← Q8×3−1+2⋊Q8 #13152727
Subgroups — Q8×3−1+2⋊Q8 #13152727
Order 1728.
671 conjugacy classes of subgroups.
The subgroup lattice has 214 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C4×Q8×3−1+2 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | Q8×3−1+2⋊C4 | C2 | 1 | 2 | yes | maximal |
| 3 | #63450⋊C3 | C2 | 1 | 6 | yes | maximal |
| 4 | #63431⋊C3 | C2 | 1 | 3 | yes | maximal |
| 5 | C4×3−1+2⋊Q8 | C2 | 1 | 3 | yes | maximal |
| 6 | C3×Q8×Dic6 | — | 3 | 1 | no | maximal |
| 7 | Q8×Dic18 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C2×Q8×3−1+2 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×3−1+2⋊Q8 | C22 | 1 | 6 | yes | — |
| 10 | C2×3−1+2⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4⋊C4×3−1+2 | C22 | 1 | 3 | yes | — |
| 12 | C42×3−1+2 | C22 | 1 | 3 | yes | — |
| 13 | C4⋊(3−1+2⋊C4) | C22 | 1 | 6 | yes | — |
| 14 | C4⋊(3−1+2⋊C4) | C22 | 1 | 3 | yes | — |
| 15 | (3−1+2⋊C4)⋊C4 | C22 | 1 | 6 | yes | — |
| 16 | C4×3−1+2⋊C4 | C22 | 1 | 6 | yes | — |
| 17 | C32×C4×Q8 | D3 | 1 | 1 | yes | characteristic |
| 18 | C3×Q8×Dic3 | — | 3 | 2 | no | — |
| 19 | C3×#888 | — | 3 | 6 | no | — |
| 20 | C3×#869 | — | 3 | 3 | no | — |
| 21 | C3×C4×Dic6 | — | 3 | 3 | no | — |
| 22 | C4×Q8×C9 | — | 2 | 1 | no | — |
| 23 | C4×Q8×C9 | C2×C3 | 1 | 1 | yes | characteristic |
| 24 | Q8×Dic9 | C2×C3 | 1 | 2 | yes | — |
| 25 | #63450 | C2×C3 | 1 | 6 | yes | — |
| 26 | #63431 | C2×C3 | 1 | 3 | yes | — |
| 27 | C4×Dic18 | C2×C3 | 1 | 3 | yes | — |
| 28 | Q8×3−1+2 | Q8 | 1 | 4 | yes | complemented |
| 29 | C2×C4×3−1+2 | C23 | 1 | 3 | yes | — |
| 30 | C2×C4×3−1+2 | C23 | 1 | 1 | yes | characteristic |
| 31 | C2×C4×3−1+2 | C23 | 1 | 3 | yes | — |
| 32 | C2×3−1+2⋊C4 | C23 | 1 | 6 | yes | — |
| 33 | C2×3−1+2⋊C4 | C23 | 1 | 2 | yes | — |
| 34 | 3−1+2⋊Q8 | — | 4 | 6 | no | — |
| 35 | 3−1+2⋊Q8 | Q8 | 1 | 4 | yes | direct product factor |
| 36 | C3×Q82 | — | 9 | 1 | no | — |
| 37 | Q8×Dic6 | — | 3 | 1 | no | — |
| 38 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 39 | C2×C3×Dic6 | — | 3 | 1 | no | — |
| 40 | C2×C3×Dic6 | — | 3 | 6 | no | — |
| 41 | C32×C4⋊C4 | C2×D3 | 1 | 3 | yes | — |
| 42 | C32×C42 | C2×D3 | 1 | 3 | yes | — |
| 43 | C3×C4⋊Dic3 | — | 3 | 6 | no | — |
| 44 | C3×C4⋊Dic3 | — | 3 | 3 | no | — |
| 45 | C3×Dic3⋊C4 | — | 3 | 6 | no | — |
| 46 | C3×C4×Dic3 | — | 3 | 6 | no | — |
| 47 | C2×Q8×C9 | — | 2 | 1 | no | — |
| 48 | C2×Q8×C9 | C22×C3 | 1 | 1 | yes | characteristic |
| 49 | C2×Dic18 | C22×C3 | 1 | 6 | yes | — |
| 50 | C2×Dic18 | C22×C3 | 1 | 1 | yes | characteristic |
| 51 | C9×C4⋊C4 | — | 2 | 3 | no | — |
| 52 | C9×C4⋊C4 | C22×C3 | 1 | 3 | yes | — |
| 53 | C42×C9 | — | 2 | 3 | no | — |
| 54 | C42×C9 | C22×C3 | 1 | 3 | yes | — |
| 55 | C4⋊Dic9 | C22×C3 | 1 | 6 | yes | — |
| 56 | C4⋊Dic9 | C22×C3 | 1 | 3 | yes | — |
| 57 | Dic9⋊C4 | C22×C3 | 1 | 6 | yes | — |
| 58 | C4×Dic9 | C22×C3 | 1 | 6 | yes | — |
| 59 | C22×3−1+2 | C24 | 1 | 1 | yes | characteristic |
| 60 | C4×3−1+2 | — | 2 | 3 | no | — |
| 61 | C4×3−1+2 | C2×Q8 | 1 | 6 | yes | — |
| 62 | C4×3−1+2 | C2×Q8 | 1 | 2 | yes | — |
| 63 | 3−1+2⋊C4 | — | 2 | 6 | no | — |
| 64 | 3−1+2⋊C4 | C2×Q8 | 1 | 4 | yes | — |
| 65 | C3×C4⋊Q8 | — | 9 | 3 | no | — |
| 66 | C3×C4⋊Q8 | — | 9 | 6 | no | — |
| 67 | C3×C4×Q8 | — | 3 | 1 | no | — |
| 68 | C3×C4×Q8 | — | 9 | 3 | no | — |
| 69 | C3×C4×Q8 | — | 9 | 2 | no | — |
| 70 | C3×C4×Q8 | C3×D3 | 1 | 1 | yes | characteristic |
| 71 | Q8×Dic3 | — | 3 | 2 | no | — |
| 72 | #888 | — | 3 | 6 | no | — |
| 73 | #869 | — | 3 | 3 | no | — |
| 74 | C4×Dic6 | — | 3 | 3 | no | — |
| 75 | C32×Q8 | Dic6 | 1 | 4 | yes | — |
| 76 | C2×C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 77 | C2×C32×C4 | C22×D3 | 1 | 3 | yes | — |
| 78 | C2×C32×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 79 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 80 | C2×C3×Dic3 | — | 3 | 6 | no | — |
| 81 | C3×Dic6 | — | 3 | 4 | no | — |
| 82 | C3×Dic6 | — | 12 | 6 | no | — |
| 83 | Q8×C9 | — | 2 | 4 | no | — |
| 84 | Q8×C9 | C3×Q8 | 1 | 4 | yes | complemented |
| 85 | C2×C4×C9 | — | 2 | 1 | no | — |
| 86 | C2×C4×C9 | — | 2 | 3 | no | — |
| 87 | C2×C4×C9 | — | 2 | 3 | no | — |
| 88 | C2×C4×C9 | C23×C3 | 1 | 3 | yes | — |
| 89 | C2×C4×C9 | C23×C3 | 1 | 3 | yes | — |
| 90 | C2×C4×C9 | C23×C3 | 1 | 1 | yes | characteristic |
| 91 | C2×Dic9 | C23×C3 | 1 | 6 | yes | — |
| 92 | C2×Dic9 | C23×C3 | 1 | 2 | yes | — |
| 93 | Dic18 | — | 4 | 6 | no | — |
| 94 | Dic18 | C3×Q8 | 1 | 4 | yes | complemented |
| 95 | Q82 | — | 9 | 1 | no | Sylow |
| 96 | C2×3−1+2 | C22×Q8 | 1 | 1 | yes | characteristic |
| 97 | C2×3−1+2 | C22×Q8 | 1 | 1 | yes | characteristic |
| 98 | C2×3−1+2 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 99 | C2×C3×Q8 | — | 3 | 1 | no | — |
| 100 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 101 | C2×C3×Q8 | — | 9 | 6 | no | — |
| 102 | C2×C3×Q8 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 103 | C2×Dic6 | — | 3 | 1 | no | — |
| 104 | C2×Dic6 | — | 3 | 6 | no | — |
| 105 | C3×C4⋊C4 | — | 3 | 3 | no | — |
| 106 | C3×C4⋊C4 | — | 9 | 6 | no | — |
| 107 | C3×C4⋊C4 | — | 9 | 3 | no | — |
| 108 | C3×C4⋊C4 | — | 9 | 6 | no | — |
| 109 | C3×C4⋊C4 | C2×C3×D3 | 1 | 3 | yes | — |
| 110 | C3×C42 | — | 3 | 3 | no | — |
| 111 | C3×C42 | — | 9 | 6 | no | — |
| 112 | C3×C42 | C2×C3×D3 | 1 | 3 | yes | — |
| 113 | C4⋊Dic3 | — | 3 | 3 | no | — |
| 114 | C4⋊Dic3 | — | 3 | 6 | no | — |
| 115 | Dic3⋊C4 | — | 3 | 6 | no | — |
| 116 | C4×Dic3 | — | 3 | 6 | no | — |
| 117 | C22×C32 | C23×D3 | 1 | 1 | yes | characteristic |
| 118 | C32×C4 | — | 2 | 3 | no | — |
| 119 | C32×C4 | C2×Dic6 | 1 | 6 | yes | — |
| 120 | C32×C4 | D3×Q8 | 1 | 2 | yes | — |
| 121 | C3×Dic3 | — | 3 | 4 | no | — |
| 122 | C3×Dic3 | — | 6 | 6 | no | — |
| 123 | C22×C9 | — | 2 | 1 | no | — |
| 124 | C22×C9 | C24×C3 | 1 | 1 | yes | characteristic |
| 125 | C4×C9 | — | 2 | 3 | no | — |
| 126 | C4×C9 | — | 2 | 6 | no | — |
| 127 | C4×C9 | — | 2 | 2 | no | — |
| 128 | C4×C9 | — | 4 | 3 | no | — |
| 129 | C4×C9 | C2×C3×Q8 | 1 | 6 | yes | — |
| 130 | C4×C9 | C2×C3×Q8 | 1 | 2 | yes | — |
| 131 | Dic9 | — | 2 | 6 | no | — |
| 132 | Dic9 | C2×C3×Q8 | 1 | 4 | yes | — |
| 133 | C4⋊Q8 | — | 9 | 6 | no | — |
| 134 | C4⋊Q8 | — | 9 | 3 | no | — |
| 135 | C4×Q8 | — | 9 | 2 | no | — |
| 136 | C4×Q8 | — | 9 | 3 | no | — |
| 137 | C4×Q8 | AGL(1,ℤ/9) | 1 | 1 | yes | characteristic |
| 138 | 3−1+2 | Q82 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 139 | C3×Q8 | — | 3 | 4 | no | — |
| 140 | C3×Q8 | — | 9 | 4 | no | — |
| 141 | C3×Q8 | — | 36 | 6 | no | — |
| 142 | C3×Q8 | C3×Dic6 | 1 | 4 | yes | — |
| 143 | C2×C3×C4 | — | 3 | 1 | no | — |
| 144 | C2×C3×C4 | — | 3 | 3 | no | — |
| 145 | C2×C3×C4 | — | 3 | 3 | no | — |
| 146 | C2×C3×C4 | — | 9 | 2 | no | — |
| 147 | C2×C3×C4 | — | 9 | 6 | no | — |
| 148 | C2×C3×C4 | C22×C3×D3 | 1 | 3 | yes | — |
| 149 | C2×C3×C4 | C22×C3×D3 | 1 | 3 | yes | — |
| 150 | C2×C3×C4 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 151 | C2×Dic3 | — | 3 | 2 | no | — |
| 152 | C2×Dic3 | — | 3 | 6 | no | — |
| 153 | Dic6 | — | 3 | 4 | no | — |
| 154 | Dic6 | — | 12 | 6 | no | — |
| 155 | C2×C32 | C22×Dic6 | 1 | 1 | yes | characteristic |
| 156 | C2×C32 | C2×D3×Q8 | 1 | 1 | yes | characteristic |
| 157 | C2×C32 | #1009 | 1 | 1 | yes | characteristic |
| 158 | C2×C9 | — | 2 | 1 | no | — |
| 159 | C2×C9 | — | 2 | 1 | no | — |
| 160 | C2×C9 | — | 2 | 1 | no | — |
| 161 | C2×C9 | C22×C3×Q8 | 1 | 1 | yes | characteristic |
| 162 | C2×C9 | C22×C3×Q8 | 1 | 1 | yes | characteristic |
| 163 | C2×C9 | C3×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 164 | C2×Q8 | — | 9 | 1 | no | — |
| 165 | C2×Q8 | — | 9 | 6 | no | — |
| 166 | C2×Q8 | C2×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic |
| 167 | C4⋊C4 | — | 9 | 6 | no | — |
| 168 | C4⋊C4 | — | 9 | 6 | no | — |
| 169 | C4⋊C4 | — | 9 | 3 | no | — |
| 170 | C4⋊C4 | C2×AGL(1,ℤ/9) | 1 | 3 | yes | — |
| 171 | C42 | — | 9 | 6 | no | — |
| 172 | C42 | C2×AGL(1,ℤ/9) | 1 | 3 | yes | — |
| 173 | C22×C3 | — | 3 | 1 | no | — |
| 174 | C22×C3 | C23×C3×D3 | 1 | 1 | yes | characteristic |
| 175 | C3×C4 | — | 2 | 3 | no | — |
| 176 | C3×C4 | — | 3 | 6 | no | — |
| 177 | C3×C4 | — | 3 | 2 | no | — |
| 178 | C3×C4 | — | 6 | 3 | no | — |
| 179 | C3×C4 | — | 9 | 4 | no | — |
| 180 | C3×C4 | — | 18 | 6 | no | — |
| 181 | C3×C4 | C2×C3×Dic6 | 1 | 6 | yes | — |
| 182 | C3×C4 | C3×D3×Q8 | 1 | 2 | yes | — |
| 183 | Dic3 | — | 3 | 4 | no | — |
| 184 | Dic3 | — | 6 | 6 | no | — |
| 185 | C32 | Q8×Dic6 | 1 | 1 | yes | characteristic |
| 186 | C9 | — | 2 | 1 | no | — |
| 187 | C9 | C3×Q82 | 1 | 1 | yes | characteristic, complemented |
| 188 | Q8 | — | 9 | 4 | no | — |
| 189 | Q8 | — | 36 | 6 | no | — |
| 190 | Q8 | 3−1+2⋊Q8 | 1 | 4 | yes | direct product factor |
| 191 | C2×C4 | — | 9 | 2 | no | — |
| 192 | C2×C4 | — | 9 | 6 | no | — |
| 193 | C2×C4 | C22×AGL(1,ℤ/9) | 1 | 3 | yes | — |
| 194 | C2×C4 | C22×AGL(1,ℤ/9) | 1 | 3 | yes | — |
| 195 | C2×C4 | C22×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic |
| 196 | C2×C3 | — | 3 | 1 | no | — |
| 197 | C2×C3 | — | 3 | 1 | no | — |
| 198 | C2×C3 | — | 3 | 1 | no | — |
| 199 | C2×C3 | C22×C3×Dic6 | 1 | 1 | yes | characteristic |
| 200 | C2×C3 | C2×C3×D3×Q8 | 1 | 1 | yes | characteristic |
| 201 | C2×C3 | C3×#1009 | 1 | 1 | yes | characteristic |
| 202 | C22 | C23×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic |
| 203 | C4 | — | 2 | 3 | no | — |
| 204 | C4 | — | 9 | 4 | no | — |
| 205 | C4 | — | 18 | 6 | no | — |
| 206 | C4 | C2×3−1+2⋊Q8 | 1 | 6 | yes | — |
| 207 | C4 | Q8×AGL(1,ℤ/9) | 1 | 2 | yes | — |
| 208 | C3 | — | 3 | 1 | no | — |
| 209 | C3 | C3×Q8×Dic6 | 1 | 1 | yes | characteristic |
| 210 | C2 | C22×3−1+2⋊Q8 | 1 | 1 | yes | characteristic |
| 211 | C2 | C2×Q8×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic |
| 212 | C2 | #11722641 | 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