← C22×D3×2−1+4 #11709075
Subgroups — C22×D3×2−1+4 #11709075
Order 768.
59806 conjugacy classes of subgroups.
The subgroup lattice has 222 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×C3×2−1+4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×D3×2−1+4 | C2 | 1 | 96 | yes | maximal, direct product factor |
| 3 | C22×D3×#41 | C2 | 1 | 10 | yes | maximal, complemented |
| 4 | C22×#1010 | C2 | 1 | 10 | yes | maximal, complemented |
| 5 | C22×#1007 | C2 | 1 | 5 | yes | maximal, complemented |
| 6 | C23×D3×Q8 | C2 | 1 | 5 | yes | maximal, complemented |
| 7 | C23×2−1+4 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C2×C3×2−1+4 | C22 | 1 | 48 | yes | complemented |
| 9 | C22×C3×#41 | C22 | 1 | 10 | yes | complemented |
| 10 | C23×C3×Q8 | C22 | 1 | 5 | yes | complemented |
| 11 | D3×2−1+4 | C22 | 1 | 1024 | yes | direct product factor |
| 12 | C2×#1010 | C22 | 1 | 480 | yes | complemented |
| 13 | C2×D3×#41 | C22 | 1 | 480 | yes | complemented |
| 14 | C2×#1007 | C22 | 1 | 240 | yes | complemented |
| 15 | C22×#239 | C22 | 1 | 10 | yes | complemented |
| 16 | C22×D3×Q8 | C22 | 1 | 10 | yes | — |
| 17 | C22×D3×Q8 | C22 | 1 | 20 | yes | complemented |
| 18 | C22×D3×Q8 | C22 | 1 | 240 | yes | complemented |
| 19 | C22×#237 | C22 | 1 | 30 | yes | complemented |
| 20 | C22×D3×D4 | C22 | 1 | 10 | yes | — |
| 21 | C22×#235 | C22 | 1 | 30 | yes | complemented |
| 22 | C23×C4×D3 | C22 | 1 | 15 | yes | — |
| 23 | C23×Dic6 | C22 | 1 | 15 | yes | complemented |
| 24 | C22×2−1+4 | — | 3 | 96 | no | — |
| 25 | C22×2−1+4 | — | 3 | 10 | no | — |
| 26 | C22×2−1+4 | — | 3 | 5 | no | — |
| 27 | C22×2−1+4 | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 28 | C23×#41 | — | 3 | 10 | no | — |
| 29 | C24×Q8 | — | 3 | 5 | no | — |
| 30 | C24×D3 | C23 | 1 | 5 | yes | — |
| 31 | C3×2−1+4 | C23 | 1 | 256 | yes | complemented |
| 32 | C2×C3×#41 | C23 | 1 | 240 | yes | complemented |
| 33 | C22×C3×Q8 | C23 | 1 | 10 | yes | — |
| 34 | C22×C3×Q8 | C23 | 1 | 120 | yes | complemented |
| 35 | C22×C3×D4 | C23 | 1 | 10 | yes | — |
| 36 | C23×C3×C4 | C23 | 1 | 15 | yes | — |
| 37 | C22×#67 | C23 | 1 | 20 | yes | — |
| 38 | C23×Dic3 | C23 | 1 | 15 | yes | — |
| 39 | #1010 | C23 | 1 | 2560 | yes | complemented |
| 40 | D3×#41 | C23 | 1 | 2560 | yes | complemented |
| 41 | #1007 | C23 | 1 | 1280 | yes | complemented |
| 42 | C2×#239 | C23 | 1 | 240 | yes | complemented |
| 43 | C2×D3×Q8 | C23 | 1 | 240 | yes | — |
| 44 | C2×D3×Q8 | C23 | 1 | 480 | yes | complemented |
| 45 | C2×D3×Q8 | C23 | 1 | 1280 | yes | complemented |
| 46 | C2×#237 | C23 | 1 | 720 | yes | complemented |
| 47 | C2×D3×D4 | C23 | 1 | 240 | yes | — |
| 48 | C2×#235 | C23 | 1 | 720 | yes | complemented |
| 49 | C22×D12 | C23 | 1 | 10 | yes | — |
| 50 | C22×C4×D3 | C23 | 1 | 10 | yes | — |
| 51 | C22×C4×D3 | C23 | 1 | 30 | yes | — |
| 52 | C22×C4×D3 | C23 | 1 | 360 | yes | — |
| 53 | C22×Dic6 | C23 | 1 | 30 | yes | — |
| 54 | C22×Dic6 | C23 | 1 | 360 | yes | complemented |
| 55 | C2×2−1+4 | — | 3 | 1024 | no | — |
| 56 | C2×2−1+4 | — | 3 | 480 | no | — |
| 57 | C2×2−1+4 | — | 3 | 240 | no | — |
| 58 | C2×2−1+4 | C2×D3 | 1 | 48 | yes | direct product factor |
| 59 | C22×#41 | — | 3 | 480 | no | — |
| 60 | C22×#41 | — | 3 | 10 | no | — |
| 61 | C22×#41 | — | 3 | 30 | no | — |
| 62 | C22×#41 | — | 3 | 30 | no | — |
| 63 | C22×#41 | C2×D3 | 1 | 10 | yes | complemented |
| 64 | C23×Q8 | — | 3 | 10 | no | — |
| 65 | C23×Q8 | — | 3 | 20 | no | — |
| 66 | C23×Q8 | — | 3 | 240 | no | — |
| 67 | C23×Q8 | — | 3 | 15 | no | — |
| 68 | C23×Q8 | C2×D3 | 1 | 5 | yes | complemented |
| 69 | C23×D4 | — | 3 | 10 | no | — |
| 70 | C24×C4 | — | 3 | 15 | no | — |
| 71 | C24×C3 | C24 | 1 | 5 | yes | — |
| 72 | C23×D3 | — | 2 | 40 | no | — |
| 73 | C23×D3 | C24 | 1 | 5 | yes | — |
| 74 | C23×D3 | C24 | 1 | 60 | yes | — |
| 75 | C23×D3 | C24 | 1 | 1 | yes | characteristic |
| 76 | C3×#41 | C24 | 1 | 640 | yes | complemented |
| 77 | C2×C3×Q8 | C24 | 1 | 120 | yes | — |
| 78 | C2×C3×Q8 | C24 | 1 | 320 | yes | complemented |
| 79 | C2×C3×D4 | C24 | 1 | 120 | yes | — |
| 80 | C22×C3×C4 | C24 | 1 | 10 | yes | — |
| 81 | C22×C3×C4 | C24 | 1 | 180 | yes | — |
| 82 | C2×#67 | C24 | 1 | 240 | yes | — |
| 83 | C22×Dic3 | C24 | 1 | 10 | yes | — |
| 84 | C22×Dic3 | C24 | 1 | 180 | yes | — |
| 85 | #239 | C24 | 1 | 640 | yes | complemented |
| 86 | D3×Q8 | C24 | 1 | 640 | yes | — |
| 87 | D3×Q8 | C24 | 1 | 1280 | yes | complemented |
| 88 | #237 | C24 | 1 | 1920 | yes | complemented |
| 89 | D3×D4 | C24 | 1 | 640 | yes | — |
| 90 | #235 | C24 | 1 | 1920 | yes | complemented |
| 91 | C2×D12 | C24 | 1 | 120 | yes | — |
| 92 | C2×C4×D3 | C24 | 1 | 120 | yes | — |
| 93 | C2×C4×D3 | C24 | 1 | 360 | yes | — |
| 94 | C2×C4×D3 | C24 | 1 | 960 | yes | — |
| 95 | C2×Dic6 | C24 | 1 | 360 | yes | — |
| 96 | C2×Dic6 | C24 | 1 | 960 | yes | complemented |
| 97 | C25 | — | 3 | 5 | no | — |
| 98 | 2−1+4 | — | 3 | 2560 | no | — |
| 99 | 2−1+4 | — | 3 | 1280 | no | — |
| 100 | 2−1+4 | C22×D3 | 1 | 256 | yes | direct product factor |
| 101 | C2×#41 | — | 3 | 2560 | no | — |
| 102 | C2×#41 | — | 3 | 240 | no | — |
| 103 | C2×#41 | — | 3 | 720 | no | — |
| 104 | C2×#41 | — | 3 | 720 | no | — |
| 105 | C2×#41 | C22×D3 | 1 | 240 | yes | complemented |
| 106 | C22×Q8 | — | 3 | 240 | no | — |
| 107 | C22×Q8 | — | 3 | 480 | no | — |
| 108 | C22×Q8 | — | 3 | 1280 | no | — |
| 109 | C22×Q8 | — | 3 | 30 | no | — |
| 110 | C22×Q8 | — | 3 | 360 | no | — |
| 111 | C22×Q8 | C22×D3 | 1 | 10 | yes | — |
| 112 | C22×Q8 | C22×D3 | 1 | 120 | yes | complemented |
| 113 | C22×D4 | — | 3 | 20 | no | — |
| 114 | C22×D4 | — | 3 | 240 | no | — |
| 115 | C22×D4 | — | 3 | 10 | no | — |
| 116 | C22×D4 | C22×D3 | 1 | 10 | yes | — |
| 117 | C23×C4 | — | 3 | 15 | no | — |
| 118 | C23×C4 | — | 3 | 10 | no | — |
| 119 | C23×C4 | — | 3 | 360 | no | — |
| 120 | C23×C4 | — | 3 | 30 | no | — |
| 121 | C23×C4 | C22×D3 | 1 | 15 | yes | — |
| 122 | C23×C3 | — | 2 | 20 | no | — |
| 123 | C23×C3 | C25 | 1 | 30 | yes | — |
| 124 | C23×C3 | C25 | 1 | 1 | yes | characteristic |
| 125 | C22×D3 | — | 2 | 20 | no | — |
| 126 | C22×D3 | — | 2 | 240 | no | — |
| 127 | C22×D3 | 2−1+4 | 1 | 8 | yes | direct product factor |
| 128 | C22×D3 | C25 | 1 | 30 | yes | — |
| 129 | C22×D3 | C25 | 1 | 80 | yes | — |
| 130 | C22×D3 | C25 | 1 | 6 | yes | — |
| 131 | C3×Q8 | C25 | 1 | 160 | yes | — |
| 132 | C3×D4 | C25 | 1 | 160 | yes | — |
| 133 | C2×C3×C4 | C25 | 1 | 60 | yes | — |
| 134 | C2×C3×C4 | C25 | 1 | 240 | yes | — |
| 135 | #67 | C25 | 1 | 320 | yes | — |
| 136 | C2×Dic3 | C25 | 1 | 60 | yes | — |
| 137 | C2×Dic3 | C25 | 1 | 240 | yes | — |
| 138 | D12 | C25 | 1 | 160 | yes | — |
| 139 | C4×D3 | C25 | 1 | 160 | yes | — |
| 140 | C4×D3 | C25 | 1 | 480 | yes | — |
| 141 | Dic6 | C25 | 1 | 480 | yes | — |
| 142 | C24 | — | 3 | 1 | no | — |
| 143 | C24 | — | 3 | 60 | no | — |
| 144 | C24 | — | 3 | 5 | no | — |
| 145 | C24 | — | 6 | 40 | no | — |
| 146 | C24 | C23×D3 | 1 | 5 | yes | — |
| 147 | #41 | — | 3 | 640 | no | — |
| 148 | #41 | — | 3 | 1920 | no | — |
| 149 | #41 | — | 3 | 1920 | no | — |
| 150 | #41 | C23×D3 | 1 | 640 | yes | complemented |
| 151 | C2×Q8 | — | 3 | 640 | no | — |
| 152 | C2×Q8 | — | 3 | 1280 | no | — |
| 153 | C2×Q8 | — | 3 | 360 | no | — |
| 154 | C2×Q8 | — | 3 | 960 | no | — |
| 155 | C2×Q8 | C23×D3 | 1 | 120 | yes | — |
| 156 | C2×Q8 | C23×D3 | 1 | 320 | yes | complemented |
| 157 | C2×D4 | — | 3 | 240 | no | — |
| 158 | C2×D4 | — | 3 | 640 | no | — |
| 159 | C2×D4 | — | 3 | 120 | no | — |
| 160 | C2×D4 | C23×D3 | 1 | 120 | yes | — |
| 161 | C22×C4 | — | 3 | 180 | no | — |
| 162 | C22×C4 | — | 3 | 10 | no | — |
| 163 | C22×C4 | — | 3 | 120 | no | — |
| 164 | C22×C4 | — | 3 | 360 | no | — |
| 165 | C22×C4 | — | 3 | 960 | no | — |
| 166 | C22×C4 | C23×D3 | 1 | 180 | yes | — |
| 167 | C22×C4 | C23×D3 | 1 | 10 | yes | — |
| 168 | C22×C3 | — | 2 | 60 | no | — |
| 169 | C22×C3 | C2×2−1+4 | 1 | 4 | yes | complemented |
| 170 | C22×C3 | C26 | 1 | 20 | yes | — |
| 171 | C22×C3 | C26 | 1 | 3 | yes | — |
| 172 | C2×D3 | — | 2 | 60 | no | — |
| 173 | C2×D3 | — | 2 | 160 | no | — |
| 174 | C2×D3 | C2×2−1+4 | 1 | 24 | yes | direct product factor |
| 175 | C2×D3 | C26 | 1 | 20 | yes | — |
| 176 | C2×D3 | C26 | 1 | 4 | yes | — |
| 177 | C3×C4 | C26 | 1 | 40 | yes | — |
| 178 | Dic3 | C26 | 1 | 40 | yes | — |
| 179 | C23 | — | 2 | 20 | no | — |
| 180 | C23 | — | 3 | 8 | no | — |
| 181 | C23 | — | 3 | 6 | no | — |
| 182 | C23 | — | 3 | 80 | no | — |
| 183 | C23 | — | 3 | 30 | no | — |
| 184 | C23 | — | 6 | 240 | no | — |
| 185 | C23 | — | 6 | 20 | no | — |
| 186 | C23 | C24×D3 | 1 | 30 | yes | — |
| 187 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 188 | Q8 | — | 3 | 480 | no | — |
| 189 | Q8 | C24×D3 | 1 | 160 | yes | — |
| 190 | D4 | — | 3 | 320 | no | — |
| 191 | D4 | — | 3 | 160 | no | — |
| 192 | D4 | C24×D3 | 1 | 160 | yes | — |
| 193 | C2×C4 | — | 3 | 60 | no | — |
| 194 | C2×C4 | — | 3 | 240 | no | — |
| 195 | C2×C4 | — | 3 | 160 | no | — |
| 196 | C2×C4 | — | 3 | 480 | no | — |
| 197 | C2×C4 | C24×D3 | 1 | 60 | yes | — |
| 198 | C2×C4 | C24×D3 | 1 | 240 | yes | — |
| 199 | C2×C3 | — | 2 | 20 | no | — |
| 200 | C2×C3 | C22×2−1+4 | 1 | 6 | yes | complemented |
| 201 | C2×C3 | C27 | 1 | 1 | yes | characteristic |
| 202 | D3 | — | 2 | 20 | no | — |
| 203 | D3 | C22×2−1+4 | 1 | 8 | yes | direct product factor |
| 204 | C22 | — | 2 | 60 | no | — |
| 205 | C22 | — | 3 | 24 | no | — |
| 206 | C22 | — | 3 | 4 | no | — |
| 207 | C22 | — | 3 | 20 | no | — |
| 208 | C22 | — | 6 | 160 | no | — |
| 209 | C22 | — | 6 | 60 | no | — |
| 210 | C22 | D3×2−1+4 | 1 | 4 | yes | direct product factor |
| 211 | C22 | C25×D3 | 1 | 20 | yes | — |
| 212 | C22 | C25×D3 | 1 | 3 | yes | — |
| 213 | C4 | — | 3 | 40 | no | — |
| 214 | C4 | C25×D3 | 1 | 40 | yes | — |
| 215 | C3 | C23×2−1+4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 216 | C2 | — | 2 | 20 | no | — |
| 217 | C2 | — | 3 | 8 | no | — |
| 218 | C2 | — | 6 | 20 | no | — |
| 219 | C2 | C2×D3×2−1+4 | 1 | 6 | yes | direct product factor |
| 220 | C2 | C26×D3 | 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