Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
432 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 41 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×#63703 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10593110 | — | 3 | 3 | no | maximal |
| 3 | #10592940 | — | 3 | 1 | no | maximal |
| 4 | C2×C3⋊((C22⋊C9)⋊C2) | — | 4 | 1 | no | maximal |
| 5 | C3⋊#3689 | — | 4 | 2 | no | maximal |
| 6 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 7 | #64158 | — | 3 | 2 | no | — |
| 8 | #64149 | — | 3 | 1 | no | — |
| 9 | #63703 | D3 | 1 | 3 | yes | complemented |
| 10 | #63664 | — | 3 | 2 | no | — |
| 11 | #63660 | — | 3 | 1 | no | — |
| 12 | C2×C3×C22⋊C9 | — | 4 | 1 | no | — |
| 13 | C2×C3×C22⋊C9 | — | 8 | 1 | no | — |
| 14 | C3⋊((C22⋊C9)⋊C2) | — | 4 | 2 | no | — |
| 15 | C3×Q8⋊C9 | — | 4 | 2 | no | — |
| 16 | #5241 | — | 9 | 3 | no | — |
| 17 | #5241 | — | 9 | 1 | no | — |
| 18 | C32×#41 | — | 3 | 2 | no | — |
| 19 | C2×C32×D4 | — | 3 | 1 | no | — |
| 20 | C2×C32×D4 | — | 6 | 1 | no | — |
| 21 | C2×#638 | — | 6 | 1 | no | — |
| 22 | C2×#638 | — | 6 | 1 | no | — |
| 23 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 24 | C2×(C22⋊C9)⋊C2 | — | 12 | 3 | no | — |
| 25 | #3755 | — | 6 | 2 | no | — |
| 26 | #3753 | — | 6 | 2 | no | — |
| 27 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 28 | #3748 | — | 3 | 2 | no | — |
| 29 | #3689 | — | 12 | 6 | no | — |
| 30 | C2×C3⋊D9 | — | 16 | 1 | no | — |
| 31 | C3×C22⋊C9 | — | 4 | 1 | no | — |
| 32 | C3×C22⋊C9 | — | 8 | 1 | no | — |
| 33 | C3×H(5,𝔽2) | D9 | 1 | 3 | yes | complemented |
| 34 | C3×H(5,𝔽2) | C3⋊D3 | 1 | 1 | yes | characteristic |
| 35 | #949 | — | 9 | 6 | no | — |
| 36 | #949 | — | 9 | 2 | no | — |
| 37 | #940 | — | 9 | 3 | no | — |
| 38 | #940 | — | 9 | 1 | no | — |
| 39 | #837 | — | 9 | 6 | no | — |
| 40 | #837 | — | 9 | 2 | no | — |
| 41 | #833 | — | 9 | 3 | no | — |
| 42 | #833 | — | 9 | 1 | no | — |
| 43 | C23×C32 | — | 2 | 1 | no | — |
| 44 | C23×C32 | — | 3 | 1 | no | — |
| 45 | C23×C32 | S4 | 1 | 1 | yes | characteristic |
| 46 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 47 | C32×Q8 | S4 | 1 | 2 | yes | — |
| 48 | C32×D4 | — | 3 | 2 | no | — |
| 49 | C32×D4 | — | 6 | 2 | no | — |
| 50 | C2×C32×C4 | — | 3 | 1 | no | — |
| 51 | C2×C32×C4 | — | 6 | 1 | no | — |
| 52 | #638 | — | 12 | 2 | no | — |
| 53 | #638 | — | 12 | 2 | no | — |
| 54 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 55 | C4⋊(C3⋊D3) | — | 6 | 2 | no | — |
| 56 | C2×C22⋊C9 | — | 4 | 3 | no | — |
| 57 | C2×C22⋊C9 | — | 8 | 3 | no | — |
| 58 | (C22⋊C9)⋊C2 | — | 12 | 6 | no | — |
| 59 | C3⋊(C3⋊C8) | — | 6 | 2 | no | — |
| 60 | Q8⋊C9 | — | 4 | 6 | no | — |
| 61 | #453 | — | 27 | 1 | no | Sylow |
| 62 | C2×C3×C9 | — | 16 | 1 | no | — |
| 63 | C3⋊D9 | — | 16 | 2 | no | — |
| 64 | C3×#41 | — | 3 | 6 | no | — |
| 65 | C3×#41 | — | 3 | 2 | no | — |
| 66 | C2×C3×D4 | — | 3 | 3 | no | — |
| 67 | C2×C3×D4 | — | 3 | 1 | no | — |
| 68 | C2×C3×D4 | — | 6 | 3 | no | — |
| 69 | C2×C3×D4 | — | 6 | 1 | no | — |
| 70 | C2×#67 | — | 18 | 1 | no | — |
| 71 | C2×#67 | — | 18 | 3 | no | — |
| 72 | C2×#67 | — | 18 | 3 | no | — |
| 73 | C2×#67 | — | 18 | 1 | no | — |
| 74 | C2×D12 | — | 9 | 1 | no | — |
| 75 | C2×D12 | — | 9 | 3 | no | — |
| 76 | #215 | — | 18 | 6 | no | — |
| 77 | #215 | — | 18 | 2 | no | — |
| 78 | #213 | — | 18 | 2 | no | — |
| 79 | #213 | — | 18 | 6 | no | — |
| 80 | C4×Dic3 | — | 9 | 1 | no | — |
| 81 | C4×Dic3 | — | 9 | 3 | no | — |
| 82 | #208 | — | 9 | 2 | no | — |
| 83 | #208 | — | 9 | 6 | no | — |
| 84 | C22×C32 | — | 3 | 1 | no | — |
| 85 | C22×C32 | — | 4 | 1 | no | — |
| 86 | C22×C32 | — | 6 | 1 | no | — |
| 87 | C22×C32 | — | 8 | 1 | no | — |
| 88 | C22×C32 | — | 12 | 1 | no | — |
| 89 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 90 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 91 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 92 | C32×C4 | — | 3 | 2 | no | — |
| 93 | C3⋊Dic3 | — | 6 | 2 | no | — |
| 94 | C2×D9 | — | 48 | 3 | no | — |
| 95 | C22⋊C9 | — | 4 | 3 | no | — |
| 96 | C22⋊C9 | — | 8 | 3 | no | — |
| 97 | H(5,𝔽2) | C3⋊D9 | 1 | 1 | yes | characteristic, complemented |
| 98 | AGL(1,ℤ/8) | — | 27 | 2 | no | — |
| 99 | #127 | — | 27 | 1 | no | — |
| 100 | #104 | — | 27 | 2 | no | — |
| 101 | #100 | — | 27 | 1 | no | — |
| 102 | C3×C9 | — | 16 | 1 | no | Sylow |
| 103 | C23×C3 | — | 2 | 1 | no | — |
| 104 | C23×C3 | — | 2 | 3 | no | — |
| 105 | C23×C3 | — | 3 | 1 | no | — |
| 106 | C23×C3 | — | 3 | 3 | no | — |
| 107 | C23×C3 | (C22⋊C9)⋊C2 | 1 | 3 | yes | — |
| 108 | C23×C3 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 109 | C22×D3 | — | 18 | 1 | no | — |
| 110 | C22×D3 | — | 18 | 3 | no | — |
| 111 | C3×Q8 | (C22⋊C9)⋊C2 | 1 | 6 | yes | complemented |
| 112 | C3×Q8 | C3⋊S4 | 1 | 2 | yes | — |
| 113 | C3×D4 | — | 3 | 2 | no | — |
| 114 | C3×D4 | — | 3 | 6 | no | — |
| 115 | C3×D4 | — | 6 | 2 | no | — |
| 116 | C3×D4 | — | 6 | 6 | no | — |
| 117 | C2×C3×C4 | — | 3 | 1 | no | — |
| 118 | C2×C3×C4 | — | 3 | 3 | no | — |
| 119 | C2×C3×C4 | — | 6 | 1 | no | — |
| 120 | C2×C3×C4 | — | 6 | 3 | no | — |
| 121 | #67 | — | 36 | 2 | no | — |
| 122 | #67 | — | 36 | 6 | no | — |
| 123 | #67 | — | 36 | 6 | no | — |
| 124 | #67 | — | 36 | 2 | no | — |
| 125 | C2×Dic3 | — | 18 | 1 | no | — |
| 126 | C2×Dic3 | — | 18 | 3 | no | — |
| 127 | D12 | — | 18 | 2 | no | — |
| 128 | D12 | — | 18 | 6 | no | — |
| 129 | C3⋊C8 | — | 18 | 2 | no | — |
| 130 | C3⋊C8 | — | 18 | 6 | no | — |
| 131 | C2×C32 | — | 6 | 1 | no | — |
| 132 | C2×C32 | — | 12 | 1 | no | — |
| 133 | C2×C32 | #1020 | 1 | 1 | yes | characteristic |
| 134 | C3⋊D3 | — | 24 | 1 | no | — |
| 135 | C2×C9 | — | 16 | 3 | no | — |
| 136 | D9 | — | 48 | 6 | no | — |
| 137 | #41 | — | 3 | 2 | no | — |
| 138 | C2×D4 | — | 3 | 1 | no | — |
| 139 | C2×D4 | — | 6 | 1 | no | — |
| 140 | C2×D4 | — | 27 | 1 | no | — |
| 141 | C2×D4 | — | 54 | 1 | no | — |
| 142 | C2×D4 | — | 54 | 1 | no | — |
| 143 | SD16 | — | 54 | 2 | no | — |
| 144 | D8 | — | 54 | 2 | no | — |
| 145 | M16 | — | 27 | 2 | no | — |
| 146 | C42 | — | 27 | 1 | no | — |
| 147 | C22×C3 | — | 3 | 1 | no | — |
| 148 | C22×C3 | — | 3 | 3 | no | — |
| 149 | C22×C3 | — | 4 | 3 | no | — |
| 150 | C22×C3 | — | 4 | 1 | no | — |
| 151 | C22×C3 | — | 6 | 1 | no | — |
| 152 | C22×C3 | — | 6 | 3 | no | — |
| 153 | C22×C3 | — | 8 | 1 | no | — |
| 154 | C22×C3 | — | 8 | 3 | no | — |
| 155 | C22×C3 | — | 12 | 3 | no | — |
| 156 | C22×C3 | — | 12 | 1 | no | — |
| 157 | C2×D3 | — | 36 | 3 | no | — |
| 158 | C2×D3 | — | 36 | 3 | no | — |
| 159 | C2×D3 | — | 36 | 1 | no | — |
| 160 | C2×D3 | — | 36 | 1 | no | — |
| 161 | C2×D3 | — | 36 | 1 | no | — |
| 162 | C2×D3 | — | 36 | 3 | no | — |
| 163 | C3×C4 | — | 3 | 2 | no | — |
| 164 | C3×C4 | — | 3 | 6 | no | — |
| 165 | Dic3 | — | 18 | 6 | no | — |
| 166 | Dic3 | — | 18 | 2 | no | — |
| 167 | C32 | #5935 | 1 | 1 | yes | characteristic |
| 168 | C9 | — | 16 | 3 | no | — |
| 169 | C23 | — | 2 | 1 | no | — |
| 170 | C23 | — | 3 | 1 | no | — |
| 171 | C23 | — | 54 | 1 | no | — |
| 172 | C23 | C3⋊((C22⋊C9)⋊C2) | 1 | 1 | yes | characteristic |
| 173 | Q8 | C3⋊((C22⋊C9)⋊C2) | 1 | 2 | yes | complemented |
| 174 | D4 | — | 3 | 2 | no | — |
| 175 | D4 | — | 6 | 2 | no | — |
| 176 | D4 | — | 54 | 2 | no | — |
| 177 | D4 | — | 108 | 2 | no | — |
| 178 | D4 | — | 108 | 2 | no | — |
| 179 | C2×C4 | — | 3 | 1 | no | — |
| 180 | C2×C4 | — | 6 | 1 | no | — |
| 181 | C2×C4 | — | 54 | 1 | no | — |
| 182 | C8 | — | 54 | 2 | no | — |
| 183 | C2×C3 | — | 6 | 3 | no | — |
| 184 | C2×C3 | — | 6 | 1 | no | — |
| 185 | C2×C3 | — | 12 | 3 | no | — |
| 186 | C2×C3 | — | 12 | 1 | no | — |
| 187 | C2×C3 | #64188 | 1 | 3 | yes | — |
| 188 | C2×C3 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 189 | D3 | — | 72 | 3 | no | — |
| 190 | D3 | — | 72 | 1 | no | — |
| 191 | C22 | — | 3 | 1 | no | — |
| 192 | C22 | — | 4 | 1 | no | — |
| 193 | C22 | — | 6 | 1 | no | — |
| 194 | C22 | — | 8 | 1 | no | — |
| 195 | C22 | — | 12 | 1 | no | — |
| 196 | C22 | — | 108 | 1 | no | — |
| 197 | C22 | — | 108 | 1 | no | — |
| 198 | C22 | — | 108 | 1 | no | — |
| 199 | C4 | — | 3 | 2 | no | — |
| 200 | C4 | — | 54 | 2 | no | — |
| 201 | C3 | #10593110 | 1 | 3 | yes | complemented |
| 202 | C3 | C3⋊#5935 | 1 | 1 | yes | characteristic |
| 203 | C2 | — | 6 | 1 | no | — |
| 204 | C2 | — | 12 | 1 | no | — |
| 205 | C2 | — | 216 | 1 | no | — |
| 206 | C2 | C3⋊#64188 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#64188 |
| commutator, cosocle, hypocenter | C3×#63703 | C2 |
| Frattini | C2×C3 | C3⋊#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| socle | C2×C32 | #1020 |
| LCS | G ▷ C3×#63703 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63703 | C2 | semidirect |
| 9. #63703 | D3 | semidirect |
| 33. C3×H(5,𝔽2) | D9 | semidirect |
| 97. H(5,𝔽2) | C3⋊D9 | semidirect |
| 111. C3×Q8 | (C22⋊C9)⋊C2 | semidirect |
| 173. Q8 | C3⋊((C22⋊C9)⋊C2) | semidirect |
| 201. C3 | #10593110 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63703 | 2 |
| #10593110 | 3 |
| #10592940 | 3 |
| C2×C3⋊((C22⋊C9)⋊C2) | 4 |
| C3⋊#3689 | 4 |