 | C24 | 1 | 6 | yes | — |
| 116 | C2×D4 | C24 | 1 | 6 | yes | — |
| 117 | C2×D4 | C24 | 1 | 24 | yes | — |
| 118 | C2×D4 | C24 | 1 | 24 | yes | — |
| 119 | C22×C4 | — | 2 | 24 | no | — |
| 120 | C22×C4 | — | 8 | 2 | no | — |
| 121 | C22×C4 | — | 8 | 12 | no | — |
| 122 | C22×C4 | #41 | 1 | 8 | yes | complemented |
| 123 | C22×C4 | C24 | 1 | 6 | yes | — |
| 124 | C22×C4 | C24 | 1 | 24 | yes | — |
| 125 | C22×C4 | C24 | 1 | 6 | yes | — |
| 126 | C22×C4 | C24 | 1 | 12 | yes | — |
| 127 | C22×C4 | C24 | 1 | 6 | yes | — |
| 128 | C22×C4 | C24 | 1 | 48 | yes | — |
| 129 | C22×C4 | C24 | 1 | 3 | yes | — |
| 130 | C22×C4 | C24 | 1 | 3 | yes | — |
| 131 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 132 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 133 | C4⋊C4 | C24 | 1 | 4 | yes | — |
| 134 | C4⋊C4 | C24 | 1 | 48 | yes | — |
| 135 | C4⋊C4 | C24 | 1 | 12 | yes | — |
| 136 | C4⋊C4 | C24 | 1 | 24 | yes | — |
| 137 | C4⋊C4 | C24 | 1 | 24 | yes | — |
| 138 | C4⋊C4 | C24 | 1 | 24 | yes | — |
| 139 | C4⋊C4 | C24 | 1 | 24 | yes | — |
| 140 | C22⋊C4 | C24 | 1 | 24 | yes | — |
| 141 | C22⋊C4 | C24 | 1 | 48 | yes | — |
| 142 | C22⋊C4 | C24 | 1 | 48 | yes | — |
| 143 | C22⋊C4 | C24 | 1 | 48 | yes | — |
| 144 | C42 | C24 | 1 | 48 | yes | — |
| 145 | C42 | C24 | 1 | 12 | yes | — |
| 146 | C42 | C24 | 1 | 12 | yes | — |
| 147 | C23 | — | 2 | 12 | no | — |
| 148 | C23 | — | 4 | 1 | no | — |
| 149 | C23 | — | 4 | 6 | no | — |
| 150 | C23 | — | 4 | 1 | no | — |
| 151 | C23 | — | 4 | 6 | no | — |
| 152 | C23 | C2×#41 | 1 | 4 | yes | — |
| 153 | C23 | C25 | 1 | 12 | yes | — |
| 154 | C23 | C25 | 1 | 3 | yes | — |
| 155 | Q8 | — | 2 | 12 | no | — |
| 156 | Q8 | — | 2 | 12 | no | — |
| 157 | Q8 | — | 4 | 8 | no | — |
| 158 | Q8 | — | 4 | 8 | no | — |
| 159 | D4 | — | 2 | 48 | no | — |
| 160 | D4 | — | 2 | 12 | no | — |
| 161 | D4 | — | 2 | 12 | no | — |
| 162 | D4 | — | 4 | 24 | no | — |
| 163 | D4 | — | 4 | 24 | no | — |
| 164 | C2×C4 | — | 2 | 12 | no | — |
| 165 | C2×C4 | — | 2 | 48 | no | — |
| 166 | C2×C4 | — | 4 | 12 | no | — |
| 167 | C2×C4 | — | 4 | 6 | no | — |
| 168 | C2×C4 | — | 4 | 48 | no | — |
| 169 | C2×C4 | — | 4 | 6 | no | — |
| 170 | C2×C4 | C2×#41 | 1 | 12 | yes | — |
| 171 | C2×C4 | C2×#41 | 1 | 12 | yes | — |
| 172 | C2×C4 | C25 | 1 | 12 | yes | — |
| 173 | C2×C4 | C25 | 1 | 8 | yes | — |
| 174 | C2×C4 | C25 | 1 | 12 | yes | — |
| 175 | C2×C4 | C25 | 1 | 12 | yes | — |
| 176 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 177 | C2×C4 | C25 | 1 | 3 | yes | — |
| 178 | C22 | — | 2 | 3 | no | — |
| 179 | C22 | — | 2 | 3 | no | — |
| 180 | C22 | — | 2 | 12 | no | — |
| 181 | C22 | — | 2 | 12 | no | — |
| 182 | C22 | — | 2 | 12 | no | — |
| 183 | C22 | — | 4 | 4 | no | — |
| 184 | C22 | — | 4 | 24 | no | — |
| 185 | C22 | C22×#41 | 1 | 6 | yes | — |
| 186 | C22 | C26 | 1 | 1 | yes | characteristic |
| 187 | C4 | — | 2 | 12 | no | — |
| 188 | C4 | — | 2 | 8 | no | — |
| 189 | C4 | — | 2 | 12 | no | — |
| 190 | C4 | — | 2 | 12 | no | — |
| 191 | C4 | C22×#41 | 1 | 6 | yes | — |
| 192 | C4 | C22×#41 | 1 | 2 | yes | — |
| 193 | C2 | — | 2 | 6 | no | — |
| 194 | C2 | — | 4 | 12 | no | — |
| 195 | C2 | C23×#41 | 1 | 1 | yes | characteristic |
| 196 | C2 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 197 | C2 | 2−1+6 | 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