← #22802
Subgroups — #22802
Order 256.
361 conjugacy classes of subgroups.
The subgroup lattice has 194 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C4×#135 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2925 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #1974 | C2 | 1 | 2 | yes | maximal |
| 4 | #1929 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #1848 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #1806 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #1772 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C8×C4⋊C4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#135 | C22 | 1 | 1 | yes | characteristic |
| 10 | C4×#41 | C22 | 1 | 2 | yes | — |
| 11 | C4×Q16 | C4 | 1 | 1 | yes | characteristic, complemented |
| 12 | C4×Q16 | C4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C4×SD16 | C4 | 1 | 2 | yes | complemented |
| 14 | C4×SD16 | C4 | 1 | 2 | yes | complemented |
| 15 | C4×D8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 16 | C4×D8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 17 | #433 | — | 2 | 4 | no | — |
| 18 | #432 | — | 2 | 2 | no | — |
| 19 | #431 | — | 2 | 2 | no | — |
| 20 | C2×#108 | C22 | 1 | 2 | yes | — |
| 21 | C2×#108 | C22 | 1 | 2 | yes | — |
| 22 | #427 | C22 | 1 | 1 | yes | characteristic |
| 23 | #423 | C22 | 1 | 2 | yes | — |
| 24 | C2×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 25 | C2×#104 | C22 | 1 | 4 | yes | — |
| 26 | #416 | C22 | 1 | 2 | yes | — |
| 27 | #408 | C22 | 1 | 4 | yes | — |
| 28 | #405 | C22 | 1 | 4 | yes | — |
| 29 | C2×C4×C8 | C22 | 1 | 2 | yes | — |
| 30 | C2×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 31 | C4×C4⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 32 | #339 | C22 | 1 | 4 | yes | — |
| 33 | #336 | C22 | 1 | 2 | yes | — |
| 34 | C2×#41 | C23 | 1 | 2 | yes | — |
| 35 | #135 | — | 2 | 2 | no | — |
| 36 | #135 | — | 2 | 2 | no | — |
| 37 | C2×Q16 | C2×C4 | 1 | 1 | yes | characteristic |
| 38 | C2×SD16 | C2×C4 | 1 | 2 | yes | — |
| 39 | C2×D8 | C2×C4 | 1 | 1 | yes | characteristic |
| 40 | C2×M16 | C23 | 1 | 4 | yes | — |
| 41 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 42 | C22×C8 | C23 | 1 | 2 | yes | — |
| 43 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | C4×Q8 | C2×C4 | 1 | 2 | yes | — |
| 45 | C4×Q8 | C2×C4 | 1 | 2 | yes | — |
| 46 | C4×D4 | — | 2 | 2 | no | — |
| 47 | C4×D4 | — | 2 | 2 | no | — |
| 48 | C4×D4 | C2×C4 | 1 | 2 | yes | — |
| 49 | C4×D4 | C2×C4 | 1 | 2 | yes | — |
| 50 | #117 | — | 2 | 2 | no | — |
| 51 | #117 | C23 | 1 | 2 | yes | — |
| 52 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 53 | C2×C42 | — | 2 | 2 | no | — |
| 54 | C2×C42 | C23 | 1 | 2 | yes | — |
| 55 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 56 | #108 | — | 2 | 4 | no | — |
| 57 | #108 | — | 4 | 2 | no | — |
| 58 | #107 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | #107 | C2×C4 | 1 | 1 | yes | characteristic |
| 60 | #106 | C2×C4 | 1 | 2 | yes | — |
| 61 | C4⋊C8 | — | 2 | 2 | no | — |
| 62 | C4⋊C8 | — | 2 | 2 | no | — |
| 63 | C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 64 | C4⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 65 | C4⋊C8 | D4 | 1 | 2 | yes | — |
| 66 | #104 | — | 4 | 4 | no | — |
| 67 | #103 | C2×C4 | 1 | 2 | yes | — |
| 68 | #103 | C2×C4 | 1 | 2 | yes | — |
| 69 | #102 | C2×C4 | 1 | 2 | yes | — |
| 70 | #102 | C2×C4 | 1 | 2 | yes | — |
| 71 | C22⋊C8 | — | 2 | 4 | no | — |
| 72 | C22⋊C8 | — | 2 | 4 | no | — |
| 73 | #97 | — | 2 | 4 | no | — |
| 74 | C4×C8 | — | 2 | 4 | no | — |
| 75 | C4×C8 | — | 2 | 4 | no | — |
| 76 | C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 77 | C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 78 | C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 79 | C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 80 | #95 | — | 2 | 1 | no | — |
| 81 | #41 | — | 2 | 2 | no | — |
| 82 | #41 | — | 2 | 2 | no | — |
| 83 | #41 | — | 2 | 2 | no | — |
| 84 | #41 | — | 2 | 2 | no | — |
| 85 | C2×Q8 | C22×C4 | 1 | 2 | yes | — |
| 86 | C2×D4 | — | 2 | 2 | no | — |
| 87 | C2×D4 | C22×C4 | 1 | 2 | yes | — |
| 88 | C22×C4 | — | 2 | 2 | no | — |
| 89 | C22×C4 | — | 2 | 2 | no | — |
| 90 | C22×C4 | — | 2 | 2 | no | — |
| 91 | C22×C4 | — | 2 | 2 | no | — |
| 92 | C22×C4 | #41 | 1 | 1 | yes | characteristic |
| 93 | C22×C4 | #41 | 1 | 1 | yes | characteristic |
| 94 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 95 | Q16 | — | 2 | 2 | no | — |
| 96 | SD16 | — | 2 | 4 | no | — |
| 97 | D8 | — | 2 | 2 | no | — |
| 98 | M16 | — | 2 | 4 | no | — |
| 99 | M16 | — | 2 | 4 | no | — |
| 100 | C2×C8 | — | 2 | 2 | no | — |
| 101 | C2×C8 | — | 2 | 1 | no | — |
| 102 | C2×C8 | — | 2 | 1 | no | — |
| 103 | C2×C8 | — | 2 | 2 | no | — |
| 104 | C2×C8 | — | 2 | 4 | no | — |
| 105 | C2×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 106 | C2×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 107 | C2×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 108 | C2×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 110 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 111 | C2×C8 | #41 | 1 | 2 | yes | — |
| 112 | C2×C8 | #41 | 1 | 2 | yes | — |
| 113 | C4⋊C4 | — | 2 | 4 | no | — |
| 114 | C4⋊C4 | — | 2 | 2 | no | — |
| 115 | C4⋊C4 | — | 2 | 2 | no | — |
| 116 | C4⋊C4 | C22×C4 | 1 | 2 | yes | — |
| 117 | C4⋊C4 | C22×C4 | 1 | 2 | yes | — |
| 118 | C22⋊C4 | — | 2 | 2 | no | — |
| 119 | C22⋊C4 | — | 2 | 2 | no | — |
| 120 | C22⋊C4 | — | 2 | 2 | no | — |
| 121 | C42 | — | 2 | 2 | no | — |
| 122 | C42 | — | 2 | 2 | no | — |
| 123 | C42 | — | 2 | 2 | no | — |
| 124 | C42 | — | 2 | 2 | no | — |
| 125 | C42 | — | 2 | 2 | no | — |
| 126 | C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 127 | C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 128 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 129 | C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 130 | C23 | — | 2 | 2 | no | — |
| 131 | C23 | C2×#41 | 1 | 1 | yes | characteristic |
| 132 | Q8 | — | 2 | 2 | no | — |
| 133 | Q8 | — | 2 | 2 | no | — |
| 134 | D4 | — | 2 | 2 | no | — |
| 135 | D4 | — | 2 | 2 | no | — |
| 136 | D4 | — | 4 | 2 | no | — |
| 137 | D4 | — | 4 | 2 | no | — |
| 138 | C2×C4 | — | 2 | 2 | no | — |
| 139 | C2×C4 | — | 2 | 4 | no | — |
| 140 | C2×C4 | — | 2 | 2 | no | — |
| 141 | C2×C4 | — | 2 | 2 | no | — |
| 142 | C2×C4 | — | 2 | 2 | no | — |
| 143 | C2×C4 | — | 4 | 1 | no | — |
| 144 | C2×C4 | — | 4 | 2 | no | — |
| 145 | C2×C4 | — | 4 | 2 | no | — |
| 146 | C2×C4 | — | 4 | 2 | no | — |
| 147 | C2×C4 | — | 4 | 2 | no | — |
| 148 | C2×C4 | — | 4 | 4 | no | — |
| 149 | C2×C4 | — | 8 | 2 | no | — |
| 150 | C2×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 151 | C2×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 152 | C2×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 153 | C2×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 154 | C2×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 155 | C2×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 156 | C2×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 157 | C2×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 158 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 159 | C2×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 160 | C8 | — | 2 | 1 | no | — |
| 161 | C8 | — | 2 | 1 | no | — |
| 162 | C8 | — | 2 | 2 | no | — |
| 163 | C8 | — | 2 | 2 | no | — |
| 164 | C8 | — | 4 | 4 | no | — |
| 165 | C8 | #117 | 1 | 2 | yes | — |
| 166 | C8 | #117 | 1 | 2 | yes | — |
| 167 | C22 | — | 2 | 1 | no | — |
| 168 | C22 | — | 2 | 1 | no | — |
| 169 | C22 | — | 4 | 2 | no | — |
| 170 | C22 | — | 4 | 2 | no | — |
| 171 | C22 | — | 4 | 2 | no | — |
| 172 | C22 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 173 | C22 | #555 | 1 | 1 | yes | characteristic |
| 174 | C22 | #556 | 1 | 1 | yes | characteristic |
| 175 | C4 | — | 2 | 1 | no | — |
| 176 | C4 | — | 2 | 1 | no | — |
| 177 | C4 | — | 2 | 1 | no | — |
| 178 | C4 | — | 4 | 4 | no | — |
| 179 | C4 | — | 4 | 2 | no | — |
| 180 | C4 | — | 4 | 2 | no | — |
| 181 | C4 | — | 4 | 2 | no | — |
| 182 | C4 | #443 | 1 | 2 | yes | — |
| 183 | C4 | C2×#117 | 1 | 1 | yes | characteristic |
| 184 | C4 | #520 | 1 | 1 | yes | characteristic |
| 185 | C4 | #547 | 1 | 1 | yes | characteristic |
| 186 | C4 | #550 | 1 | 1 | yes | characteristic |
| 187 | C2 | — | 2 | 1 | no | — |
| 188 | C2 | — | 2 | 1 | no | — |
| 189 | C2 | — | 8 | 2 | no | — |
| 190 | C2 | #2354 | 1 | 1 | yes | characteristic |
| 191 | C2 | C2×#443 | 1 | 1 | yes | characteristic |
| 192 | C2 | #2957 | 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