Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 22 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 | #127⋊ΓL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D7⋊(C42⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10620797 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D7×#127 | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Hall |
| 5 | C2×D7⋊A4 | — | 4 | 1 | no | maximal |
| 6 | C7⋊(C42⋊C3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×D4×D7 | — | 3 | 1 | no | — |
| 8 | C2×D4×D7 | — | 3 | 1 | no | — |
| 9 | C7×#127 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #6503 | — | 3 | 1 | no | — |
| 11 | #6503 | — | 3 | 1 | no | — |
| 12 | #6437 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C42×D7 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×#127⋊C3 | — | 7 | 1 | no | maximal, Hall |
| 15 | C2×C22⋊ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 16 | D7⋊A4 | — | 4 | 1 | no | — |
| 17 | D7⋊A4 | — | 4 | 1 | no | — |
| 18 | C23×D7 | — | 3 | 1 | no | — |
| 19 | C23×D7 | A4 | 1 | 1 | yes | characteristic |
| 20 | C2×C7×D4 | — | 3 | 1 | no | — |
| 21 | C2×C7×D4 | — | 3 | 1 | no | — |
| 22 | C2×C22⋊D7 | — | 3 | 2 | no | — |
| 23 | C2×C22⋊D7 | — | 3 | 1 | no | — |
| 24 | C2×C22⋊D7 | — | 3 | 1 | no | — |
| 25 | D4×D7 | — | 6 | 2 | no | — |
| 26 | D4×D7 | — | 6 | 2 | no | — |
| 27 | D4×D7 | — | 6 | 2 | no | — |
| 28 | D4×D7 | — | 6 | 2 | no | — |
| 29 | C2×D28 | — | 3 | 1 | no | — |
| 30 | C2×D28 | — | 3 | 1 | no | — |
| 31 | C2×C4×D7 | — | 3 | 1 | no | — |
| 32 | C42×C7 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 33 | C4×Dic7 | — | 3 | 1 | no | — |
| 34 | #127⋊C3 | — | 7 | 1 | no | — |
| 35 | #127⋊C3 | — | 7 | 1 | no | — |
| 36 | C2×C42⋊C3 | — | 7 | 1 | no | — |
| 37 | C22⋊ΓL(1,𝔽8) | — | 4 | 1 | no | — |
| 38 | C2×AGL(1,𝔽7) | — | 16 | 1 | no | — |
| 39 | C2×#127 | — | 7 | 1 | no | Sylow |
| 40 | C23×C7 | — | 3 | 1 | no | — |
| 41 | C23×C7 | C2×A4 | 1 | 1 | yes | characteristic |
| 42 | C22×D7 | — | 3 | 1 | no | — |
| 43 | C22×D7 | — | 6 | 1 | no | — |
| 44 | C22×D7 | — | 6 | 1 | no | — |
| 45 | C22×D7 | — | 6 | 2 | no | — |
| 46 | C22×D7 | — | 6 | 2 | no | — |
| 47 | C22×D7 | — | 6 | 1 | no | — |
| 48 | C22×D7 | — | 6 | 1 | no | — |
| 49 | C22×D7 | C2×A4 | 1 | 1 | yes | characteristic |
| 50 | C22×D7 | C2×A4 | 1 | 1 | yes | characteristic |
| 51 | C7×D4 | — | 6 | 2 | no | — |
| 52 | C7×D4 | — | 6 | 2 | no | — |
| 53 | C2×C4×C7 | — | 3 | 1 | no | — |
| 54 | C22⋊D7 | — | 6 | 2 | no | — |
| 55 | C22⋊D7 | — | 6 | 2 | no | — |
| 56 | C22⋊D7 | — | 6 | 4 | no | — |
| 57 | C2×Dic7 | — | 3 | 1 | no | — |
| 58 | D28 | — | 6 | 2 | no | — |
| 59 | D28 | — | 6 | 2 | no | — |
| 60 | C4×D7 | — | 3 | 2 | no | — |
| 61 | C4×D7 | — | 3 | 2 | no | — |
| 62 | C22×A4 | — | 28 | 1 | no | — |
| 63 | C42⋊C3 | — | 7 | 1 | no | — |
| 64 | C2×ΓL(1,𝔽8) | — | 16 | 1 | no | — |
| 65 | AGL(1,𝔽7) | — | 16 | 1 | no | — |
| 66 | AGL(1,𝔽7) | — | 16 | 1 | no | — |
| 67 | C22×D4 | — | 21 | 1 | no | — |
| 68 | C22×D4 | — | 21 | 1 | no | — |
| 69 | #127 | — | 7 | 1 | no | — |
| 70 | #127 | — | 21 | 1 | no | — |
| 71 | #127 | — | 21 | 1 | no | — |
| 72 | #127 | AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 73 | C2×C42 | — | 7 | 1 | no | — |
| 74 | C22×C7 | — | 6 | 1 | no | — |
| 75 | C22×C7 | — | 6 | 1 | no | — |
| 76 | C22×C7 | — | 6 | 2 | no | — |
| 77 | C22×C7 | C22×A4 | 1 | 1 | yes | characteristic |
| 78 | C2×D7 | — | 3 | 1 | no | — |
| 79 | C2×D7 | — | 3 | 1 | no | — |
| 80 | C2×D7 | — | 4 | 1 | no | — |
| 81 | C2×D7 | — | 6 | 1 | no | — |
| 82 | C2×D7 | — | 6 | 1 | no | — |
| 83 | C2×D7 | — | 6 | 2 | no | — |
| 84 | C2×D7 | — | 12 | 1 | no | — |
| 85 | C2×D7 | — | 12 | 1 | no | — |
| 86 | C2×D7 | — | 12 | 2 | no | — |
| 87 | C2×D7 | — | 12 | 1 | no | — |
| 88 | C4×C7 | — | 3 | 2 | no | — |
| 89 | Dic7 | — | 3 | 2 | no | — |
| 90 | C2×A4 | — | 28 | 1 | no | — |
| 91 | C2×A4 | — | 28 | 1 | no | — |
| 92 | C2×A4 | — | 28 | 1 | no | — |
| 93 | ΓL(1,𝔽8) | — | 16 | 1 | no | Hall |
| 94 | C24 | — | 7 | 1 | no | — |
| 95 | C24 | — | 21 | 1 | no | — |
| 96 | C2×D4 | — | 3 | 1 | no | — |
| 97 | C2×D4 | — | 3 | 1 | no | — |
| 98 | C2×D4 | — | 21 | 1 | no | — |
| 99 | C2×D4 | — | 21 | 1 | no | — |
| 100 | C2×D4 | — | 21 | 1 | no | — |
| 101 | C2×D4 | — | 21 | 2 | no | — |
| 102 | C2×D4 | — | 21 | 1 | no | — |
| 103 | C2×D4 | — | 42 | 2 | no | — |
| 104 | C2×D4 | — | 42 | 2 | no | — |
| 105 | C2×D4 | — | 42 | 2 | no | — |
| 106 | C2×D4 | — | 42 | 2 | no | — |
| 107 | C22×C4 | — | 21 | 1 | no | — |
| 108 | C42 | — | 21 | 1 | no | — |
| 109 | C42 | C2×AGL(1,𝔽7) | 1 | 1 | yes | characteristic, complemented |
| 110 | C2×C7 | — | 3 | 1 | no | — |
| 111 | C2×C7 | — | 4 | 1 | no | — |
| 112 | C2×C7 | — | 12 | 1 | no | — |
| 113 | D7 | — | 3 | 1 | no | — |
| 114 | D7 | — | 4 | 1 | no | — |
| 115 | D7 | — | 12 | 1 | no | — |
| 116 | D7 | #127⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 117 | C22×C3 | — | 112 | 1 | no | — |
| 118 | A4 | — | 28 | 1 | no | — |
| 119 | C23 | — | 3 | 1 | no | — |
| 120 | C23 | — | 7 | 1 | no | — |
| 121 | C23 | — | 7 | 1 | no | — |
| 122 | C23 | — | 21 | 1 | no | — |
| 123 | C23 | — | 42 | 1 | no | — |
| 124 | C23 | — | 42 | 1 | no | — |
| 125 | C23 | — | 42 | 1 | no | — |
| 126 | C23 | — | 42 | 2 | no | — |
| 127 | C23 | — | 42 | 1 | no | — |
| 128 | C23 | — | 42 | 2 | no | — |
| 129 | C23 | D7⋊A4 | 1 | 1 | yes | characteristic |
| 130 | D4 | — | 6 | 2 | no | — |
| 131 | D4 | — | 6 | 2 | no | — |
| 132 | D4 | — | 42 | 2 | no | — |
| 133 | D4 | — | 42 | 2 | no | — |
| 134 | D4 | — | 42 | 2 | no | — |
| 135 | D4 | — | 42 | 2 | no | — |
| 136 | D4 | — | 42 | 4 | no | — |
| 137 | C2×C4 | — | 3 | 1 | no | — |
| 138 | C2×C4 | — | 21 | 2 | no | — |
| 139 | C2×C4 | — | 21 | 2 | no | — |
| 140 | C2×C4 | — | 21 | 1 | no | — |
| 141 | C7 | C2×#127⋊C3 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 142 | C2×C3 | — | 112 | 1 | no | — |
| 143 | C2×C3 | — | 112 | 1 | no | — |
| 144 | C2×C3 | — | 112 | 1 | no | — |
| 145 | C22 | — | 6 | 1 | no | — |
| 146 | C22 | — | 6 | 1 | no | — |
| 147 | C22 | — | 6 | 2 | no | — |
| 148 | C22 | — | 21 | 1 | no | — |
| 149 | C22 | — | 21 | 1 | no | — |
| 150 | C22 | — | 28 | 1 | no | — |
| 151 | C22 | — | 42 | 1 | no | — |
| 152 | C22 | — | 42 | 1 | no | — |
| 153 | C22 | — | 42 | 2 | no | — |
| 154 | C22 | — | 84 | 1 | no | — |
| 155 | C22 | — | 84 | 1 | no | — |
| 156 | C22 | — | 84 | 1 | no | — |
| 157 | C22 | — | 84 | 2 | no | — |
| 158 | C22 | C2×D7⋊A4 | 1 | 1 | yes | characteristic |
| 159 | C4 | — | 3 | 2 | no | — |
| 160 | C4 | — | 21 | 2 | no | — |
| 161 | C3 | — | 112 | 1 | no | Sylow |
| 162 | C2 | — | 3 | 1 | no | — |
| 163 | C2 | — | 4 | 1 | no | — |
| 164 | C2 | — | 7 | 1 | no | — |
| 165 | C2 | — | 12 | 1 | no | — |
| 166 | C2 | — | 21 | 1 | no | — |
| 167 | C2 | — | 28 | 1 | no | — |
| 168 | C2 | — | 84 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42×C7 | C22×C3 |
| Frattini | C22 | C2×D7⋊A4 |
| Fitting | C7×#127 | C2×C3 |
| socle | C22×C7 | C22×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #127⋊ΓL(1,𝔽8) | C2 | semidirect |
| 2. D7⋊(C42⋊C3) | C2 | semidirect |
| 3. #10620797 | C2 | semidirect |
| 4. D7×#127 | C3 | semidirect |
| 6. C7⋊(C42⋊C3) | C22 | semidirect |
| 9. C7×#127 | C2×C3 | semidirect |
| 12. #6437 | C2×C3 | semidirect |
| 13. C42×D7 | C2×C3 | semidirect |
| 32. C42×C7 | C22×C3 | semidirect |
| 72. #127 | AGL(1,𝔽7) | semidirect |
| 109. C42 | C2×AGL(1,𝔽7) | semidirect |
| 116. D7 | #127⋊C3 | semidirect |
| 141. C7 | C2×#127⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #127⋊ΓL(1,𝔽8) | 2 |
| D7⋊(C42⋊C3) | 2 |
| #10620797 | 2 |
| D7×#127 | 3 |
| C2×D7⋊A4 | 4 |
| C2×#127⋊C3 | 7 |