← #13138091
Subgroups — #13138091
Order 1728.
280 conjugacy classes of subgroups.
The subgroup lattice has 212 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C9×#901 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3⋊(C9⋊C16) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#63371 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11720237 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | D3×C9⋊C16 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10590622 | — | 3 | 1 | no | maximal |
| 7 | #10588820 | — | 3 | 1 | no | maximal |
| 8 | C9×#235 | C4 | 1 | 1 | yes | characteristic |
| 9 | C2×C3×C8×C9 | C22 | 1 | 1 | yes | characteristic |
| 10 | C9×#208 | C4 | 1 | 1 | yes | characteristic |
| 11 | C9×ΓL(1,𝔽25) | C4 | 1 | 2 | yes | — |
| 12 | D3×C8×C9 | C22 | 1 | 2 | yes | — |
| 13 | C3⋊(C9⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 14 | C3⋊(C9⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 15 | C3×C9⋊C16 | C22 | 1 | 2 | yes | — |
| 16 | C3×#901 | D3 | 1 | 1 | yes | characteristic |
| 17 | C2×C3⋊(C3⋊C16) | — | 3 | 1 | no | — |
| 18 | C3×#812 | — | 3 | 1 | no | — |
| 19 | #63543 | — | 3 | 2 | no | — |
| 20 | D3×C3⋊C16 | — | 3 | 2 | no | — |
| 21 | C9×#131 | — | 3 | 1 | no | — |
| 22 | #63371 | — | 3 | 2 | no | — |
| 23 | #63371 | D3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×C9⋊C16 | — | 3 | 1 | no | — |
| 25 | C2×C9⋊C16 | — | 3 | 2 | no | — |
| 26 | C2×C9⋊C16 | — | 6 | 1 | no | — |
| 27 | C2×C3×C4×C9 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | C9×#67 | C8 | 1 | 2 | yes | — |
| 29 | C9×D12 | C8 | 1 | 1 | yes | characteristic |
| 30 | C4×D3×C9 | C2×C4 | 1 | 2 | yes | — |
| 31 | C9×Dic6 | C8 | 1 | 1 | yes | characteristic |
| 32 | C3×C8×C9 | C2×C4 | 1 | 1 | yes | characteristic |
| 33 | C3×C8×C9 | C23 | 1 | 1 | yes | characteristic |
| 34 | C9×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 35 | #5140 | — | 9 | 1 | no | — |
| 36 | #4907 | — | 9 | 1 | no | — |
| 37 | C3×#235 | Dic3 | 1 | 1 | yes | characteristic |
| 38 | C2×C32×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 39 | C3×#208 | Dic3 | 1 | 1 | yes | characteristic |
| 40 | C3×ΓL(1,𝔽25) | Dic3 | 1 | 2 | yes | — |
| 41 | C3×D3×C8 | C2×D3 | 1 | 2 | yes | — |
| 42 | C9×#41 | — | 3 | 1 | no | — |
| 43 | C3⋊(C3⋊C16) | — | 3 | 1 | no | — |
| 44 | C3⋊(C3⋊C16) | — | 3 | 1 | no | — |
| 45 | C3×C3⋊C16 | — | 3 | 2 | no | — |
| 46 | C9×M16 | — | 3 | 1 | no | — |
| 47 | C9×M16 | — | 3 | 2 | no | — |
| 48 | C2×C8×C9 | — | 2 | 1 | no | — |
| 49 | C2×C8×C9 | — | 3 | 2 | no | — |
| 50 | C2×C8×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 51 | C9⋊C16 | — | 3 | 1 | no | — |
| 52 | C9⋊C16 | — | 3 | 1 | no | — |
| 53 | C9⋊C16 | — | 6 | 1 | no | — |
| 54 | C9⋊C16 | — | 6 | 1 | no | — |
| 55 | C9⋊C16 | C2×D3 | 1 | 2 | yes | — |
| 56 | C22×C3×C9 | C2×C8 | 1 | 1 | yes | characteristic |
| 57 | C2×D3×C9 | C2×C8 | 1 | 2 | yes | — |
| 58 | C3×C4×C9 | C2×C8 | 1 | 1 | yes | characteristic |
| 59 | C3×C4×C9 | C22×C4 | 1 | 1 | yes | characteristic |
| 60 | C9×Dic3 | C2×C8 | 1 | 2 | yes | — |
| 61 | C3×#131 | — | 3 | 1 | no | — |
| 62 | #901 | D9 | 1 | 1 | yes | characteristic |
| 63 | C3×M32 | — | 9 | 1 | no | — |
| 64 | #812 | — | 3 | 1 | no | — |
| 65 | #812 | — | 9 | 2 | no | — |
| 66 | C2×C3⋊C16 | — | 9 | 2 | no | — |
| 67 | C2×C3⋊C16 | — | 9 | 1 | no | — |
| 68 | C2×C3⋊C16 | — | 9 | 1 | no | — |
| 69 | C2×C3⋊C16 | — | 18 | 1 | no | — |
| 70 | #798 | — | 9 | 2 | no | — |
| 71 | D3×C16 | — | 9 | 2 | no | — |
| 72 | C2×C32×C4 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 73 | C3×#67 | C3⋊C8 | 1 | 2 | yes | — |
| 74 | C3×D12 | C3⋊C8 | 1 | 1 | yes | characteristic |
| 75 | C3×C4×D3 | C2×Dic3 | 1 | 2 | yes | — |
| 76 | C3×Dic6 | C3⋊C8 | 1 | 1 | yes | characteristic |
| 77 | C32×C8 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 78 | C32×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 79 | C3×C3⋊C8 | C2×Dic3 | 1 | 2 | yes | — |
| 80 | Q8×C9 | — | 3 | 1 | no | — |
| 81 | D4×C9 | — | 3 | 2 | no | — |
| 82 | D4×C9 | — | 3 | 1 | no | — |
| 83 | C2×C4×C9 | — | 2 | 1 | no | — |
| 84 | C2×C4×C9 | — | 3 | 2 | no | — |
| 85 | C2×C4×C9 | C4×D3 | 1 | 1 | yes | characteristic |
| 86 | C8×C9 | — | 2 | 1 | no | — |
| 87 | C8×C9 | — | 2 | 1 | no | — |
| 88 | C8×C9 | — | 3 | 2 | no | — |
| 89 | C8×C9 | C4×D3 | 1 | 1 | yes | characteristic |
| 90 | C8×C9 | C22×D3 | 1 | 1 | yes | characteristic |
| 91 | #504 | — | 27 | 1 | no | Sylow |
| 92 | C2×C3×C9 | — | 2 | 1 | no | — |
| 93 | C2×C3×C9 | C22×C8 | 1 | 1 | yes | characteristic |
| 94 | D3×C9 | — | 2 | 2 | no | — |
| 95 | C3×#41 | — | 3 | 1 | no | — |
| 96 | #235 | Dic9 | 1 | 1 | yes | characteristic |
| 97 | C3×M16 | — | 3 | 1 | no | — |
| 98 | C3×M16 | — | 3 | 2 | no | — |
| 99 | C2×C3×C8 | — | 2 | 1 | no | — |
| 100 | C2×C3×C8 | — | 3 | 2 | no | — |
| 101 | C2×C3×C8 | C2×D9 | 1 | 1 | yes | characteristic |
| 102 | C2×C3×C8 | D32 | 1 | 1 | yes | characteristic |
| 103 | #208 | Dic9 | 1 | 1 | yes | characteristic |
| 104 | ΓL(1,𝔽25) | Dic9 | 1 | 2 | yes | — |
| 105 | D3×C8 | C2×D9 | 1 | 2 | yes | — |
| 106 | C3×C16 | — | 9 | 2 | no | — |
| 107 | C3⋊C16 | — | 3 | 2 | no | — |
| 108 | C3⋊C16 | — | 9 | 1 | no | — |
| 109 | C3⋊C16 | — | 9 | 1 | no | — |
| 110 | C3⋊C16 | — | 9 | 1 | no | — |
| 111 | C3⋊C16 | — | 9 | 1 | no | — |
| 112 | C3⋊C16 | — | 18 | 1 | no | — |
| 113 | C3⋊C16 | — | 18 | 1 | no | — |
| 114 | C22×C32 | C2×C3⋊C8 | 1 | 1 | yes | characteristic |
| 115 | C2×C3×D3 | C2×C3⋊C8 | 1 | 2 | yes | — |
| 116 | C32×C4 | C2×C3⋊C8 | 1 | 1 | yes | characteristic |
| 117 | C32×C4 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 118 | C3×Dic3 | C2×C3⋊C8 | 1 | 2 | yes | — |
| 119 | C22×C9 | — | 2 | 1 | no | — |
| 120 | C22×C9 | — | 3 | 2 | no | — |
| 121 | C22×C9 | D3×C8 | 1 | 1 | yes | characteristic |
| 122 | C4×C9 | — | 2 | 1 | no | — |
| 123 | C4×C9 | — | 2 | 1 | no | — |
| 124 | C4×C9 | — | 3 | 2 | no | — |
| 125 | C4×C9 | D3×C8 | 1 | 1 | yes | characteristic |
| 126 | C4×C9 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 127 | #131 | — | 3 | 1 | no | — |
| 128 | M32 | — | 9 | 1 | no | — |
| 129 | M32 | — | 27 | 2 | no | — |
| 130 | C2×C16 | — | 27 | 2 | no | — |
| 131 | C2×C16 | — | 27 | 1 | no | — |
| 132 | C3×C9 | #504 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 133 | C3×Q8 | — | 3 | 1 | no | — |
| 134 | C3×D4 | — | 3 | 2 | no | — |
| 135 | C3×D4 | — | 3 | 1 | no | — |
| 136 | C2×C3×C4 | — | 2 | 1 | no | — |
| 137 | C2×C3×C4 | — | 3 | 2 | no | — |
| 138 | C2×C3×C4 | C2×Dic9 | 1 | 1 | yes | characteristic |
| 139 | C2×C3×C4 | D3×Dic3 | 1 | 1 | yes | characteristic |
| 140 | #67 | C9⋊C8 | 1 | 2 | yes | — |
| 141 | D12 | C9⋊C8 | 1 | 1 | yes | characteristic |
| 142 | C4×D3 | C2×Dic9 | 1 | 2 | yes | — |
| 143 | Dic6 | C9⋊C8 | 1 | 1 | yes | characteristic |
| 144 | C3×C8 | — | 2 | 1 | no | — |
| 145 | C3×C8 | — | 2 | 1 | no | — |
| 146 | C3×C8 | — | 3 | 2 | no | — |
| 147 | C3×C8 | C2×Dic9 | 1 | 1 | yes | characteristic |
| 148 | C3×C8 | C22×D9 | 1 | 1 | yes | characteristic |
| 149 | C3×C8 | D3×Dic3 | 1 | 1 | yes | characteristic |
| 150 | C3×C8 | C2×D32 | 1 | 1 | yes | characteristic |
| 151 | C3⋊C8 | C2×Dic9 | 1 | 2 | yes | — |
| 152 | C2×C32 | — | 2 | 1 | no | — |
| 153 | C2×C32 | C22×C3⋊C8 | 1 | 1 | yes | characteristic |
| 154 | C3×D3 | — | 2 | 2 | no | — |
| 155 | C2×C9 | — | 2 | 2 | no | — |
| 156 | C2×C9 | — | 2 | 1 | no | — |
| 157 | C2×C9 | — | 2 | 1 | no | — |
| 158 | C2×C9 | — | 6 | 2 | no | — |
| 159 | C2×C9 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 160 | #41 | — | 3 | 1 | no | — |
| 161 | M16 | — | 3 | 1 | no | — |
| 162 | M16 | — | 3 | 2 | no | — |
| 163 | C2×C8 | — | 3 | 2 | no | — |
| 164 | C2×C8 | D3×D9 | 1 | 1 | yes | characteristic |
| 165 | C16 | — | 9 | 2 | no | — |
| 166 | C16 | — | 27 | 1 | no | — |
| 167 | C16 | — | 27 | 1 | no | — |
| 168 | C22×C3 | — | 2 | 1 | no | — |
| 169 | C22×C3 | — | 3 | 2 | no | — |
| 170 | C22×C3 | C2×C9⋊C8 | 1 | 1 | yes | characteristic |
| 171 | C22×C3 | D3×C3⋊C8 | 1 | 1 | yes | characteristic |
| 172 | C2×D3 | C2×C9⋊C8 | 1 | 2 | yes | — |
| 173 | C3×C4 | — | 2 | 1 | no | — |
| 174 | C3×C4 | — | 2 | 1 | no | — |
| 175 | C3×C4 | — | 3 | 2 | no | — |
| 176 | C3×C4 | C2×C9⋊C8 | 1 | 1 | yes | characteristic |
| 177 | C3×C4 | C22×Dic9 | 1 | 1 | yes | characteristic |
| 178 | C3×C4 | D3×C3⋊C8 | 1 | 1 | yes | characteristic |
| 179 | C3×C4 | C2×D3×Dic3 | 1 | 1 | yes | characteristic |
| 180 | Dic3 | C2×C9⋊C8 | 1 | 2 | yes | — |
| 181 | C32 | #5140 | 1 | 1 | yes | characteristic |
| 182 | C9 | — | 2 | 1 | no | — |
| 183 | C9 | #4907 | 1 | 1 | yes | characteristic, complemented |
| 184 | Q8 | — | 3 | 1 | no | — |
| 185 | D4 | — | 3 | 2 | no | — |
| 186 | D4 | — | 3 | 1 | no | — |
| 187 | C2×C4 | — | 3 | 2 | no | — |
| 188 | C2×C4 | D3×Dic9 | 1 | 1 | yes | characteristic |
| 189 | C8 | — | 3 | 2 | no | — |
| 190 | C8 | D3×Dic9 | 1 | 1 | yes | characteristic |
| 191 | C8 | C2×D3×D9 | 1 | 1 | yes | characteristic |
| 192 | C2×C3 | — | 2 | 2 | no | — |
| 193 | C2×C3 | — | 2 | 1 | no | — |
| 194 | C2×C3 | — | 2 | 1 | no | — |
| 195 | C2×C3 | — | 2 | 1 | no | — |
| 196 | C2×C3 | — | 6 | 2 | no | — |
| 197 | C2×C3 | C22×C9⋊C8 | 1 | 1 | yes | characteristic |
| 198 | C2×C3 | C2×D3×C3⋊C8 | 1 | 1 | yes | characteristic |
| 199 | D3 | — | 2 | 2 | no | — |
| 200 | C22 | — | 3 | 2 | no | — |
| 201 | C22 | D3×C9⋊C8 | 1 | 1 | yes | characteristic |
| 202 | C4 | — | 3 | 2 | no | — |
| 203 | C4 | D3×C9⋊C8 | 1 | 1 | yes | characteristic |
| 204 | C4 | C2×D3×Dic9 | 1 | 1 | yes | characteristic |
| 205 | C3 | — | 2 | 1 | no | — |
| 206 | C3 | #10588820 | 1 | 1 | yes | characteristic, complemented |
| 207 | C3 | #10590622 | 1 | 1 | yes | characteristic |
| 208 | C2 | — | 2 | 1 | no | — |
| 209 | C2 | — | 6 | 2 | no | — |
| 210 | C2 | C2×D3×C9⋊C8 | 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