← C2×#82012 #11701654
Subgroups — C2×#82012 #11701654
Order 768.
2074 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×D3×#131 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×#504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#5140 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #82012 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 5 | C2×#4907 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C2×D3×M32 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×#4901 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | C22×#798 | C2 | 1 | 3 | yes | maximal, complemented |
| 9 | C2×#3407 | — | 3 | 1 | no | maximal, Sylow |
| 10 | C2×D3×#41 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×#131 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#948 | C4 | 1 | 1 | yes | characteristic |
| 13 | D3×#131 | C4 | 1 | 8 | yes | — |
| 14 | D3×#131 | C22 | 1 | 8 | yes | — |
| 15 | C2×#907 | C4 | 1 | 3 | yes | — |
| 16 | C2×D3×M16 | C4 | 1 | 3 | yes | — |
| 17 | C2×#901 | C22 | 1 | 3 | yes | — |
| 18 | C22×D3×C8 | C22 | 1 | 3 | yes | — |
| 19 | C3×#504 | C22 | 1 | 8 | yes | complemented |
| 20 | C2×C3×M32 | C22 | 1 | 3 | yes | complemented |
| 21 | C22×C3×C16 | C22 | 1 | 3 | yes | complemented |
| 22 | #5140 | C22 | 1 | 8 | yes | complemented |
| 23 | C2×#812 | C22 | 1 | 3 | yes | complemented |
| 24 | C22×C3⋊C16 | C22 | 1 | 3 | yes | complemented |
| 25 | #4907 | C22 | 1 | 24 | yes | complemented |
| 26 | D3×M32 | C22 | 1 | 24 | yes | complemented |
| 27 | #4901 | C22 | 1 | 24 | yes | complemented |
| 28 | C2×#798 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×#798 | C22 | 1 | 3 | yes | — |
| 30 | C2×#798 | C22 | 1 | 24 | yes | complemented |
| 31 | C2×#798 | C22 | 1 | 3 | yes | complemented |
| 32 | C2×#798 | C22 | 1 | 3 | yes | complemented |
| 33 | C2×D3×C16 | C22 | 1 | 6 | yes | complemented |
| 34 | C22×#131 | — | 3 | 1 | no | — |
| 35 | #3407 | — | 3 | 16 | no | — |
| 36 | C2×#504 | — | 3 | 3 | no | — |
| 37 | C2×#504 | — | 3 | 3 | no | — |
| 38 | C2×#504 | — | 3 | 1 | no | — |
| 39 | C2×#504 | D3 | 1 | 1 | yes | characteristic, complemented |
| 40 | C22×M32 | — | 3 | 3 | no | — |
| 41 | C22×M32 | — | 3 | 3 | no | — |
| 42 | C2×C3×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 43 | D3×#41 | C8 | 1 | 8 | yes | — |
| 44 | D3×#41 | C2×C4 | 1 | 8 | yes | — |
| 45 | C2×#239 | C8 | 1 | 1 | yes | characteristic |
| 46 | C2×D3×Q8 | C8 | 1 | 1 | yes | characteristic |
| 47 | C2×#237 | C8 | 1 | 3 | yes | — |
| 48 | C2×D3×D4 | C8 | 1 | 3 | yes | — |
| 49 | C2×#235 | C2×C4 | 1 | 3 | yes | — |
| 50 | C22×C4×D3 | C2×C4 | 1 | 3 | yes | — |
| 51 | C3×#131 | C2×C4 | 1 | 4 | yes | — |
| 52 | C3×#131 | C23 | 1 | 4 | yes | — |
| 53 | C2×C3×M16 | C2×C4 | 1 | 3 | yes | — |
| 54 | C22×C3×C8 | C23 | 1 | 3 | yes | — |
| 55 | #948 | C2×C4 | 1 | 8 | yes | — |
| 56 | C2×#208 | C2×C4 | 1 | 3 | yes | — |
| 57 | C22×C3⋊C8 | C2×C4 | 1 | 3 | yes | — |
| 58 | #907 | C2×C4 | 1 | 24 | yes | — |
| 59 | D3×M16 | C2×C4 | 1 | 24 | yes | — |
| 60 | #901 | C2×C4 | 1 | 12 | yes | — |
| 61 | #901 | C23 | 1 | 12 | yes | — |
| 62 | C2×ΓL(1,𝔽25) | C2×C4 | 1 | 6 | yes | — |
| 63 | C2×D3×C8 | C2×C4 | 1 | 3 | yes | — |
| 64 | C2×D3×C8 | C2×C4 | 1 | 3 | yes | — |
| 65 | C2×D3×C8 | C2×C4 | 1 | 12 | yes | — |
| 66 | C2×D3×C8 | C23 | 1 | 3 | yes | — |
| 67 | C2×D3×C8 | C23 | 1 | 1 | yes | characteristic |
| 68 | C2×D3×C8 | C23 | 1 | 12 | yes | — |
| 69 | C3×M32 | C23 | 1 | 12 | yes | complemented |
| 70 | C2×C3×C16 | C23 | 1 | 1 | yes | characteristic |
| 71 | C2×C3×C16 | C23 | 1 | 3 | yes | — |
| 72 | C2×C3×C16 | C23 | 1 | 12 | yes | complemented |
| 73 | #812 | C23 | 1 | 12 | yes | complemented |
| 74 | C2×C3⋊C16 | C23 | 1 | 3 | yes | — |
| 75 | C2×C3⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 76 | C2×C3⋊C16 | C23 | 1 | 12 | yes | complemented |
| 77 | #798 | C23 | 1 | 4 | yes | — |
| 78 | #798 | C23 | 1 | 12 | yes | — |
| 79 | #798 | C23 | 1 | 12 | yes | complemented |
| 80 | #798 | C23 | 1 | 12 | yes | complemented |
| 81 | D3×C16 | C23 | 1 | 24 | yes | complemented |
| 82 | C22×#41 | — | 3 | 1 | no | — |
| 83 | C2×#131 | — | 3 | 1 | no | — |
| 84 | C2×#131 | — | 3 | 8 | no | — |
| 85 | C2×#131 | — | 3 | 8 | no | — |
| 86 | C2×#131 | — | 3 | 3 | no | — |
| 87 | C2×#131 | — | 3 | 3 | no | — |
| 88 | C2×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 89 | C22×M16 | — | 3 | 3 | no | — |
| 90 | C23×C8 | — | 3 | 3 | no | — |
| 91 | #504 | — | 3 | 8 | no | — |
| 92 | #504 | — | 3 | 24 | no | — |
| 93 | #504 | — | 3 | 24 | no | — |
| 94 | #504 | C2×D3 | 1 | 8 | yes | complemented |
| 95 | C2×M32 | — | 3 | 3 | no | — |
| 96 | C2×M32 | — | 3 | 24 | no | — |
| 97 | C2×M32 | — | 3 | 3 | no | — |
| 98 | C2×M32 | — | 3 | 1 | no | — |
| 99 | C2×M32 | — | 3 | 3 | no | — |
| 100 | C2×M32 | — | 3 | 3 | no | — |
| 101 | C2×M32 | — | 3 | 24 | no | — |
| 102 | C2×M32 | C2×D3 | 1 | 3 | yes | complemented |
| 103 | C22×C16 | — | 3 | 3 | no | — |
| 104 | C22×C16 | — | 3 | 6 | no | — |
| 105 | C22×C16 | C2×D3 | 1 | 3 | yes | complemented |
| 106 | C23×D3 | C2×C8 | 1 | 3 | yes | — |
| 107 | C3×#41 | C2×C8 | 1 | 4 | yes | — |
| 108 | C3×#41 | C22×C4 | 1 | 4 | yes | — |
| 109 | C2×C3×Q8 | C2×C8 | 1 | 1 | yes | characteristic |
| 110 | C2×C3×D4 | C2×C8 | 1 | 3 | yes | — |
| 111 | C22×C3×C4 | C22×C4 | 1 | 3 | yes | — |
| 112 | C2×#67 | C2×C8 | 1 | 6 | yes | — |
| 113 | C22×Dic3 | C2×C8 | 1 | 3 | yes | — |
| 114 | #239 | C2×C8 | 1 | 8 | yes | — |
| 115 | D3×Q8 | C2×C8 | 1 | 8 | yes | — |
| 116 | #237 | C2×C8 | 1 | 24 | yes | — |
| 117 | D3×D4 | C2×C8 | 1 | 24 | yes | — |
| 118 | #235 | C2×C8 | 1 | 12 | yes | — |
| 119 | #235 | C22×C4 | 1 | 12 | yes | — |
| 120 | C2×D12 | C2×C8 | 1 | 3 | yes | — |
| 121 | C2×C4×D3 | C2×C8 | 1 | 3 | yes | — |
| 122 | C2×C4×D3 | C2×C8 | 1 | 3 | yes | — |
| 123 | C2×C4×D3 | C2×C8 | 1 | 12 | yes | — |
| 124 | C2×C4×D3 | C22×C4 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×D3 | C22×C4 | 1 | 3 | yes | — |
| 126 | C2×C4×D3 | C22×C4 | 1 | 12 | yes | — |
| 127 | C2×Dic6 | C2×C8 | 1 | 3 | yes | — |
| 128 | C3×M16 | C22×C4 | 1 | 12 | yes | — |
| 129 | C2×C3×C8 | C22×C4 | 1 | 3 | yes | — |
| 130 | C2×C3×C8 | C22×C4 | 1 | 6 | yes | — |
| 131 | C2×C3×C8 | C24 | 1 | 6 | yes | — |
| 132 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 133 | #208 | C22×C4 | 1 | 12 | yes | — |
| 134 | C2×C3⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 135 | C2×C3⋊C8 | C22×C4 | 1 | 3 | yes | — |
| 136 | C2×C3⋊C8 | C22×C4 | 1 | 12 | yes | — |
| 137 | ΓL(1,𝔽25) | C22×C4 | 1 | 24 | yes | — |
| 138 | D3×C8 | C22×C4 | 1 | 2 | yes | — |
| 139 | D3×C8 | C22×C4 | 1 | 6 | yes | — |
| 140 | D3×C8 | C22×C4 | 1 | 12 | yes | — |
| 141 | D3×C8 | C22×C4 | 1 | 12 | yes | — |
| 142 | D3×C8 | C24 | 1 | 6 | yes | — |
| 143 | D3×C8 | C24 | 1 | 2 | yes | — |
| 144 | C3×C16 | C24 | 1 | 6 | yes | — |
| 145 | C3×C16 | C24 | 1 | 2 | yes | — |
| 146 | C3⋊C16 | C24 | 1 | 6 | yes | — |
| 147 | C3⋊C16 | C24 | 1 | 2 | yes | — |
| 148 | C2×#41 | — | 3 | 8 | no | — |
| 149 | C2×#41 | — | 3 | 8 | no | — |
| 150 | C2×#41 | — | 3 | 1 | no | — |
| 151 | C2×#41 | — | 3 | 3 | no | — |
| 152 | C2×#41 | — | 3 | 3 | no | — |
| 153 | C2×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 154 | C22×Q8 | — | 3 | 1 | no | — |
| 155 | C22×D4 | — | 3 | 3 | no | — |
| 156 | C23×C4 | — | 3 | 3 | no | — |
| 157 | #131 | — | 3 | 8 | no | — |
| 158 | #131 | — | 3 | 24 | no | — |
| 159 | #131 | — | 3 | 12 | no | — |
| 160 | #131 | — | 3 | 12 | no | — |
| 161 | #131 | C4×D3 | 1 | 4 | yes | — |
| 162 | #131 | C22×D3 | 1 | 4 | yes | — |
| 163 | C2×M16 | — | 3 | 3 | no | — |
| 164 | C2×M16 | — | 3 | 24 | no | — |
| 165 | C2×M16 | — | 3 | 6 | no | — |
| 166 | C2×M16 | C4×D3 | 1 | 3 | yes | — |
| 167 | C22×C8 | — | 3 | 3 | no | — |
| 168 | C22×C8 | — | 3 | 3 | no | — |
| 169 | C22×C8 | — | 3 | 3 | no | — |
| 170 | C22×C8 | — | 3 | 12 | no | — |
| 171 | C22×C8 | — | 3 | 12 | no | — |
| 172 | C22×C8 | — | 3 | 1 | no | — |
| 173 | C22×C8 | — | 3 | 3 | no | — |
| 174 | C22×C8 | C22×D3 | 1 | 3 | yes | — |
| 175 | M32 | — | 3 | 12 | no | — |
| 176 | M32 | — | 3 | 12 | no | — |
| 177 | M32 | — | 3 | 4 | no | — |
| 178 | M32 | — | 3 | 12 | no | — |
| 179 | M32 | — | 3 | 12 | no | — |
| 180 | M32 | C22×D3 | 1 | 12 | yes | complemented |
| 181 | C2×C16 | — | 3 | 12 | no | — |
| 182 | C2×C16 | — | 3 | 3 | no | — |
| 183 | C2×C16 | — | 3 | 1 | no | — |
| 184 | C2×C16 | — | 3 | 24 | no | — |
| 185 | C2×C16 | C22×D3 | 1 | 3 | yes | — |
| 186 | C2×C16 | C22×D3 | 1 | 1 | yes | characteristic |
| 187 | C2×C16 | C22×D3 | 1 | 12 | yes | complemented |
| 188 | C23×C3 | C22×C8 | 1 | 3 | yes | — |
| 189 | C22×D3 | — | 4 | 6 | no | — |
| 190 | C22×D3 | C22×C8 | 1 | 1 | yes | characteristic |
| 191 | C22×D3 | C22×C8 | 1 | 3 | yes | — |
| 192 | C22×D3 | C22×C8 | 1 | 12 | yes | — |
| 193 | C3×Q8 | C22×C8 | 1 | 4 | yes | — |
| 194 | C3×D4 | C22×C8 | 1 | 12 | yes | — |
| 195 | C2×C3×C4 | C22×C8 | 1 | 3 | yes | — |
| 196 | C2×C3×C4 | C22×C8 | 1 | 6 | yes | — |
| 197 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 198 | C2×C3×C4 | C23×C4 | 1 | 6 | yes | — |
| 199 | #67 | C22×C8 | 1 | 24 | yes | — |
| 200 | C2×Dic3 | C22×C8 | 1 | 3 | yes | — |
| 201 | C2×Dic3 | C22×C8 | 1 | 1 | yes | characteristic |
| 202 | C2×Dic3 | C22×C8 | 1 | 12 | yes | — |
| 203 | D12 | C22×C8 | 1 | 12 | yes | — |
| 204 | C4×D3 | C22×C8 | 1 | 2 | yes | — |
| 205 | C4×D3 | C22×C8 | 1 | 6 | yes | — |
| 206 | C4×D3 | C22×C8 | 1 | 12 | yes | — |
| 207 | C4×D3 | C22×C8 | 1 | 12 | yes | — |
| 208 | C4×D3 | C23×C4 | 1 | 2 | yes | — |
| 209 | C4×D3 | C23×C4 | 1 | 6 | yes | — |
| 210 | Dic6 | C22×C8 | 1 | 12 | yes | — |
| 211 | C3×C8 | C23×C4 | 1 | 1 | yes | characteristic |
| 212 | C3×C8 | C23×C4 | 1 | 6 | yes | — |
| 213 | C3×C8 | C25 | 1 | 1 | yes | characteristic |
| 214 | C3⋊C8 | C23×C4 | 1 | 2 | yes | — |
| 215 | C3⋊C8 | C23×C4 | 1 | 6 | yes | — |
| 216 | C24 | — | 3 | 3 | no | — |
| 217 | #41 | — | 3 | 8 | no | — |
| 218 | #41 | — | 3 | 24 | no | — |
| 219 | #41 | — | 3 | 12 | no | — |
| 220 | #41 | — | 3 | 12 | no | — |
| 221 | #41 | D3×C8 | 1 | 4 | yes | — |
| 222 | #41 | C2×C4×D3 | 1 | 4 | yes | — |
| 223 | C2×Q8 | — | 3 | 8 | no | — |
| 224 | C2×Q8 | — | 3 | 3 | no | — |
| 225 | C2×Q8 | D3×C8 | 1 | 1 | yes | characteristic |
| 226 | C2×D4 | — | 3 | 6 | no | — |
| 227 | C2×D4 | — | 3 | 24 | no | — |
| 228 | C2×D4 | — | 3 | 3 | no | — |
| 229 | C2×D4 | D3×C8 | 1 | 3 | yes | — |
| 230 | C22×C4 | — | 3 | 3 | no | — |
| 231 | C22×C4 | — | 3 | 3 | no | — |
| 232 | C22×C4 | — | 3 | 3 | no | — |
| 233 | C22×C4 | — | 3 | 12 | no | — |
| 234 | C22×C4 | — | 3 | 1 | no | — |
| 235 | C22×C4 | — | 3 | 3 | no | — |
| 236 | C22×C4 | — | 3 | 12 | no | — |
| 237 | C22×C4 | C2×C4×D3 | 1 | 3 | yes | — |
| 238 | M16 | — | 3 | 12 | no | — |
| 239 | M16 | — | 3 | 24 | no | — |
| 240 | M16 | C2×C4×D3 | 1 | 12 | yes | — |
| 241 | C2×C8 | — | 3 | 1 | no | — |
| 242 | C2×C8 | — | 3 | 3 | no | — |
| 243 | C2×C8 | — | 3 | 12 | no | — |
| 244 | C2×C8 | — | 3 | 2 | no | — |
| 245 | C2×C8 | — | 3 | 12 | no | — |
| 246 | C2×C8 | — | 3 | 12 | no | — |
| 247 | C2×C8 | — | 3 | 6 | no | — |
| 248 | C2×C8 | — | 3 | 6 | no | — |
| 249 | C2×C8 | — | 3 | 2 | no | — |
| 250 | C2×C8 | C2×C4×D3 | 1 | 3 | yes | — |
| 251 | C2×C8 | C2×C4×D3 | 1 | 6 | yes | — |
| 252 | C2×C8 | C23×D3 | 1 | 6 | yes | — |
| 253 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 254 | C16 | — | 3 | 2 | no | — |
| 255 | C16 | — | 3 | 6 | no | — |
| 256 | C16 | C23×D3 | 1 | 6 | yes | — |
| 257 | C16 | C23×D3 | 1 | 2 | yes | — |
| 258 | C22×C3 | — | 2 | 6 | no | — |
| 259 | C22×C3 | C23×C8 | 1 | 6 | yes | — |
| 260 | C22×C3 | C23×C8 | 1 | 1 | yes | characteristic |
| 261 | C2×D3 | — | 2 | 2 | no | — |
| 262 | C2×D3 | — | 2 | 6 | no | — |
| 263 | C2×D3 | — | 4 | 12 | no | — |
| 264 | C2×D3 | C23×C8 | 1 | 2 | yes | — |
| 265 | C2×D3 | C23×C8 | 1 | 6 | yes | — |
| 266 | C3×C4 | C23×C8 | 1 | 6 | yes | — |
| 267 | C3×C4 | C23×C8 | 1 | 1 | yes | characteristic |
| 268 | C3×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 269 | Dic3 | C23×C8 | 1 | 6 | yes | — |
| 270 | Dic3 | C23×C8 | 1 | 2 | yes | — |
| 271 | C23 | — | 3 | 1 | no | — |
| 272 | C23 | — | 3 | 3 | no | — |
| 273 | C23 | — | 3 | 12 | no | — |
| 274 | C23 | — | 12 | 6 | no | — |
| 275 | C23 | C2×D3×C8 | 1 | 3 | yes | — |
| 276 | Q8 | — | 3 | 12 | no | — |
| 277 | Q8 | C2×D3×C8 | 1 | 4 | yes | — |
| 278 | D4 | — | 3 | 24 | no | — |
| 279 | D4 | — | 3 | 12 | no | — |
| 280 | D4 | C2×D3×C8 | 1 | 12 | yes | — |
| 281 | C2×C4 | — | 3 | 1 | no | — |
| 282 | C2×C4 | — | 3 | 3 | no | — |
| 283 | C2×C4 | — | 3 | 12 | no | — |
| 284 | C2×C4 | — | 3 | 2 | no | — |
| 285 | C2×C4 | — | 3 | 6 | no | — |
| 286 | C2×C4 | — | 3 | 2 | no | — |
| 287 | C2×C4 | — | 3 | 12 | no | — |
| 288 | C2×C4 | — | 3 | 12 | no | — |
| 289 | C2×C4 | — | 3 | 6 | no | — |
| 290 | C2×C4 | C2×D3×C8 | 1 | 3 | yes | — |
| 291 | C2×C4 | C2×D3×C8 | 1 | 6 | yes | — |
| 292 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C22×C4×D3 | 1 | 6 | yes | — |
| 294 | C8 | — | 3 | 2 | no | — |
| 295 | C8 | — | 3 | 6 | no | — |
| 296 | C8 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 297 | C8 | C22×C4×D3 | 1 | 6 | yes | — |
| 298 | C8 | C24×D3 | 1 | 1 | yes | characteristic |
| 299 | C2×C3 | — | 2 | 6 | no | — |
| 300 | C2×C3 | #3407 | 1 | 2 | yes | complemented |
| 301 | C2×C3 | C24×C8 | 1 | 1 | yes | characteristic |
| 302 | D3 | — | 2 | 2 | no | — |
| 303 | D3 | — | 2 | 6 | no | — |
| 304 | C22 | — | 2 | 6 | no | — |
| 305 | C22 | — | 3 | 2 | no | — |
| 306 | C22 | — | 3 | 6 | no | — |
| 307 | C22 | — | 6 | 2 | no | — |
| 308 | C22 | — | 6 | 6 | no | — |
| 309 | C22 | — | 12 | 12 | no | — |
| 310 | C22 | C22×D3×C8 | 1 | 6 | yes | — |
| 311 | C22 | C22×D3×C8 | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 3 | 2 | no | — |
| 313 | C4 | — | 3 | 6 | no | — |
| 314 | C4 | C22×D3×C8 | 1 | 1 | yes | characteristic |
| 315 | C4 | C22×D3×C8 | 1 | 6 | yes | — |
| 316 | C4 | C23×C4×D3 | 1 | 1 | yes | characteristic |
| 317 | C3 | C2×#3407 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 318 | C2 | — | 2 | 6 | no | — |
| 319 | C2 | — | 6 | 2 | no | — |
| 320 | C2 | — | 6 | 6 | no | — |
| 321 | C2 | #82012 | 1 | 2 | yes | direct product factor |
| 322 | C2 | C23×D3×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