Finite Group DatabasebetaFinite Group DatabasebetaOrder 2916.
4336 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 4336 conjugacy classes of subgroups — too many to draw. Only its 206 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⋊#92404 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊#92392 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#92392 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11758087 | — | 3 | 9 | no | maximal |
| 5 | C3⋊D3×#272 | — | 3 | 4 | no | maximal |
| 6 | D3×C3⋊#272 | — | 3 | 4 | no | maximal |
| 7 | C3×H(5,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | #92404 | — | 3 | 9 | no | — |
| 9 | #92404 | — | 6 | 36 | no | — |
| 10 | C3⋊(C3⋊#272) | — | 3 | 4 | no | — |
| 11 | C3⋊(C3⋊#272) | — | 3 | 4 | no | — |
| 12 | C3⋊(C3⋊#272) | — | 6 | 16 | no | — |
| 13 | C3×C3⋊#272 | D3 | 1 | 4 | yes | complemented |
| 14 | #92392 | — | 3 | 9 | no | — |
| 15 | #92392 | D3 | 1 | 9 | yes | complemented |
| 16 | C3⋊D3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 17 | C32×#272 | D3 | 1 | 4 | yes | complemented |
| 18 | C3×D3×H(3,𝔽3) | — | 3 | 4 | no | — |
| 19 | C3×D3×C3⋊D3 | — | 9 | 16 | no | — |
| 20 | C2×C3⋊#272 | — | 9 | 1 | no | — |
| 21 | D3×#272 | — | 9 | 36 | no | — |
| 22 | D3×#272 | — | 9 | 1 | no | — |
| 23 | D3×#272 | — | 9 | 12 | no | — |
| 24 | H(5,𝔽3) | — | 2 | 36 | no | — |
| 25 | H(5,𝔽3) | C2×D3 | 1 | 9 | yes | complemented |
| 26 | C32×H(3,𝔽3) | — | 2 | 16 | no | — |
| 27 | C32×H(3,𝔽3) | C2×D3 | 1 | 4 | yes | complemented |
| 28 | C32×H(3,𝔽3) | C2×D3 | 1 | 4 | yes | complemented |
| 29 | C3×C3⋊(C3⋊D3) | — | 9 | 16 | no | — |
| 30 | C3×C3⋊(C3⋊D3) | — | 18 | 12 | no | — |
| 31 | C32×C3⋊D3 | — | 3 | 16 | no | — |
| 32 | C33×D3 | — | 3 | 16 | no | — |
| 33 | C2×C3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 34 | C3⋊#272 | — | 9 | 1 | no | — |
| 35 | C3⋊#272 | — | 9 | 36 | no | — |
| 36 | C3⋊#272 | — | 9 | 1 | no | — |
| 37 | C3⋊#272 | — | 9 | 12 | no | — |
| 38 | C3⋊#272 | — | 18 | 36 | no | — |
| 39 | C3⋊#272 | — | 18 | 16 | no | — |
| 40 | C3⋊#272 | — | 18 | 16 | no | — |
| 41 | C3⋊#272 | — | 18 | 48 | no | — |
| 42 | C3⋊#272 | — | 18 | 144 | no | — |
| 43 | C3⋊#272 | — | 18 | 12 | no | — |
| 44 | C3⋊#272 | — | 18 | 288 | no | — |
| 45 | C3⋊#272 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 46 | C3×#272 | — | 3 | 36 | no | — |
| 47 | C3×#272 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 48 | C3×#272 | C3⋊D3 | 1 | 12 | yes | complemented |
| 49 | D3×H(3,𝔽3) | — | 3 | 36 | no | — |
| 50 | D3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 51 | D3×H(3,𝔽3) | — | 9 | 12 | no | — |
| 52 | C2×C3×C3⋊D3 | — | 27 | 4 | no | — |
| 53 | D3×C3⋊D3 | — | 9 | 16 | no | — |
| 54 | C3×D32 | — | 27 | 48 | no | — |
| 55 | C3×D32 | — | 27 | 4 | no | — |
| 56 | C2×#272 | — | 27 | 9 | no | — |
| 57 | C2×#272 | — | 27 | 1 | no | — |
| 58 | C34 | — | 2 | 12 | no | — |
| 59 | C34 | D32 | 1 | 16 | yes | complemented |
| 60 | C3×H(3,𝔽3) | — | 2 | 12 | no | — |
| 61 | C3×H(3,𝔽3) | — | 2 | 288 | no | — |
| 62 | C3×H(3,𝔽3) | — | 2 | 16 | no | — |
| 63 | C3×H(3,𝔽3) | — | 2 | 16 | no | — |
| 64 | C3×H(3,𝔽3) | — | 2 | 144 | no | — |
| 65 | C3×H(3,𝔽3) | — | 2 | 48 | no | — |
| 66 | C3×H(3,𝔽3) | — | 2 | 36 | no | — |
| 67 | C3×H(3,𝔽3) | D32 | 1 | 36 | yes | complemented |
| 68 | C3×H(3,𝔽3) | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 69 | C3×H(3,𝔽3) | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 70 | C3×H(3,𝔽3) | C2×C3⋊D3 | 1 | 12 | yes | complemented |
| 71 | C2×C33 | — | 9 | 4 | no | — |
| 72 | C3⋊(C3⋊D3) | — | 9 | 16 | no | — |
| 73 | C3⋊(C3⋊D3) | — | 18 | 12 | no | — |
| 74 | C3×C3⋊D3 | — | 3 | 4 | no | — |
| 75 | C3×C3⋊D3 | — | 9 | 16 | no | — |
| 76 | C3×C3⋊D3 | — | 27 | 4 | no | — |
| 77 | C3×C3⋊D3 | — | 27 | 4 | no | — |
| 78 | C3×C3⋊D3 | — | 27 | 48 | no | — |
| 79 | C3×C3⋊D3 | — | 54 | 48 | no | — |
| 80 | C3×C3⋊D3 | — | 54 | 16 | no | — |
| 81 | C3×C3⋊D3 | — | 54 | 108 | no | — |
| 82 | C32×D3 | — | 3 | 4 | no | — |
| 83 | C32×D3 | — | 3 | 48 | no | — |
| 84 | C32×D3 | — | 9 | 16 | no | — |
| 85 | C32×D3 | — | 9 | 4 | no | — |
| 86 | C32×D3 | — | 9 | 48 | no | — |
| 87 | C32×D3 | — | 18 | 16 | no | — |
| 88 | C2×H(3,𝔽3) | — | 9 | 9 | no | — |
| 89 | C2×H(3,𝔽3) | — | 27 | 1 | no | — |
| 90 | #272 | — | 3 | 9 | no | — |
| 91 | #272 | — | 27 | 9 | no | — |
| 92 | #272 | — | 27 | 1 | no | — |
| 93 | #272 | — | 54 | 4 | no | — |
| 94 | #272 | — | 54 | 48 | no | — |
| 95 | #272 | — | 54 | 144 | no | — |
| 96 | #272 | — | 54 | 96 | no | — |
| 97 | #272 | — | 54 | 108 | no | — |
| 98 | #272 | C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 99 | C2×C3⋊D3 | — | 27 | 4 | no | — |
| 100 | C2×C3×D3 | — | 81 | 4 | no | — |
| 101 | C2×C3×D3 | — | 81 | 12 | no | — |
| 102 | C2×C3×D3 | — | 81 | 1 | no | — |
| 103 | D32 | — | 27 | 4 | no | — |
| 104 | D32 | — | 27 | 48 | no | — |
| 105 | C33 | — | 2 | 108 | no | — |
| 106 | C33 | — | 2 | 16 | no | — |
| 107 | C33 | — | 2 | 48 | no | — |
| 108 | C33 | — | 9 | 16 | no | — |
| 109 | C33 | — | 18 | 12 | no | — |
| 110 | C33 | — | 18 | 16 | no | — |
| 111 | C33 | — | 36 | 12 | no | — |
| 112 | C33 | D3×C3⋊D3 | 1 | 4 | yes | — |
| 113 | C33 | D3×C3⋊D3 | 1 | 4 | yes | — |
| 114 | C33 | D3×C3⋊D3 | 1 | 48 | yes | complemented |
| 115 | H(3,𝔽3) | — | 2 | 96 | no | — |
| 116 | H(3,𝔽3) | — | 2 | 108 | no | — |
| 117 | H(3,𝔽3) | — | 2 | 4 | no | — |
| 118 | H(3,𝔽3) | — | 2 | 48 | no | — |
| 119 | H(3,𝔽3) | — | 2 | 144 | no | — |
| 120 | H(3,𝔽3) | D3×C3⋊D3 | 1 | 9 | yes | — |
| 121 | H(3,𝔽3) | C2×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 122 | C2×C32 | — | 9 | 1 | no | — |
| 123 | C2×C32 | — | 9 | 12 | no | — |
| 124 | C2×C32 | — | 27 | 4 | no | — |
| 125 | C2×C32 | — | 27 | 4 | no | — |
| 126 | C2×C32 | — | 54 | 4 | no | — |
| 127 | C3⋊D3 | — | 3 | 4 | no | — |
| 128 | C3⋊D3 | — | 27 | 4 | no | — |
| 129 | C3⋊D3 | — | 27 | 4 | no | — |
| 130 | C3⋊D3 | — | 27 | 48 | no | — |
| 131 | C3⋊D3 | — | 54 | 16 | no | — |
| 132 | C3⋊D3 | — | 54 | 48 | no | — |
| 133 | C3⋊D3 | — | 54 | 108 | no | — |
| 134 | C3×D3 | — | 3 | 4 | no | — |
| 135 | C3×D3 | — | 3 | 4 | no | — |
| 136 | C3×D3 | — | 6 | 4 | no | — |
| 137 | C3×D3 | — | 9 | 48 | no | — |
| 138 | C3×D3 | — | 9 | 12 | no | — |
| 139 | C3×D3 | — | 9 | 1 | no | — |
| 140 | C3×D3 | — | 27 | 4 | no | — |
| 141 | C3×D3 | — | 27 | 48 | no | — |
| 142 | C3×D3 | — | 81 | 4 | no | — |
| 143 | C3×D3 | — | 81 | 12 | no | — |
| 144 | C3×D3 | — | 81 | 1 | no | — |
| 145 | C3×D3 | — | 162 | 4 | no | — |
| 146 | C3×D3 | — | 162 | 48 | no | — |
| 147 | C22×C3 | — | 243 | 1 | no | — |
| 148 | C2×D3 | — | 81 | 4 | no | — |
| 149 | C2×D3 | — | 81 | 1 | no | — |
| 150 | C2×D3 | — | 81 | 12 | no | — |
| 151 | C32 | — | 2 | 4 | no | — |
| 152 | C32 | — | 2 | 48 | no | — |
| 153 | C32 | — | 3 | 4 | no | — |
| 154 | C32 | — | 3 | 4 | no | — |
| 155 | C32 | — | 6 | 4 | no | — |
| 156 | C32 | — | 6 | 4 | no | — |
| 157 | C32 | — | 6 | 16 | no | — |
| 158 | C32 | — | 9 | 48 | no | — |
| 159 | C32 | — | 12 | 16 | no | — |
| 160 | C32 | — | 18 | 48 | no | — |
| 161 | C32 | — | 18 | 108 | no | — |
| 162 | C32 | D3×C3⋊(C3⋊D3) | 1 | 4 | yes | — |
| 163 | C32 | C3⋊D32 | 1 | 1 | yes | characteristic |
| 164 | C32 | C3⋊D32 | 1 | 12 | yes | — |
| 165 | C2×C3 | — | 9 | 1 | no | — |
| 166 | C2×C3 | — | 9 | 1 | no | — |
| 167 | C2×C3 | — | 18 | 1 | no | — |
| 168 | C2×C3 | — | 27 | 12 | no | — |
| 169 | C2×C3 | — | 27 | 1 | no | — |
| 170 | C2×C3 | — | 81 | 4 | no | — |
| 171 | C2×C3 | — | 243 | 1 | no | — |
| 172 | D3 | — | 3 | 4 | no | — |
| 173 | D3 | — | 9 | 12 | no | — |
| 174 | D3 | — | 9 | 1 | no | — |
| 175 | D3 | — | 81 | 4 | no | — |
| 176 | D3 | — | 81 | 12 | no | — |
| 177 | D3 | — | 81 | 1 | no | — |
| 178 | D3 | — | 162 | 4 | no | — |
| 179 | D3 | — | 162 | 48 | no | — |
| 180 | C22 | — | 243 | 1 | no | Sylow |
| 181 | C3 | — | 2 | 1 | no | — |
| 182 | C3 | — | 3 | 4 | no | — |
| 183 | C3 | — | 3 | 12 | no | — |
| 184 | C3 | — | 6 | 4 | no | — |
| 185 | C3 | — | 6 | 48 | no | — |
| 186 | C3 | #11758087 | 1 | 1 | yes | characteristic, complemented |
| 187 | C3 | C3⋊D3×C3⋊(C3⋊D3) | 1 | 1 | yes | characteristic |
| 188 | C2 | — | 9 | 1 | no | — |
| 189 | C2 | — | 27 | 1 | no | — |
| 190 | C2 | — | 243 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | C3⋊D3×C3⋊(C3⋊D3) |
| commutator, Fitting, cosocle, hypocenter | C3×H(5,𝔽3) | C22 |
| socle | C32 | C3⋊D32 |
| LCS | G ▷ C3×H(5,𝔽3) | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊#92404 | C2 | semidirect |
| 2. C3⋊#92392 | C2 | semidirect |
| 3. C3×#92392 | C2 | semidirect |
| 7. C3×H(5,𝔽3) | C22 | semidirect |
| 13. C3×C3⋊#272 | D3 | semidirect |
| 15. #92392 | D3 | semidirect |
| 17. C32×#272 | D3 | semidirect |
| 25. H(5,𝔽3) | C2×D3 | semidirect |
| 27. C32×H(3,𝔽3) | C2×D3 | semidirect |
| 28. C32×H(3,𝔽3) | C2×D3 | semidirect |
| 45. C3⋊#272 | C3⋊D3 | semidirect |
| 47. C3×#272 | C3⋊D3 | semidirect |
| 48. C3×#272 | C3⋊D3 | semidirect |
| 59. C34 | D32 | semidirect |
| 67. C3×H(3,𝔽3) | D32 | semidirect |
| 70. C3×H(3,𝔽3) | C2×C3⋊D3 | semidirect |
| 98. #272 | C3⋊(C3⋊D3) | semidirect |
| 114. C33 | D3×C3⋊D3 | semidirect |
| 186. C3 | #11758087 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊#92404 | 2 |
| C3⋊#92392 | 2 |
| C3×#92392 | 2 |
| #11758087 | 3 |
| C3⋊D3×#272 | 3 |
| D3×C3⋊#272 | 3 |