Finite Group DatabasebetaFinite Group DatabasebetaOrder 1458.
330 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 32 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 | #10624748 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | #92404 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊#4178 | — | 3 | 1 | no | maximal |
| 4 | C3⋊#4176 | — | 3 | 1 | no | maximal |
| 5 | #92305 | — | 3 | 2 | no | maximal |
| 6 | H(5,𝔽3) | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | #6910 | — | 2 | 1 | no | — |
| 8 | #6909 | — | 2 | 2 | no | — |
| 9 | #6909 | — | 2 | 1 | no | — |
| 10 | C3×#732 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3×#731 | D3 | 1 | 1 | yes | characteristic, complemented |
| 12 | #6893 | D3 | 1 | 2 | yes | complemented |
| 13 | C3⋊#272 | — | 3 | 1 | no | — |
| 14 | C3⋊#272 | — | 3 | 1 | no | — |
| 15 | C3⋊#272 | — | 3 | 2 | no | — |
| 16 | C3⋊#272 | — | 9 | 3 | no | — |
| 17 | C3⋊#272 | — | 9 | 3 | no | — |
| 18 | C3⋊#272 | — | 9 | 6 | no | — |
| 19 | C9×C3⋊D3 | — | 9 | 1 | no | — |
| 20 | #4178 | — | 9 | 3 | no | — |
| 21 | #4176 | — | 9 | 3 | no | — |
| 22 | (C3⋊D3)⋊C9 | — | 9 | 6 | no | — |
| 23 | C3×3−1+2 | — | 2 | 2 | no | — |
| 24 | C3×3−1+2 | — | 2 | 1 | no | — |
| 25 | C3×H(3,𝔽3) | — | 2 | 1 | no | — |
| 26 | C3×H(3,𝔽3) | — | 3 | 6 | no | — |
| 27 | C3×H(3,𝔽3) | — | 3 | 3 | no | — |
| 28 | C3×H(3,𝔽3) | — | 3 | 3 | no | — |
| 29 | C3×H(3,𝔽3) | C3×D3 | 1 | 2 | yes | — |
| 30 | C3×H(3,𝔽3) | C3×D3 | 1 | 1 | yes | characteristic |
| 31 | C3×H(3,𝔽3) | C3×D3 | 1 | 1 | yes | characteristic |
| 32 | C32×C9 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 33 | #732 | — | 3 | 3 | no | — |
| 34 | #731 | — | 3 | 3 | no | — |
| 35 | #731 | — | 6 | 3 | no | — |
| 36 | AΣL(1,𝔽27) | — | 6 | 6 | no | — |
| 37 | AΣL(1,𝔽27) | — | 6 | 3 | no | — |
| 38 | C32⋊C9 | — | 3 | 6 | no | — |
| 39 | C3×C3⋊D3 | — | 9 | 1 | no | — |
| 40 | C3×C3⋊D3 | — | 9 | 6 | no | — |
| 41 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 42 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 43 | C3×C3⋊D3 | — | 27 | 9 | no | — |
| 44 | #272 | — | 9 | 3 | no | — |
| 45 | #272 | — | 9 | 3 | no | — |
| 46 | #272 | — | 27 | 2 | no | — |
| 47 | #272 | — | 27 | 2 | no | — |
| 48 | #272 | — | 27 | 2 | no | — |
| 49 | #272 | — | 27 | 2 | no | — |
| 50 | #272 | — | 27 | 2 | no | — |
| 51 | #272 | — | 27 | 18 | no | — |
| 52 | D3×C9 | — | 27 | 2 | no | — |
| 53 | D3×C9 | — | 27 | 1 | no | — |
| 54 | D3×C9 | — | 27 | 1 | no | — |
| 55 | C33 | — | 3 | 9 | no | — |
| 56 | C33 | — | 3 | 1 | no | — |
| 57 | C33 | — | 3 | 1 | no | — |
| 58 | C33 | — | 6 | 1 | no | — |
| 59 | C33 | (C3⋊D3)⋊C3 | 1 | 6 | yes | — |
| 60 | C33 | C3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 61 | 3−1+2 | — | 6 | 2 | no | — |
| 62 | 3−1+2 | — | 6 | 2 | no | — |
| 63 | 3−1+2 | — | 6 | 2 | no | — |
| 64 | 3−1+2 | — | 6 | 1 | no | — |
| 65 | 3−1+2 | — | 6 | 2 | no | — |
| 66 | H(3,𝔽3) | — | 3 | 18 | no | — |
| 67 | H(3,𝔽3) | — | 3 | 2 | no | — |
| 68 | H(3,𝔽3) | — | 3 | 2 | no | — |
| 69 | H(3,𝔽3) | — | 3 | 2 | no | — |
| 70 | H(3,𝔽3) | — | 3 | 2 | no | — |
| 71 | H(3,𝔽3) | — | 3 | 2 | no | — |
| 72 | H(3,𝔽3) | — | 6 | 2 | no | — |
| 73 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 74 | H(3,𝔽3) | (C3⋊D3)⋊C3 | 1 | 3 | yes | — |
| 75 | H(3,𝔽3) | (C3⋊D3)⋊C3 | 1 | 3 | yes | — |
| 76 | C3×C9 | — | 3 | 2 | no | — |
| 77 | C3×C9 | — | 3 | 1 | no | — |
| 78 | C3×C9 | — | 3 | 1 | no | — |
| 79 | C3×C9 | — | 6 | 2 | no | — |
| 80 | C3×C9 | — | 6 | 1 | no | — |
| 81 | C3⋊D3 | — | 9 | 1 | no | — |
| 82 | C3⋊D3 | — | 9 | 6 | no | — |
| 83 | C3⋊D3 | — | 27 | 1 | no | — |
| 84 | C3⋊D3 | — | 27 | 1 | no | — |
| 85 | C3⋊D3 | — | 27 | 9 | no | — |
| 86 | C3×D3 | — | 27 | 1 | no | — |
| 87 | C3×D3 | — | 27 | 1 | no | — |
| 88 | C3×D3 | — | 27 | 2 | no | — |
| 89 | C3×D3 | — | 81 | 3 | no | — |
| 90 | C3×D3 | — | 81 | 3 | no | — |
| 91 | C3×D3 | — | 81 | 6 | no | — |
| 92 | C2×C9 | — | 81 | 1 | no | — |
| 93 | C32 | — | 3 | 6 | no | — |
| 94 | C32 | — | 3 | 3 | no | — |
| 95 | C32 | — | 3 | 3 | no | — |
| 96 | C32 | — | 9 | 1 | no | — |
| 97 | C32 | — | 9 | 6 | no | — |
| 98 | C32 | — | 18 | 1 | no | — |
| 99 | C32 | — | 27 | 1 | no | — |
| 100 | C32 | — | 27 | 1 | no | — |
| 101 | C32 | — | 27 | 9 | no | — |
| 102 | C32 | — | 54 | 1 | no | — |
| 103 | C32 | C3⋊((C3⋊D3)⋊C3) | 1 | 2 | yes | — |
| 104 | C32 | C3⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 105 | C32 | C3⋊((C3⋊D3)⋊C3) | 1 | 1 | yes | characteristic |
| 106 | C9 | — | 9 | 1 | no | — |
| 107 | C9 | — | 18 | 2 | no | — |
| 108 | C9 | — | 18 | 1 | no | — |
| 109 | C2×C3 | — | 81 | 1 | no | — |
| 110 | D3 | — | 27 | 2 | no | — |
| 111 | D3 | — | 27 | 1 | no | — |
| 112 | D3 | — | 27 | 1 | no | — |
| 113 | D3 | — | 81 | 6 | no | — |
| 114 | D3 | — | 81 | 3 | no | — |
| 115 | D3 | — | 81 | 3 | no | — |
| 116 | C3 | — | 3 | 2 | no | — |
| 117 | C3 | — | 3 | 1 | no | — |
| 118 | C3 | — | 3 | 1 | no | — |
| 119 | C3 | — | 9 | 6 | no | — |
| 120 | C3 | — | 9 | 3 | no | — |
| 121 | C3 | — | 9 | 3 | no | — |
| 122 | C3 | — | 54 | 1 | no | — |
| 123 | C3 | #92301 | 1 | 1 | yes | characteristic |
| 124 | C2 | — | 81 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C3 | #92301 |
| commutator, cosocle, hypocenter | H(5,𝔽3) | C2×C3 |
| Frattini | C33 | C3×C3⋊D3 |
| Fitting | #10624748 | C2 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624748 | C2 | semidirect |
| 2. #92404 | C3 | semidirect |
| 10. C3×#732 | D3 | semidirect |
| 11. C3×#731 | D3 | semidirect |
| 12. #6893 | D3 | semidirect |
| 32. C32×C9 | C3⋊D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624748 | 2 |
| #92404 | 3 |
| C3⋊#4178 | 3 |
| C3⋊#4176 | 3 |
| #92305 | 3 |