Finite Group DatabasebetaFinite Group DatabasebetaOrder 1944.
290 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 20 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 | C3×#1113⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 2 | C2×C3×#1113 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#1113⋊C3 | C3 | 1 | 3 | yes | maximal, direct product factor |
| 4 | C2×C3×AΣL(1,𝔽27) | — | 4 | 1 | no | maximal |
| 5 | C2×C32×C3⋊D3 | — | 3 | 1 | no | — |
| 6 | C3×#1113 | — | 3 | 1 | no | — |
| 7 | C3×#1113 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | #1113⋊C3 | C2×C3 | 1 | 3 | yes | direct product factor |
| 9 | C3×AΣL(1,𝔽27) | — | 4 | 1 | no | Sylow |
| 10 | C2×#1113 | C32 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C3×D32 | — | 9 | 1 | no | — |
| 12 | C2×C34 | A4 | 1 | 1 | yes | characteristic, complemented |
| 13 | C32×C3⋊D3 | — | 3 | 1 | no | — |
| 14 | C32×C3⋊D3 | — | 3 | 1 | no | — |
| 15 | C2×C3×3−1+2 | — | 4 | 2 | no | — |
| 16 | C2×C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 17 | C2×AΣL(1,𝔽27) | — | 4 | 6 | no | — |
| 18 | C2×AΣL(1,𝔽27) | — | 4 | 3 | no | — |
| 19 | C2×C3×C3⋊D3 | — | 3 | 1 | no | — |
| 20 | C2×C3×C3⋊D3 | — | 3 | 1 | no | — |
| 21 | C2×C3×C3⋊D3 | — | 6 | 1 | no | — |
| 22 | C2×C32×D3 | — | 9 | 2 | no | — |
| 23 | C2×C32×D3 | — | 18 | 1 | no | — |
| 24 | #1113 | — | 3 | 1 | no | — |
| 25 | #1113 | C2×C32 | 1 | 1 | yes | characteristic, complemented |
| 26 | C3×D32 | — | 9 | 1 | no | — |
| 27 | C3×D32 | — | 9 | 2 | no | — |
| 28 | C3×D32 | — | 9 | 1 | no | — |
| 29 | C34 | C2×A4 | 1 | 1 | yes | characteristic, complemented |
| 30 | C3×3−1+2 | — | 4 | 2 | no | — |
| 31 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 32 | AΣL(1,𝔽27) | — | 4 | 6 | no | — |
| 33 | AΣL(1,𝔽27) | — | 4 | 3 | no | — |
| 34 | C22×C3×D3 | — | 27 | 1 | no | — |
| 35 | C2×C3×A4 | — | 27 | 1 | no | maximal |
| 36 | C2×D32 | — | 9 | 1 | no | — |
| 37 | C2×C33 | — | 3 | 1 | no | — |
| 38 | C2×C33 | — | 4 | 1 | no | — |
| 39 | C2×C33 | — | 4 | 2 | no | — |
| 40 | C2×C33 | — | 6 | 1 | no | — |
| 41 | C2×C33 | — | 6 | 1 | no | — |
| 42 | C2×C33 | — | 12 | 1 | no | — |
| 43 | C2×C33 | — | 12 | 1 | no | — |
| 44 | C2×C33 | C3×A4 | 1 | 1 | yes | characteristic, complemented |
| 45 | C3×C3⋊D3 | — | 3 | 1 | no | — |
| 46 | C3×C3⋊D3 | — | 3 | 1 | no | — |
| 47 | C3×C3⋊D3 | — | 3 | 1 | no | — |
| 48 | C3×C3⋊D3 | — | 3 | 1 | no | — |
| 49 | C3×C3⋊D3 | — | 6 | 1 | no | — |
| 50 | C3×C3⋊D3 | — | 6 | 1 | no | — |
| 51 | C32×D3 | — | 9 | 2 | no | — |
| 52 | C32×D3 | — | 9 | 2 | no | — |
| 53 | C32×D3 | — | 18 | 1 | no | — |
| 54 | C32×D3 | — | 18 | 1 | no | — |
| 55 | C2×3−1+2 | — | 4 | 6 | no | — |
| 56 | C2×3−1+2 | — | 12 | 4 | no | — |
| 57 | C2×H(3,𝔽3) | — | 4 | 3 | no | — |
| 58 | C2×H(3,𝔽3) | — | 12 | 2 | no | — |
| 59 | C2×C3×C9 | — | 12 | 2 | no | — |
| 60 | C22×C32 | — | 27 | 1 | no | — |
| 61 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 62 | C2×C3×D3 | — | 9 | 2 | no | — |
| 63 | C2×C3×D3 | — | 9 | 2 | no | — |
| 64 | C2×C3×D3 | — | 18 | 2 | no | — |
| 65 | C2×C3×D3 | — | 18 | 1 | no | — |
| 66 | C2×C3×D3 | — | 18 | 1 | no | — |
| 67 | C2×C3×D3 | — | 18 | 2 | no | — |
| 68 | C2×C3×D3 | — | 27 | 1 | no | — |
| 69 | C2×C3×D3 | — | 27 | 2 | no | — |
| 70 | C2×C3×D3 | — | 27 | 1 | no | — |
| 71 | C3×A4 | — | 27 | 1 | no | — |
| 72 | D32 | — | 9 | 2 | no | — |
| 73 | D32 | — | 9 | 1 | no | — |
| 74 | D32 | — | 9 | 1 | no | — |
| 75 | C33 | — | 3 | 1 | no | — |
| 76 | C33 | — | 4 | 1 | no | — |
| 77 | C33 | — | 4 | 2 | no | — |
| 78 | C33 | — | 6 | 1 | no | — |
| 79 | C33 | — | 6 | 1 | no | — |
| 80 | C33 | — | 12 | 1 | no | — |
| 81 | C33 | — | 12 | 1 | no | — |
| 82 | C33 | C2×C3×A4 | 1 | 1 | yes | characteristic, complemented |
| 83 | 3−1+2 | — | 4 | 6 | no | — |
| 84 | 3−1+2 | — | 12 | 4 | no | — |
| 85 | H(3,𝔽3) | — | 4 | 3 | no | — |
| 86 | H(3,𝔽3) | — | 12 | 2 | no | — |
| 87 | C3×C9 | — | 12 | 2 | no | — |
| 88 | C23×C3 | — | 27 | 1 | no | — |
| 89 | C22×D3 | — | 27 | 1 | no | — |
| 90 | C2×A4 | — | 27 | 3 | no | — |
| 91 | C2×C32 | — | 3 | 1 | no | — |
| 92 | C2×C32 | — | 3 | 1 | no | — |
| 93 | C2×C32 | — | 4 | 1 | no | — |
| 94 | C2×C32 | — | 4 | 1 | no | — |
| 95 | C2×C32 | — | 4 | 2 | no | — |
| 96 | C2×C32 | — | 6 | 1 | no | — |
| 97 | C2×C32 | — | 6 | 2 | no | — |
| 98 | C2×C32 | — | 6 | 2 | no | — |
| 99 | C2×C32 | — | 6 | 1 | no | — |
| 100 | C2×C32 | — | 12 | 1 | no | — |
| 101 | C2×C32 | — | 12 | 2 | no | — |
| 102 | C2×C32 | — | 12 | 2 | no | — |
| 103 | C2×C32 | — | 12 | 3 | no | — |
| 104 | C2×C32 | — | 12 | 1 | no | — |
| 105 | C2×C32 | — | 27 | 1 | no | — |
| 106 | C2×C32 | — | 27 | 1 | no | — |
| 107 | C2×C32 | — | 36 | 2 | no | — |
| 108 | C2×C32 | — | 36 | 1 | no | — |
| 109 | C3⋊D3 | — | 3 | 1 | no | — |
| 110 | C3⋊D3 | — | 3 | 1 | no | — |
| 111 | C3×D3 | — | 9 | 2 | no | — |
| 112 | C3×D3 | — | 9 | 2 | no | — |
| 113 | C3×D3 | — | 9 | 2 | no | — |
| 114 | C3×D3 | — | 9 | 2 | no | — |
| 115 | C3×D3 | — | 18 | 2 | no | — |
| 116 | C3×D3 | — | 18 | 2 | no | — |
| 117 | C3×D3 | — | 18 | 1 | no | — |
| 118 | C3×D3 | — | 18 | 1 | no | — |
| 119 | C3×D3 | — | 18 | 1 | no | — |
| 120 | C3×D3 | — | 18 | 1 | no | — |
| 121 | C3×D3 | — | 18 | 2 | no | — |
| 122 | C3×D3 | — | 18 | 2 | no | — |
| 123 | C2×C9 | — | 12 | 6 | no | — |
| 124 | C22×C3 | — | 27 | 1 | no | — |
| 125 | C22×C3 | — | 27 | 1 | no | — |
| 126 | C22×C3 | — | 27 | 1 | no | — |
| 127 | C22×C3 | — | 54 | 1 | no | — |
| 128 | C22×C3 | — | 81 | 1 | no | — |
| 129 | C2×D3 | — | 9 | 2 | no | — |
| 130 | C2×D3 | — | 18 | 1 | no | — |
| 131 | C2×D3 | — | 27 | 1 | no | — |
| 132 | C2×D3 | — | 27 | 2 | no | — |
| 133 | C2×D3 | — | 27 | 1 | no | — |
| 134 | A4 | — | 27 | 3 | no | — |
| 135 | C32 | — | 3 | 1 | no | — |
| 136 | C32 | — | 3 | 1 | no | — |
| 137 | C32 | — | 4 | 1 | no | — |
| 138 | C32 | — | 4 | 1 | no | — |
| 139 | C32 | — | 4 | 2 | no | — |
| 140 | C32 | — | 6 | 1 | no | — |
| 141 | C32 | — | 6 | 2 | no | — |
| 142 | C32 | — | 6 | 2 | no | — |
| 143 | C32 | — | 6 | 1 | no | — |
| 144 | C32 | — | 12 | 1 | no | — |
| 145 | C32 | — | 12 | 2 | no | — |
| 146 | C32 | — | 12 | 2 | no | — |
| 147 | C32 | — | 12 | 3 | no | — |
| 148 | C32 | — | 12 | 1 | no | — |
| 149 | C32 | — | 36 | 2 | no | — |
| 150 | C32 | — | 36 | 1 | no | — |
| 151 | C9 | — | 12 | 6 | no | — |
| 152 | C23 | — | 27 | 1 | no | Sylow |
| 153 | C2×C3 | — | 3 | 1 | no | — |
| 154 | C2×C3 | — | 4 | 1 | no | — |
| 155 | C2×C3 | — | 4 | 2 | no | — |
| 156 | C2×C3 | — | 6 | 1 | no | — |
| 157 | C2×C3 | — | 6 | 1 | no | — |
| 158 | C2×C3 | — | 12 | 1 | no | — |
| 159 | C2×C3 | — | 27 | 1 | no | — |
| 160 | C2×C3 | — | 27 | 1 | no | — |
| 161 | C2×C3 | — | 27 | 1 | no | — |
| 162 | C2×C3 | — | 27 | 1 | no | — |
| 163 | C2×C3 | — | 36 | 3 | no | — |
| 164 | C2×C3 | — | 54 | 1 | no | — |
| 165 | C2×C3 | — | 54 | 1 | no | — |
| 166 | C2×C3 | #1113⋊C3 | 1 | 1 | yes | characteristic, direct product factor |
| 167 | D3 | — | 9 | 2 | no | — |
| 168 | D3 | — | 9 | 2 | no | — |
| 169 | D3 | — | 18 | 1 | no | — |
| 170 | D3 | — | 18 | 1 | no | — |
| 171 | C22 | — | 27 | 1 | no | — |
| 172 | C22 | — | 27 | 1 | no | — |
| 173 | C22 | — | 81 | 1 | no | — |
| 174 | C3 | — | 3 | 1 | no | — |
| 175 | C3 | — | 4 | 1 | no | — |
| 176 | C3 | — | 4 | 2 | no | — |
| 177 | C3 | — | 6 | 1 | no | — |
| 178 | C3 | — | 6 | 1 | no | — |
| 179 | C3 | — | 12 | 1 | no | — |
| 180 | C3 | — | 36 | 3 | no | — |
| 181 | C3 | C2×#1113⋊C3 | 1 | 1 | yes | characteristic, direct product factor |
| 182 | C2 | — | 27 | 1 | no | — |
| 183 | C2 | — | 27 | 1 | no | — |
| 184 | C2 | C3×#1113⋊C3 | 1 | 1 | yes | characteristic, direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #1113⋊C3 |
| commutator, cosocle, hypocenter | #1113 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | A4 |
| LCS | G ▷ #1113 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#1113⋊C3 | C2 | direct (N × H) |
| 2. C2×C3×#1113 | C3 | semidirect |
| 3. C2×#1113⋊C3 | C3 | direct (N × H) |
| 7. C3×#1113 | C2×C3 | semidirect |
| 8. #1113⋊C3 | C2×C3 | direct (N × H) |
| 10. C2×#1113 | C32 | semidirect |
| 12. C2×C34 | A4 | semidirect |
| 25. #1113 | C2×C32 | semidirect |
| 29. C34 | C2×A4 | semidirect |
| 44. C2×C33 | C3×A4 | semidirect |
| 82. C33 | C2×C3×A4 | semidirect |
| 166. C2×C3 | #1113⋊C3 | direct (N × H) |
| 181. C3 | C2×#1113⋊C3 | direct (N × H) |
| 184. C2 | C3×#1113⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#1113⋊C3 | 2 |
| C2×C3×#1113 | 3 |
| C2×#1113⋊C3 | 3 |
| C2×C3×AΣL(1,𝔽27) | 4 |
| C2×C3×A4 | 27 |