Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
207 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 28 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 | #11721335 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11720744 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720743 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5720⋊C3 | — | 3 | 1 | no | maximal |
| 5 | #10589822 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×D9⋊A4 | — | 4 | 1 | no | maximal |
| 7 | C9⋊(C42⋊C3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3×#127⋊C3 | D3 | 1 | 1 | yes | characteristic |
| 9 | C3⋊(#126⋊C3) | — | 3 | 1 | no | — |
| 10 | C3⋊(#126⋊C3) | — | 3 | 1 | no | — |
| 11 | C9×#127 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 12 | #63501 | — | 3 | 1 | no | — |
| 13 | #63501 | — | 3 | 1 | no | — |
| 14 | #63499 | — | 3 | 1 | no | — |
| 15 | #63497 | — | 3 | 1 | no | — |
| 16 | #63438 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 17 | #63438 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 18 | #127⋊C9 | — | 2 | 1 | no | — |
| 19 | C2×C9⋊A4 | — | 4 | 1 | no | — |
| 20 | D9⋊A4 | — | 4 | 1 | no | — |
| 21 | D9⋊A4 | — | 4 | 1 | no | — |
| 22 | #5720 | — | 3 | 1 | no | — |
| 23 | #560⋊C3 | — | 9 | 1 | no | — |
| 24 | C2×D3×A4 | — | 12 | 1 | no | — |
| 25 | C23×D9 | A4 | 1 | 1 | yes | characteristic |
| 26 | C3×C42⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 27 | C2×D4×C9 | — | 3 | 1 | no | — |
| 28 | C2×D4×C9 | — | 3 | 1 | no | — |
| 29 | C2×#611 | — | 3 | 1 | no | — |
| 30 | C2×#611 | — | 3 | 1 | no | — |
| 31 | C22×Dic9 | — | 3 | 1 | no | — |
| 32 | C42×C9 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 33 | #3676 | — | 3 | 1 | no | — |
| 34 | #3676 | — | 3 | 1 | no | — |
| 35 | (C2×D9)⋊C4 | — | 3 | 1 | no | — |
| 36 | (C2×D9)⋊C4 | — | 3 | 1 | no | — |
| 37 | Dic9⋊C4 | — | 3 | 1 | no | — |
| 38 | Dic9⋊C4 | — | 3 | 1 | no | — |
| 39 | C42⋊C9 | — | 2 | 1 | no | — |
| 40 | C2×AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 41 | C9⋊A4 | — | 4 | 1 | no | — |
| 42 | C3×#127 | C3×D3 | 1 | 1 | yes | characteristic |
| 43 | #939 | — | 9 | 1 | no | — |
| 44 | #939 | — | 9 | 1 | no | — |
| 45 | #937 | — | 9 | 1 | no | — |
| 46 | #935 | — | 9 | 1 | no | — |
| 47 | #876 | — | 3 | 1 | no | — |
| 48 | #876 | — | 3 | 1 | no | — |
| 49 | #127⋊C3 | — | 3 | 1 | no | — |
| 50 | #126⋊C3 | — | 9 | 1 | no | — |
| 51 | #126⋊C3 | — | 9 | 1 | no | — |
| 52 | C2×C3×A4 | — | 4 | 1 | no | — |
| 53 | D3×A4 | — | 12 | 1 | no | — |
| 54 | D3×A4 | — | 12 | 1 | no | — |
| 55 | C23×C9 | — | 3 | 1 | no | — |
| 56 | C23×C9 | C2×A4 | 1 | 1 | yes | characteristic |
| 57 | C22×D9 | — | 12 | 1 | no | — |
| 58 | C22×D9 | C2×A4 | 1 | 1 | yes | characteristic |
| 59 | C22×D9 | C2×A4 | 1 | 1 | yes | characteristic |
| 60 | C2×C22⋊C9 | — | 8 | 1 | no | — |
| 61 | D4×C9 | — | 12 | 1 | no | — |
| 62 | D4×C9 | — | 12 | 1 | no | — |
| 63 | C2×C4×C9 | — | 3 | 1 | no | — |
| 64 | #611 | — | 12 | 1 | no | — |
| 65 | #611 | — | 12 | 1 | no | — |
| 66 | C2×Dic9 | — | 3 | 1 | no | — |
| 67 | C2×Dic9 | — | 3 | 1 | no | — |
| 68 | C2×Dic9 | — | 12 | 1 | no | — |
| 69 | #560 | — | 9 | 1 | no | Sylow |
| 70 | C2×3−1+2 | — | 16 | 1 | no | — |
| 71 | AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 72 | AGL(1,ℤ/9) | — | 16 | 1 | no | — |
| 73 | C23×D3 | — | 3 | 1 | no | — |
| 74 | C22×A4 | — | 36 | 1 | no | — |
| 75 | C2×C3×D4 | — | 3 | 1 | no | — |
| 76 | C2×C3×D4 | — | 3 | 1 | no | — |
| 77 | C2×#67 | — | 9 | 1 | no | — |
| 78 | C2×#67 | — | 9 | 1 | no | — |
| 79 | C22×Dic3 | — | 9 | 1 | no | — |
| 80 | C3×C42 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 81 | #217 | — | 9 | 1 | no | — |
| 82 | #217 | — | 9 | 1 | no | — |
| 83 | (C2×D3)⋊C4 | — | 9 | 1 | no | — |
| 84 | (C2×D3)⋊C4 | — | 9 | 1 | no | — |
| 85 | Dic3⋊C4 | — | 9 | 1 | no | — |
| 86 | Dic3⋊C4 | — | 9 | 1 | no | — |
| 87 | C42⋊C3 | — | 3 | 1 | no | — |
| 88 | C2×C3×D3 | — | 48 | 1 | no | — |
| 89 | C3×A4 | — | 4 | 1 | no | — |
| 90 | C22×C9 | — | 6 | 1 | no | — |
| 91 | C22×C9 | — | 6 | 1 | no | — |
| 92 | C22×C9 | — | 6 | 1 | no | — |
| 93 | C22×C9 | — | 6 | 1 | no | — |
| 94 | C22×C9 | C22×A4 | 1 | 1 | yes | characteristic |
| 95 | C2×D9 | — | 4 | 1 | no | — |
| 96 | C2×D9 | — | 6 | 1 | no | — |
| 97 | C2×D9 | — | 6 | 1 | no | — |
| 98 | C2×D9 | — | 12 | 1 | no | — |
| 99 | C22⋊C9 | — | 8 | 1 | no | — |
| 100 | C4×C9 | — | 6 | 1 | no | — |
| 101 | Dic9 | — | 6 | 1 | no | — |
| 102 | Dic9 | — | 6 | 1 | no | — |
| 103 | #127 | AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 104 | #126 | — | 9 | 1 | no | — |
| 105 | #126 | — | 9 | 1 | no | — |
| 106 | #123 | — | 27 | 1 | no | — |
| 107 | #121 | — | 27 | 1 | no | — |
| 108 | #121 | — | 27 | 1 | no | — |
| 109 | #120 | — | 27 | 1 | no | — |
| 110 | 3−1+2 | — | 16 | 1 | no | Sylow |
| 111 | C23×C3 | — | 3 | 1 | no | — |
| 112 | C23×C3 | D3×A4 | 1 | 1 | yes | characteristic |
| 113 | C22×D3 | — | 3 | 1 | no | — |
| 114 | C22×D3 | — | 3 | 1 | no | — |
| 115 | C22×D3 | — | 36 | 1 | no | — |
| 116 | C2×A4 | — | 12 | 1 | no | — |
| 117 | C2×A4 | — | 36 | 1 | no | — |
| 118 | C2×A4 | — | 36 | 1 | no | — |
| 119 | C3×D4 | — | 12 | 1 | no | — |
| 120 | C3×D4 | — | 12 | 1 | no | — |
| 121 | C2×C3×C4 | — | 3 | 1 | no | — |
| 122 | #67 | — | 36 | 1 | no | — |
| 123 | #67 | — | 36 | 1 | no | — |
| 124 | C2×Dic3 | — | 9 | 1 | no | — |
| 125 | C2×Dic3 | — | 9 | 1 | no | — |
| 126 | C2×Dic3 | — | 36 | 1 | no | — |
| 127 | C2×C32 | — | 16 | 1 | no | — |
| 128 | C3×D3 | — | 48 | 1 | no | — |
| 129 | C3×D3 | — | 48 | 1 | no | — |
| 130 | C2×C9 | — | 3 | 1 | no | — |
| 131 | C2×C9 | — | 4 | 1 | no | — |
| 132 | C2×C9 | — | 12 | 1 | no | — |
| 133 | C2×C9 | — | 32 | 1 | no | — |
| 134 | D9 | — | 4 | 1 | no | — |
| 135 | D9 | — | 4 | 1 | no | — |
| 136 | C24 | — | 9 | 1 | no | — |
| 137 | C2×D4 | — | 3 | 1 | no | — |
| 138 | C2×D4 | — | 3 | 1 | no | — |
| 139 | C2×D4 | — | 27 | 1 | no | — |
| 140 | C2×D4 | — | 27 | 1 | no | — |
| 141 | C22×C4 | — | 27 | 1 | no | — |
| 142 | C4⋊C4 | — | 27 | 1 | no | — |
| 143 | C4⋊C4 | — | 27 | 1 | no | — |
| 144 | C22⋊C4 | — | 27 | 1 | no | — |
| 145 | C22⋊C4 | — | 27 | 1 | no | — |
| 146 | C22⋊C4 | — | 27 | 1 | no | — |
| 147 | C22⋊C4 | — | 27 | 1 | no | — |
| 148 | C42 | C2×AGL(1,ℤ/9) | 1 | 1 | yes | characteristic, complemented |
| 149 | C22×C3 | — | 6 | 1 | no | — |
| 150 | C22×C3 | — | 6 | 1 | no | — |
| 151 | C22×C3 | — | 6 | 1 | no | — |
| 152 | C22×C3 | — | 6 | 1 | no | — |
| 153 | C22×C3 | — | 144 | 1 | no | — |
| 154 | C22×C3 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 155 | C2×D3 | — | 12 | 1 | no | — |
| 156 | C2×D3 | — | 18 | 1 | no | — |
| 157 | C2×D3 | — | 18 | 1 | no | — |
| 158 | C2×D3 | — | 36 | 1 | no | — |
| 159 | A4 | — | 12 | 1 | no | — |
| 160 | C3×C4 | — | 6 | 1 | no | — |
| 161 | Dic3 | — | 18 | 1 | no | — |
| 162 | Dic3 | — | 18 | 1 | no | — |
| 163 | C32 | — | 16 | 1 | no | — |
| 164 | C9 | — | 32 | 1 | no | — |
| 165 | C9 | #560⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 166 | C23 | — | 3 | 1 | no | — |
| 167 | C23 | — | 9 | 1 | no | — |
| 168 | C23 | — | 9 | 1 | no | — |
| 169 | C23 | — | 108 | 1 | no | — |
| 170 | C23 | D9⋊A4 | 1 | 1 | yes | characteristic |
| 171 | D4 | — | 12 | 1 | no | — |
| 172 | D4 | — | 12 | 1 | no | — |
| 173 | D4 | — | 108 | 1 | no | — |
| 174 | D4 | — | 108 | 1 | no | — |
| 175 | C2×C4 | — | 3 | 1 | no | — |
| 176 | C2×C4 | — | 27 | 1 | no | — |
| 177 | C2×C4 | — | 27 | 1 | no | — |
| 178 | C2×C4 | — | 108 | 1 | no | — |
| 179 | C2×C3 | — | 3 | 1 | no | — |
| 180 | C2×C3 | — | 4 | 1 | no | — |
| 181 | C2×C3 | — | 12 | 1 | no | — |
| 182 | C2×C3 | — | 48 | 1 | no | — |
| 183 | C2×C3 | — | 144 | 1 | no | — |
| 184 | C2×C3 | — | 144 | 1 | no | — |
| 185 | D3 | — | 12 | 1 | no | — |
| 186 | D3 | — | 12 | 1 | no | — |
| 187 | C22 | — | 6 | 1 | no | — |
| 188 | C22 | — | 6 | 1 | no | — |
| 189 | C22 | — | 6 | 1 | no | — |
| 190 | C22 | — | 6 | 1 | no | — |
| 191 | C22 | — | 36 | 1 | no | — |
| 192 | C22 | — | 54 | 1 | no | — |
| 193 | C22 | — | 54 | 1 | no | — |
| 194 | C22 | — | 108 | 1 | no | — |
| 195 | C22 | C2×D9⋊A4 | 1 | 1 | yes | characteristic |
| 196 | C4 | — | 6 | 1 | no | — |
| 197 | C4 | — | 54 | 1 | no | — |
| 198 | C4 | — | 54 | 1 | no | — |
| 199 | C3 | — | 48 | 1 | no | — |
| 200 | C3 | #5720⋊C3 | 1 | 1 | yes | characteristic |
| 201 | C2 | — | 3 | 1 | no | — |
| 202 | C2 | — | 4 | 1 | no | — |
| 203 | C2 | — | 12 | 1 | no | — |
| 204 | C2 | — | 36 | 1 | no | — |
| 205 | C2 | — | 36 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C42×C9 | C22×C3 |
| Frattini, socle | C22×C3 | C2×D3×A4 |
| Fitting | C9×#127 | C2×C3 |
| LCS | G ▷ C42×C9 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11721335 | C2 | semidirect |
| 2. #11720744 | C2 | semidirect |
| 3. #11720743 | C2 | semidirect |
| 5. #10589822 | C3 | semidirect |
| 7. C9⋊(C42⋊C3) | C22 | semidirect |
| 11. C9×#127 | C2×C3 | semidirect |
| 16. #63438 | C2×C3 | semidirect |
| 17. #63438 | C2×C3 | semidirect |
| 32. C42×C9 | C22×C3 | semidirect |
| 103. #127 | AGL(1,ℤ/9) | semidirect |
| 148. C42 | C2×AGL(1,ℤ/9) | semidirect |
| 165. C9 | #560⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11721335 | 2 |
| #11720744 | 2 |
| #11720743 | 2 |
| #5720⋊C3 | 3 |
| #10589822 | 3 |
| C2×D9⋊A4 | 4 |