Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
292 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 43 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×#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10621504 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #10621503 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #90526 | — | 3 | 1 | no | maximal, Hall |
| 5 | #66732 | — | 4 | 1 | no | maximal |
| 6 | C7×#41⋊C3 | C22 | 1 | 3 | yes | complemented |
| 7 | C7⋊GL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 8 | C7⋊#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | C7×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #6553 | — | 3 | 1 | no | — |
| 11 | #6513 | — | 3 | 3 | no | — |
| 12 | #6512 | — | 3 | 3 | no | — |
| 13 | #6511 | — | 3 | 1 | no | — |
| 14 | #6495 | — | 3 | 3 | no | — |
| 15 | C2×#1141 | — | 3 | 3 | no | — |
| 16 | #5925 | — | 7 | 1 | no | maximal, Hall |
| 17 | C3×C7×Q8 | — | 4 | 1 | no | — |
| 18 | D84 | — | 4 | 3 | no | — |
| 19 | C4×D21 | — | 4 | 3 | no | — |
| 20 | C7×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 21 | C7×#41 | — | 3 | 1 | no | — |
| 22 | C7×#41 | C2×D3 | 1 | 3 | yes | — |
| 23 | C2×C7×D4 | — | 3 | 3 | no | — |
| 24 | #1161 | — | 6 | 1 | no | — |
| 25 | D4×D7 | — | 6 | 3 | no | — |
| 26 | #1157 | — | 3 | 3 | no | — |
| 27 | C2×D28 | — | 3 | 3 | no | — |
| 28 | #1144 | — | 3 | 1 | no | — |
| 29 | #1143 | — | 3 | 3 | no | — |
| 30 | #1142 | — | 3 | 3 | no | — |
| 31 | #1141 | — | 3 | 3 | no | — |
| 32 | #1141 | — | 3 | 6 | no | — |
| 33 | #1136 | — | 3 | 3 | no | — |
| 34 | C2×C7⋊C8 | — | 3 | 3 | no | — |
| 35 | #994 | D7 | 1 | 1 | yes | characteristic, complemented |
| 36 | #986 | — | 7 | 3 | no | — |
| 37 | #985 | — | 7 | 3 | no | — |
| 38 | C2×D21 | — | 4 | 3 | no | — |
| 39 | C3×C4×C7 | — | 4 | 3 | no | — |
| 40 | Dic21 | — | 4 | 1 | no | — |
| 41 | #576 | — | 21 | 1 | no | Sylow |
| 42 | C23×C7 | — | 6 | 1 | no | — |
| 43 | C22×D7 | — | 6 | 3 | no | — |
| 44 | C7×Q8 | S4 | 1 | 1 | yes | characteristic |
| 45 | C7×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 46 | C7×D4 | — | 3 | 3 | no | — |
| 47 | C7×D4 | — | 3 | 3 | no | — |
| 48 | C2×C4×C7 | — | 3 | 3 | no | — |
| 49 | C22⋊D7 | — | 6 | 3 | no | — |
| 50 | D28 | — | 3 | 3 | no | — |
| 51 | D28 | — | 6 | 3 | no | — |
| 52 | C4×D7 | — | 6 | 3 | no | — |
| 53 | Dic14 | — | 3 | 1 | no | — |
| 54 | C7⋊C8 | — | 3 | 1 | no | — |
| 55 | C7⋊C8 | — | 3 | 3 | no | — |
| 56 | #239 | — | 28 | 1 | no | — |
| 57 | #41⋊C3 | C2×D7 | 1 | 3 | yes | complemented |
| 58 | GL(2,𝔽3) | — | 7 | 3 | no | — |
| 59 | #226 | — | 7 | 1 | no | — |
| 60 | C2×C3×C7 | — | 4 | 1 | no | — |
| 61 | D21 | — | 8 | 3 | no | — |
| 62 | H(5,𝔽2) | — | 21 | 1 | no | — |
| 63 | H(5,𝔽2) | D21 | 1 | 1 | yes | characteristic, complemented |
| 64 | AGL(1,ℤ/8) | — | 21 | 3 | no | — |
| 65 | AGL(1,ℤ/8) | — | 21 | 3 | no | — |
| 66 | #135 | — | 21 | 3 | no | — |
| 67 | C2×D8 | — | 21 | 3 | no | — |
| 68 | #131 | — | 21 | 1 | no | — |
| 69 | C22×C7 | — | 3 | 3 | no | — |
| 70 | C22×C7 | — | 24 | 1 | no | — |
| 71 | C2×D7 | — | 6 | 3 | no | — |
| 72 | C2×D7 | — | 24 | 3 | no | — |
| 73 | C4×C7 | — | 3 | 1 | no | — |
| 74 | C4×C7 | C2×S4 | 1 | 3 | yes | — |
| 75 | Dic7 | — | 6 | 1 | no | — |
| 76 | C3×Q8 | — | 4 | 1 | no | — |
| 77 | D12 | — | 28 | 3 | no | — |
| 78 | C4×D3 | — | 28 | 3 | no | — |
| 79 | SL(2,𝔽3) | C22×D7 | 1 | 1 | yes | characteristic |
| 80 | C3×C7 | — | 4 | 1 | no | Hall |
| 81 | #41 | — | 3 | 1 | no | — |
| 82 | #41 | — | 21 | 3 | no | — |
| 83 | #41 | — | 42 | 1 | no | — |
| 84 | #41 | C2×D21 | 1 | 3 | yes | — |
| 85 | C2×D4 | — | 3 | 3 | no | — |
| 86 | C2×D4 | — | 21 | 3 | no | — |
| 87 | C2×D4 | — | 42 | 3 | no | — |
| 88 | Q16 | — | 21 | 1 | no | — |
| 89 | SD16 | — | 21 | 3 | no | — |
| 90 | SD16 | — | 21 | 3 | no | — |
| 91 | D8 | — | 21 | 3 | no | — |
| 92 | D8 | — | 21 | 6 | no | — |
| 93 | M16 | — | 21 | 3 | no | — |
| 94 | C2×C8 | — | 21 | 3 | no | — |
| 95 | C2×C7 | — | 6 | 3 | no | — |
| 96 | C2×C7 | C22×S4 | 1 | 1 | yes | characteristic |
| 97 | D7 | — | 12 | 3 | no | — |
| 98 | C2×D3 | — | 28 | 3 | no | — |
| 99 | C3×C4 | — | 4 | 3 | no | — |
| 100 | Dic3 | — | 28 | 1 | no | — |
| 101 | C23 | — | 6 | 1 | no | — |
| 102 | C23 | — | 42 | 3 | no | — |
| 103 | Q8 | — | 21 | 1 | no | — |
| 104 | Q8 | C7⋊S4 | 1 | 1 | yes | characteristic |
| 105 | Q8 | C22×D21 | 1 | 1 | yes | characteristic |
| 106 | D4 | — | 3 | 3 | no | — |
| 107 | D4 | — | 3 | 3 | no | — |
| 108 | D4 | — | 21 | 3 | no | — |
| 109 | D4 | — | 42 | 3 | no | — |
| 110 | D4 | — | 42 | 3 | no | — |
| 111 | C2×C4 | — | 3 | 3 | no | — |
| 112 | C2×C4 | — | 42 | 3 | no | — |
| 113 | C8 | — | 21 | 1 | no | — |
| 114 | C8 | — | 21 | 3 | no | — |
| 115 | C7 | #5925 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 116 | C2×C3 | — | 4 | 1 | no | — |
| 117 | D3 | — | 56 | 3 | no | — |
| 118 | C22 | — | 3 | 3 | no | — |
| 119 | C22 | — | 24 | 1 | no | — |
| 120 | C22 | — | 42 | 3 | no | — |
| 121 | C22 | — | 168 | 3 | no | — |
| 122 | C4 | — | 3 | 1 | no | — |
| 123 | C4 | — | 42 | 1 | no | — |
| 124 | C4 | C2×C7⋊S4 | 1 | 3 | yes | — |
| 125 | C3 | — | 4 | 1 | no | Sylow |
| 126 | C2 | — | 6 | 3 | no | — |
| 127 | C2 | — | 84 | 3 | no | — |
| 128 | C2 | C22×C7⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×C7⋊S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Fitting | C7×H(5,𝔽2) | D3 |
| socle | C2×C7 | C22×S4 |
| hypercenter | Q8 | C7⋊S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#994 | C2 | semidirect |
| 2. #10621504 | C2 | semidirect |
| 3. #10621503 | C2 | semidirect |
| 6. C7×#41⋊C3 | C22 | semidirect |
| 7. C7⋊GL(2,𝔽3) | C22 | semidirect |
| 8. C7⋊#226 | C22 | semidirect |
| 9. C7×H(5,𝔽2) | D3 | semidirect |
| 35. #994 | D7 | semidirect |
| 57. #41⋊C3 | C2×D7 | semidirect |
| 63. H(5,𝔽2) | D21 | semidirect |
| 115. C7 | #5925 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#994 | 2 |
| #10621504 | 2 |
| #10621503 | 2 |
| #90526 | 3 |
| #66732 | 4 |
| #5925 | 7 |