← #13135888
Subgroups — #13135888
Order 1728.
627 conjugacy classes of subgroups.
The subgroup lattice has 245 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C4⋊C4×H(3,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C4⋊#1084 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C22×#6180 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C3×#4529 | — | 3 | 4 | no | maximal |
| 5 | C22×C4×H(3,𝔽3) | — | 2 | 1 | no | — |
| 6 | C22×C4×H(3,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | C22×#1084 | — | 2 | 1 | no | — |
| 8 | C4⋊C4×H(3,𝔽3) | — | 2 | 1 | no | — |
| 9 | C4⋊C4×H(3,𝔽3) | C4 | 1 | 2 | yes | complemented |
| 10 | C4⋊#1084 | — | 2 | 1 | no | — |
| 11 | C4⋊#1084 | C4 | 1 | 2 | yes | complemented |
| 12 | C2×#6180 | — | 2 | 1 | no | — |
| 13 | C2×#6180 | — | 2 | 1 | no | — |
| 14 | C2×#6180 | C4 | 1 | 2 | yes | complemented |
| 15 | C2×C32×C4⋊C4 | D3 | 1 | 4 | yes | — |
| 16 | C2×C3×C4⋊Dic3 | — | 3 | 4 | no | — |
| 17 | C22×C3×C3⋊C8 | — | 3 | 4 | no | — |
| 18 | C23×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 19 | C2×C4×H(3,𝔽3) | — | 2 | 2 | no | — |
| 20 | C2×C4×H(3,𝔽3) | — | 4 | 1 | no | — |
| 21 | C2×C4×H(3,𝔽3) | — | 4 | 1 | no | — |
| 22 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 24 | C2×C4×H(3,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 25 | C2×C4×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 26 | C2×C4×H(3,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 27 | C2×C4×H(3,𝔽3) | Q8 | 1 | 1 | yes | characteristic |
| 28 | C2×#1084 | — | 2 | 2 | no | — |
| 29 | C2×#1084 | — | 4 | 1 | no | — |
| 30 | C2×#1084 | — | 4 | 1 | no | — |
| 31 | #6180 | — | 2 | 2 | no | — |
| 32 | C3×#340 | — | 9 | 1 | no | — |
| 33 | #4529 | — | 3 | 4 | no | — |
| 34 | C22×C32×C4 | — | 2 | 4 | no | — |
| 35 | C22×C32×C4 | C2×D3 | 1 | 4 | yes | — |
| 36 | C22×C3×Dic3 | — | 6 | 4 | no | — |
| 37 | C32×C4⋊C4 | — | 2 | 4 | no | — |
| 38 | C32×C4⋊C4 | Dic3 | 1 | 8 | yes | complemented |
| 39 | C3×C4⋊Dic3 | — | 3 | 8 | no | — |
| 40 | C3×C4⋊Dic3 | — | 6 | 4 | no | — |
| 41 | C2×C3×C3⋊C8 | — | 3 | 8 | no | — |
| 42 | C2×C3×C3⋊C8 | — | 6 | 4 | no | — |
| 43 | C2×C3×C3⋊C8 | — | 6 | 4 | no | — |
| 44 | C22×H(3,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 45 | C22×H(3,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 46 | C22×H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic |
| 47 | C22×H(3,𝔽3) | D8 | 1 | 1 | yes | characteristic |
| 48 | C22×H(3,𝔽3) | SD16 | 1 | 1 | yes | characteristic |
| 49 | C22×H(3,𝔽3) | SD16 | 1 | 1 | yes | characteristic |
| 50 | C22×H(3,𝔽3) | Q16 | 1 | 1 | yes | characteristic |
| 51 | C4×H(3,𝔽3) | — | 4 | 2 | no | — |
| 52 | C4×H(3,𝔽3) | C42 | 1 | 1 | yes | characteristic |
| 53 | C4×H(3,𝔽3) | C22⋊C4 | 1 | 1 | yes | characteristic |
| 54 | C4×H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic |
| 55 | C4×H(3,𝔽3) | C4⋊C4 | 1 | 1 | yes | characteristic |
| 56 | #1084 | — | 4 | 2 | no | — |
| 57 | C22×C3×C8 | — | 9 | 1 | no | — |
| 58 | C2×C3×C4⋊C4 | — | 3 | 4 | no | — |
| 59 | C2×C3×C4⋊C4 | — | 9 | 1 | no | — |
| 60 | C2×C3×C4⋊C4 | C3⋊D3 | 1 | 1 | yes | characteristic |
| 61 | C2×C4⋊Dic3 | — | 3 | 4 | no | — |
| 62 | C22×C3⋊C8 | — | 3 | 4 | no | — |
| 63 | C23×C32 | #67 | 1 | 4 | yes | — |
| 64 | C2×C32×C4 | — | 2 | 8 | no | — |
| 65 | C2×C32×C4 | — | 4 | 4 | no | — |
| 66 | C2×C32×C4 | — | 4 | 4 | no | — |
| 67 | C2×C32×C4 | Dic6 | 1 | 4 | yes | — |
| 68 | C2×C32×C4 | C4×D3 | 1 | 4 | yes | — |
| 69 | C2×C32×C4 | C4×D3 | 1 | 4 | yes | — |
| 70 | C2×C32×C4 | D12 | 1 | 4 | yes | — |
| 71 | C2×C32×C4 | C2×Dic3 | 1 | 4 | yes | — |
| 72 | C2×C32×C4 | #67 | 1 | 4 | yes | — |
| 73 | C2×C3×Dic3 | — | 6 | 8 | no | — |
| 74 | C2×C3×Dic3 | — | 12 | 4 | no | — |
| 75 | C2×C3×Dic3 | — | 12 | 4 | no | — |
| 76 | C3×C3⋊C8 | — | 6 | 8 | no | — |
| 77 | #340 | — | 9 | 1 | no | Sylow |
| 78 | C2×H(3,𝔽3) | #95 | 1 | 1 | yes | characteristic |
| 79 | C2×H(3,𝔽3) | #102 | 1 | 1 | yes | characteristic |
| 80 | C2×H(3,𝔽3) | #102 | 1 | 1 | yes | characteristic |
| 81 | C2×H(3,𝔽3) | #103 | 1 | 1 | yes | characteristic |
| 82 | C2×H(3,𝔽3) | #103 | 1 | 1 | yes | characteristic |
| 83 | C2×H(3,𝔽3) | #106 | 1 | 1 | yes | characteristic |
| 84 | C2×H(3,𝔽3) | #107 | 1 | 1 | yes | characteristic |
| 85 | C22×C3×C4 | — | 2 | 1 | no | — |
| 86 | C22×C3×C4 | — | 3 | 4 | no | — |
| 87 | C22×C3×C4 | — | 6 | 4 | no | — |
| 88 | C22×C3×C4 | — | 18 | 1 | no | — |
| 89 | C22×C3×C4 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 90 | C22×Dic3 | — | 6 | 4 | no | — |
| 91 | C2×C3×C8 | — | 9 | 2 | no | — |
| 92 | C2×C3×C8 | — | 18 | 1 | no | — |
| 93 | C2×C3×C8 | — | 18 | 1 | no | — |
| 94 | C3×C4⋊C4 | — | 2 | 1 | no | — |
| 95 | C3×C4⋊C4 | — | 3 | 8 | no | — |
| 96 | C3×C4⋊C4 | — | 6 | 4 | no | — |
| 97 | C3×C4⋊C4 | — | 9 | 2 | no | — |
| 98 | C3×C4⋊C4 | — | 18 | 1 | no | — |
| 99 | C3×C4⋊C4 | C3⋊Dic3 | 1 | 2 | yes | — |
| 100 | C4⋊Dic3 | — | 3 | 8 | no | — |
| 101 | C4⋊Dic3 | — | 6 | 4 | no | — |
| 102 | C2×C3⋊C8 | — | 3 | 8 | no | — |
| 103 | C2×C3⋊C8 | — | 6 | 4 | no | — |
| 104 | C2×C3⋊C8 | — | 6 | 4 | no | — |
| 105 | C22×C32 | Dic3⋊C4 | 1 | 4 | yes | — |
| 106 | C22×C32 | (C2×D3)⋊C4 | 1 | 4 | yes | — |
| 107 | C22×C32 | #213 | 1 | 4 | yes | — |
| 108 | C22×C32 | #214 | 1 | 4 | yes | — |
| 109 | C22×C32 | #215 | 1 | 4 | yes | — |
| 110 | C22×C32 | #216 | 1 | 4 | yes | — |
| 111 | C22×C32 | #217 | 1 | 4 | yes | — |
| 112 | C32×C4 | — | 4 | 8 | no | — |
| 113 | C32×C4 | C4×Dic3 | 1 | 4 | yes | — |
| 114 | C32×C4 | Dic3⋊C4 | 1 | 4 | yes | — |
| 115 | C32×C4 | C4⋊Dic3 | 1 | 4 | yes | — |
| 116 | C32×C4 | (C2×D3)⋊C4 | 1 | 4 | yes | — |
| 117 | C3×Dic3 | — | 12 | 8 | no | — |
| 118 | C22×C8 | — | 9 | 1 | no | — |
| 119 | C2×C4⋊C4 | — | 9 | 1 | no | — |
| 120 | C2×C4⋊C4 | #272 | 1 | 1 | yes | characteristic |
| 121 | H(3,𝔽3) | #340 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 122 | C23×C3 | — | 3 | 4 | no | — |
| 123 | C23×C3 | #638 | 1 | 1 | yes | characteristic |
| 124 | C2×C3×C4 | — | 2 | 2 | no | — |
| 125 | C2×C3×C4 | — | 3 | 4 | no | — |
| 126 | C2×C3×C4 | — | 3 | 4 | no | — |
| 127 | C2×C3×C4 | — | 3 | 4 | no | — |
| 128 | C2×C3×C4 | — | 3 | 4 | no | — |
| 129 | C2×C3×C4 | — | 3 | 4 | no | — |
| 130 | C2×C3×C4 | — | 3 | 4 | no | — |
| 131 | C2×C3×C4 | — | 4 | 1 | no | — |
| 132 | C2×C3×C4 | — | 4 | 1 | no | — |
| 133 | C2×C3×C4 | — | 6 | 8 | no | — |
| 134 | C2×C3×C4 | — | 12 | 4 | no | — |
| 135 | C2×C3×C4 | — | 12 | 4 | no | — |
| 136 | C2×C3×C4 | — | 18 | 2 | no | — |
| 137 | C2×C3×C4 | — | 36 | 1 | no | — |
| 138 | C2×C3×C4 | — | 36 | 1 | no | — |
| 139 | C2×C3×C4 | #634 | 1 | 1 | yes | characteristic |
| 140 | C2×C3×C4 | C4×C3⋊D3 | 1 | 1 | yes | characteristic |
| 141 | C2×C3×C4 | C4×C3⋊D3 | 1 | 1 | yes | characteristic |
| 142 | C2×C3×C4 | C4⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 143 | C2×C3×C4 | C2×C3⋊Dic3 | 1 | 1 | yes | characteristic |
| 144 | C2×C3×C4 | #638 | 1 | 1 | yes | characteristic |
| 145 | C2×Dic3 | — | 6 | 8 | no | — |
| 146 | C2×Dic3 | — | 12 | 4 | no | — |
| 147 | C2×Dic3 | — | 12 | 4 | no | — |
| 148 | C3×C8 | — | 18 | 2 | no | — |
| 149 | C3⋊C8 | — | 6 | 8 | no | — |
| 150 | C2×C32 | #807 | 1 | 4 | yes | — |
| 151 | C2×C32 | #808 | 1 | 4 | yes | — |
| 152 | C2×C32 | #809 | 1 | 4 | yes | — |
| 153 | C2×C32 | #810 | 1 | 4 | yes | — |
| 154 | C2×C32 | C3⋊#95 | 1 | 4 | yes | — |
| 155 | C2×C32 | #832 | 1 | 4 | yes | — |
| 156 | C2×C32 | #835 | 1 | 4 | yes | — |
| 157 | C22×C4 | — | 2 | 1 | no | — |
| 158 | C22×C4 | — | 18 | 1 | no | — |
| 159 | C22×C4 | C2×#272 | 1 | 1 | yes | characteristic |
| 160 | C2×C8 | — | 9 | 2 | no | — |
| 161 | C2×C8 | — | 18 | 1 | no | — |
| 162 | C2×C8 | — | 18 | 1 | no | — |
| 163 | C4⋊C4 | — | 2 | 1 | no | — |
| 164 | C4⋊C4 | — | 9 | 2 | no | — |
| 165 | C4⋊C4 | — | 18 | 1 | no | — |
| 166 | C4⋊C4 | #1084 | 1 | 2 | yes | complemented |
| 167 | C22×C3 | — | 3 | 4 | no | — |
| 168 | C22×C3 | — | 3 | 4 | no | — |
| 169 | C22×C3 | — | 3 | 4 | no | — |
| 170 | C22×C3 | — | 3 | 4 | no | — |
| 171 | C22×C3 | — | 3 | 4 | no | — |
| 172 | C22×C3 | — | 3 | 4 | no | — |
| 173 | C22×C3 | — | 3 | 4 | no | — |
| 174 | C22×C3 | #3750 | 1 | 1 | yes | characteristic |
| 175 | C22×C3 | #3752 | 1 | 1 | yes | characteristic |
| 176 | C22×C3 | #3753 | 1 | 1 | yes | characteristic |
| 177 | C22×C3 | #3754 | 1 | 1 | yes | characteristic |
| 178 | C22×C3 | #3755 | 1 | 1 | yes | characteristic |
| 179 | C22×C3 | #3756 | 1 | 1 | yes | characteristic |
| 180 | C22×C3 | #3757 | 1 | 1 | yes | characteristic |
| 181 | C3×C4 | — | 3 | 4 | no | — |
| 182 | C3×C4 | — | 3 | 4 | no | — |
| 183 | C3×C4 | — | 3 | 4 | no | — |
| 184 | C3×C4 | — | 3 | 4 | no | — |
| 185 | C3×C4 | — | 4 | 2 | no | — |
| 186 | C3×C4 | — | 12 | 8 | no | — |
| 187 | C3×C4 | — | 36 | 2 | no | — |
| 188 | C3×C4 | C4×C3⋊Dic3 | 1 | 1 | yes | characteristic |
| 189 | C3×C4 | #3750 | 1 | 1 | yes | characteristic |
| 190 | C3×C4 | C4⋊(C3⋊Dic3) | 1 | 1 | yes | characteristic |
| 191 | C3×C4 | #3752 | 1 | 1 | yes | characteristic |
| 192 | Dic3 | — | 12 | 8 | no | — |
| 193 | C32 | #4529 | 1 | 4 | yes | complemented |
| 194 | C23 | #6235 | 1 | 1 | yes | characteristic |
| 195 | C2×C4 | — | 2 | 2 | no | — |
| 196 | C2×C4 | — | 4 | 1 | no | — |
| 197 | C2×C4 | — | 4 | 1 | no | — |
| 198 | C2×C4 | — | 18 | 2 | no | — |
| 199 | C2×C4 | — | 36 | 1 | no | — |
| 200 | C2×C4 | — | 36 | 1 | no | — |
| 201 | C2×C4 | #6229 | 1 | 1 | yes | characteristic |
| 202 | C2×C4 | C4×#272 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | C4×#272 | 1 | 1 | yes | characteristic |
| 204 | C2×C4 | C4⋊#272 | 1 | 1 | yes | characteristic |
| 205 | C2×C4 | C2×#1084 | 1 | 1 | yes | characteristic |
| 206 | C2×C4 | #6235 | 1 | 1 | yes | characteristic |
| 207 | C8 | — | 18 | 2 | no | — |
| 208 | C2×C3 | — | 3 | 4 | no | — |
| 209 | C2×C3 | — | 3 | 4 | no | — |
| 210 | C2×C3 | — | 3 | 4 | no | — |
| 211 | C2×C3 | — | 3 | 4 | no | — |
| 212 | C2×C3 | — | 3 | 4 | no | — |
| 213 | C2×C3 | — | 3 | 4 | no | — |
| 214 | C2×C3 | — | 3 | 4 | no | — |
| 215 | C2×C3 | #63634 | 1 | 1 | yes | characteristic |
| 216 | C2×C3 | #63635 | 1 | 1 | yes | characteristic |
| 217 | C2×C3 | #63636 | 1 | 1 | yes | characteristic |
| 218 | C2×C3 | #63637 | 1 | 1 | yes | characteristic |
| 219 | C2×C3 | #63658 | 1 | 1 | yes | characteristic |
| 220 | C2×C3 | #63659 | 1 | 1 | yes | characteristic |
| 221 | C2×C3 | #63662 | 1 | 1 | yes | characteristic |
| 222 | C22 | #88540 | 1 | 1 | yes | characteristic |
| 223 | C22 | #88542 | 1 | 1 | yes | characteristic |
| 224 | C22 | #88545 | 1 | 1 | yes | characteristic |
| 225 | C22 | #88546 | 1 | 1 | yes | characteristic |
| 226 | C22 | #88549 | 1 | 1 | yes | characteristic |
| 227 | C22 | #88550 | 1 | 1 | yes | characteristic |
| 228 | C22 | #88552 | 1 | 1 | yes | characteristic |
| 229 | C4 | — | 4 | 2 | no | — |
| 230 | C4 | — | 36 | 2 | no | — |
| 231 | C4 | C4×#1084 | 1 | 1 | yes | characteristic |
| 232 | C4 | #88540 | 1 | 1 | yes | characteristic |
| 233 | C4 | C4⋊#1084 | 1 | 1 | yes | characteristic |
| 234 | C4 | #88542 | 1 | 1 | yes | characteristic |
| 235 | C3 | — | 3 | 4 | no | — |
| 236 | C3 | #10589395 | 1 | 1 | yes | characteristic |
| 237 | C2 | #11720529 | 1 | 1 | yes | characteristic |
| 238 | C2 | #11720530 | 1 | 1 | yes | characteristic |
| 239 | C2 | #11720531 | 1 | 1 | yes | characteristic |
| 240 | C2 | #11720532 | 1 | 1 | yes | characteristic |
| 241 | C2 | #11720574 | 1 | 1 | yes | characteristic |
| 242 | C2 | #11720578 | 1 | 1 | yes | characteristic |
| 243 | C2 | #11720583 | 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