← C3×Q8×2−1+4 #10681604
Subgroups — C3×Q8×2−1+4 #10681604
Order 768.
6850 conjugacy classes of subgroups.
The subgroup lattice has 182 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×#3503 | C2 | 1 | 30 | yes | maximal, complemented |
| 2 | C3×#3502 | C2 | 1 | 15 | yes | maximal, complemented |
| 3 | C3×Q8×#41 | C2 | 1 | 10 | yes | maximal, complemented |
| 4 | C2×C3×Q82 | C2 | 1 | 5 | yes | maximal, complemented |
| 5 | C3×C4×2−1+4 | C2 | 1 | 3 | yes | maximal |
| 6 | Q8×2−1+4 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor, Sylow |
| 7 | C2×C3×2−1+4 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×Q82 | C22 | 1 | 10 | yes | — |
| 9 | C3×Q82 | C22 | 1 | 60 | yes | — |
| 10 | C3×#557 | C22 | 1 | 30 | yes | — |
| 11 | C3×#557 | C22 | 1 | 60 | yes | — |
| 12 | C3×#554 | C22 | 1 | 90 | yes | — |
| 13 | C3×D4×Q8 | C22 | 1 | 10 | yes | — |
| 14 | C3×D4×Q8 | C22 | 1 | 60 | yes | — |
| 15 | C3×#544 | C22 | 1 | 90 | yes | — |
| 16 | C3×#533 | C22 | 1 | 90 | yes | — |
| 17 | C2×C3×C4⋊Q8 | C22 | 1 | 45 | yes | — |
| 18 | C3×#520 | C22 | 1 | 30 | yes | — |
| 19 | C3×#519 | C22 | 1 | 15 | yes | — |
| 20 | C3×C4×#41 | C22 | 1 | 30 | yes | — |
| 21 | C2×C3×C4×Q8 | C22 | 1 | 15 | yes | — |
| 22 | C2×C3×C4×Q8 | C22 | 1 | 15 | yes | — |
| 23 | #3503 | C2×C3 | 1 | 30 | yes | complemented |
| 24 | #3502 | C2×C3 | 1 | 15 | yes | complemented |
| 25 | Q8×#41 | C2×C3 | 1 | 10 | yes | complemented |
| 26 | C2×Q82 | C2×C3 | 1 | 5 | yes | complemented |
| 27 | C4×2−1+4 | C2×C3 | 1 | 3 | yes | — |
| 28 | C3×2−1+4 | Q8 | 1 | 16 | yes | direct product factor |
| 29 | C2×C3×#41 | C23 | 1 | 10 | yes | — |
| 30 | C22×C3×Q8 | C23 | 1 | 5 | yes | — |
| 31 | C22×C3×Q8 | C23 | 1 | 30 | yes | — |
| 32 | C22×C3×Q8 | C23 | 1 | 5 | yes | — |
| 33 | C3×C4⋊Q8 | C23 | 1 | 90 | yes | — |
| 34 | C3×C4⋊Q8 | C23 | 1 | 180 | yes | — |
| 35 | C3×C4⋊Q8 | C23 | 1 | 30 | yes | — |
| 36 | C3×#125 | C23 | 1 | 180 | yes | — |
| 37 | C3×C22⋊Q8 | C23 | 1 | 60 | yes | — |
| 38 | C3×C22⋊Q8 | C23 | 1 | 180 | yes | — |
| 39 | C3×C4×Q8 | C23 | 1 | 10 | yes | — |
| 40 | C3×C4×Q8 | C23 | 1 | 30 | yes | — |
| 41 | C3×C4×Q8 | C23 | 1 | 60 | yes | — |
| 42 | C3×C4×Q8 | C23 | 1 | 90 | yes | — |
| 43 | C3×C4×Q8 | C23 | 1 | 30 | yes | — |
| 44 | C3×C4×Q8 | C23 | 1 | 60 | yes | — |
| 45 | C3×C4×D4 | C23 | 1 | 30 | yes | — |
| 46 | C3×C4×D4 | C23 | 1 | 90 | yes | — |
| 47 | C3×#117 | C23 | 1 | 90 | yes | — |
| 48 | C2×C3×C4⋊C4 | C23 | 1 | 45 | yes | — |
| 49 | C2×C3×C4⋊C4 | C23 | 1 | 45 | yes | — |
| 50 | C2×C3×C42 | C23 | 1 | 45 | yes | — |
| 51 | C2×2−1+4 | C22×C3 | 1 | 1 | yes | characteristic |
| 52 | Q82 | C22×C3 | 1 | 10 | yes | — |
| 53 | Q82 | C22×C3 | 1 | 60 | yes | — |
| 54 | #557 | C22×C3 | 1 | 30 | yes | — |
| 55 | #557 | C22×C3 | 1 | 60 | yes | — |
| 56 | #554 | C22×C3 | 1 | 90 | yes | — |
| 57 | D4×Q8 | C22×C3 | 1 | 10 | yes | — |
| 58 | D4×Q8 | C22×C3 | 1 | 60 | yes | — |
| 59 | #544 | C22×C3 | 1 | 90 | yes | — |
| 60 | #533 | C22×C3 | 1 | 90 | yes | — |
| 61 | C2×C4⋊Q8 | C22×C3 | 1 | 45 | yes | — |
| 62 | #520 | C22×C3 | 1 | 30 | yes | — |
| 63 | #519 | C22×C3 | 1 | 15 | yes | — |
| 64 | C4×#41 | C22×C3 | 1 | 30 | yes | — |
| 65 | C2×C4×Q8 | C22×C3 | 1 | 15 | yes | — |
| 66 | C2×C4×Q8 | C22×C3 | 1 | 15 | yes | — |
| 67 | C3×#41 | C2×Q8 | 1 | 80 | yes | complemented |
| 68 | C2×C3×Q8 | — | 2 | 20 | no | — |
| 69 | C2×C3×Q8 | — | 8 | 30 | no | — |
| 70 | C2×C3×Q8 | C2×Q8 | 1 | 40 | yes | complemented |
| 71 | C2×C3×Q8 | C24 | 1 | 15 | yes | — |
| 72 | C2×C3×Q8 | C24 | 1 | 60 | yes | — |
| 73 | C2×C3×Q8 | C24 | 1 | 10 | yes | — |
| 74 | C2×C3×Q8 | C24 | 1 | 1 | yes | characteristic |
| 75 | C2×C3×D4 | C24 | 1 | 10 | yes | — |
| 76 | C22×C3×C4 | C24 | 1 | 15 | yes | — |
| 77 | C22×C3×C4 | C24 | 1 | 45 | yes | — |
| 78 | C22×C3×C4 | C24 | 1 | 15 | yes | — |
| 79 | C3×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 80 | C3×C4⋊C4 | C24 | 1 | 90 | yes | — |
| 81 | C3×C4⋊C4 | C24 | 1 | 90 | yes | — |
| 82 | C3×C4⋊C4 | C24 | 1 | 30 | yes | — |
| 83 | C3×C4⋊C4 | C24 | 1 | 60 | yes | — |
| 84 | C3×C22⋊C4 | C24 | 1 | 60 | yes | — |
| 85 | C3×C42 | C24 | 1 | 30 | yes | — |
| 86 | C3×C42 | C24 | 1 | 90 | yes | — |
| 87 | 2−1+4 | C3×Q8 | 1 | 16 | yes | direct product factor |
| 88 | C2×#41 | C23×C3 | 1 | 10 | yes | — |
| 89 | C22×Q8 | C23×C3 | 1 | 5 | yes | — |
| 90 | C22×Q8 | C23×C3 | 1 | 30 | yes | — |
| 91 | C22×Q8 | C23×C3 | 1 | 5 | yes | — |
| 92 | C4⋊Q8 | C23×C3 | 1 | 90 | yes | — |
| 93 | C4⋊Q8 | C23×C3 | 1 | 180 | yes | — |
| 94 | C4⋊Q8 | C23×C3 | 1 | 30 | yes | — |
| 95 | #125 | C23×C3 | 1 | 180 | yes | — |
| 96 | C22⋊Q8 | C23×C3 | 1 | 60 | yes | — |
| 97 | C22⋊Q8 | C23×C3 | 1 | 180 | yes | — |
| 98 | C4×Q8 | C23×C3 | 1 | 10 | yes | — |
| 99 | C4×Q8 | C23×C3 | 1 | 30 | yes | — |
| 100 | C4×Q8 | C23×C3 | 1 | 60 | yes | — |
| 101 | C4×Q8 | C23×C3 | 1 | 90 | yes | — |
| 102 | C4×Q8 | C23×C3 | 1 | 30 | yes | — |
| 103 | C4×Q8 | C23×C3 | 1 | 60 | yes | — |
| 104 | C4×D4 | C23×C3 | 1 | 30 | yes | — |
| 105 | C4×D4 | C23×C3 | 1 | 90 | yes | — |
| 106 | #117 | C23×C3 | 1 | 90 | yes | — |
| 107 | C2×C4⋊C4 | C23×C3 | 1 | 45 | yes | — |
| 108 | C2×C4⋊C4 | C23×C3 | 1 | 45 | yes | — |
| 109 | C2×C42 | C23×C3 | 1 | 45 | yes | — |
| 110 | C23×C3 | C25 | 1 | 5 | yes | — |
| 111 | C3×Q8 | — | 2 | 30 | no | — |
| 112 | C3×Q8 | — | 4 | 60 | no | — |
| 113 | C3×Q8 | C22×Q8 | 1 | 40 | yes | — |
| 114 | C3×Q8 | 2−1+4 | 1 | 4 | yes | direct product factor |
| 115 | C3×D4 | C22×Q8 | 1 | 40 | yes | — |
| 116 | C2×C3×C4 | — | 2 | 30 | no | — |
| 117 | C2×C3×C4 | — | 4 | 45 | no | — |
| 118 | C2×C3×C4 | C22×Q8 | 1 | 60 | yes | — |
| 119 | C2×C3×C4 | C25 | 1 | 15 | yes | — |
| 120 | C2×C3×C4 | C25 | 1 | 30 | yes | — |
| 121 | C2×C3×C4 | C25 | 1 | 10 | yes | — |
| 122 | C2×C3×C4 | C25 | 1 | 3 | yes | — |
| 123 | #41 | C2×C3×Q8 | 1 | 80 | yes | complemented |
| 124 | C2×Q8 | — | 2 | 20 | no | — |
| 125 | C2×Q8 | — | 8 | 30 | no | — |
| 126 | C2×Q8 | C2×C3×Q8 | 1 | 40 | yes | complemented |
| 127 | C2×Q8 | C24×C3 | 1 | 15 | yes | — |
| 128 | C2×Q8 | C24×C3 | 1 | 60 | yes | — |
| 129 | C2×Q8 | C24×C3 | 1 | 10 | yes | — |
| 130 | C2×Q8 | C24×C3 | 1 | 1 | yes | characteristic |
| 131 | C2×D4 | C24×C3 | 1 | 10 | yes | — |
| 132 | C22×C4 | C24×C3 | 1 | 15 | yes | — |
| 133 | C22×C4 | C24×C3 | 1 | 45 | yes | — |
| 134 | C22×C4 | C24×C3 | 1 | 15 | yes | — |
| 135 | C4⋊C4 | C24×C3 | 1 | 30 | yes | — |
| 136 | C4⋊C4 | C24×C3 | 1 | 90 | yes | — |
| 137 | C4⋊C4 | C24×C3 | 1 | 90 | yes | — |
| 138 | C4⋊C4 | C24×C3 | 1 | 30 | yes | — |
| 139 | C4⋊C4 | C24×C3 | 1 | 60 | yes | — |
| 140 | C22⋊C4 | C24×C3 | 1 | 60 | yes | — |
| 141 | C42 | C24×C3 | 1 | 30 | yes | — |
| 142 | C42 | C24×C3 | 1 | 90 | yes | — |
| 143 | C22×C3 | — | 2 | 5 | no | — |
| 144 | C22×C3 | — | 2 | 5 | no | — |
| 145 | C22×C3 | C23×Q8 | 1 | 10 | yes | — |
| 146 | C22×C3 | C26 | 1 | 1 | yes | characteristic |
| 147 | C3×C4 | — | 2 | 15 | no | — |
| 148 | C3×C4 | — | 2 | 30 | no | — |
| 149 | C3×C4 | C23×Q8 | 1 | 20 | yes | — |
| 150 | C3×C4 | C2×2−1+4 | 1 | 6 | yes | — |
| 151 | C23 | C25×C3 | 1 | 5 | yes | — |
| 152 | Q8 | — | 2 | 30 | no | — |
| 153 | Q8 | — | 4 | 60 | no | — |
| 154 | Q8 | C22×C3×Q8 | 1 | 40 | yes | — |
| 155 | Q8 | C3×2−1+4 | 1 | 4 | yes | direct product factor |
| 156 | D4 | C22×C3×Q8 | 1 | 40 | yes | — |
| 157 | C2×C4 | — | 2 | 30 | no | — |
| 158 | C2×C4 | — | 4 | 45 | no | — |
| 159 | C2×C4 | C22×C3×Q8 | 1 | 60 | yes | — |
| 160 | C2×C4 | C25×C3 | 1 | 15 | yes | — |
| 161 | C2×C4 | C25×C3 | 1 | 30 | yes | — |
| 162 | C2×C4 | C25×C3 | 1 | 10 | yes | — |
| 163 | C2×C4 | C25×C3 | 1 | 3 | yes | — |
| 164 | C2×C3 | — | 2 | 10 | no | — |
| 165 | C2×C3 | C24×Q8 | 1 | 1 | yes | characteristic |
| 166 | C2×C3 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 167 | C2×C3 | H(7,𝔽2) | 1 | 1 | yes | characteristic |
| 168 | C22 | — | 2 | 5 | no | — |
| 169 | C22 | — | 2 | 5 | no | — |
| 170 | C22 | C23×C3×Q8 | 1 | 10 | yes | — |
| 171 | C22 | C26×C3 | 1 | 1 | yes | characteristic |
| 172 | C4 | — | 2 | 15 | no | — |
| 173 | C4 | — | 2 | 30 | no | — |
| 174 | C4 | C23×C3×Q8 | 1 | 20 | yes | — |
| 175 | C4 | C2×C3×2−1+4 | 1 | 6 | yes | — |
| 176 | C3 | Q8×2−1+4 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 177 | C2 | — | 2 | 10 | no | — |
| 178 | C2 | C24×C3×Q8 | 1 | 1 | yes | characteristic |
| 179 | C2 | C22×C3×2−1+4 | 1 | 1 | yes | characteristic |
| 180 | C2 | C3×H(7,𝔽2) | 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