Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
174 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×D4×A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊(C7⋊S4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7⋊(A4⋊C8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #89989 | — | 3 | 1 | no | maximal, Hall |
| 5 | C22×C7×A4 | — | 2 | 1 | no | — |
| 6 | C2×C7⋊S4 | — | 2 | 1 | no | — |
| 7 | C4×C7×A4 | C22 | 1 | 1 | yes | characteristic |
| 8 | #66632 | — | 4 | 1 | no | maximal |
| 9 | C22×C7×D4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×#1141 | — | 3 | 1 | no | — |
| 11 | C2×#1141 | — | 3 | 1 | no | — |
| 12 | #6493 | — | 3 | 1 | no | — |
| 13 | #6406 | — | 3 | 1 | no | — |
| 14 | #6406 | — | 3 | 1 | no | — |
| 15 | C22⋊(C7⋊C8) | — | 3 | 1 | no | — |
| 16 | #5415 | — | 7 | 1 | no | maximal, Hall |
| 17 | C2×C7×A4 | — | 4 | 1 | no | — |
| 18 | C2×C7×A4 | D4 | 1 | 1 | yes | characteristic |
| 19 | C7⋊S4 | — | 4 | 1 | no | — |
| 20 | C3×C7×D4 | — | 4 | 1 | no | — |
| 21 | D84 | — | 4 | 1 | no | — |
| 22 | C3⋊(C7⋊C8) | — | 4 | 1 | no | — |
| 23 | C24×C7 | — | 2 | 1 | no | — |
| 24 | C2×C7×D4 | — | 3 | 1 | no | — |
| 25 | C2×C7×D4 | — | 3 | 1 | no | — |
| 26 | C2×C7×D4 | — | 6 | 1 | no | — |
| 27 | C22×C4×C7 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | C2×C22⋊D7 | — | 6 | 1 | no | — |
| 29 | C2×D28 | — | 3 | 1 | no | — |
| 30 | #1141 | — | 6 | 2 | no | — |
| 31 | #1141 | — | 6 | 1 | no | — |
| 32 | #1141 | — | 6 | 1 | no | — |
| 33 | (C2×D7)⋊C4 | — | 6 | 1 | no | — |
| 34 | C4⋊Dic7 | — | 3 | 1 | no | — |
| 35 | C2×C7⋊C8 | — | 3 | 1 | no | — |
| 36 | C2×C7⋊C8 | — | 3 | 1 | no | — |
| 37 | D4×A4 | D7 | 1 | 1 | yes | characteristic, complemented |
| 38 | C4⋊S4 | — | 7 | 1 | no | — |
| 39 | A4⋊C8 | — | 7 | 1 | no | — |
| 40 | C22×C3×C7 | — | 8 | 1 | no | — |
| 41 | C2×D21 | — | 8 | 1 | no | — |
| 42 | C7×A4 | D8 | 1 | 1 | yes | characteristic, complemented |
| 43 | C3×C4×C7 | — | 4 | 1 | no | — |
| 44 | #447 | — | 21 | 1 | no | Sylow |
| 45 | C23×C7 | — | 4 | 1 | no | — |
| 46 | C23×C7 | — | 6 | 1 | no | — |
| 47 | C23×C7 | — | 6 | 1 | no | — |
| 48 | C23×C7 | — | 12 | 1 | no | — |
| 49 | C23×C7 | #67 | 1 | 1 | yes | characteristic |
| 50 | C22×D7 | — | 6 | 1 | no | — |
| 51 | C7×D4 | — | 3 | 2 | no | — |
| 52 | C7×D4 | — | 3 | 1 | no | — |
| 53 | C7×D4 | — | 6 | 1 | no | — |
| 54 | C7×D4 | S4 | 1 | 1 | yes | characteristic, complemented |
| 55 | C2×C4×C7 | — | 3 | 1 | no | — |
| 56 | C2×C4×C7 | — | 3 | 1 | no | — |
| 57 | C22⋊D7 | — | 12 | 1 | no | — |
| 58 | C22⋊D7 | — | 12 | 1 | no | — |
| 59 | C2×Dic7 | — | 6 | 1 | no | — |
| 60 | D28 | — | 6 | 1 | no | — |
| 61 | D28 | — | 6 | 1 | no | — |
| 62 | C7⋊C8 | — | 6 | 1 | no | — |
| 63 | C7⋊C8 | — | 6 | 1 | no | — |
| 64 | C22×A4 | — | 2 | 1 | no | — |
| 65 | C2×S4 | — | 14 | 1 | no | — |
| 66 | C4×A4 | C2×D7 | 1 | 1 | yes | characteristic |
| 67 | #213 | — | 28 | 1 | no | — |
| 68 | C2×C3×C7 | — | 4 | 1 | no | — |
| 69 | C2×C3×C7 | — | 16 | 1 | no | — |
| 70 | D21 | — | 16 | 1 | no | — |
| 71 | C22×D4 | D21 | 1 | 1 | yes | characteristic, complemented |
| 72 | C2×D8 | — | 21 | 1 | no | — |
| 73 | C2×D8 | — | 21 | 1 | no | — |
| 74 | #121 | — | 21 | 1 | no | — |
| 75 | #102 | — | 21 | 1 | no | — |
| 76 | #102 | — | 21 | 1 | no | — |
| 77 | C22⋊C8 | — | 21 | 1 | no | — |
| 78 | C22×C7 | — | 2 | 1 | no | — |
| 79 | C22×C7 | — | 3 | 1 | no | — |
| 80 | C22×C7 | — | 3 | 1 | no | — |
| 81 | C22×C7 | — | 6 | 1 | no | — |
| 82 | C22×C7 | — | 12 | 1 | no | — |
| 83 | C22×C7 | — | 12 | 1 | no | — |
| 84 | C22×C7 | — | 12 | 1 | no | — |
| 85 | C22×C7 | — | 12 | 1 | no | — |
| 86 | C22×C7 | #213 | 1 | 1 | yes | characteristic, complemented |
| 87 | C2×D7 | — | 12 | 1 | no | — |
| 88 | C2×D7 | — | 12 | 1 | no | — |
| 89 | C2×D7 | — | 12 | 1 | no | — |
| 90 | C4×C7 | — | 3 | 1 | no | — |
| 91 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 92 | Dic7 | — | 12 | 1 | no | — |
| 93 | C2×A4 | — | 4 | 1 | no | — |
| 94 | C2×A4 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 95 | S4 | — | 28 | 1 | no | — |
| 96 | C3×D4 | — | 4 | 1 | no | — |
| 97 | D12 | — | 28 | 1 | no | — |
| 98 | C3⋊C8 | — | 28 | 1 | no | — |
| 99 | C3×C7 | — | 4 | 1 | no | Hall |
| 100 | C24 | — | 2 | 1 | no | — |
| 101 | C2×D4 | — | 3 | 1 | no | — |
| 102 | C2×D4 | — | 3 | 1 | no | — |
| 103 | C2×D4 | — | 6 | 1 | no | — |
| 104 | C2×D4 | — | 21 | 1 | no | — |
| 105 | C2×D4 | — | 42 | 1 | no | — |
| 106 | C22×C4 | C2×D21 | 1 | 1 | yes | characteristic |
| 107 | D8 | — | 42 | 2 | no | — |
| 108 | D8 | — | 42 | 1 | no | — |
| 109 | D8 | — | 42 | 1 | no | — |
| 110 | C2×C8 | — | 21 | 1 | no | — |
| 111 | C2×C8 | — | 21 | 1 | no | — |
| 112 | C4⋊C4 | — | 21 | 1 | no | — |
| 113 | C22⋊C4 | — | 42 | 1 | no | — |
| 114 | C2×C7 | — | 3 | 1 | no | — |
| 115 | C2×C7 | — | 3 | 1 | no | — |
| 116 | C2×C7 | — | 4 | 1 | no | — |
| 117 | C2×C7 | — | 12 | 1 | no | — |
| 118 | C2×C7 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 119 | D7 | — | 24 | 1 | no | — |
| 120 | C22×C3 | — | 8 | 1 | no | — |
| 121 | C2×D3 | — | 56 | 1 | no | — |
| 122 | A4 | #1141 | 1 | 1 | yes | characteristic, complemented |
| 123 | C3×C4 | — | 4 | 1 | no | — |
| 124 | C23 | — | 4 | 1 | no | — |
| 125 | C23 | — | 6 | 1 | no | — |
| 126 | C23 | — | 6 | 1 | no | — |
| 127 | C23 | — | 12 | 1 | no | — |
| 128 | C23 | — | 42 | 1 | no | — |
| 129 | C23 | #4268 | 1 | 1 | yes | characteristic |
| 130 | D4 | — | 3 | 2 | no | — |
| 131 | D4 | — | 3 | 1 | no | — |
| 132 | D4 | — | 6 | 1 | no | — |
| 133 | D4 | — | 42 | 1 | no | — |
| 134 | D4 | — | 42 | 1 | no | — |
| 135 | D4 | — | 84 | 1 | no | — |
| 136 | D4 | — | 84 | 1 | no | — |
| 137 | D4 | C7⋊S4 | 1 | 1 | yes | characteristic, complemented |
| 138 | C2×C4 | — | 3 | 1 | no | — |
| 139 | C2×C4 | — | 3 | 1 | no | — |
| 140 | C2×C4 | — | 42 | 1 | no | — |
| 141 | C8 | — | 42 | 1 | no | — |
| 142 | C8 | — | 42 | 1 | no | — |
| 143 | C7 | #5415 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 144 | C2×C3 | — | 4 | 1 | no | — |
| 145 | C2×C3 | — | 16 | 1 | no | — |
| 146 | D3 | — | 112 | 1 | no | — |
| 147 | C22 | — | 2 | 1 | no | — |
| 148 | C22 | — | 3 | 1 | no | — |
| 149 | C22 | — | 3 | 1 | no | — |
| 150 | C22 | — | 6 | 1 | no | — |
| 151 | C22 | — | 12 | 1 | no | — |
| 152 | C22 | — | 12 | 1 | no | — |
| 153 | C22 | — | 12 | 1 | no | — |
| 154 | C22 | — | 12 | 1 | no | — |
| 155 | C22 | — | 84 | 1 | no | — |
| 156 | C22 | — | 84 | 1 | no | — |
| 157 | C22 | — | 84 | 1 | no | — |
| 158 | C22 | #66632 | 1 | 1 | yes | characteristic, complemented |
| 159 | C4 | — | 3 | 1 | no | — |
| 160 | C4 | — | 84 | 1 | no | — |
| 161 | C4 | C2×C7⋊S4 | 1 | 1 | yes | characteristic |
| 162 | C3 | — | 4 | 1 | no | Sylow |
| 163 | C2 | — | 3 | 1 | no | — |
| 164 | C2 | — | 3 | 1 | no | — |
| 165 | C2 | — | 4 | 1 | no | — |
| 166 | C2 | — | 12 | 1 | no | — |
| 167 | C2 | — | 168 | 1 | no | — |
| 168 | C2 | #10621506 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10621506 |
| commutator, cosocle | C4×C7×A4 | C22 |
| Frattini | C4 | C2×C7⋊S4 |
| Fitting | C22×C7×D4 | D3 |
| socle | C23×C7 | #67 |
| hypercenter | D4 | C7⋊S4 |
| hypocenter | C7×A4 | D8 |
| LCS | G ▷ C4×C7×A4 ▷ C2×C7×A4 ▷ C7×A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×D4×A4 | C2 | semidirect |
| 2. C4⋊(C7⋊S4) | C2 | semidirect |
| 3. C7⋊(A4⋊C8) | C2 | semidirect |
| 9. C22×C7×D4 | D3 | semidirect |
| 37. D4×A4 | D7 | semidirect |
| 42. C7×A4 | D8 | semidirect |
| 54. C7×D4 | S4 | semidirect |
| 71. C22×D4 | D21 | semidirect |
| 86. C22×C7 | #213 | semidirect |
| 122. A4 | #1141 | semidirect |
| 137. D4 | C7⋊S4 | semidirect |
| 143. C7 | #5415 | semidirect |
| 158. C22 | #66632 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×D4×A4 | 2 |
| C4⋊(C7⋊S4) | 2 |
| C7⋊(A4⋊C8) | 2 |
| #89989 | 3 |
| #66632 | 4 |
| #5415 | 7 |