← #22728
Subgroups — #22728
Order 256.
387 conjugacy classes of subgroups.
The subgroup lattice has 206 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#446 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2355 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | Q8×C4⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #1949 | C2 | 1 | 2 | yes | maximal |
| 5 | #1948 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #1947 | C2 | 1 | 2 | yes | maximal |
| 7 | #1947 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #1847 | C2 | 1 | 2 | yes | maximal |
| 9 | #1845 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #1776 | C2 | 1 | 2 | yes | maximal |
| 11 | C4×#106 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | C2×C4×Q8 | — | 2 | 1 | no | — |
| 13 | C2×C4×Q8 | C22 | 1 | 1 | yes | characteristic |
| 14 | #446 | C4 | 1 | 8 | yes | complemented |
| 15 | C2×#107 | C22 | 1 | 2 | yes | — |
| 16 | C2×#107 | C22 | 1 | 2 | yes | — |
| 17 | C2×#107 | C22 | 1 | 2 | yes | — |
| 18 | C2×#106 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×#106 | C22 | 1 | 2 | yes | — |
| 20 | C2×#106 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C4⋊C8 | C22 | 1 | 2 | yes | — |
| 23 | C2×#97 | C22 | 1 | 2 | yes | — |
| 24 | C2×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 25 | #391 | — | 2 | 1 | no | — |
| 26 | #389 | — | 2 | 1 | no | — |
| 27 | #389 | — | 2 | 2 | no | — |
| 28 | #387 | — | 2 | 2 | no | — |
| 29 | #384 | C22 | 1 | 1 | yes | characteristic |
| 30 | #384 | C22 | 1 | 2 | yes | — |
| 31 | #384 | C22 | 1 | 1 | yes | characteristic |
| 32 | #382 | C22 | 1 | 2 | yes | — |
| 33 | C4×C4⋊C4 | — | 2 | 1 | no | — |
| 34 | C4×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 35 | C4×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 36 | C4×C4⋊C4 | C22 | 1 | 2 | yes | — |
| 37 | C4×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 38 | #340 | C22 | 1 | 1 | yes | characteristic |
| 39 | #340 | C22 | 1 | 2 | yes | — |
| 40 | #340 | C22 | 1 | 2 | yes | — |
| 41 | #340 | C22 | 1 | 1 | yes | characteristic |
| 42 | C22×Q8 | D4 | 1 | 1 | yes | characteristic |
| 43 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C22×C8 | C23 | 1 | 2 | yes | — |
| 45 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 46 | C4×Q8 | — | 4 | 4 | no | — |
| 47 | C4×Q8 | C2×C4 | 1 | 4 | yes | — |
| 48 | C4×Q8 | Q8 | 1 | 4 | yes | — |
| 49 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 50 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 51 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 52 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 53 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 54 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 55 | C2×C4⋊C4 | D4 | 1 | 1 | yes | characteristic |
| 56 | C2×C4⋊C4 | D4 | 1 | 2 | yes | — |
| 57 | C2×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 58 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 59 | C2×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 60 | C2×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 61 | C2×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 62 | C2×C4⋊C4 | C23 | 1 | 1 | yes | characteristic |
| 63 | C2×C42 | — | 2 | 2 | no | — |
| 64 | C2×C42 | — | 2 | 1 | no | — |
| 65 | C2×C42 | — | 2 | 1 | no | — |
| 66 | C2×C42 | — | 2 | 2 | no | — |
| 67 | C2×C42 | C23 | 1 | 2 | yes | — |
| 68 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 69 | #107 | — | 2 | 4 | no | — |
| 70 | #107 | — | 4 | 2 | no | — |
| 71 | #107 | — | 4 | 2 | no | — |
| 72 | #106 | — | 2 | 2 | no | — |
| 73 | #106 | — | 4 | 2 | no | — |
| 74 | #106 | Q8 | 1 | 4 | yes | complemented |
| 75 | C4⋊C8 | C2×C4 | 1 | 4 | yes | — |
| 76 | C4⋊C8 | C2×C4 | 1 | 8 | yes | — |
| 77 | #97 | C2×C4 | 1 | 8 | yes | — |
| 78 | C4×C8 | C2×C4 | 1 | 4 | yes | — |
| 79 | #95 | — | 2 | 2 | no | — |
| 80 | #95 | — | 2 | 1 | no | — |
| 81 | #95 | — | 2 | 2 | no | — |
| 82 | #95 | — | 2 | 2 | no | — |
| 83 | #95 | — | 2 | 1 | no | — |
| 84 | #95 | — | 2 | 2 | no | — |
| 85 | C2×Q8 | — | 2 | 2 | no | — |
| 86 | C2×Q8 | — | 2 | 2 | no | — |
| 87 | C2×Q8 | C4⋊C4 | 1 | 4 | yes | — |
| 88 | C22×C4 | — | 2 | 1 | no | — |
| 89 | C22×C4 | — | 2 | 1 | no | — |
| 90 | C22×C4 | — | 2 | 1 | no | — |
| 91 | C22×C4 | — | 2 | 2 | no | — |
| 92 | C22×C4 | — | 2 | 1 | no | — |
| 93 | C22×C4 | — | 2 | 2 | no | — |
| 94 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 95 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 96 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 97 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 98 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 99 | C2×C8 | — | 2 | 2 | no | — |
| 100 | C2×C8 | — | 2 | 1 | no | — |
| 101 | C2×C8 | — | 2 | 1 | no | — |
| 102 | C2×C8 | — | 2 | 2 | no | — |
| 103 | C2×C8 | C22×C4 | 1 | 2 | yes | — |
| 104 | C2×C8 | C22×C4 | 1 | 4 | yes | — |
| 105 | C2×C8 | C22×C4 | 1 | 2 | yes | — |
| 106 | C2×C8 | C2×Q8 | 1 | 2 | yes | — |
| 107 | C2×C8 | #41 | 1 | 2 | yes | — |
| 108 | C4⋊C4 | — | 2 | 1 | no | — |
| 109 | C4⋊C4 | — | 2 | 1 | no | — |
| 110 | C4⋊C4 | — | 2 | 1 | no | — |
| 111 | C4⋊C4 | — | 2 | 1 | no | — |
| 112 | C4⋊C4 | — | 2 | 2 | no | — |
| 113 | C4⋊C4 | — | 2 | 2 | no | — |
| 114 | C4⋊C4 | — | 2 | 1 | no | — |
| 115 | C4⋊C4 | — | 2 | 2 | no | — |
| 116 | C4⋊C4 | — | 2 | 2 | no | — |
| 117 | C4⋊C4 | — | 2 | 1 | no | — |
| 118 | C4⋊C4 | — | 4 | 4 | no | — |
| 119 | C4⋊C4 | — | 4 | 2 | no | — |
| 120 | C4⋊C4 | — | 4 | 2 | no | — |
| 121 | C4⋊C4 | — | 8 | 2 | no | — |
| 122 | C4⋊C4 | — | 8 | 1 | no | — |
| 123 | C4⋊C4 | — | 8 | 1 | no | — |
| 124 | C4⋊C4 | C4⋊C4 | 1 | 8 | yes | — |
| 125 | C4⋊C4 | C4⋊C4 | 1 | 4 | yes | — |
| 126 | C4⋊C4 | C2×Q8 | 1 | 2 | yes | — |
| 127 | C4⋊C4 | C2×Q8 | 1 | 2 | yes | — |
| 128 | C42 | — | 4 | 4 | no | — |
| 129 | C42 | — | 4 | 2 | no | — |
| 130 | C42 | — | 4 | 2 | no | — |
| 131 | C42 | — | 8 | 2 | no | — |
| 132 | C42 | C22×C4 | 1 | 2 | yes | — |
| 133 | C42 | C22×C4 | 1 | 4 | yes | — |
| 134 | C42 | C2×Q8 | 1 | 4 | yes | — |
| 135 | C42 | C2×Q8 | 1 | 2 | yes | — |
| 136 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 137 | Q8 | — | 2 | 8 | no | — |
| 138 | C2×C4 | — | 2 | 2 | no | — |
| 139 | C2×C4 | — | 2 | 2 | no | — |
| 140 | C2×C4 | — | 2 | 2 | no | — |
| 141 | C2×C4 | — | 2 | 1 | no | — |
| 142 | C2×C4 | — | 2 | 2 | no | — |
| 143 | C2×C4 | — | 2 | 1 | no | — |
| 144 | C2×C4 | — | 2 | 2 | no | — |
| 145 | C2×C4 | — | 2 | 2 | no | — |
| 146 | C2×C4 | — | 4 | 1 | no | — |
| 147 | C2×C4 | — | 4 | 1 | no | — |
| 148 | C2×C4 | — | 4 | 1 | no | — |
| 149 | C2×C4 | — | 4 | 1 | no | — |
| 150 | C2×C4 | — | 4 | 1 | no | — |
| 151 | C2×C4 | — | 4 | 1 | no | — |
| 152 | C2×C4 | — | 4 | 2 | no | — |
| 153 | C2×C4 | — | 4 | 2 | no | — |
| 154 | C2×C4 | — | 4 | 2 | no | — |
| 155 | C2×C4 | — | 4 | 1 | no | — |
| 156 | C2×C4 | — | 4 | 2 | no | — |
| 157 | C2×C4 | — | 4 | 1 | no | — |
| 158 | C2×C4 | — | 4 | 2 | no | — |
| 159 | C2×C4 | — | 4 | 1 | no | — |
| 160 | C2×C4 | — | 4 | 2 | no | — |
| 161 | C2×C4 | — | 4 | 1 | no | — |
| 162 | C2×C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 163 | C2×C4 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 164 | C2×C4 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 165 | C2×C4 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 166 | C2×C4 | AGL(1,ℤ/8) | 1 | 2 | yes | — |
| 167 | C2×C4 | #137 | 1 | 2 | yes | — |
| 168 | C2×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 169 | C2×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 170 | C2×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 171 | C2×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 172 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 173 | C2×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 174 | C8 | — | 2 | 4 | no | — |
| 175 | C8 | — | 2 | 2 | no | — |
| 176 | C8 | C4×Q8 | 1 | 4 | yes | complemented |
| 177 | C22 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 178 | C22 | D4×Q8 | 1 | 1 | yes | characteristic |
| 179 | C22 | #550 | 1 | 1 | yes | characteristic |
| 180 | C22 | C2×AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 181 | C22 | C2×#137 | 1 | 1 | yes | characteristic |
| 182 | C22 | #576 | 1 | 1 | yes | characteristic |
| 183 | C22 | #578 | 1 | 1 | yes | characteristic |
| 184 | C4 | — | 2 | 4 | no | — |
| 185 | C4 | — | 2 | 4 | no | — |
| 186 | C4 | — | 2 | 2 | no | — |
| 187 | C4 | — | 4 | 2 | no | — |
| 188 | C4 | — | 4 | 2 | no | — |
| 189 | C4 | — | 8 | 1 | no | — |
| 190 | C4 | — | 8 | 1 | no | — |
| 191 | C4 | — | 8 | 2 | no | — |
| 192 | C4 | — | 8 | 2 | no | — |
| 193 | C4 | #428 | 1 | 4 | yes | — |
| 194 | C4 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 195 | C4 | #520 | 1 | 1 | yes | characteristic |
| 196 | C4 | #557 | 1 | 1 | yes | characteristic |
| 197 | C4 | Q82 | 1 | 1 | yes | characteristic |
| 198 | C2 | Q8×C4⋊C4 | 1 | 1 | yes | characteristic |
| 199 | C2 | C2×#428 | 1 | 1 | yes | characteristic |
| 200 | C2 | #2913 | 1 | 1 | yes | characteristic |
| 201 | C2 | #3385 | 1 | 1 | yes | characteristic |
| 202 | C2 | #3386 | 1 | 1 | yes | characteristic |
| 203 | C2 | #3398 | 1 | 1 | yes | characteristic |
| 204 | C2 | #3399 | 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