Finite Group DatabasebetaFinite Group DatabasebetaOrder 1458.
335 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 40 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 | #10624555 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | C3⋊D3×H(3,𝔽3) | — | 3 | 1 | no | maximal |
| 3 | C32×C3⋊D9 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C3×#4169 | C3 | 1 | 3 | yes | maximal, complemented |
| 5 | C32×H(3,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 6 | C33×C9 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3×C32⋊C9 | — | 2 | 3 | no | — |
| 8 | C3×C32⋊C9 | — | 2 | 1 | no | — |
| 9 | C3×C32⋊C9 | C2×C3 | 1 | 3 | yes | complemented |
| 10 | C32×C3⋊D3 | — | 3 | 3 | no | — |
| 11 | C32×C3⋊D3 | — | 3 | 1 | no | — |
| 12 | C3×C3⋊D9 | — | 3 | 1 | no | — |
| 13 | C3×C3⋊D9 | C32 | 1 | 1 | yes | characteristic |
| 14 | D3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 15 | D3×H(3,𝔽3) | — | 9 | 2 | no | — |
| 16 | D3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 17 | C32×D9 | — | 9 | 1 | no | — |
| 18 | #4169 | — | 3 | 3 | no | — |
| 19 | C34 | C3×D3 | 1 | 1 | yes | characteristic |
| 20 | C34 | C3×D3 | 1 | 3 | yes | — |
| 21 | C3×H(3,𝔽3) | — | 2 | 1 | no | — |
| 22 | C3×H(3,𝔽3) | — | 2 | 3 | no | — |
| 23 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 24 | C3×H(3,𝔽3) | — | 3 | 2 | no | — |
| 25 | C3×H(3,𝔽3) | — | 6 | 1 | no | — |
| 26 | C3×H(3,𝔽3) | — | 6 | 2 | no | — |
| 27 | C3×H(3,𝔽3) | D9 | 1 | 1 | yes | characteristic, complemented |
| 28 | C32×C9 | — | 2 | 1 | no | — |
| 29 | C32×C9 | — | 2 | 3 | no | — |
| 30 | C32×C9 | — | 3 | 1 | no | — |
| 31 | C32×C9 | — | 3 | 1 | no | — |
| 32 | C32×C9 | — | 6 | 3 | no | — |
| 33 | C32×C9 | — | 6 | 1 | no | — |
| 34 | C32×C9 | — | 6 | 1 | no | — |
| 35 | C32×C9 | C2×C32 | 1 | 1 | yes | characteristic |
| 36 | C32⋊C9 | — | 3 | 3 | no | — |
| 37 | C32⋊C9 | — | 6 | 9 | no | — |
| 38 | C32⋊C9 | — | 6 | 3 | no | — |
| 39 | C32⋊C9 | — | 6 | 2 | no | — |
| 40 | C32⋊C9 | — | 6 | 1 | no | — |
| 41 | C3×C3⋊D3 | — | 3 | 1 | no | — |
| 42 | C3×C3⋊D3 | — | 9 | 3 | no | — |
| 43 | C3×C3⋊D3 | — | 9 | 1 | no | — |
| 44 | C32×D3 | — | 9 | 3 | no | — |
| 45 | C32×D3 | — | 9 | 6 | no | — |
| 46 | C32×D3 | — | 9 | 1 | no | — |
| 47 | C32×D3 | — | 9 | 3 | no | — |
| 48 | C32×D3 | — | 9 | 1 | no | — |
| 49 | C32×D3 | — | 9 | 2 | no | — |
| 50 | C2×H(3,𝔽3) | — | 27 | 1 | no | — |
| 51 | C3⋊D9 | H(3,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 52 | C3×D9 | — | 9 | 1 | no | — |
| 53 | C3×D9 | — | 9 | 3 | no | — |
| 54 | C33 | — | 2 | 1 | no | — |
| 55 | C33 | — | 2 | 1 | no | — |
| 56 | C33 | — | 2 | 2 | no | — |
| 57 | C33 | — | 2 | 3 | no | — |
| 58 | C33 | — | 3 | 1 | no | — |
| 59 | C33 | — | 3 | 3 | no | — |
| 60 | C33 | — | 3 | 3 | no | — |
| 61 | C33 | — | 3 | 6 | no | — |
| 62 | C33 | — | 6 | 1 | no | — |
| 63 | C33 | — | 6 | 1 | no | — |
| 64 | C33 | — | 6 | 2 | no | — |
| 65 | C33 | — | 6 | 3 | no | — |
| 66 | C33 | — | 6 | 3 | no | — |
| 67 | C33 | — | 6 | 6 | no | — |
| 68 | C33 | C3×D9 | 1 | 1 | yes | characteristic |
| 69 | C33 | C3×D9 | 1 | 3 | yes | complemented |
| 70 | C33 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 71 | C33 | AGL(1,ℤ/9) | 1 | 2 | yes | — |
| 72 | C33 | C32×D3 | 1 | 1 | yes | characteristic |
| 73 | H(3,𝔽3) | — | 3 | 1 | no | — |
| 74 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 75 | H(3,𝔽3) | — | 6 | 3 | no | — |
| 76 | H(3,𝔽3) | — | 6 | 6 | no | — |
| 77 | H(3,𝔽3) | — | 6 | 2 | no | — |
| 78 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 79 | C3×C9 | — | 2 | 3 | no | — |
| 80 | C3×C9 | — | 2 | 1 | no | — |
| 81 | C3×C9 | — | 3 | 3 | no | — |
| 82 | C3×C9 | — | 3 | 1 | no | — |
| 83 | C3×C9 | — | 6 | 3 | no | — |
| 84 | C3×C9 | — | 6 | 1 | no | — |
| 85 | C3×C9 | — | 6 | 1 | no | — |
| 86 | C3×C9 | — | 6 | 3 | no | — |
| 87 | C3×C9 | — | 6 | 3 | no | — |
| 88 | C3×C9 | — | 6 | 1 | no | — |
| 89 | C3×C9 | — | 6 | 1 | no | — |
| 90 | C3×C9 | — | 6 | 3 | no | — |
| 91 | C3×C9 | — | 6 | 6 | no | — |
| 92 | C3×C9 | — | 6 | 3 | no | — |
| 93 | C3×C9 | — | 6 | 3 | no | — |
| 94 | C3×C9 | C2×H(3,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 95 | C2×C32 | — | 27 | 3 | no | — |
| 96 | C2×C32 | — | 27 | 1 | no | — |
| 97 | C3⋊D3 | — | 3 | 1 | no | — |
| 98 | C3×D3 | — | 9 | 1 | no | — |
| 99 | C3×D3 | — | 9 | 1 | no | — |
| 100 | C3×D3 | — | 9 | 2 | no | — |
| 101 | C3×D3 | — | 27 | 1 | no | — |
| 102 | C3×D3 | — | 27 | 1 | no | — |
| 103 | C3×D3 | — | 27 | 3 | no | — |
| 104 | C3×D3 | — | 27 | 2 | no | — |
| 105 | C3×D3 | — | 27 | 3 | no | — |
| 106 | C3×D3 | — | 27 | 6 | no | — |
| 107 | D9 | — | 9 | 1 | no | — |
| 108 | C32 | — | 2 | 1 | no | — |
| 109 | C32 | — | 2 | 1 | no | — |
| 110 | C32 | — | 2 | 2 | no | — |
| 111 | C32 | — | 2 | 1 | no | — |
| 112 | C32 | — | 2 | 1 | no | — |
| 113 | C32 | — | 2 | 2 | no | — |
| 114 | C32 | — | 3 | 1 | no | — |
| 115 | C32 | — | 3 | 1 | no | — |
| 116 | C32 | — | 3 | 2 | no | — |
| 117 | C32 | — | 3 | 3 | no | — |
| 118 | C32 | — | 3 | 3 | no | — |
| 119 | C32 | — | 6 | 1 | no | — |
| 120 | C32 | — | 6 | 1 | no | — |
| 121 | C32 | — | 6 | 2 | no | — |
| 122 | C32 | — | 6 | 2 | no | — |
| 123 | C32 | — | 6 | 2 | no | — |
| 124 | C32 | — | 6 | 4 | no | — |
| 125 | C32 | — | 6 | 3 | no | — |
| 126 | C32 | — | 6 | 6 | no | — |
| 127 | C32 | — | 6 | 1 | no | — |
| 128 | C32 | — | 6 | 1 | no | — |
| 129 | C32 | — | 6 | 2 | no | — |
| 130 | C32 | — | 6 | 3 | no | — |
| 131 | C32 | — | 6 | 3 | no | — |
| 132 | C32 | — | 9 | 3 | no | — |
| 133 | C32 | — | 9 | 6 | no | — |
| 134 | C32 | — | 18 | 3 | no | — |
| 135 | C32 | — | 18 | 6 | no | — |
| 136 | C32 | #4169 | 1 | 1 | yes | characteristic, complemented |
| 137 | C32 | C32×D9 | 1 | 1 | yes | characteristic |
| 138 | C32 | C3×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 139 | C32 | D3×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 140 | C32 | C3×AGL(1,ℤ/9) | 1 | 2 | yes | — |
| 141 | C9 | — | 3 | 1 | no | — |
| 142 | C9 | — | 6 | 3 | no | — |
| 143 | C9 | — | 6 | 1 | no | — |
| 144 | C9 | — | 18 | 3 | no | — |
| 145 | C2×C3 | — | 27 | 1 | no | — |
| 146 | C2×C3 | — | 81 | 1 | no | — |
| 147 | C2×C3 | — | 81 | 3 | no | — |
| 148 | D3 | — | 9 | 1 | no | — |
| 149 | D3 | — | 9 | 1 | no | — |
| 150 | D3 | — | 9 | 2 | no | — |
| 151 | C3 | — | 2 | 1 | no | — |
| 152 | C3 | — | 2 | 1 | no | — |
| 153 | C3 | — | 2 | 2 | no | — |
| 154 | C3 | — | 3 | 1 | no | — |
| 155 | C3 | — | 6 | 1 | no | — |
| 156 | C3 | — | 6 | 1 | no | — |
| 157 | C3 | — | 6 | 2 | no | — |
| 158 | C3 | — | 9 | 3 | no | — |
| 159 | C3 | — | 18 | 3 | no | — |
| 160 | C3 | C3×#4169 | 1 | 1 | yes | characteristic |
| 161 | C3 | D9×H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 162 | C3 | (C3⋊D3)⋊H(3,𝔽3) | 1 | 1 | yes | characteristic |
| 163 | C3 | D9⋊H(3,𝔽3) | 1 | 2 | yes | — |
| 164 | C2 | — | 27 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×#4169 |
| commutator, cosocle | C32×C9 | C2×C32 |
| Frattini, socle | C33 | C32×D3 |
| Fitting | #10624555 | C2 |
| hypercenter | C32 | #4169 |
| hypocenter | C3×C9 | C2×H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C3 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624555 | C2 | semidirect |
| 3. C32×C3⋊D9 | C3 | semidirect |
| 4. C3×#4169 | C3 | semidirect |
| 6. C33×C9 | C2×C3 | semidirect |
| 9. C3×C32⋊C9 | C2×C3 | semidirect |
| 27. C3×H(3,𝔽3) | D9 | semidirect |
| 51. C3⋊D9 | H(3,𝔽3) | semidirect |
| 69. C33 | C3×D9 | semidirect |
| 94. C3×C9 | C2×H(3,𝔽3) | semidirect |
| 136. C32 | #4169 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624555 | 2 |
| C3⋊D3×H(3,𝔽3) | 3 |
| C32×C3⋊D9 | 3 |
| C3×#4169 | 3 |