← #60253
Subgroups — #60253
Order 256.
851 conjugacy classes of subgroups.
The subgroup lattice has 214 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | Q8×#41 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2986 | C2 | 1 | 6 | yes | maximal, complemented |
| 3 | #2985 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #2984 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #2982 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | Q8×M16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #2962 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #2960 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | C2×C8×Q8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #2925 | C2 | 1 | 3 | yes | maximal |
| 11 | #2922 | C2 | 1 | 3 | yes | maximal |
| 12 | Q82 | C4 | 1 | 1 | yes | characteristic |
| 13 | #557 | C4 | 1 | 3 | yes | — |
| 14 | #554 | C4 | 1 | 6 | yes | — |
| 15 | #554 | C4 | 1 | 3 | yes | — |
| 16 | D4×Q8 | C4 | 1 | 1 | yes | characteristic |
| 17 | D4×Q8 | C4 | 1 | 2 | yes | — |
| 18 | #533 | C22 | 1 | 3 | yes | — |
| 19 | #533 | C22 | 1 | 6 | yes | — |
| 20 | C4×#41 | C22 | 1 | 3 | yes | — |
| 21 | C2×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4×Q8 | C22 | 1 | 2 | yes | — |
| 23 | #446 | C22 | 1 | 6 | yes | — |
| 24 | #446 | C22 | 1 | 6 | yes | — |
| 25 | #446 | C22 | 1 | 6 | yes | — |
| 26 | #446 | C22 | 1 | 12 | yes | — |
| 27 | #446 | C22 | 1 | 6 | yes | — |
| 28 | C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 29 | C8×Q8 | C22 | 1 | 1 | yes | characteristic |
| 30 | C8×Q8 | C22 | 1 | 3 | yes | — |
| 31 | C8×Q8 | C22 | 1 | 3 | yes | — |
| 32 | C8×Q8 | C22 | 1 | 6 | yes | — |
| 33 | C8×Q8 | C22 | 1 | 6 | yes | — |
| 34 | C8×Q8 | C22 | 1 | 6 | yes | — |
| 35 | C8×Q8 | C22 | 1 | 2 | yes | — |
| 36 | #433 | C22 | 1 | 6 | yes | — |
| 37 | #433 | C22 | 1 | 6 | yes | — |
| 38 | #433 | C22 | 1 | 12 | yes | — |
| 39 | #432 | C22 | 1 | 6 | yes | — |
| 40 | #431 | C22 | 1 | 6 | yes | — |
| 41 | #431 | C22 | 1 | 6 | yes | — |
| 42 | #431 | C22 | 1 | 6 | yes | — |
| 43 | #424 | C22 | 1 | 3 | yes | — |
| 44 | #424 | C22 | 1 | 3 | yes | — |
| 45 | #423 | C22 | 1 | 3 | yes | — |
| 46 | C2×C4⋊C8 | C22 | 1 | 3 | yes | — |
| 47 | #408 | C22 | 1 | 1 | yes | characteristic |
| 48 | #408 | C22 | 1 | 3 | yes | — |
| 49 | #405 | C22 | 1 | 6 | yes | — |
| 50 | C4×M16 | C22 | 1 | 3 | yes | — |
| 51 | C2×C4×C8 | C22 | 1 | 3 | yes | — |
| 52 | C2×#41 | C23 | 1 | 1 | yes | characteristic |
| 53 | C22×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C22×Q8 | C2×C4 | 1 | 2 | yes | — |
| 55 | C2×M16 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×M16 | C23 | 1 | 3 | yes | — |
| 57 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 58 | C22×C8 | C23 | 1 | 3 | yes | — |
| 59 | C4⋊Q8 | C2×C4 | 1 | 3 | yes | — |
| 60 | C4⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 61 | C4⋊Q8 | C2×C4 | 1 | 3 | yes | — |
| 62 | C4⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 63 | #125 | C2×C4 | 1 | 6 | yes | — |
| 64 | #125 | C2×C4 | 1 | 12 | yes | — |
| 65 | C22⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 66 | C22⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 67 | C22⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 68 | C22⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 69 | C22⋊Q8 | C2×C4 | 1 | 12 | yes | — |
| 70 | C4×Q8 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | C4×Q8 | C2×C4 | 1 | 2 | yes | — |
| 72 | C4×Q8 | C2×C4 | 1 | 3 | yes | — |
| 73 | C4×Q8 | C2×C4 | 1 | 6 | yes | — |
| 74 | C4×Q8 | C2×C4 | 1 | 3 | yes | — |
| 75 | C4×Q8 | C2×C4 | 1 | 3 | yes | — |
| 76 | C4×Q8 | C23 | 1 | 1 | yes | characteristic |
| 77 | C4×Q8 | C23 | 1 | 3 | yes | — |
| 78 | C4×Q8 | C23 | 1 | 6 | yes | — |
| 79 | C4×Q8 | C23 | 1 | 6 | yes | — |
| 80 | C4×D4 | C2×C4 | 1 | 6 | yes | — |
| 81 | C4×D4 | C2×C4 | 1 | 3 | yes | — |
| 82 | C4×D4 | C2×C4 | 1 | 6 | yes | — |
| 83 | C4×D4 | C2×C4 | 1 | 3 | yes | — |
| 84 | #117 | C23 | 1 | 3 | yes | — |
| 85 | #117 | C23 | 1 | 6 | yes | — |
| 86 | C2×C4⋊C4 | C2×C4 | 1 | 3 | yes | — |
| 87 | C2×C4⋊C4 | C2×C4 | 1 | 6 | yes | — |
| 88 | C2×C42 | C23 | 1 | 3 | yes | — |
| 89 | C2×C42 | C23 | 1 | 6 | yes | — |
| 90 | C4⋊C8 | C23 | 1 | 3 | yes | — |
| 91 | C4⋊C8 | C23 | 1 | 3 | yes | — |
| 92 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 93 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 94 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 95 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 96 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 97 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 98 | C4⋊C8 | C23 | 1 | 6 | yes | — |
| 99 | C22⋊C8 | C23 | 1 | 2 | yes | — |
| 100 | C22⋊C8 | C23 | 1 | 2 | yes | — |
| 101 | C22⋊C8 | C23 | 1 | 6 | yes | — |
| 102 | C22⋊C8 | C23 | 1 | 6 | yes | — |
| 103 | #97 | C23 | 1 | 6 | yes | — |
| 104 | #97 | C23 | 1 | 12 | yes | — |
| 105 | C4×C8 | C23 | 1 | 3 | yes | — |
| 106 | C4×C8 | C23 | 1 | 3 | yes | — |
| 107 | C4×C8 | C23 | 1 | 3 | yes | — |
| 108 | C4×C8 | C23 | 1 | 3 | yes | — |
| 109 | C4×C8 | C23 | 1 | 6 | yes | — |
| 110 | C4×C8 | C23 | 1 | 12 | yes | — |
| 111 | #41 | — | 2 | 2 | no | — |
| 112 | #41 | — | 2 | 2 | no | — |
| 113 | C2×Q8 | — | 2 | 4 | no | — |
| 114 | C2×Q8 | — | 2 | 8 | no | — |
| 115 | C2×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 116 | C2×Q8 | C22×C4 | 1 | 3 | yes | — |
| 117 | C2×Q8 | C22×C4 | 1 | 6 | yes | — |
| 118 | C2×Q8 | C22×C4 | 1 | 6 | yes | — |
| 119 | C2×Q8 | C22×C4 | 1 | 1 | yes | characteristic |
| 120 | C2×D4 | C22×C4 | 1 | 2 | yes | — |
| 121 | C2×D4 | C22×C4 | 1 | 1 | yes | characteristic |
| 122 | C22×C4 | C22×C4 | 1 | 6 | yes | — |
| 123 | C22×C4 | C22×C4 | 1 | 3 | yes | — |
| 124 | C22×C4 | C22×C4 | 1 | 6 | yes | — |
| 125 | C22×C4 | C22×C4 | 1 | 3 | yes | — |
| 126 | C22×C4 | C24 | 1 | 2 | yes | — |
| 127 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 128 | M16 | — | 2 | 2 | no | — |
| 129 | M16 | — | 2 | 6 | no | — |
| 130 | C2×C8 | — | 2 | 6 | no | — |
| 131 | C2×C8 | — | 2 | 2 | no | — |
| 132 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 133 | C2×C8 | C24 | 1 | 3 | yes | — |
| 134 | C2×C8 | C24 | 1 | 1 | yes | characteristic |
| 135 | C2×C8 | C24 | 1 | 2 | yes | — |
| 136 | C2×C8 | C24 | 1 | 6 | yes | — |
| 137 | C2×C8 | C24 | 1 | 3 | yes | — |
| 138 | C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 139 | C4⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 140 | C4⋊C4 | C22×C4 | 1 | 12 | yes | — |
| 141 | C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 142 | C4⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 143 | C4⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 144 | C4⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 145 | C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 146 | C4⋊C4 | C22×C4 | 1 | 3 | yes | — |
| 147 | C4⋊C4 | C22×C4 | 1 | 12 | yes | — |
| 148 | C4⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 149 | C22⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 150 | C22⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 151 | C22⋊C4 | C22×C4 | 1 | 6 | yes | — |
| 152 | C42 | C22×C4 | 1 | 6 | yes | — |
| 153 | C42 | C22×C4 | 1 | 3 | yes | — |
| 154 | C42 | C22×C4 | 1 | 6 | yes | — |
| 155 | C42 | C22×C4 | 1 | 3 | yes | — |
| 156 | C42 | C24 | 1 | 3 | yes | — |
| 157 | C42 | C24 | 1 | 3 | yes | — |
| 158 | C42 | C24 | 1 | 6 | yes | — |
| 159 | C23 | C23×C4 | 1 | 2 | yes | — |
| 160 | C23 | C23×C4 | 1 | 1 | yes | characteristic |
| 161 | Q8 | — | 2 | 2 | no | — |
| 162 | Q8 | — | 2 | 6 | no | — |
| 163 | Q8 | — | 2 | 12 | no | — |
| 164 | Q8 | — | 4 | 6 | no | — |
| 165 | Q8 | #131 | 1 | 4 | yes | — |
| 166 | D4 | — | 2 | 4 | no | — |
| 167 | D4 | — | 2 | 2 | no | — |
| 168 | C2×C4 | — | 2 | 6 | no | — |
| 169 | C2×C4 | — | 2 | 12 | no | — |
| 170 | C2×C4 | — | 2 | 2 | no | — |
| 171 | C2×C4 | — | 2 | 2 | no | — |
| 172 | C2×C4 | — | 4 | 3 | no | — |
| 173 | C2×C4 | — | 4 | 6 | no | — |
| 174 | C2×C4 | C23×C4 | 1 | 2 | yes | — |
| 175 | C2×C4 | C23×C4 | 1 | 3 | yes | — |
| 176 | C2×C4 | C23×C4 | 1 | 6 | yes | — |
| 177 | C2×C4 | C23×C4 | 1 | 3 | yes | — |
| 178 | C2×C4 | C23×C4 | 1 | 6 | yes | — |
| 179 | C2×C4 | C23×C4 | 1 | 3 | yes | — |
| 180 | C2×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 181 | C2×C4 | C23×C4 | 1 | 3 | yes | — |
| 182 | C2×C4 | 2−1+4 | 1 | 2 | yes | — |
| 183 | C2×C4 | 2−1+4 | 1 | 2 | yes | — |
| 184 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 185 | C8 | — | 2 | 3 | no | — |
| 186 | C8 | — | 2 | 1 | no | — |
| 187 | C8 | — | 2 | 1 | no | — |
| 188 | C8 | — | 2 | 3 | no | — |
| 189 | C8 | — | 2 | 2 | no | — |
| 190 | C8 | — | 2 | 6 | no | — |
| 191 | C22 | — | 2 | 1 | no | — |
| 192 | C22 | — | 2 | 1 | no | — |
| 193 | C22 | — | 2 | 2 | no | — |
| 194 | C22 | — | 2 | 2 | no | — |
| 195 | C22 | — | 2 | 2 | no | — |
| 196 | C22 | #519 | 1 | 2 | yes | — |
| 197 | C22 | C24×C4 | 1 | 1 | yes | characteristic |
| 198 | C4 | — | 2 | 2 | no | — |
| 199 | C4 | — | 2 | 3 | no | — |
| 200 | C4 | — | 2 | 6 | no | — |
| 201 | C4 | — | 2 | 3 | no | — |
| 202 | C4 | — | 2 | 3 | no | — |
| 203 | C4 | — | 2 | 6 | no | — |
| 204 | C4 | #519 | 1 | 2 | yes | — |
| 205 | C4 | C2×#131 | 1 | 6 | yes | — |
| 206 | C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 207 | C4 | C2×2−1+4 | 1 | 1 | yes | characteristic |
| 208 | C2 | — | 2 | 2 | no | — |
| 209 | C2 | — | 4 | 2 | no | — |
| 210 | C2 | C2×#519 | 1 | 1 | yes | characteristic |
| 211 | C2 | C22×#131 | 1 | 1 | yes | characteristic |
| 212 | C2 | #3573 | 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