Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
256 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 24 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 | C9×#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#5935 | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C9×#453 | — | 3 | 1 | no | maximal |
| 4 | C2×C9×S4 | — | 4 | 1 | no | maximal |
| 5 | C9×GL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 6 | C3×#997 | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | C9×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | #63703 | — | 2 | 1 | no | — |
| 9 | C9×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 10 | C9×#127 | — | 3 | 1 | no | — |
| 11 | C9×#104 | — | 3 | 2 | no | — |
| 12 | C9×#100 | — | 3 | 1 | no | — |
| 13 | C2×C9×A4 | — | 4 | 1 | no | — |
| 14 | C2×C9×A4 | — | 8 | 1 | no | — |
| 15 | C9×S4 | — | 4 | 2 | no | — |
| 16 | C9×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 17 | #5935 | C9 | 1 | 1 | yes | characteristic, direct product factor |
| 18 | C3×#453 | — | 3 | 1 | no | — |
| 19 | C2×C3×S4 | — | 4 | 1 | no | — |
| 20 | C3×GL(2,𝔽3) | — | 4 | 2 | no | — |
| 21 | C9×#41 | — | 3 | 2 | no | — |
| 22 | C2×D4×C9 | — | 3 | 1 | no | — |
| 23 | C2×D4×C9 | — | 3 | 1 | no | — |
| 24 | C2×D4×C9 | — | 6 | 1 | no | — |
| 25 | C2×D4×C9 | — | 6 | 1 | no | — |
| 26 | C2×D4×C9 | — | 6 | 1 | no | — |
| 27 | C9×SD16 | — | 6 | 2 | no | — |
| 28 | C9×D8 | — | 6 | 2 | no | — |
| 29 | C9×M16 | — | 3 | 2 | no | — |
| 30 | C42×C9 | — | 3 | 1 | no | — |
| 31 | C2×D3×C9 | — | 16 | 1 | no | — |
| 32 | C9×A4 | — | 4 | 1 | no | — |
| 33 | C9×A4 | — | 8 | 1 | no | — |
| 34 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 35 | #997 | — | 2 | 1 | no | — |
| 36 | #997 | C2×C9 | 1 | 1 | yes | characteristic, complemented |
| 37 | C3×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 38 | C3×#127 | — | 3 | 1 | no | — |
| 39 | C3×#104 | — | 3 | 2 | no | — |
| 40 | C3×#100 | — | 3 | 1 | no | — |
| 41 | C2×C3×A4 | — | 4 | 1 | no | — |
| 42 | C2×C3×A4 | — | 8 | 1 | no | — |
| 43 | C3×S4 | — | 4 | 2 | no | — |
| 44 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 45 | C23×C9 | — | 2 | 1 | no | — |
| 46 | C23×C9 | — | 3 | 1 | no | — |
| 47 | C23×C9 | — | 6 | 1 | no | — |
| 48 | C23×C9 | S4 | 1 | 1 | yes | characteristic |
| 49 | C2×C22⋊C9 | — | 8 | 2 | no | — |
| 50 | C2×C22⋊C9 | — | 8 | 1 | no | — |
| 51 | Q8×C9 | S4 | 1 | 2 | yes | complemented |
| 52 | D4×C9 | — | 3 | 2 | no | — |
| 53 | D4×C9 | — | 6 | 2 | no | — |
| 54 | D4×C9 | — | 6 | 2 | no | — |
| 55 | D4×C9 | — | 12 | 2 | no | — |
| 56 | D4×C9 | — | 12 | 2 | no | — |
| 57 | C2×C4×C9 | — | 3 | 1 | no | — |
| 58 | C2×C4×C9 | — | 6 | 1 | no | — |
| 59 | C2×C4×C9 | — | 6 | 1 | no | — |
| 60 | Q8⋊C9 | — | 8 | 2 | no | — |
| 61 | C8×C9 | — | 6 | 2 | no | — |
| 62 | #453 | — | 3 | 1 | no | Sylow |
| 63 | C2×C3×C9 | — | 16 | 1 | no | — |
| 64 | D3×C9 | — | 16 | 2 | no | — |
| 65 | C2×S4 | — | 4 | 1 | no | — |
| 66 | C3×#41 | — | 3 | 2 | no | — |
| 67 | C2×C3×D4 | — | 3 | 1 | no | — |
| 68 | C2×C3×D4 | — | 3 | 1 | no | — |
| 69 | C2×C3×D4 | — | 6 | 1 | no | — |
| 70 | C2×C3×D4 | — | 6 | 1 | no | — |
| 71 | C2×C3×D4 | — | 6 | 1 | no | — |
| 72 | GL(2,𝔽3) | — | 4 | 2 | no | — |
| 73 | C3×SD16 | — | 6 | 2 | no | — |
| 74 | C3×D8 | — | 6 | 2 | no | — |
| 75 | C3×M16 | — | 3 | 2 | no | — |
| 76 | C3×C42 | — | 3 | 1 | no | — |
| 77 | C2×C3×D3 | — | 16 | 1 | no | — |
| 78 | C3×A4 | — | 4 | 1 | no | — |
| 79 | C3×A4 | — | 8 | 1 | no | — |
| 80 | C22×C9 | — | 3 | 1 | no | — |
| 81 | C22×C9 | — | 4 | 1 | no | — |
| 82 | C22×C9 | — | 6 | 1 | no | — |
| 83 | C22×C9 | — | 8 | 1 | no | — |
| 84 | C22×C9 | — | 12 | 1 | no | — |
| 85 | C22×C9 | — | 12 | 1 | no | — |
| 86 | C22×C9 | — | 12 | 1 | no | — |
| 87 | C22×C9 | — | 12 | 1 | no | — |
| 88 | C22⋊C9 | — | 8 | 1 | no | — |
| 89 | C22⋊C9 | — | 8 | 2 | no | — |
| 90 | C4×C9 | — | 3 | 2 | no | — |
| 91 | C4×C9 | — | 6 | 2 | no | — |
| 92 | H(5,𝔽2) | D3×C9 | 1 | 1 | yes | characteristic, complemented |
| 93 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 94 | #127 | — | 3 | 1 | no | — |
| 95 | #104 | — | 3 | 2 | no | — |
| 96 | #100 | — | 3 | 1 | no | — |
| 97 | C3×C9 | — | 16 | 1 | no | Sylow |
| 98 | C23×C3 | — | 2 | 1 | no | — |
| 99 | C23×C3 | — | 3 | 1 | no | — |
| 100 | C23×C3 | — | 6 | 1 | no | — |
| 101 | C23×C3 | C3×S4 | 1 | 1 | yes | characteristic |
| 102 | C2×A4 | — | 4 | 1 | no | — |
| 103 | C2×A4 | — | 8 | 1 | no | — |
| 104 | C2×A4 | — | 8 | 2 | no | — |
| 105 | C2×A4 | — | 8 | 1 | no | — |
| 106 | S4 | — | 4 | 2 | no | — |
| 107 | C3×Q8 | C3×S4 | 1 | 2 | yes | — |
| 108 | C3×D4 | — | 3 | 2 | no | — |
| 109 | C3×D4 | — | 6 | 2 | no | — |
| 110 | C3×D4 | — | 6 | 2 | no | — |
| 111 | C3×D4 | — | 12 | 2 | no | — |
| 112 | C3×D4 | — | 12 | 2 | no | — |
| 113 | C2×C3×C4 | — | 3 | 1 | no | — |
| 114 | C2×C3×C4 | — | 6 | 1 | no | — |
| 115 | C2×C3×C4 | — | 6 | 1 | no | — |
| 116 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 117 | SL(2,𝔽3) | — | 8 | 2 | no | — |
| 118 | C3×C8 | — | 6 | 2 | no | — |
| 119 | C2×C32 | — | 16 | 1 | no | — |
| 120 | C3×D3 | — | 16 | 2 | no | — |
| 121 | C2×C9 | — | 6 | 1 | no | — |
| 122 | C2×C9 | — | 12 | 1 | no | — |
| 123 | C2×C9 | — | 24 | 1 | no | — |
| 124 | C2×C9 | — | 32 | 1 | no | — |
| 125 | C2×C9 | #1020 | 1 | 1 | yes | characteristic |
| 126 | #41 | — | 3 | 2 | no | — |
| 127 | C2×D4 | — | 3 | 1 | no | — |
| 128 | C2×D4 | — | 3 | 1 | no | — |
| 129 | C2×D4 | — | 6 | 1 | no | — |
| 130 | C2×D4 | — | 6 | 1 | no | — |
| 131 | C2×D4 | — | 6 | 1 | no | — |
| 132 | SD16 | — | 6 | 2 | no | — |
| 133 | D8 | — | 6 | 2 | no | — |
| 134 | M16 | — | 3 | 2 | no | — |
| 135 | C42 | — | 3 | 1 | no | — |
| 136 | C22×C3 | — | 3 | 1 | no | — |
| 137 | C22×C3 | — | 4 | 1 | no | — |
| 138 | C22×C3 | — | 6 | 1 | no | — |
| 139 | C22×C3 | — | 8 | 1 | no | — |
| 140 | C22×C3 | — | 12 | 1 | no | — |
| 141 | C22×C3 | — | 12 | 1 | no | — |
| 142 | C22×C3 | — | 12 | 1 | no | — |
| 143 | C22×C3 | — | 12 | 1 | no | — |
| 144 | C2×D3 | — | 16 | 1 | no | — |
| 145 | A4 | — | 4 | 1 | no | — |
| 146 | A4 | — | 8 | 1 | no | — |
| 147 | A4 | — | 8 | 1 | no | — |
| 148 | A4 | — | 8 | 2 | no | — |
| 149 | C3×C4 | — | 3 | 2 | no | — |
| 150 | C3×C4 | — | 6 | 2 | no | — |
| 151 | C32 | — | 16 | 1 | no | — |
| 152 | C9 | — | 32 | 1 | no | — |
| 153 | C9 | #5935 | 1 | 1 | yes | characteristic, direct product factor |
| 154 | C23 | — | 2 | 1 | no | — |
| 155 | C23 | — | 3 | 1 | no | — |
| 156 | C23 | — | 6 | 1 | no | — |
| 157 | C23 | C9×S4 | 1 | 1 | yes | characteristic |
| 158 | Q8 | C9×S4 | 1 | 2 | yes | complemented |
| 159 | D4 | — | 3 | 2 | no | — |
| 160 | D4 | — | 6 | 2 | no | — |
| 161 | D4 | — | 6 | 2 | no | — |
| 162 | D4 | — | 12 | 2 | no | — |
| 163 | D4 | — | 12 | 2 | no | — |
| 164 | C2×C4 | — | 3 | 1 | no | — |
| 165 | C2×C4 | — | 6 | 1 | no | — |
| 166 | C2×C4 | — | 6 | 1 | no | — |
| 167 | C8 | — | 6 | 2 | no | — |
| 168 | C2×C3 | — | 6 | 1 | no | — |
| 169 | C2×C3 | — | 12 | 1 | no | — |
| 170 | C2×C3 | — | 16 | 1 | no | — |
| 171 | C2×C3 | — | 24 | 1 | no | — |
| 172 | C2×C3 | — | 32 | 1 | no | — |
| 173 | C2×C3 | C3×#1020 | 1 | 1 | yes | characteristic |
| 174 | D3 | — | 16 | 2 | no | — |
| 175 | C22 | — | 3 | 1 | no | — |
| 176 | C22 | — | 4 | 1 | no | — |
| 177 | C22 | — | 6 | 1 | no | — |
| 178 | C22 | — | 8 | 1 | no | — |
| 179 | C22 | — | 12 | 1 | no | — |
| 180 | C22 | — | 12 | 1 | no | — |
| 181 | C22 | — | 12 | 1 | no | — |
| 182 | C22 | — | 12 | 1 | no | — |
| 183 | C4 | — | 3 | 2 | no | — |
| 184 | C4 | — | 6 | 2 | no | — |
| 185 | C3 | — | 16 | 1 | no | — |
| 186 | C3 | — | 32 | 1 | no | — |
| 187 | C3 | C3×#5935 | 1 | 1 | yes | characteristic |
| 188 | C2 | — | 6 | 1 | no | — |
| 189 | C2 | — | 12 | 1 | no | — |
| 190 | C2 | — | 24 | 1 | no | — |
| 191 | C2 | C9×#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | #1020 |
| commutator, hypocenter | #997 | C2×C9 |
| Frattini, socle | C2×C3 | C3×#1020 |
| Fitting | C9×H(5,𝔽2) | D3 |
| prime radical | C3×#5935 | C3 |
| cosocle | C3×#997 | C2×C3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#997 | C2 | semidirect |
| 7. C9×H(5,𝔽2) | D3 | semidirect |
| 17. #5935 | C9 | direct (N × H) |
| 36. #997 | C2×C9 | semidirect |
| 51. Q8×C9 | S4 | semidirect |
| 92. H(5,𝔽2) | D3×C9 | semidirect |
| 153. C9 | #5935 | direct (N × H) |
| 158. Q8 | C9×S4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#997 | 2 |
| C3×#5935 | 3 |
| C9×#453 | 3 |
| C2×C9×S4 | 4 |
| C9×GL(2,𝔽3) | 4 |