← M16×(C3⋊Dic3)⋊C3 #13143201
Order 1728.
565 conjugacy classes of subgroups.
The subgroup lattice has 299 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×M16×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C4×(C3⋊Dic3)⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#3748⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #63642⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C8×(C3⋊Dic3)⋊C3 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | M16×C3⋊Dic3 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C3×Dic3×M16 | — | 3 | 1 | no | maximal |
| 8 | C22×C4×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 9 | C22×(C3⋊Dic3)⋊C3 | C4 | 1 | 2 | yes | — |
| 10 | M16×H(3,𝔽3) | C4 | 1 | 4 | yes | complemented |
| 11 | C2×C8×H(3,𝔽3) | C22 | 1 | 2 | yes | — |
| 12 | C4×(C3⋊Dic3)⋊C3 | C4 | 1 | 2 | yes | — |
| 13 | C4×(C3⋊Dic3)⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×(C3⋊Dic3)⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 15 | #3748⋊C3 | C4 | 1 | 4 | yes | complemented |
| 16 | C2×(C3⋊(C3⋊C8))⋊C3 | C22 | 1 | 2 | yes | — |
| 17 | C2×C32×M16 | — | 2 | 1 | no | — |
| 18 | C2×C32×M16 | D3 | 1 | 1 | yes | characteristic |
| 19 | C2×C32×M16 | C2×C3 | 1 | 1 | yes | characteristic |
| 20 | C2×C4×C3⋊Dic3 | C2×C3 | 1 | 1 | yes | characteristic |
| 21 | C2×#3748 | C2×C3 | 1 | 1 | yes | characteristic |
| 22 | C2×C3×C4×Dic3 | — | 3 | 1 | no | — |
| 23 | C2×C3×#208 | — | 3 | 1 | no | — |
| 24 | #63642 | C2×C3 | 1 | 2 | yes | complemented |
| 25 | C8×C3⋊Dic3 | C2×C3 | 1 | 2 | yes | complemented |
| 26 | C3×#815 | — | 3 | 2 | no | — |
| 27 | C3×C8×Dic3 | — | 3 | 2 | no | — |
| 28 | C23×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 29 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 2 | yes | — |
| 31 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 2 | yes | — |
| 32 | C2×C4×H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×(C3⋊Dic3)⋊C3 | — | 2 | 4 | no | — |
| 34 | C2×(C3⋊Dic3)⋊C3 | C2×C4 | 1 | 2 | yes | — |
| 35 | C2×(C3⋊Dic3)⋊C3 | C2×C4 | 1 | 2 | yes | — |
| 36 | C8×H(3,𝔽3) | C2×C4 | 1 | 4 | yes | complemented |
| 37 | (C3⋊(C3⋊C8))⋊C3 | C2×C4 | 1 | 4 | yes | complemented |
| 38 | C3×C4×M16 | — | 9 | 1 | no | — |
| 39 | Dic3×M16 | — | 3 | 1 | no | — |
| 40 | Dic3×M16 | — | 9 | 1 | no | — |
| 41 | C22×C32×C4 | — | 2 | 1 | no | — |
| 42 | C22×C32×C4 | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | C22×C32×C4 | C22×C3 | 1 | 1 | yes | characteristic |
| 44 | C22×C3⋊Dic3 | C3×C4 | 1 | 2 | yes | — |
| 45 | C22×C3×Dic3 | — | 3 | 2 | no | — |
| 46 | C32×M16 | — | 2 | 4 | no | — |
| 47 | C32×M16 | Dic3 | 1 | 4 | yes | complemented |
| 48 | C32×M16 | C3×C4 | 1 | 4 | yes | complemented |
| 49 | C2×C32×C8 | — | 2 | 2 | no | — |
| 50 | C2×C32×C8 | C2×D3 | 1 | 2 | yes | — |
| 51 | C2×C32×C8 | C22×C3 | 1 | 2 | yes | — |
| 52 | C4×C3⋊Dic3 | C3×C4 | 1 | 2 | yes | — |
| 53 | C4×C3⋊Dic3 | C22×C3 | 1 | 1 | yes | characteristic |
| 54 | C4×C3⋊Dic3 | C22×C3 | 1 | 1 | yes | characteristic |
| 55 | #3748 | C3×C4 | 1 | 4 | yes | complemented |
| 56 | C2×C3⋊(C3⋊C8) | C22×C3 | 1 | 2 | yes | — |
| 57 | C3×C4×Dic3 | — | 3 | 2 | no | — |
| 58 | C3×C4×Dic3 | — | 3 | 1 | no | — |
| 59 | C3×C4×Dic3 | — | 3 | 1 | no | — |
| 60 | C3×#208 | — | 3 | 4 | no | — |
| 61 | C2×C3×C3⋊C8 | — | 3 | 2 | no | — |
| 62 | C22×H(3,𝔽3) | — | 2 | 2 | no | — |
| 63 | C22×H(3,𝔽3) | C42 | 1 | 2 | yes | — |
| 64 | C22×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 65 | C4×H(3,𝔽3) | C42 | 1 | 2 | yes | — |
| 66 | C4×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 67 | C4×H(3,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 68 | (C3⋊Dic3)⋊C3 | — | 2 | 2 | no | — |
| 69 | (C3⋊Dic3)⋊C3 | M16 | 1 | 4 | yes | direct product factor |
| 70 | C2×C3×M16 | — | 3 | 1 | no | — |
| 71 | C2×C3×M16 | — | 3 | 1 | no | — |
| 72 | C2×C3×M16 | — | 6 | 1 | no | — |
| 73 | C2×C3×M16 | — | 9 | 1 | no | — |
| 74 | C2×C3×M16 | C3×D3 | 1 | 1 | yes | characteristic |
| 75 | C2×C3×C42 | — | 9 | 1 | no | — |
| 76 | C2×C4×Dic3 | — | 3 | 1 | no | — |
| 77 | C2×C4×Dic3 | — | 9 | 1 | no | — |
| 78 | C2×#208 | — | 3 | 1 | no | — |
| 79 | C2×#208 | — | 9 | 1 | no | — |
| 80 | C3×#97 | — | 9 | 2 | no | — |
| 81 | C3×C4×C8 | — | 9 | 2 | no | — |
| 82 | #815 | — | 3 | 2 | no | — |
| 83 | #815 | — | 9 | 2 | no | — |
| 84 | C8×Dic3 | — | 3 | 2 | no | — |
| 85 | C8×Dic3 | — | 9 | 2 | no | — |
| 86 | C23×C32 | — | 2 | 1 | no | — |
| 87 | C23×C32 | C4×D3 | 1 | 1 | yes | characteristic |
| 88 | C23×C32 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C32×C4 | — | 2 | 1 | no | — |
| 90 | C2×C32×C4 | — | 2 | 2 | no | — |
| 91 | C2×C32×C4 | — | 2 | 2 | no | — |
| 92 | C2×C32×C4 | — | 2 | 1 | no | — |
| 93 | C2×C32×C4 | C4×D3 | 1 | 1 | yes | characteristic |
| 94 | C2×C32×C4 | C4×D3 | 1 | 2 | yes | — |
| 95 | C2×C32×C4 | C2×Dic3 | 1 | 2 | yes | — |
| 96 | C2×C32×C4 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C32×C4 | C2×C3×C4 | 1 | 2 | yes | — |
| 98 | C2×C32×C4 | C2×C3×C4 | 1 | 2 | yes | — |
| 99 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 100 | C2×C32×C4 | C23×C3 | 1 | 1 | yes | characteristic |
| 101 | C2×C3⋊Dic3 | — | 2 | 4 | no | — |
| 102 | C2×C3⋊Dic3 | C2×C3×C4 | 1 | 2 | yes | — |
| 103 | C2×C3⋊Dic3 | C2×C3×C4 | 1 | 2 | yes | — |
| 104 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 105 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 106 | C2×C3×Dic3 | — | 6 | 4 | no | — |
| 107 | C32×C8 | — | 2 | 4 | no | — |
| 108 | C32×C8 | C2×Dic3 | 1 | 4 | yes | complemented |
| 109 | C32×C8 | C2×C3×C4 | 1 | 4 | yes | complemented |
| 110 | C3⋊(C3⋊C8) | C2×C3×C4 | 1 | 4 | yes | complemented |
| 111 | C3×C3⋊C8 | — | 3 | 4 | no | — |
| 112 | C4×M16 | — | 9 | 1 | no | Sylow |
| 113 | C2×H(3,𝔽3) | — | 2 | 2 | no | — |
| 114 | C2×H(3,𝔽3) | C2×C42 | 1 | 1 | yes | characteristic |
| 115 | C2×H(3,𝔽3) | C2×M16 | 1 | 2 | yes | — |
| 116 | C22×C3×C4 | — | 3 | 1 | no | — |
| 117 | C22×C3×C4 | — | 3 | 1 | no | — |
| 118 | C22×C3×C4 | — | 6 | 1 | no | — |
| 119 | C22×C3×C4 | — | 9 | 2 | no | — |
| 120 | C22×C3×C4 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 121 | C22×Dic3 | — | 3 | 2 | no | — |
| 122 | C22×Dic3 | — | 9 | 2 | no | — |
| 123 | C3×M16 | — | 3 | 4 | no | — |
| 124 | C3×M16 | — | 3 | 4 | no | — |
| 125 | C3×M16 | — | 6 | 4 | no | — |
| 126 | C3×M16 | — | 9 | 4 | no | — |
| 127 | C3×M16 | C3×Dic3 | 1 | 4 | yes | — |
| 128 | C2×C3×C8 | — | 3 | 2 | no | — |
| 129 | C2×C3×C8 | — | 3 | 2 | no | — |
| 130 | C2×C3×C8 | — | 6 | 2 | no | — |
| 131 | C2×C3×C8 | — | 9 | 2 | no | — |
| 132 | C2×C3×C8 | C2×C3×D3 | 1 | 2 | yes | — |
| 133 | C3×C42 | — | 9 | 1 | no | — |
| 134 | C3×C42 | — | 9 | 1 | no | — |
| 135 | C3×C42 | — | 9 | 2 | no | — |
| 136 | C4×Dic3 | — | 3 | 2 | no | — |
| 137 | C4×Dic3 | — | 3 | 1 | no | — |
| 138 | C4×Dic3 | — | 3 | 1 | no | — |
| 139 | C4×Dic3 | — | 9 | 1 | no | — |
| 140 | C4×Dic3 | — | 9 | 1 | no | — |
| 141 | C4×Dic3 | — | 9 | 2 | no | — |
| 142 | #208 | — | 3 | 4 | no | — |
| 143 | #208 | — | 9 | 4 | no | — |
| 144 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 145 | C2×C3⋊C8 | — | 9 | 2 | no | — |
| 146 | C22×C32 | — | 2 | 2 | no | — |
| 147 | C22×C32 | — | 2 | 2 | no | — |
| 148 | C22×C32 | — | 2 | 2 | no | — |
| 149 | C22×C32 | — | 2 | 1 | no | — |
| 150 | C22×C32 | — | 4 | 2 | no | — |
| 151 | C22×C32 | C4×Dic3 | 1 | 2 | yes | — |
| 152 | C22×C32 | C3×C42 | 1 | 2 | yes | — |
| 153 | C22×C32 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 154 | C22×C32 | C22×C3×C4 | 1 | 1 | yes | characteristic |
| 155 | C32×C4 | — | 2 | 2 | no | — |
| 156 | C32×C4 | — | 2 | 1 | no | — |
| 157 | C32×C4 | — | 2 | 1 | no | — |
| 158 | C32×C4 | C4×Dic3 | 1 | 2 | yes | — |
| 159 | C32×C4 | C3×C42 | 1 | 2 | yes | — |
| 160 | C32×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 161 | C32×C4 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 162 | C32×C4 | C22×C3×C4 | 1 | 1 | yes | characteristic |
| 163 | C32×C4 | C22×C3×C4 | 1 | 1 | yes | characteristic |
| 164 | C3⋊Dic3 | — | 2 | 2 | no | — |
| 165 | C3⋊Dic3 | C3×M16 | 1 | 4 | yes | complemented |
| 166 | C3×Dic3 | — | 3 | 4 | no | — |
| 167 | C3×Dic3 | — | 6 | 2 | no | — |
| 168 | C2×M16 | — | 9 | 1 | no | — |
| 169 | C2×M16 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 170 | C2×C42 | — | 9 | 1 | no | — |
| 171 | #97 | — | 9 | 2 | no | — |
| 172 | C4×C8 | — | 9 | 2 | no | — |
| 173 | H(3,𝔽3) | C4×M16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 174 | C23×C3 | — | 3 | 1 | no | — |
| 175 | C23×C3 | — | 3 | 1 | no | — |
| 176 | C23×C3 | — | 6 | 1 | no | — |
| 177 | C23×C3 | C3×C4×D3 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | — | 3 | 1 | no | — |
| 179 | C2×C3×C4 | — | 3 | 2 | no | — |
| 180 | C2×C3×C4 | — | 3 | 2 | no | — |
| 181 | C2×C3×C4 | — | 3 | 1 | no | — |
| 182 | C2×C3×C4 | — | 3 | 2 | no | — |
| 183 | C2×C3×C4 | — | 3 | 2 | no | — |
| 184 | C2×C3×C4 | — | 3 | 1 | no | — |
| 185 | C2×C3×C4 | — | 3 | 1 | no | — |
| 186 | C2×C3×C4 | — | 6 | 2 | no | — |
| 187 | C2×C3×C4 | — | 6 | 1 | no | — |
| 188 | C2×C3×C4 | — | 6 | 2 | no | — |
| 189 | C2×C3×C4 | — | 6 | 1 | no | — |
| 190 | C2×C3×C4 | — | 9 | 2 | no | — |
| 191 | C2×C3×C4 | — | 9 | 2 | no | — |
| 192 | C2×C3×C4 | — | 18 | 4 | no | — |
| 193 | C2×C3×C4 | C3×C4×D3 | 1 | 1 | yes | characteristic |
| 194 | C2×C3×C4 | C3×C4×D3 | 1 | 2 | yes | — |
| 195 | C2×C3×C4 | C2×C3×Dic3 | 1 | 2 | yes | — |
| 196 | C2×C3×C4 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 197 | C2×Dic3 | — | 3 | 2 | no | — |
| 198 | C2×Dic3 | — | 3 | 2 | no | — |
| 199 | C2×Dic3 | — | 6 | 4 | no | — |
| 200 | C2×Dic3 | — | 9 | 2 | no | — |
| 201 | C2×Dic3 | — | 9 | 2 | no | — |
| 202 | C2×Dic3 | — | 18 | 4 | no | — |
| 203 | C3×C8 | — | 3 | 4 | no | — |
| 204 | C3×C8 | — | 3 | 4 | no | — |
| 205 | C3×C8 | — | 6 | 4 | no | — |
| 206 | C3×C8 | — | 9 | 4 | no | — |
| 207 | C3×C8 | C2×C3×Dic3 | 1 | 4 | yes | — |
| 208 | C3⋊C8 | — | 3 | 4 | no | — |
| 209 | C3⋊C8 | — | 9 | 4 | no | — |
| 210 | C2×C32 | — | 2 | 2 | no | — |
| 211 | C2×C32 | — | 2 | 2 | no | — |
| 212 | C2×C32 | — | 2 | 2 | no | — |
| 213 | C2×C32 | — | 2 | 1 | no | — |
| 214 | C2×C32 | — | 4 | 2 | no | — |
| 215 | C2×C32 | D3×M16 | 1 | 2 | yes | — |
| 216 | C2×C32 | C2×C4×Dic3 | 1 | 1 | yes | characteristic |
| 217 | C2×C32 | C2×C3×C42 | 1 | 1 | yes | characteristic |
| 218 | C2×C32 | C2×C3×M16 | 1 | 2 | yes | — |
| 219 | C22×C4 | — | 9 | 2 | no | — |
| 220 | C22×C4 | C2×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 221 | M16 | — | 9 | 4 | no | — |
| 222 | M16 | (C3⋊Dic3)⋊C3 | 1 | 4 | yes | direct product factor |
| 223 | C2×C8 | — | 9 | 2 | no | — |
| 224 | C2×C8 | C2×(C3⋊D3)⋊C3 | 1 | 2 | yes | — |
| 225 | C42 | — | 9 | 2 | no | — |
| 226 | C42 | — | 9 | 1 | no | — |
| 227 | C42 | — | 9 | 1 | no | — |
| 228 | C22×C3 | — | 2 | 2 | no | — |
| 229 | C22×C3 | — | 3 | 1 | no | — |
| 230 | C22×C3 | — | 3 | 1 | no | — |
| 231 | C22×C3 | — | 3 | 2 | no | — |
| 232 | C22×C3 | — | 3 | 2 | no | — |
| 233 | C22×C3 | — | 6 | 2 | no | — |
| 234 | C22×C3 | — | 6 | 2 | no | — |
| 235 | C22×C3 | — | 6 | 1 | no | — |
| 236 | C22×C3 | — | 6 | 2 | no | — |
| 237 | C22×C3 | — | 12 | 2 | no | — |
| 238 | C22×C3 | C3×C4×Dic3 | 1 | 2 | yes | — |
| 239 | C22×C3 | C2×C3×C4×D3 | 1 | 1 | yes | characteristic |
| 240 | C3×C4 | — | 3 | 1 | no | — |
| 241 | C3×C4 | — | 3 | 1 | no | — |
| 242 | C3×C4 | — | 3 | 1 | no | — |
| 243 | C3×C4 | — | 3 | 1 | no | — |
| 244 | C3×C4 | — | 3 | 2 | no | — |
| 245 | C3×C4 | — | 3 | 2 | no | — |
| 246 | C3×C4 | — | 6 | 1 | no | — |
| 247 | C3×C4 | — | 6 | 1 | no | — |
| 248 | C3×C4 | — | 6 | 2 | no | — |
| 249 | C3×C4 | — | 9 | 4 | no | — |
| 250 | C3×C4 | — | 18 | 2 | no | — |
| 251 | C3×C4 | C3×C4×Dic3 | 1 | 2 | yes | — |
| 252 | C3×C4 | C2×C3×C4×D3 | 1 | 1 | yes | characteristic |
| 253 | C3×C4 | C22×C3×Dic3 | 1 | 1 | yes | characteristic |
| 254 | Dic3 | — | 3 | 4 | no | — |
| 255 | Dic3 | — | 6 | 2 | no | — |
| 256 | Dic3 | — | 9 | 4 | no | — |
| 257 | Dic3 | — | 18 | 2 | no | — |
| 258 | C32 | — | 2 | 1 | no | — |
| 259 | C32 | Dic3×M16 | 1 | 1 | yes | characteristic, complemented |
| 260 | C32 | C3×C4×M16 | 1 | 1 | yes | characteristic, complemented |
| 261 | C23 | C4×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 262 | C2×C4 | — | 9 | 2 | no | — |
| 263 | C2×C4 | — | 9 | 2 | no | — |
| 264 | C2×C4 | — | 18 | 4 | no | — |
| 265 | C2×C4 | C4×(C3⋊D3)⋊C3 | 1 | 2 | yes | — |
| 266 | C2×C4 | C4×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | C2×(C3⋊Dic3)⋊C3 | 1 | 2 | yes | — |
| 268 | C2×C4 | C22×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 269 | C8 | — | 9 | 4 | no | — |
| 270 | C8 | C2×(C3⋊Dic3)⋊C3 | 1 | 4 | yes | complemented |
| 271 | C2×C3 | — | 2 | 2 | no | — |
| 272 | C2×C3 | — | 3 | 2 | no | — |
| 273 | C2×C3 | — | 3 | 2 | no | — |
| 274 | C2×C3 | — | 3 | 1 | no | — |
| 275 | C2×C3 | — | 3 | 1 | no | — |
| 276 | C2×C3 | — | 6 | 2 | no | — |
| 277 | C2×C3 | — | 6 | 2 | no | — |
| 278 | C2×C3 | — | 6 | 2 | no | — |
| 279 | C2×C3 | — | 6 | 1 | no | — |
| 280 | C2×C3 | — | 12 | 2 | no | — |
| 281 | C2×C3 | C3×D3×M16 | 1 | 2 | yes | — |
| 282 | C2×C3 | C2×C3×C4×Dic3 | 1 | 1 | yes | characteristic |
| 283 | C22 | — | 2 | 2 | no | — |
| 284 | C22 | C4×(C3⋊Dic3)⋊C3 | 1 | 2 | yes | — |
| 285 | C22 | C2×C4×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 286 | C4 | — | 9 | 4 | no | — |
| 287 | C4 | — | 18 | 2 | no | — |
| 288 | C4 | C4×(C3⋊Dic3)⋊C3 | 1 | 2 | yes | — |
| 289 | C4 | C2×C4×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 290 | C4 | C22×(C3⋊Dic3)⋊C3 | 1 | 1 | yes | characteristic |
| 291 | C3 | — | 3 | 1 | no | — |
| 292 | C3 | — | 3 | 1 | no | — |
| 293 | C3 | — | 6 | 1 | no | — |
| 294 | C3 | C3×Dic3×M16 | 1 | 1 | yes | characteristic |
| 295 | C2 | — | 2 | 2 | no | — |
| 296 | C2 | M16×(C3⋊D3)⋊C3 | 1 | 2 | yes | — |
| 297 | C2 | C2×C4×(C3⋊Dic3)⋊C3 | 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