← C22×#4907 #11701652
Subgroups — C22×#4907 #11701652
Order 768.
2118 conjugacy classes of subgroups.
The subgroup lattice has 218 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C22×#901 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×C3×M32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C23×C3⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#4907 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 5 | C22×#798 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C22×D3×C16 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C22×#504 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C22×#235 | C4 | 1 | 1 | yes | characteristic |
| 9 | C23×C3×C8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×#208 | C4 | 1 | 1 | yes | characteristic |
| 11 | C2×#901 | C4 | 1 | 12 | yes | — |
| 12 | C2×#901 | C22 | 1 | 12 | yes | — |
| 13 | C22×ΓL(1,𝔽25) | C4 | 1 | 2 | yes | — |
| 14 | C22×D3×C8 | C22 | 1 | 2 | yes | — |
| 15 | C2×C3×M32 | C22 | 1 | 12 | yes | complemented |
| 16 | C22×C3×C16 | C22 | 1 | 2 | yes | — |
| 17 | C22×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 19 | C22×C3⋊C16 | C22 | 1 | 12 | yes | complemented |
| 20 | #4907 | C22 | 1 | 64 | yes | direct product factor |
| 21 | C2×#798 | C22 | 1 | 24 | yes | complemented |
| 22 | C2×D3×C16 | C22 | 1 | 24 | yes | complemented |
| 23 | C22×#131 | — | 3 | 1 | no | — |
| 24 | C2×#504 | — | 3 | 24 | no | — |
| 25 | C22×M32 | — | 3 | 2 | no | — |
| 26 | C22×M32 | D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C23×C16 | — | 3 | 2 | no | — |
| 28 | C23×C16 | — | 3 | 1 | no | — |
| 29 | C23×C3×C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 30 | C22×#67 | C8 | 1 | 2 | yes | — |
| 31 | C2×#235 | C8 | 1 | 12 | yes | — |
| 32 | C2×#235 | C2×C4 | 1 | 12 | yes | — |
| 33 | C22×D12 | C8 | 1 | 1 | yes | characteristic |
| 34 | C22×C4×D3 | C2×C4 | 1 | 2 | yes | — |
| 35 | C22×Dic6 | C8 | 1 | 1 | yes | characteristic |
| 36 | C22×C3×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 37 | C22×C3×C8 | C2×C4 | 1 | 6 | yes | — |
| 38 | C22×C3×C8 | C23 | 1 | 6 | yes | — |
| 39 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×#208 | C2×C4 | 1 | 12 | yes | — |
| 41 | C22×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 42 | #901 | C2×C4 | 1 | 48 | yes | — |
| 43 | #901 | C23 | 1 | 16 | yes | — |
| 44 | C2×ΓL(1,𝔽25) | C2×C4 | 1 | 24 | yes | — |
| 45 | C2×D3×C8 | C2×C4 | 1 | 12 | yes | — |
| 46 | C2×D3×C8 | C23 | 1 | 12 | yes | — |
| 47 | C3×M32 | C23 | 1 | 16 | yes | complemented |
| 48 | C2×C3×C16 | C23 | 1 | 12 | yes | — |
| 49 | C2×C3⋊C16 | C23 | 1 | 6 | yes | — |
| 50 | C2×C3⋊C16 | C23 | 1 | 6 | yes | — |
| 51 | C2×C3⋊C16 | C23 | 1 | 16 | yes | complemented |
| 52 | #798 | C23 | 1 | 32 | yes | complemented |
| 53 | D3×C16 | C23 | 1 | 32 | yes | complemented |
| 54 | C22×#41 | — | 3 | 1 | no | — |
| 55 | C2×#131 | — | 3 | 12 | no | — |
| 56 | C2×#131 | — | 3 | 12 | no | — |
| 57 | C22×M16 | — | 3 | 1 | no | — |
| 58 | C22×M16 | — | 3 | 2 | no | — |
| 59 | C23×C8 | — | 3 | 2 | no | — |
| 60 | C23×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | #504 | — | 3 | 64 | no | — |
| 62 | C2×M32 | — | 3 | 24 | no | — |
| 63 | C2×M32 | C2×D3 | 1 | 12 | yes | complemented |
| 64 | C22×C16 | — | 3 | 12 | no | — |
| 65 | C22×C16 | — | 3 | 1 | no | — |
| 66 | C22×C16 | — | 3 | 1 | no | — |
| 67 | C22×C16 | — | 3 | 24 | no | — |
| 68 | C22×C16 | C2×D3 | 1 | 2 | yes | — |
| 69 | C24×C3 | C2×C8 | 1 | 1 | yes | characteristic |
| 70 | C23×D3 | C2×C8 | 1 | 2 | yes | — |
| 71 | C22×C3×C4 | C2×C8 | 1 | 1 | yes | characteristic |
| 72 | C22×C3×C4 | C2×C8 | 1 | 6 | yes | — |
| 73 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | C22×C3×C4 | C22×C4 | 1 | 6 | yes | — |
| 75 | C2×#67 | C2×C8 | 1 | 24 | yes | — |
| 76 | C22×Dic3 | C2×C8 | 1 | 2 | yes | — |
| 77 | #235 | C2×C8 | 1 | 48 | yes | — |
| 78 | #235 | C22×C4 | 1 | 16 | yes | — |
| 79 | C2×D12 | C2×C8 | 1 | 12 | yes | — |
| 80 | C2×C4×D3 | C2×C8 | 1 | 12 | yes | — |
| 81 | C2×C4×D3 | C22×C4 | 1 | 12 | yes | — |
| 82 | C2×Dic6 | C2×C8 | 1 | 12 | yes | — |
| 83 | C2×C3×C8 | C22×C4 | 1 | 3 | yes | — |
| 84 | C2×C3×C8 | C22×C4 | 1 | 6 | yes | — |
| 85 | C2×C3×C8 | C22×C4 | 1 | 12 | yes | — |
| 86 | C2×C3×C8 | C24 | 1 | 3 | yes | — |
| 87 | C2×C3×C8 | C24 | 1 | 4 | yes | — |
| 88 | #208 | C22×C4 | 1 | 16 | yes | — |
| 89 | C2×C3⋊C8 | C22×C4 | 1 | 12 | yes | — |
| 90 | ΓL(1,𝔽25) | C22×C4 | 1 | 32 | yes | — |
| 91 | D3×C8 | C22×C4 | 1 | 24 | yes | — |
| 92 | D3×C8 | C24 | 1 | 8 | yes | — |
| 93 | C3×C16 | C24 | 1 | 8 | yes | — |
| 94 | C3⋊C16 | C24 | 1 | 4 | yes | — |
| 95 | C3⋊C16 | C24 | 1 | 4 | yes | — |
| 96 | C2×#41 | — | 3 | 12 | no | — |
| 97 | C2×#41 | — | 3 | 12 | no | — |
| 98 | C22×Q8 | — | 3 | 1 | no | — |
| 99 | C22×D4 | — | 3 | 2 | no | — |
| 100 | C22×D4 | — | 3 | 1 | no | — |
| 101 | C23×C4 | — | 3 | 2 | no | — |
| 102 | C23×C4 | C4×D3 | 1 | 1 | yes | characteristic |
| 103 | #131 | — | 3 | 48 | no | — |
| 104 | #131 | — | 3 | 16 | no | — |
| 105 | C2×M16 | — | 3 | 12 | no | — |
| 106 | C2×M16 | — | 3 | 24 | no | — |
| 107 | C22×C8 | — | 3 | 2 | no | — |
| 108 | C22×C8 | — | 3 | 12 | no | — |
| 109 | C22×C8 | — | 3 | 12 | no | — |
| 110 | C22×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 111 | C22×C8 | C4×D3 | 1 | 6 | yes | — |
| 112 | C22×C8 | C22×D3 | 1 | 6 | yes | — |
| 113 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 114 | M32 | — | 3 | 32 | no | — |
| 115 | M32 | C22×D3 | 1 | 16 | yes | complemented |
| 116 | C2×C16 | — | 3 | 6 | no | — |
| 117 | C2×C16 | — | 3 | 6 | no | — |
| 118 | C2×C16 | — | 3 | 16 | no | — |
| 119 | C2×C16 | — | 3 | 32 | no | — |
| 120 | C2×C16 | C22×D3 | 1 | 12 | yes | — |
| 121 | C23×C3 | — | 2 | 4 | no | — |
| 122 | C23×C3 | C22×C8 | 1 | 6 | yes | — |
| 123 | C23×C3 | C22×C8 | 1 | 1 | yes | characteristic |
| 124 | C22×D3 | — | 2 | 8 | no | — |
| 125 | C22×D3 | C22×C8 | 1 | 12 | yes | — |
| 126 | C2×C3×C4 | C22×C8 | 1 | 6 | yes | — |
| 127 | C2×C3×C4 | C22×C8 | 1 | 3 | yes | — |
| 128 | C2×C3×C4 | C22×C8 | 1 | 12 | yes | — |
| 129 | C2×C3×C4 | C23×C4 | 1 | 3 | yes | — |
| 130 | C2×C3×C4 | C23×C4 | 1 | 4 | yes | — |
| 131 | #67 | C22×C8 | 1 | 32 | yes | — |
| 132 | C2×Dic3 | C22×C8 | 1 | 12 | yes | — |
| 133 | D12 | C22×C8 | 1 | 16 | yes | — |
| 134 | C4×D3 | C22×C8 | 1 | 24 | yes | — |
| 135 | C4×D3 | C23×C4 | 1 | 8 | yes | — |
| 136 | Dic6 | C22×C8 | 1 | 16 | yes | — |
| 137 | C3×C8 | C23×C4 | 1 | 3 | yes | — |
| 138 | C3×C8 | C23×C4 | 1 | 4 | yes | — |
| 139 | C3×C8 | C25 | 1 | 1 | yes | characteristic |
| 140 | C3⋊C8 | C23×C4 | 1 | 8 | yes | — |
| 141 | C24 | — | 3 | 2 | no | — |
| 142 | C24 | D3×C8 | 1 | 1 | yes | characteristic |
| 143 | #41 | — | 3 | 48 | no | — |
| 144 | #41 | — | 3 | 16 | no | — |
| 145 | C2×Q8 | — | 3 | 12 | no | — |
| 146 | C2×D4 | — | 3 | 24 | no | — |
| 147 | C2×D4 | — | 3 | 12 | no | — |
| 148 | C22×C4 | — | 3 | 2 | no | — |
| 149 | C22×C4 | — | 3 | 12 | no | — |
| 150 | C22×C4 | — | 3 | 12 | no | — |
| 151 | C22×C4 | D3×C8 | 1 | 1 | yes | characteristic |
| 152 | C22×C4 | D3×C8 | 1 | 6 | yes | — |
| 153 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 154 | C22×C4 | C2×C4×D3 | 1 | 6 | yes | — |
| 155 | M16 | — | 3 | 16 | no | — |
| 156 | M16 | — | 3 | 32 | no | — |
| 157 | C2×C8 | — | 3 | 12 | no | — |
| 158 | C2×C8 | — | 3 | 24 | no | — |
| 159 | C2×C8 | — | 3 | 8 | no | — |
| 160 | C2×C8 | C2×C4×D3 | 1 | 3 | yes | — |
| 161 | C2×C8 | C2×C4×D3 | 1 | 6 | yes | — |
| 162 | C2×C8 | C2×C4×D3 | 1 | 12 | yes | — |
| 163 | C2×C8 | C23×D3 | 1 | 3 | yes | — |
| 164 | C2×C8 | C23×D3 | 1 | 4 | yes | — |
| 165 | C16 | — | 3 | 4 | no | — |
| 166 | C16 | — | 3 | 4 | no | — |
| 167 | C16 | C23×D3 | 1 | 8 | yes | — |
| 168 | C22×C3 | — | 2 | 12 | no | — |
| 169 | C22×C3 | #504 | 1 | 4 | yes | complemented |
| 170 | C22×C3 | C23×C8 | 1 | 4 | yes | — |
| 171 | C22×C3 | C23×C8 | 1 | 3 | yes | — |
| 172 | C2×D3 | — | 2 | 24 | no | — |
| 173 | C2×D3 | C23×C8 | 1 | 8 | yes | — |
| 174 | C3×C4 | C23×C8 | 1 | 4 | yes | — |
| 175 | C3×C4 | C23×C8 | 1 | 3 | yes | — |
| 176 | C3×C4 | C24×C4 | 1 | 1 | yes | characteristic |
| 177 | Dic3 | C23×C8 | 1 | 8 | yes | — |
| 178 | C23 | — | 2 | 4 | no | — |
| 179 | C23 | — | 3 | 12 | no | — |
| 180 | C23 | — | 6 | 8 | no | — |
| 181 | C23 | C2×D3×C8 | 1 | 6 | yes | — |
| 182 | C23 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 183 | Q8 | — | 3 | 16 | no | — |
| 184 | D4 | — | 3 | 32 | no | — |
| 185 | D4 | — | 3 | 16 | no | — |
| 186 | C2×C4 | — | 3 | 12 | no | — |
| 187 | C2×C4 | — | 3 | 8 | no | — |
| 188 | C2×C4 | — | 3 | 24 | no | — |
| 189 | C2×C4 | C2×D3×C8 | 1 | 3 | yes | — |
| 190 | C2×C4 | C2×D3×C8 | 1 | 6 | yes | — |
| 191 | C2×C4 | C2×D3×C8 | 1 | 12 | yes | — |
| 192 | C2×C4 | C22×C4×D3 | 1 | 3 | yes | — |
| 193 | C2×C4 | C22×C4×D3 | 1 | 4 | yes | — |
| 194 | C8 | — | 3 | 8 | no | — |
| 195 | C8 | C22×C4×D3 | 1 | 3 | yes | — |
| 196 | C8 | C22×C4×D3 | 1 | 4 | yes | — |
| 197 | C8 | C24×D3 | 1 | 1 | yes | characteristic |
| 198 | C2×C3 | — | 2 | 4 | no | — |
| 199 | C2×C3 | C2×#504 | 1 | 6 | yes | complemented |
| 200 | C2×C3 | C24×C8 | 1 | 1 | yes | characteristic |
| 201 | D3 | — | 2 | 8 | no | — |
| 202 | C22 | — | 2 | 12 | no | — |
| 203 | C22 | — | 3 | 8 | no | — |
| 204 | C22 | — | 6 | 24 | no | — |
| 205 | C22 | #4907 | 1 | 4 | yes | direct product factor |
| 206 | C22 | C22×D3×C8 | 1 | 4 | yes | — |
| 207 | C22 | C22×D3×C8 | 1 | 3 | yes | — |
| 208 | C4 | — | 3 | 8 | no | — |
| 209 | C4 | C22×D3×C8 | 1 | 3 | yes | — |
| 210 | C4 | C22×D3×C8 | 1 | 4 | yes | — |
| 211 | C4 | C23×C4×D3 | 1 | 1 | yes | characteristic |
| 212 | C3 | C22×#504 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 213 | C2 | — | 2 | 4 | no | — |
| 214 | C2 | — | 6 | 8 | no | — |
| 215 | C2 | C2×#4907 | 1 | 6 | yes | direct product factor |
| 216 | 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