Finite Group DatabasebetaFinite Group DatabasebetaOrder 648.
134 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 18 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 | C2×#4175 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#4174 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #66330 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #6291 | — | 3 | 1 | no | maximal |
| 5 | #6200 | — | 3 | 1 | no | maximal |
| 6 | C2×AΣL(1,𝔽27) | C22 | 1 | 1 | yes | characteristic |
| 7 | #4175 | — | 2 | 1 | no | — |
| 8 | #4174 | — | 2 | 1 | no | — |
| 9 | C2×C3⋊(C3⋊D3) | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×C32×D3 | — | 3 | 1 | no | — |
| 11 | C3×C3⋊Dic3 | — | 3 | 1 | no | — |
| 12 | C2×#272 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×(C3⋊D3)⋊C3 | — | 3 | 1 | no | — |
| 14 | 3−1+2⋊C4 | — | 6 | 1 | no | — |
| 15 | (C3⋊Dic3)⋊C3 | — | 3 | 1 | no | — |
| 16 | AΣL(1,𝔽27) | D4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 17 | #638 | — | 9 | 1 | no | — |
| 18 | #626 | — | 9 | 1 | no | — |
| 19 | #626 | — | 9 | 1 | no | — |
| 20 | #626 | — | 9 | 1 | no | — |
| 21 | #626 | — | 9 | 1 | no | — |
| 22 | #625 | — | 9 | 1 | no | — |
| 23 | C2×C33 | C2×D3 | 1 | 1 | yes | characteristic |
| 24 | C3⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 25 | C32×D3 | — | 6 | 1 | no | — |
| 26 | C2×3−1+2 | — | 2 | 1 | no | — |
| 27 | C2×H(3,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | #272 | — | 2 | 1 | no | — |
| 29 | (C3⋊D3)⋊C3 | — | 6 | 1 | no | — |
| 30 | C22×C32 | — | 9 | 1 | no | — |
| 31 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 32 | C2×C3⋊D3 | — | 9 | 1 | no | — |
| 33 | C2×C3⋊D3 | — | 9 | 1 | no | — |
| 34 | C2×C3⋊D3 | — | 18 | 1 | no | — |
| 35 | C2×C3⋊D3 | — | 18 | 1 | no | — |
| 36 | C2×C3×D3 | — | 3 | 1 | no | — |
| 37 | C2×C3×D3 | — | 3 | 1 | no | — |
| 38 | C2×C3×D3 | — | 3 | 1 | no | — |
| 39 | C2×C3×D3 | — | 3 | 1 | no | — |
| 40 | C2×C3×D3 | — | 9 | 1 | no | — |
| 41 | C2×C3×D3 | — | 18 | 1 | no | — |
| 42 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 43 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 44 | C3×Dic3 | — | 9 | 1 | no | — |
| 45 | C3×Dic3 | — | 9 | 1 | no | — |
| 46 | C3×Dic3 | — | 9 | 1 | no | — |
| 47 | C3×Dic3 | — | 9 | 1 | no | — |
| 48 | Dic9 | — | 6 | 1 | no | — |
| 49 | C33 | #67 | 1 | 1 | yes | characteristic, complemented |
| 50 | 3−1+2 | — | 2 | 1 | no | — |
| 51 | H(3,𝔽3) | #67 | 1 | 1 | yes | characteristic, complemented |
| 52 | #67 | — | 27 | 1 | no | — |
| 53 | #67 | — | 27 | 1 | no | — |
| 54 | #67 | — | 27 | 1 | no | — |
| 55 | #67 | — | 27 | 1 | no | — |
| 56 | #67 | — | 27 | 1 | no | — |
| 57 | D12 | — | 27 | 1 | no | — |
| 58 | C2×C32 | — | 3 | 1 | no | — |
| 59 | C2×C32 | — | 3 | 1 | no | — |
| 60 | C2×C32 | — | 6 | 1 | no | — |
| 61 | C2×C32 | — | 6 | 1 | no | — |
| 62 | C2×C32 | — | 6 | 1 | no | — |
| 63 | C2×C32 | — | 18 | 1 | no | — |
| 64 | C2×C32 | D32 | 1 | 1 | yes | characteristic |
| 65 | C3⋊D3 | — | 6 | 1 | no | — |
| 66 | C3⋊D3 | — | 18 | 1 | no | — |
| 67 | C3⋊D3 | — | 18 | 1 | no | — |
| 68 | C3⋊D3 | — | 18 | 1 | no | — |
| 69 | C3⋊D3 | — | 18 | 1 | no | — |
| 70 | C3×D3 | — | 6 | 1 | no | — |
| 71 | C3×D3 | — | 6 | 1 | no | — |
| 72 | C3×D3 | — | 6 | 1 | no | — |
| 73 | C3×D3 | — | 6 | 1 | no | — |
| 74 | C3×D3 | — | 18 | 1 | no | — |
| 75 | C3×D3 | — | 18 | 1 | no | — |
| 76 | C2×C9 | — | 6 | 1 | no | — |
| 77 | C22×C3 | — | 9 | 1 | no | — |
| 78 | C22×C3 | — | 9 | 1 | no | — |
| 79 | C22×C3 | — | 9 | 1 | no | — |
| 80 | C22×C3 | — | 9 | 1 | no | — |
| 81 | C22×C3 | — | 27 | 1 | no | — |
| 82 | C2×D3 | — | 3 | 1 | no | — |
| 83 | C2×D3 | — | 9 | 1 | no | — |
| 84 | C2×D3 | — | 18 | 1 | no | — |
| 85 | C2×D3 | — | 27 | 1 | no | — |
| 86 | C2×D3 | — | 54 | 1 | no | — |
| 87 | C2×D3 | — | 54 | 1 | no | — |
| 88 | C2×D3 | — | 54 | 1 | no | — |
| 89 | C3×C4 | — | 27 | 1 | no | — |
| 90 | Dic3 | — | 9 | 1 | no | — |
| 91 | Dic3 | — | 9 | 1 | no | — |
| 92 | Dic3 | — | 9 | 1 | no | — |
| 93 | Dic3 | — | 9 | 1 | no | — |
| 94 | Dic3 | — | 27 | 1 | no | — |
| 95 | C32 | — | 3 | 1 | no | — |
| 96 | C32 | — | 3 | 1 | no | — |
| 97 | C32 | — | 6 | 1 | no | — |
| 98 | C32 | — | 6 | 1 | no | — |
| 99 | C32 | — | 6 | 1 | no | — |
| 100 | C32 | #625 | 1 | 1 | yes | characteristic |
| 101 | C9 | — | 6 | 1 | no | — |
| 102 | D4 | — | 81 | 1 | no | Sylow |
| 103 | C2×C3 | — | 3 | 1 | no | — |
| 104 | C2×C3 | — | 6 | 1 | no | — |
| 105 | C2×C3 | — | 6 | 1 | no | — |
| 106 | C2×C3 | — | 6 | 1 | no | — |
| 107 | C2×C3 | — | 18 | 1 | no | — |
| 108 | C2×C3 | — | 18 | 1 | no | — |
| 109 | C2×C3 | — | 18 | 1 | no | — |
| 110 | C2×C3 | — | 18 | 1 | no | — |
| 111 | C2×C3 | — | 18 | 1 | no | — |
| 112 | C2×C3 | — | 54 | 1 | no | — |
| 113 | C2×C3 | ((C3⋊D3)⋊C3)⋊C2 | 1 | 1 | yes | characteristic |
| 114 | D3 | — | 6 | 1 | no | — |
| 115 | D3 | — | 18 | 1 | no | — |
| 116 | D3 | — | 18 | 1 | no | — |
| 117 | D3 | — | 54 | 1 | no | — |
| 118 | D3 | — | 54 | 1 | no | — |
| 119 | D3 | — | 54 | 1 | no | — |
| 120 | D3 | — | 54 | 1 | no | — |
| 121 | C22 | — | 9 | 1 | no | — |
| 122 | C22 | — | 27 | 1 | no | — |
| 123 | C4 | — | 27 | 1 | no | — |
| 124 | C3 | — | 3 | 1 | no | — |
| 125 | C3 | — | 6 | 1 | no | — |
| 126 | C3 | — | 6 | 1 | no | — |
| 127 | C3 | — | 6 | 1 | no | — |
| 128 | C3 | — | 18 | 1 | no | — |
| 129 | C3 | #6198 | 1 | 1 | yes | characteristic |
| 130 | C2 | — | 18 | 1 | no | — |
| 131 | C2 | — | 54 | 1 | no | — |
| 132 | C2 | #4174⋊C2 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4174⋊C2 |
| commutator, Fitting, cosocle | C2×AΣL(1,𝔽27) | C22 |
| Frattini | C2×C32 | D32 |
| socle | C2×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| hypocenter | AΣL(1,𝔽27) | D4 |
| LCS | G ▷ C2×AΣL(1,𝔽27) ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#4175 | C2 | semidirect |
| 2. C2×#4174 | C2 | semidirect |
| 3. #66330 | C2 | semidirect |
| 9. C2×C3⋊(C3⋊D3) | D3 | semidirect |
| 12. C2×#272 | D3 | semidirect |
| 16. AΣL(1,𝔽27) | D4 | semidirect |
| 49. C33 | #67 | semidirect |
| 51. H(3,𝔽3) | #67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#4175 | 2 |
| C2×#4174 | 2 |
| #66330 | 2 |
| #6291 | 3 |
| #6200 | 3 |