Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
576 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 576 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×#5464 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D7×H(3,𝔽4) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 3 | D7×C22⋊A4 | — | 4 | 2 | no | maximal |
| 4 | D7×C42⋊C3 | — | 4 | 3 | no | maximal |
| 5 | C7×H(3,𝔽4) | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #90374 | — | 3 | 3 | no | — |
| 7 | D7×#124 | — | 3 | 3 | no | — |
| 8 | #90285 | — | 3 | 2 | no | — |
| 9 | D7×#120 | — | 3 | 2 | no | — |
| 10 | C2×#5464 | — | 7 | 1 | no | maximal, Hall |
| 11 | C7×C22⋊A4 | — | 4 | 2 | no | — |
| 12 | C7×C42⋊C3 | — | 4 | 3 | no | — |
| 13 | C24×D7 | A4 | 1 | 2 | yes | complemented |
| 14 | C2×Q8×D7 | — | 3 | 1 | no | — |
| 15 | C2×D4×D7 | — | 3 | 3 | no | — |
| 16 | C7×#124 | — | 3 | 3 | no | — |
| 17 | C7×#120 | — | 3 | 2 | no | — |
| 18 | #6516 | — | 3 | 2 | no | — |
| 19 | #6510 | — | 3 | 3 | no | — |
| 20 | #6500 | — | 3 | 6 | no | — |
| 21 | #6499 | — | 3 | 3 | no | — |
| 22 | #6448 | — | 3 | 12 | no | — |
| 23 | #6445 | — | 3 | 6 | no | — |
| 24 | D7×C22⋊C4 | — | 3 | 6 | no | — |
| 25 | #6438 | — | 3 | 3 | no | — |
| 26 | C42×D7 | A4 | 1 | 3 | yes | complemented |
| 27 | #5464 | D7 | 1 | 1 | yes | characteristic, direct product factor |
| 28 | A4×D7 | — | 16 | 8 | no | — |
| 29 | A4×D7 | — | 16 | 1 | no | — |
| 30 | C2×H(3,𝔽4) | — | 7 | 1 | no | Sylow |
| 31 | C24×C7 | C2×A4 | 1 | 2 | yes | complemented |
| 32 | C23×D7 | — | 3 | 2 | no | — |
| 33 | C23×D7 | — | 3 | 2 | no | — |
| 34 | C23×D7 | — | 12 | 2 | no | — |
| 35 | C23×D7 | — | 12 | 2 | no | — |
| 36 | C2×C7×Q8 | — | 3 | 1 | no | — |
| 37 | C2×C7×D4 | — | 3 | 3 | no | — |
| 38 | C2×C22⋊D7 | — | 3 | 6 | no | — |
| 39 | Q8×D7 | — | 12 | 1 | no | — |
| 40 | Q8×D7 | — | 12 | 1 | no | — |
| 41 | D4×D7 | — | 12 | 3 | no | — |
| 42 | D4×D7 | — | 12 | 3 | no | — |
| 43 | C2×D28 | — | 3 | 3 | no | — |
| 44 | C2×C4×D7 | — | 3 | 3 | no | — |
| 45 | C2×Dic14 | — | 3 | 3 | no | — |
| 46 | C7×C22⋊C4 | — | 3 | 6 | no | — |
| 47 | C42×C7 | C2×A4 | 1 | 3 | yes | complemented |
| 48 | C22⋊Dic7 | — | 3 | 6 | no | — |
| 49 | (C2×D7)⋊C4 | — | 3 | 12 | no | — |
| 50 | C4×Dic7 | — | 3 | 3 | no | — |
| 51 | C2×C22⋊A4 | — | 28 | 2 | no | — |
| 52 | C2×C42⋊C3 | — | 28 | 3 | no | — |
| 53 | C7×A4 | — | 16 | 8 | no | — |
| 54 | C7×A4 | — | 16 | 1 | no | — |
| 55 | H(3,𝔽4) | — | 21 | 2 | no | — |
| 56 | H(3,𝔽4) | — | 21 | 3 | no | — |
| 57 | H(3,𝔽4) | C3×D7 | 1 | 1 | yes | characteristic, complemented |
| 58 | C2×#124 | — | 21 | 3 | no | — |
| 59 | C2×#120 | — | 21 | 2 | no | — |
| 60 | C23×C7 | — | 3 | 2 | no | — |
| 61 | C23×C7 | — | 12 | 2 | no | — |
| 62 | C22×D7 | — | 3 | 2 | no | — |
| 63 | C22×D7 | — | 4 | 8 | no | — |
| 64 | C22×D7 | — | 6 | 6 | no | — |
| 65 | C22×D7 | — | 6 | 6 | no | — |
| 66 | C22×D7 | — | 12 | 6 | no | — |
| 67 | C22×D7 | — | 12 | 8 | no | — |
| 68 | C22×D7 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 69 | C7×Q8 | — | 12 | 1 | no | — |
| 70 | C7×D4 | — | 12 | 3 | no | — |
| 71 | C2×C4×C7 | — | 3 | 3 | no | — |
| 72 | C22⋊D7 | — | 12 | 6 | no | — |
| 73 | C2×Dic7 | — | 3 | 3 | no | — |
| 74 | D28 | — | 12 | 3 | no | — |
| 75 | C4×D7 | — | 6 | 3 | no | — |
| 76 | C4×D7 | — | 6 | 3 | no | — |
| 77 | Dic14 | — | 12 | 3 | no | — |
| 78 | C22⋊A4 | — | 4 | 2 | no | — |
| 79 | C42⋊C3 | — | 4 | 3 | no | — |
| 80 | C3×D7 | — | 64 | 1 | no | — |
| 81 | C25 | — | 7 | 2 | no | — |
| 82 | C22×Q8 | — | 21 | 1 | no | — |
| 83 | C22×D4 | — | 21 | 3 | no | — |
| 84 | #124 | — | 3 | 3 | no | — |
| 85 | #124 | — | 21 | 12 | no | — |
| 86 | #124 | — | 21 | 3 | no | — |
| 87 | #124 | — | 21 | 3 | no | — |
| 88 | #124 | — | 21 | 3 | no | — |
| 89 | #120 | — | 3 | 2 | no | — |
| 90 | #120 | — | 21 | 6 | no | — |
| 91 | #120 | — | 21 | 6 | no | — |
| 92 | #120 | — | 21 | 2 | no | — |
| 93 | C2×C22⋊C4 | — | 21 | 6 | no | — |
| 94 | C2×C42 | — | 7 | 3 | no | — |
| 95 | C22×C7 | — | 4 | 8 | no | — |
| 96 | C22×C7 | — | 6 | 6 | no | — |
| 97 | C22×C7 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 98 | C2×D7 | — | 3 | 1 | no | — |
| 99 | C2×D7 | — | 3 | 1 | no | — |
| 100 | C2×D7 | — | 6 | 6 | no | — |
| 101 | C2×D7 | — | 12 | 6 | no | — |
| 102 | C2×D7 | — | 12 | 2 | no | — |
| 103 | C2×D7 | — | 12 | 8 | no | — |
| 104 | C4×C7 | — | 6 | 3 | no | — |
| 105 | Dic7 | — | 6 | 3 | no | — |
| 106 | C2×A4 | — | 112 | 1 | no | — |
| 107 | C2×A4 | — | 112 | 8 | no | — |
| 108 | C3×C7 | — | 64 | 1 | no | Hall |
| 109 | C24 | — | 21 | 2 | no | — |
| 110 | C24 | — | 21 | 2 | no | — |
| 111 | C24 | — | 84 | 2 | no | — |
| 112 | C24 | — | 84 | 2 | no | — |
| 113 | C24 | A4×D7 | 1 | 2 | yes | complemented |
| 114 | C2×Q8 | — | 3 | 1 | no | — |
| 115 | C2×Q8 | — | 21 | 3 | no | — |
| 116 | C2×Q8 | — | 84 | 1 | no | — |
| 117 | C2×Q8 | — | 84 | 1 | no | — |
| 118 | C2×D4 | — | 3 | 3 | no | — |
| 119 | C2×D4 | — | 21 | 3 | no | — |
| 120 | C2×D4 | — | 21 | 6 | no | — |
| 121 | C2×D4 | — | 84 | 3 | no | — |
| 122 | C2×D4 | — | 84 | 3 | no | — |
| 123 | C22×C4 | — | 21 | 3 | no | — |
| 124 | C22⋊C4 | — | 3 | 6 | no | — |
| 125 | C22⋊C4 | — | 21 | 12 | no | — |
| 126 | C22⋊C4 | — | 21 | 6 | no | — |
| 127 | C42 | — | 21 | 3 | no | — |
| 128 | C42 | A4×D7 | 1 | 3 | yes | complemented |
| 129 | C2×C7 | — | 3 | 1 | no | — |
| 130 | C2×C7 | — | 12 | 2 | no | — |
| 131 | D7 | — | 3 | 1 | no | — |
| 132 | D7 | — | 12 | 2 | no | — |
| 133 | D7 | #5464 | 1 | 1 | yes | characteristic, direct product factor |
| 134 | A4 | — | 16 | 8 | no | — |
| 135 | A4 | — | 16 | 1 | no | — |
| 136 | C23 | — | 3 | 2 | no | — |
| 137 | C23 | — | 7 | 1 | no | — |
| 138 | C23 | — | 12 | 2 | no | — |
| 139 | C23 | — | 21 | 2 | no | — |
| 140 | C23 | — | 28 | 8 | no | — |
| 141 | C23 | — | 42 | 6 | no | — |
| 142 | C23 | — | 42 | 6 | no | — |
| 143 | C23 | — | 84 | 8 | no | — |
| 144 | C23 | — | 84 | 6 | no | — |
| 145 | Q8 | — | 12 | 1 | no | — |
| 146 | Q8 | — | 84 | 3 | no | — |
| 147 | D4 | — | 12 | 3 | no | — |
| 148 | D4 | — | 84 | 3 | no | — |
| 149 | D4 | — | 84 | 6 | no | — |
| 150 | C2×C4 | — | 3 | 3 | no | — |
| 151 | C2×C4 | — | 21 | 3 | no | — |
| 152 | C2×C4 | — | 42 | 3 | no | — |
| 153 | C2×C4 | — | 42 | 3 | no | — |
| 154 | C7 | C2×#5464 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 155 | C2×C3 | — | 448 | 1 | no | — |
| 156 | C22 | — | 4 | 8 | no | — |
| 157 | C22 | — | 6 | 6 | no | — |
| 158 | C22 | — | 21 | 1 | no | — |
| 159 | C22 | — | 21 | 1 | no | — |
| 160 | C22 | — | 42 | 6 | no | — |
| 161 | C22 | — | 84 | 2 | no | — |
| 162 | C22 | — | 84 | 6 | no | — |
| 163 | C22 | — | 84 | 8 | no | — |
| 164 | C22 | D7×C22⋊A4 | 1 | 1 | yes | characteristic |
| 165 | C4 | — | 6 | 3 | no | — |
| 166 | C4 | — | 42 | 3 | no | — |
| 167 | C3 | — | 64 | 1 | no | Sylow |
| 168 | C2 | — | 3 | 1 | no | — |
| 169 | C2 | — | 7 | 1 | no | — |
| 170 | C2 | — | 12 | 2 | no | — |
| 171 | C2 | — | 21 | 1 | no | — |
| 172 | C2 | — | 84 | 2 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×H(3,𝔽4) | C2×C3 |
| Frattini | C22 | D7×C22⋊A4 |
| socle | C22×C7 | C2×C22⋊A4 |
| LCS | G ▷ C7×H(3,𝔽4) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#5464 | C2 | semidirect |
| 2. D7×H(3,𝔽4) | C3 | semidirect |
| 5. C7×H(3,𝔽4) | C2×C3 | semidirect |
| 13. C24×D7 | A4 | semidirect |
| 26. C42×D7 | A4 | semidirect |
| 27. #5464 | D7 | direct (N × H) |
| 31. C24×C7 | C2×A4 | semidirect |
| 47. C42×C7 | C2×A4 | semidirect |
| 57. H(3,𝔽4) | C3×D7 | semidirect |
| 113. C24 | A4×D7 | semidirect |
| 128. C42 | A4×D7 | semidirect |
| 133. D7 | #5464 | direct (N × H) |
| 154. C7 | C2×#5464 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#5464 | 2 |
| D7×H(3,𝔽4) | 3 |
| D7×C22⋊A4 | 4 |
| D7×C42⋊C3 | 4 |
| C2×#5464 | 7 |