← C2×C4⋊(C3⋊(C7⋊C8)) #13056560
Order 1344.
276 conjugacy classes of subgroups.
The subgroup lattice has 140 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×C3×C42×C7 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×C3⋊(C7⋊C8) | C2 | 1 | 2 | yes | maximal |
| 3 | C4⋊(C3⋊(C7⋊C8)) | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C2×C4⋊(C7⋊C8) | — | 3 | 1 | no | maximal, Hall |
| 5 | C22×C3×C4×C7 | C4 | 1 | 1 | yes | characteristic |
| 6 | C22×C3×C4×C7 | C4 | 1 | 1 | yes | characteristic |
| 7 | C22×C3×C4×C7 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×C42×C7 | C4 | 1 | 2 | yes | — |
| 9 | C3×C42×C7 | C22 | 1 | 2 | yes | — |
| 10 | C2×C3⋊(C7⋊C8) | — | 2 | 4 | no | — |
| 11 | C2×C3⋊(C7⋊C8) | C22 | 1 | 4 | yes | — |
| 12 | C2×C42×C7 | D3 | 1 | 1 | yes | characteristic |
| 13 | C22×C7⋊C8 | — | 3 | 2 | no | — |
| 14 | C4⋊(C7⋊C8) | — | 3 | 4 | no | — |
| 15 | C2×C4⋊(C3⋊C8) | — | 7 | 1 | no | maximal, Hall |
| 16 | C23×C3×C7 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | C2×C3×C4×C7 | — | 2 | 2 | no | — |
| 18 | C2×C3×C4×C7 | C8 | 1 | 4 | yes | complemented |
| 19 | C2×C3×C4×C7 | C2×C4 | 1 | 2 | yes | — |
| 20 | C2×C3×C4×C7 | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C2×C3×C4×C7 | C2×C4 | 1 | 2 | yes | — |
| 22 | C2×C3×C4×C7 | D4 | 1 | 2 | yes | — |
| 23 | C2×C3×C4×C7 | Q8 | 1 | 2 | yes | — |
| 24 | C2×C3×C4×C7 | C23 | 1 | 1 | yes | characteristic |
| 25 | C3⋊(C7⋊C8) | — | 2 | 4 | no | — |
| 26 | C22×C4×C7 | Dic3 | 1 | 1 | yes | characteristic |
| 27 | C22×C4×C7 | Dic3 | 1 | 1 | yes | characteristic |
| 28 | C22×C4×C7 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C42×C7 | Dic3 | 1 | 2 | yes | — |
| 30 | C42×C7 | C2×D3 | 1 | 2 | yes | — |
| 31 | C2×C7⋊C8 | — | 3 | 4 | no | — |
| 32 | C2×C7⋊C8 | — | 6 | 4 | no | — |
| 33 | C2×C3×C42 | D7 | 1 | 1 | yes | characteristic |
| 34 | C22×C3⋊C8 | — | 7 | 2 | no | — |
| 35 | C4⋊(C3⋊C8) | — | 7 | 4 | no | — |
| 36 | C22×C3×C7 | C4⋊C4 | 1 | 2 | yes | — |
| 37 | C22×C3×C7 | C2×C8 | 1 | 2 | yes | — |
| 38 | C22×C3×C7 | M16 | 1 | 2 | yes | — |
| 39 | C22×C3×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 40 | C3×C4×C7 | — | 2 | 2 | no | — |
| 41 | C3×C4×C7 | C4⋊C4 | 1 | 2 | yes | — |
| 42 | C3×C4×C7 | C2×C8 | 1 | 4 | yes | complemented |
| 43 | C3×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 44 | C3×C4×C7 | C2×Q8 | 1 | 1 | yes | characteristic |
| 45 | C2×C4⋊C8 | — | 21 | 1 | no | Sylow |
| 46 | C23×C7 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 47 | C2×C4×C7 | — | 2 | 2 | no | — |
| 48 | C2×C4×C7 | C3⋊C8 | 1 | 4 | yes | complemented |
| 49 | C2×C4×C7 | Dic6 | 1 | 2 | yes | — |
| 50 | C2×C4×C7 | D12 | 1 | 2 | yes | — |
| 51 | C2×C4×C7 | C2×Dic3 | 1 | 2 | yes | — |
| 52 | C2×C4×C7 | C2×Dic3 | 1 | 2 | yes | — |
| 53 | C2×C4×C7 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 54 | C2×C4×C7 | C22×D3 | 1 | 1 | yes | characteristic |
| 55 | C7⋊C8 | — | 6 | 4 | no | — |
| 56 | C22×C3×C4 | Dic7 | 1 | 1 | yes | characteristic |
| 57 | C22×C3×C4 | Dic7 | 1 | 1 | yes | characteristic |
| 58 | C22×C3×C4 | C2×D7 | 1 | 1 | yes | characteristic |
| 59 | C3×C42 | Dic7 | 1 | 2 | yes | — |
| 60 | C3×C42 | C2×D7 | 1 | 2 | yes | — |
| 61 | C2×C3⋊C8 | — | 7 | 4 | no | — |
| 62 | C2×C3⋊C8 | — | 14 | 4 | no | — |
| 63 | C2×C3×C7 | C4⋊C8 | 1 | 4 | yes | complemented |
| 64 | C2×C3×C7 | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 65 | C2×C3×C7 | C22×C8 | 1 | 1 | yes | characteristic |
| 66 | C2×C3×C7 | C2×M16 | 1 | 1 | yes | characteristic |
| 67 | C22×C8 | — | 21 | 2 | no | — |
| 68 | C2×C42 | D21 | 1 | 1 | yes | characteristic |
| 69 | C4⋊C8 | — | 21 | 4 | no | — |
| 70 | C22×C7 | C2×C3⋊C8 | 1 | 2 | yes | — |
| 71 | C22×C7 | #208 | 1 | 2 | yes | — |
| 72 | C22×C7 | C4⋊Dic3 | 1 | 2 | yes | — |
| 73 | C22×C7 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 74 | C4×C7 | — | 2 | 2 | no | — |
| 75 | C4×C7 | C2×C3⋊C8 | 1 | 4 | yes | complemented |
| 76 | C4×C7 | C4⋊Dic3 | 1 | 2 | yes | — |
| 77 | C4×C7 | C2×Dic6 | 1 | 1 | yes | characteristic |
| 78 | C4×C7 | C2×D12 | 1 | 1 | yes | characteristic |
| 79 | C23×C3 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 80 | C2×C3×C4 | — | 2 | 2 | no | — |
| 81 | C2×C3×C4 | C7⋊C8 | 1 | 4 | yes | complemented |
| 82 | C2×C3×C4 | Dic14 | 1 | 2 | yes | — |
| 83 | C2×C3×C4 | D28 | 1 | 2 | yes | — |
| 84 | C2×C3×C4 | C2×Dic7 | 1 | 2 | yes | — |
| 85 | C2×C3×C4 | C2×Dic7 | 1 | 2 | yes | — |
| 86 | C2×C3×C4 | C2×Dic7 | 1 | 1 | yes | characteristic |
| 87 | C2×C3×C4 | C22×D7 | 1 | 1 | yes | characteristic |
| 88 | C3⋊C8 | — | 14 | 4 | no | — |
| 89 | C3×C7 | C2×C4⋊C8 | 1 | 1 | yes | characteristic, complemented, Hall |
| 90 | C22×C4 | Dic21 | 1 | 1 | yes | characteristic |
| 91 | C22×C4 | Dic21 | 1 | 1 | yes | characteristic |
| 92 | C22×C4 | C2×D21 | 1 | 1 | yes | characteristic |
| 93 | C2×C8 | — | 21 | 4 | no | — |
| 94 | C2×C8 | — | 42 | 4 | no | — |
| 95 | C42 | Dic21 | 1 | 2 | yes | — |
| 96 | C42 | C2×D21 | 1 | 2 | yes | — |
| 97 | C2×C7 | C4⋊(C3⋊C8) | 1 | 4 | yes | complemented |
| 98 | C2×C7 | C22×C3⋊C8 | 1 | 1 | yes | characteristic |
| 99 | C2×C7 | C2×#208 | 1 | 1 | yes | characteristic |
| 100 | C2×C7 | C2×C4⋊Dic3 | 1 | 1 | yes | characteristic |
| 101 | C22×C3 | C2×C7⋊C8 | 1 | 2 | yes | — |
| 102 | C22×C3 | #1136 | 1 | 2 | yes | — |
| 103 | C22×C3 | C4⋊Dic7 | 1 | 2 | yes | — |
| 104 | C22×C3 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 105 | C3×C4 | — | 2 | 2 | no | — |
| 106 | C3×C4 | C2×C7⋊C8 | 1 | 4 | yes | complemented |
| 107 | C3×C4 | C4⋊Dic7 | 1 | 2 | yes | — |
| 108 | C3×C4 | C2×Dic14 | 1 | 1 | yes | characteristic |
| 109 | C3×C4 | C2×D28 | 1 | 1 | yes | characteristic |
| 110 | C23 | C2×Dic21 | 1 | 1 | yes | characteristic |
| 111 | C2×C4 | — | 2 | 2 | no | — |
| 112 | C2×C4 | C3⋊(C7⋊C8) | 1 | 4 | yes | complemented |
| 113 | C2×C4 | Dic42 | 1 | 2 | yes | — |
| 114 | C2×C4 | D84 | 1 | 2 | yes | — |
| 115 | C2×C4 | C2×Dic21 | 1 | 2 | yes | — |
| 116 | C2×C4 | C2×Dic21 | 1 | 2 | yes | — |
| 117 | C2×C4 | C2×Dic21 | 1 | 1 | yes | characteristic |
| 118 | C2×C4 | C22×D21 | 1 | 1 | yes | characteristic |
| 119 | C8 | — | 42 | 4 | no | — |
| 120 | C7 | C2×C4⋊(C3⋊C8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 121 | C2×C3 | C4⋊(C7⋊C8) | 1 | 4 | yes | complemented |
| 122 | C2×C3 | C22×C7⋊C8 | 1 | 1 | yes | characteristic |
| 123 | C2×C3 | C2×#1136 | 1 | 1 | yes | characteristic |
| 124 | C2×C3 | C2×C4⋊Dic7 | 1 | 1 | yes | characteristic |
| 125 | C22 | C2×C3⋊(C7⋊C8) | 1 | 2 | yes | — |
| 126 | C22 | #66627 | 1 | 2 | yes | — |
| 127 | C22 | C4⋊Dic21 | 1 | 2 | yes | — |
| 128 | C22 | C22×Dic21 | 1 | 1 | yes | characteristic |
| 129 | C4 | — | 2 | 2 | no | — |
| 130 | C4 | C2×C3⋊(C7⋊C8) | 1 | 4 | yes | complemented |
| 131 | C4 | C4⋊Dic21 | 1 | 2 | yes | — |
| 132 | C4 | C2×Dic42 | 1 | 1 | yes | characteristic |
| 133 | C4 | C2×D84 | 1 | 1 | yes | characteristic |
| 134 | C3 | C2×C4⋊(C7⋊C8) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 135 | C2 | C4⋊(C3⋊(C7⋊C8)) | 1 | 4 | yes | direct product factor |
| 136 | C2 | C22×C3⋊(C7⋊C8) | 1 | 1 | yes | characteristic |
| 137 | C2 | C2×#66627 | 1 | 1 | yes | characteristic |
| 138 | C2 | C2×C4⋊Dic21 | 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