← C22×#5752 #11708999
Subgroups — C22×#5752 #11708999
Order 768.
11462 conjugacy classes of subgroups.
The subgroup lattice has 260 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×#1009 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C22×C3×#131 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#5752 | C2 | 1 | 48 | yes | maximal, direct product factor |
| 4 | C22×#908 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | C22×#904 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C23×D24 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C22×#576 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C22×C3×#41 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#1009 | — | 2 | 24 | no | — |
| 10 | C2×#1009 | C22 | 1 | 48 | yes | complemented |
| 11 | C22×#239 | — | 2 | 2 | no | — |
| 12 | C22×D3×D4 | — | 2 | 6 | no | — |
| 13 | C22×#235 | C22 | 1 | 6 | yes | complemented |
| 14 | C23×D12 | C22 | 1 | 6 | yes | complemented |
| 15 | C2×C3×#131 | C22 | 1 | 24 | yes | complemented |
| 16 | C22×C3×M16 | C22 | 1 | 3 | yes | complemented |
| 17 | C23×C3×C8 | C22 | 1 | 3 | yes | complemented |
| 18 | #5752 | C22 | 1 | 256 | yes | direct product factor |
| 19 | C2×#908 | C22 | 1 | 144 | yes | complemented |
| 20 | C22×Dic12 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | C2×#904 | C22 | 1 | 72 | yes | complemented |
| 22 | C22×D24 | C22 | 1 | 3 | yes | complemented |
| 23 | C22×D24 | C22 | 1 | 6 | yes | complemented |
| 24 | C22×D24 | C22 | 1 | 72 | yes | complemented |
| 25 | C22×#204 | C22 | 1 | 6 | yes | complemented |
| 26 | C22×H(5,𝔽2) | — | 3 | 2 | no | — |
| 27 | C2×#576 | — | 3 | 48 | no | — |
| 28 | C22×AGL(1,ℤ/8) | — | 3 | 6 | no | — |
| 29 | C22×#135 | — | 3 | 3 | no | — |
| 30 | C23×D8 | — | 3 | 3 | no | — |
| 31 | C22×#131 | D3 | 1 | 1 | yes | characteristic, complemented |
| 32 | C24×D3 | — | 2 | 6 | no | — |
| 33 | C2×C3×#41 | D4 | 1 | 12 | yes | — |
| 34 | C2×C3×#41 | C23 | 1 | 12 | yes | — |
| 35 | C22×C3×Q8 | D4 | 1 | 1 | yes | characteristic |
| 36 | C22×C3×D4 | D4 | 1 | 3 | yes | — |
| 37 | C23×C3×C4 | C23 | 1 | 3 | yes | — |
| 38 | C22×#67 | — | 2 | 6 | no | — |
| 39 | #1009 | — | 2 | 192 | no | — |
| 40 | #1009 | C23 | 1 | 128 | yes | complemented |
| 41 | C2×#239 | — | 2 | 48 | no | — |
| 42 | C2×D3×D4 | — | 2 | 144 | no | — |
| 43 | C2×#235 | — | 2 | 36 | no | — |
| 44 | C2×#235 | C23 | 1 | 72 | yes | complemented |
| 45 | C22×D12 | — | 2 | 6 | no | — |
| 46 | C22×D12 | — | 2 | 36 | no | — |
| 47 | C22×D12 | C23 | 1 | 6 | yes | — |
| 48 | C22×D12 | C23 | 1 | 72 | yes | complemented |
| 49 | C22×C4×D3 | — | 2 | 6 | no | — |
| 50 | C22×Dic6 | C23 | 1 | 2 | yes | — |
| 51 | C3×#131 | C23 | 1 | 64 | yes | complemented |
| 52 | C2×C3×M16 | C23 | 1 | 36 | yes | complemented |
| 53 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 54 | C22×C3×C8 | C23 | 1 | 3 | yes | — |
| 55 | C22×C3×C8 | C23 | 1 | 36 | yes | complemented |
| 56 | #908 | C23 | 1 | 384 | yes | complemented |
| 57 | C2×Dic12 | C23 | 1 | 12 | yes | complemented |
| 58 | #904 | C23 | 1 | 192 | yes | complemented |
| 59 | C2×D24 | C23 | 1 | 36 | yes | complemented |
| 60 | C2×D24 | C23 | 1 | 72 | yes | complemented |
| 61 | C2×D24 | C23 | 1 | 192 | yes | complemented |
| 62 | C2×#204 | C23 | 1 | 72 | yes | complemented |
| 63 | C2×H(5,𝔽2) | — | 3 | 48 | no | — |
| 64 | C2×H(5,𝔽2) | — | 6 | 24 | no | — |
| 65 | C22×#41 | — | 3 | 6 | no | — |
| 66 | C22×#41 | — | 6 | 2 | no | — |
| 67 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 68 | C23×D4 | — | 3 | 6 | no | — |
| 69 | C23×D4 | — | 6 | 6 | no | — |
| 70 | #576 | — | 3 | 256 | no | — |
| 71 | C2×AGL(1,ℤ/8) | — | 3 | 144 | no | — |
| 72 | C2×#135 | — | 3 | 72 | no | — |
| 73 | C22×Q16 | — | 3 | 1 | no | — |
| 74 | C22×SD16 | — | 3 | 6 | no | — |
| 75 | C22×D8 | — | 3 | 3 | no | — |
| 76 | C22×D8 | — | 3 | 6 | no | — |
| 77 | C22×D8 | — | 3 | 72 | no | — |
| 78 | C2×#131 | C2×D3 | 1 | 24 | yes | complemented |
| 79 | C22×M16 | C2×D3 | 1 | 3 | yes | complemented |
| 80 | C23×C8 | C2×D3 | 1 | 3 | yes | complemented |
| 81 | C24×C3 | C2×D4 | 1 | 3 | yes | — |
| 82 | C23×D3 | — | 2 | 6 | no | — |
| 83 | C23×D3 | — | 2 | 72 | no | — |
| 84 | C23×D3 | — | 8 | 24 | no | — |
| 85 | C3×#41 | C2×D4 | 1 | 48 | yes | — |
| 86 | C3×#41 | C24 | 1 | 16 | yes | — |
| 87 | C2×C3×Q8 | C2×D4 | 1 | 12 | yes | — |
| 88 | C2×C3×D4 | C2×D4 | 1 | 36 | yes | — |
| 89 | C22×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 90 | C22×C3×C4 | C2×D4 | 1 | 18 | yes | — |
| 91 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 92 | C22×C3×C4 | C24 | 1 | 18 | yes | — |
| 93 | C2×#67 | — | 2 | 72 | no | — |
| 94 | C22×Dic3 | — | 2 | 2 | no | — |
| 95 | #239 | — | 2 | 128 | no | — |
| 96 | D3×D4 | — | 2 | 384 | no | — |
| 97 | #235 | — | 2 | 144 | no | — |
| 98 | #235 | C24 | 1 | 96 | yes | complemented |
| 99 | C2×D12 | — | 2 | 18 | no | — |
| 100 | C2×D12 | — | 2 | 72 | no | — |
| 101 | C2×D12 | — | 2 | 144 | no | — |
| 102 | C2×D12 | C24 | 1 | 36 | yes | — |
| 103 | C2×D12 | C24 | 1 | 96 | yes | complemented |
| 104 | C2×C4×D3 | — | 2 | 72 | no | — |
| 105 | C2×Dic6 | — | 2 | 6 | no | — |
| 106 | C2×Dic6 | C24 | 1 | 12 | yes | — |
| 107 | C3×M16 | C24 | 1 | 48 | yes | complemented |
| 108 | C2×C3×C8 | C24 | 1 | 18 | yes | — |
| 109 | C2×C3×C8 | C24 | 1 | 6 | yes | — |
| 110 | C2×C3×C8 | C24 | 1 | 48 | yes | complemented |
| 111 | Dic12 | C24 | 1 | 16 | yes | complemented |
| 112 | D24 | C24 | 1 | 48 | yes | complemented |
| 113 | D24 | C24 | 1 | 96 | yes | complemented |
| 114 | #204 | C24 | 1 | 96 | yes | complemented |
| 115 | C25 | — | 6 | 6 | no | — |
| 116 | H(5,𝔽2) | — | 3 | 128 | no | — |
| 117 | H(5,𝔽2) | — | 6 | 192 | no | — |
| 118 | C2×#41 | — | 3 | 72 | no | — |
| 119 | C2×#41 | — | 6 | 48 | no | — |
| 120 | C2×#41 | — | 6 | 36 | no | — |
| 121 | C2×#41 | D12 | 1 | 12 | yes | — |
| 122 | C2×#41 | C22×D3 | 1 | 12 | yes | — |
| 123 | C22×Q8 | — | 3 | 2 | no | — |
| 124 | C22×Q8 | D12 | 1 | 1 | yes | characteristic |
| 125 | C22×D4 | — | 3 | 6 | no | — |
| 126 | C22×D4 | — | 3 | 72 | no | — |
| 127 | C22×D4 | — | 6 | 6 | no | — |
| 128 | C22×D4 | — | 6 | 144 | no | — |
| 129 | C22×D4 | — | 6 | 6 | no | — |
| 130 | C22×D4 | — | 6 | 36 | no | — |
| 131 | C22×D4 | D12 | 1 | 3 | yes | — |
| 132 | C23×C4 | — | 6 | 6 | no | — |
| 133 | C23×C4 | C22×D3 | 1 | 3 | yes | — |
| 134 | AGL(1,ℤ/8) | — | 3 | 384 | no | — |
| 135 | #135 | — | 3 | 192 | no | — |
| 136 | C2×Q16 | — | 3 | 12 | no | — |
| 137 | C2×SD16 | — | 3 | 72 | no | — |
| 138 | C2×D8 | — | 3 | 36 | no | — |
| 139 | C2×D8 | — | 3 | 72 | no | — |
| 140 | C2×D8 | — | 3 | 192 | no | — |
| 141 | #131 | C22×D3 | 1 | 64 | yes | complemented |
| 142 | C2×M16 | C22×D3 | 1 | 36 | yes | complemented |
| 143 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 144 | C22×C8 | C22×D3 | 1 | 3 | yes | — |
| 145 | C22×C8 | C22×D3 | 1 | 36 | yes | complemented |
| 146 | C23×C3 | — | 2 | 12 | no | — |
| 147 | C23×C3 | C22×D4 | 1 | 18 | yes | — |
| 148 | C23×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 149 | C22×D3 | — | 2 | 36 | no | — |
| 150 | C22×D3 | — | 2 | 96 | no | — |
| 151 | C22×D3 | — | 4 | 24 | no | — |
| 152 | C22×D3 | — | 8 | 144 | no | — |
| 153 | C3×Q8 | C22×D4 | 1 | 16 | yes | — |
| 154 | C3×D4 | C22×D4 | 1 | 48 | yes | — |
| 155 | C2×C3×C4 | C22×D4 | 1 | 3 | yes | — |
| 156 | C2×C3×C4 | C22×D4 | 1 | 18 | yes | — |
| 157 | C2×C3×C4 | C22×D4 | 1 | 36 | yes | — |
| 158 | C2×C3×C4 | C25 | 1 | 3 | yes | — |
| 159 | C2×C3×C4 | C25 | 1 | 12 | yes | — |
| 160 | #67 | — | 2 | 96 | no | — |
| 161 | C2×Dic3 | — | 2 | 12 | no | — |
| 162 | D12 | — | 2 | 36 | no | — |
| 163 | D12 | — | 2 | 96 | no | — |
| 164 | D12 | C25 | 1 | 24 | yes | — |
| 165 | C4×D3 | — | 2 | 96 | no | — |
| 166 | Dic6 | — | 2 | 12 | no | — |
| 167 | Dic6 | C25 | 1 | 8 | yes | — |
| 168 | C3×C8 | C25 | 1 | 12 | yes | — |
| 169 | C3×C8 | C25 | 1 | 4 | yes | — |
| 170 | C24 | — | 6 | 6 | no | — |
| 171 | C24 | — | 6 | 72 | no | — |
| 172 | C24 | — | 24 | 24 | no | — |
| 173 | C24 | C2×D12 | 1 | 3 | yes | — |
| 174 | #41 | — | 3 | 96 | no | — |
| 175 | #41 | — | 6 | 128 | no | — |
| 176 | #41 | — | 6 | 144 | no | — |
| 177 | #41 | C2×D12 | 1 | 48 | yes | — |
| 178 | #41 | C23×D3 | 1 | 16 | yes | — |
| 179 | C2×Q8 | — | 3 | 12 | no | — |
| 180 | C2×Q8 | — | 6 | 6 | no | — |
| 181 | C2×Q8 | C2×D12 | 1 | 12 | yes | — |
| 182 | C2×D4 | — | 3 | 36 | no | — |
| 183 | C2×D4 | — | 3 | 96 | no | — |
| 184 | C2×D4 | — | 6 | 72 | no | — |
| 185 | C2×D4 | — | 6 | 384 | no | — |
| 186 | C2×D4 | — | 6 | 72 | no | — |
| 187 | C2×D4 | — | 6 | 18 | no | — |
| 188 | C2×D4 | — | 6 | 144 | no | — |
| 189 | C2×D4 | C2×D12 | 1 | 36 | yes | — |
| 190 | C22×C4 | — | 6 | 2 | no | — |
| 191 | C22×C4 | — | 6 | 72 | no | — |
| 192 | C22×C4 | C2×D12 | 1 | 18 | yes | — |
| 193 | C22×C4 | C2×D12 | 1 | 3 | yes | — |
| 194 | C22×C4 | C23×D3 | 1 | 18 | yes | — |
| 195 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 196 | Q16 | — | 3 | 16 | no | — |
| 197 | SD16 | — | 3 | 96 | no | — |
| 198 | D8 | — | 3 | 48 | no | — |
| 199 | D8 | — | 3 | 96 | no | — |
| 200 | M16 | C23×D3 | 1 | 48 | yes | complemented |
| 201 | C2×C8 | C23×D3 | 1 | 6 | yes | — |
| 202 | C2×C8 | C23×D3 | 1 | 18 | yes | — |
| 203 | C2×C8 | C23×D3 | 1 | 48 | yes | complemented |
| 204 | C22×C3 | — | 2 | 36 | no | — |
| 205 | C22×C3 | #576 | 1 | 4 | yes | complemented |
| 206 | C22×C3 | C23×D4 | 1 | 12 | yes | — |
| 207 | C22×C3 | C23×D4 | 1 | 3 | yes | — |
| 208 | C2×D3 | — | 2 | 24 | no | — |
| 209 | C2×D3 | — | 4 | 72 | no | — |
| 210 | C2×D3 | — | 8 | 96 | no | — |
| 211 | C3×C4 | C23×D4 | 1 | 3 | yes | — |
| 212 | C3×C4 | C23×D4 | 1 | 12 | yes | — |
| 213 | C3×C4 | C26 | 1 | 1 | yes | characteristic |
| 214 | Dic3 | — | 2 | 8 | no | — |
| 215 | C23 | — | 2 | 12 | no | — |
| 216 | C23 | — | 6 | 36 | no | — |
| 217 | C23 | — | 6 | 96 | no | — |
| 218 | C23 | — | 12 | 24 | no | — |
| 219 | C23 | — | 24 | 144 | no | — |
| 220 | C23 | C22×D12 | 1 | 18 | yes | — |
| 221 | C23 | C22×D12 | 1 | 1 | yes | characteristic |
| 222 | Q8 | — | 3 | 8 | no | — |
| 223 | Q8 | — | 6 | 12 | no | — |
| 224 | Q8 | C22×D12 | 1 | 16 | yes | — |
| 225 | D4 | — | 3 | 24 | no | — |
| 226 | D4 | — | 6 | 96 | no | — |
| 227 | D4 | — | 6 | 96 | no | — |
| 228 | D4 | — | 6 | 36 | no | — |
| 229 | D4 | C22×D12 | 1 | 48 | yes | — |
| 230 | C2×C4 | — | 6 | 12 | no | — |
| 231 | C2×C4 | — | 6 | 96 | no | — |
| 232 | C2×C4 | C22×D12 | 1 | 18 | yes | — |
| 233 | C2×C4 | C22×D12 | 1 | 3 | yes | — |
| 234 | C2×C4 | C22×D12 | 1 | 36 | yes | — |
| 235 | C2×C4 | C24×D3 | 1 | 3 | yes | — |
| 236 | C2×C4 | C24×D3 | 1 | 12 | yes | — |
| 237 | C8 | C24×D3 | 1 | 4 | yes | — |
| 238 | C8 | C24×D3 | 1 | 12 | yes | — |
| 239 | C2×C3 | — | 2 | 12 | no | — |
| 240 | C2×C3 | C2×#576 | 1 | 6 | yes | complemented |
| 241 | C2×C3 | C24×D4 | 1 | 1 | yes | characteristic |
| 242 | D3 | — | 4 | 24 | no | — |
| 243 | C22 | — | 2 | 36 | no | — |
| 244 | C22 | — | 6 | 24 | no | — |
| 245 | C22 | — | 12 | 72 | no | — |
| 246 | C22 | — | 24 | 96 | no | — |
| 247 | C22 | #5752 | 1 | 4 | yes | direct product factor |
| 248 | C22 | C23×D12 | 1 | 12 | yes | — |
| 249 | C22 | C23×D12 | 1 | 3 | yes | — |
| 250 | C4 | — | 6 | 8 | no | — |
| 251 | C4 | C23×D12 | 1 | 12 | yes | — |
| 252 | C4 | C23×D12 | 1 | 3 | yes | — |
| 253 | C4 | C25×D3 | 1 | 1 | yes | characteristic |
| 254 | C3 | C22×#576 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 255 | C2 | — | 2 | 12 | no | — |
| 256 | C2 | — | 12 | 24 | no | — |
| 257 | C2 | C2×#5752 | 1 | 6 | yes | direct product factor |
| 258 | C2 | C24×D12 | 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