Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
248 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 29 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 | #11722412 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C9×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C9⋊GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593233 | — | 3 | 1 | no | maximal |
| 5 | #10588879 | — | 3 | 1 | no | maximal |
| 6 | C9×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | #88615 | — | 2 | 1 | no | — |
| 8 | C9×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 9 | #88423 | — | 4 | 1 | no | maximal |
| 10 | C3⋊#994 | — | 3 | 1 | no | — |
| 11 | C3×#986 | D3 | 1 | 1 | yes | characteristic |
| 12 | C3⋊GU(2,𝔽3) | — | 3 | 1 | no | — |
| 13 | #63715 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C9×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 15 | #63502 | — | 3 | 1 | no | — |
| 16 | #63495 | — | 3 | 1 | no | — |
| 17 | #63422 | — | 6 | 1 | no | — |
| 18 | #63396 | — | 3 | 1 | no | — |
| 19 | #63384 | — | 3 | 1 | no | — |
| 20 | #63383 | — | 3 | 1 | no | — |
| 21 | C9×D12 | — | 4 | 1 | no | — |
| 22 | C3×Dic18 | — | 4 | 1 | no | — |
| 23 | C9×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic, complemented |
| 24 | C3⋊(C9⋊C8) | — | 4 | 1 | no | — |
| 25 | #5429 | — | 9 | 1 | no | — |
| 26 | #5198 | — | 9 | 1 | no | — |
| 27 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | C3⋊(#41⋊C3) | — | 6 | 1 | no | — |
| 29 | C3×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 30 | #3715 | — | 12 | 1 | no | — |
| 31 | C9×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 32 | C2×D4×C9 | — | 3 | 1 | no | — |
| 33 | C2×#611 | — | 6 | 2 | no | — |
| 34 | #3701 | — | 2 | 1 | no | — |
| 35 | D4×D9 | — | 6 | 1 | no | — |
| 36 | #3697 | — | 3 | 1 | no | — |
| 37 | C2×D36 | — | 3 | 1 | no | — |
| 38 | #41⋊C9 | — | 2 | 1 | no | — |
| 39 | C9×SD16 | — | 6 | 1 | no | — |
| 40 | C9×D8 | — | 6 | 1 | no | — |
| 41 | C9×M16 | — | 3 | 1 | no | — |
| 42 | #3673 | — | 6 | 1 | no | — |
| 43 | #3672 | — | 6 | 1 | no | — |
| 44 | C4×Dic9 | — | 3 | 1 | no | — |
| 45 | #3667 | — | 3 | 1 | no | — |
| 46 | C2×D3×C9 | — | 8 | 1 | no | — |
| 47 | C3×C4×C9 | — | 4 | 1 | no | — |
| 48 | C3×Dic9 | — | 8 | 1 | no | — |
| 49 | #1009 | — | 3 | 1 | no | — |
| 50 | #994 | — | 9 | 1 | no | — |
| 51 | #986 | D9 | 1 | 1 | yes | characteristic, complemented |
| 52 | C3×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 53 | #940 | — | 9 | 1 | no | — |
| 54 | #932 | — | 9 | 1 | no | — |
| 55 | GU(2,𝔽3) | — | 9 | 1 | no | — |
| 56 | GU(2,𝔽3) | — | 18 | 1 | no | — |
| 57 | #837 | — | 9 | 1 | no | — |
| 58 | #825 | — | 9 | 1 | no | — |
| 59 | #823 | — | 9 | 1 | no | — |
| 60 | C3×D12 | — | 4 | 1 | no | — |
| 61 | C3×Dic6 | — | 12 | 1 | no | — |
| 62 | C3×SL(2,𝔽3) | #67 | 1 | 1 | yes | characteristic |
| 63 | C23×C9 | — | 6 | 1 | no | — |
| 64 | C22×D9 | — | 3 | 2 | no | — |
| 65 | C3⋊(C3⋊C8) | — | 12 | 1 | no | — |
| 66 | Q8×C9 | #67 | 1 | 1 | yes | characteristic |
| 67 | D4×C9 | — | 3 | 1 | no | — |
| 68 | D4×C9 | — | 6 | 1 | no | — |
| 69 | D4×C9 | — | 6 | 1 | no | — |
| 70 | C2×C4×C9 | — | 3 | 1 | no | — |
| 71 | #611 | — | 6 | 1 | no | — |
| 72 | #611 | — | 12 | 4 | no | — |
| 73 | C2×Dic9 | — | 6 | 1 | no | — |
| 74 | D36 | — | 3 | 1 | no | — |
| 75 | D36 | — | 6 | 1 | no | — |
| 76 | C4×D9 | — | 6 | 1 | no | — |
| 77 | Dic18 | S4 | 1 | 1 | yes | characteristic |
| 78 | Q8⋊C9 | — | 2 | 1 | no | — |
| 79 | C8×C9 | — | 6 | 1 | no | — |
| 80 | C9⋊C8 | — | 6 | 1 | no | — |
| 81 | C9⋊C8 | — | 24 | 1 | no | — |
| 82 | #453 | — | 27 | 1 | no | Sylow |
| 83 | C2×C3×C9 | — | 4 | 1 | no | — |
| 84 | D3×C9 | — | 16 | 1 | no | — |
| 85 | C3×#41 | D32 | 1 | 1 | yes | characteristic |
| 86 | C2×C3×D4 | — | 3 | 1 | no | — |
| 87 | C2×#67 | — | 18 | 2 | no | — |
| 88 | #239 | — | 6 | 1 | no | — |
| 89 | D3×D4 | — | 18 | 1 | no | — |
| 90 | #235 | — | 9 | 1 | no | — |
| 91 | C2×D12 | — | 9 | 1 | no | — |
| 92 | #41⋊C3 | — | 2 | 1 | no | — |
| 93 | #41⋊C3 | — | 18 | 1 | no | — |
| 94 | #41⋊C3 | C2×D9 | 1 | 1 | yes | characteristic |
| 95 | GL(2,𝔽3) | — | 2 | 1 | no | — |
| 96 | C3×SD16 | — | 6 | 1 | no | — |
| 97 | C3×D8 | — | 6 | 1 | no | — |
| 98 | C3×M16 | — | 3 | 1 | no | — |
| 99 | #215 | — | 36 | 1 | no | — |
| 100 | #214 | — | 18 | 1 | no | — |
| 101 | #213 | — | 18 | 1 | no | — |
| 102 | C4×Dic3 | — | 9 | 1 | no | — |
| 103 | #208 | — | 9 | 1 | no | — |
| 104 | C2×C3×D3 | — | 8 | 1 | no | — |
| 105 | C32×C4 | — | 4 | 1 | no | — |
| 106 | C3×Dic3 | — | 24 | 1 | no | — |
| 107 | C22×C9 | — | 3 | 1 | no | — |
| 108 | C22×C9 | — | 12 | 2 | no | — |
| 109 | C22×C9 | — | 12 | 1 | no | — |
| 110 | C2×D9 | — | 6 | 1 | no | — |
| 111 | C2×D9 | — | 12 | 2 | no | — |
| 112 | C4×C9 | — | 3 | 1 | no | — |
| 113 | C4×C9 | — | 8 | 1 | no | — |
| 114 | C4×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 115 | Dic9 | — | 2 | 1 | no | — |
| 116 | Dic9 | — | 6 | 2 | no | — |
| 117 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 118 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 119 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 120 | #127 | — | 27 | 1 | no | — |
| 121 | #104 | — | 27 | 1 | no | — |
| 122 | #104 | — | 27 | 1 | no | — |
| 123 | #100 | — | 27 | 1 | no | — |
| 124 | C3×C9 | — | 4 | 1 | no | Sylow |
| 125 | C23×C3 | — | 6 | 1 | no | — |
| 126 | C22×D3 | — | 9 | 2 | no | — |
| 127 | C3×Q8 | — | 36 | 1 | no | — |
| 128 | C3×Q8 | #625 | 1 | 1 | yes | characteristic |
| 129 | C3×D4 | — | 3 | 1 | no | — |
| 130 | C3×D4 | — | 6 | 1 | no | — |
| 131 | C3×D4 | — | 6 | 1 | no | — |
| 132 | C2×C3×C4 | — | 3 | 1 | no | — |
| 133 | #67 | — | 18 | 1 | no | — |
| 134 | #67 | — | 36 | 4 | no | — |
| 135 | C2×Dic3 | — | 18 | 1 | no | — |
| 136 | D12 | — | 4 | 1 | no | — |
| 137 | D12 | — | 9 | 1 | no | — |
| 138 | D12 | — | 18 | 1 | no | — |
| 139 | C4×D3 | — | 18 | 1 | no | — |
| 140 | Dic6 | — | 3 | 1 | no | — |
| 141 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 142 | SL(2,𝔽3) | #611 | 1 | 1 | yes | characteristic, complemented |
| 143 | C3×C8 | — | 6 | 1 | no | — |
| 144 | C3⋊C8 | — | 18 | 1 | no | — |
| 145 | C3⋊C8 | — | 36 | 1 | no | — |
| 146 | C3⋊C8 | — | 72 | 1 | no | — |
| 147 | C2×C32 | — | 4 | 1 | no | — |
| 148 | C3×D3 | — | 16 | 1 | no | — |
| 149 | C2×C9 | — | 6 | 1 | no | — |
| 150 | C2×C9 | — | 8 | 1 | no | — |
| 151 | C2×C9 | — | 24 | 1 | no | — |
| 152 | C2×C9 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 153 | D9 | — | 12 | 1 | no | — |
| 154 | #41 | — | 18 | 1 | no | — |
| 155 | #41 | — | 27 | 1 | no | — |
| 156 | #41 | D3×D9 | 1 | 1 | yes | characteristic |
| 157 | C2×D4 | — | 3 | 1 | no | — |
| 158 | C2×D4 | — | 27 | 1 | no | — |
| 159 | C2×D4 | — | 54 | 1 | no | — |
| 160 | C2×D4 | — | 54 | 2 | no | — |
| 161 | SD16 | — | 6 | 1 | no | — |
| 162 | SD16 | — | 54 | 1 | no | — |
| 163 | D8 | — | 6 | 1 | no | — |
| 164 | D8 | — | 54 | 1 | no | — |
| 165 | M16 | — | 3 | 1 | no | — |
| 166 | M16 | — | 27 | 1 | no | — |
| 167 | C42 | — | 27 | 1 | no | — |
| 168 | C22×C3 | — | 3 | 1 | no | — |
| 169 | C22×C3 | — | 12 | 2 | no | — |
| 170 | C22×C3 | — | 12 | 1 | no | — |
| 171 | C2×D3 | — | 8 | 1 | no | — |
| 172 | C2×D3 | — | 18 | 1 | no | — |
| 173 | C2×D3 | — | 36 | 2 | no | — |
| 174 | C3×C4 | — | 3 | 1 | no | — |
| 175 | C3×C4 | — | 4 | 1 | no | — |
| 176 | C3×C4 | — | 8 | 1 | no | — |
| 177 | C3×C4 | — | 72 | 1 | no | — |
| 178 | C3×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 179 | Dic3 | — | 6 | 1 | no | — |
| 180 | Dic3 | — | 18 | 2 | no | — |
| 181 | C32 | — | 4 | 1 | no | — |
| 182 | C9 | — | 8 | 1 | no | — |
| 183 | C9 | #5429 | 1 | 1 | yes | characteristic, complemented |
| 184 | C23 | — | 6 | 1 | no | — |
| 185 | C23 | — | 27 | 2 | no | — |
| 186 | Q8 | — | 9 | 1 | no | — |
| 187 | Q8 | #6194 | 1 | 1 | yes | characteristic |
| 188 | D4 | — | 3 | 1 | no | — |
| 189 | D4 | — | 6 | 1 | no | — |
| 190 | D4 | — | 6 | 1 | no | — |
| 191 | D4 | — | 27 | 1 | no | — |
| 192 | D4 | — | 54 | 1 | no | — |
| 193 | D4 | — | 54 | 1 | no | — |
| 194 | D4 | — | 108 | 4 | no | — |
| 195 | C2×C4 | — | 3 | 1 | no | — |
| 196 | C2×C4 | — | 54 | 1 | no | — |
| 197 | C2×C4 | — | 54 | 1 | no | — |
| 198 | C8 | — | 6 | 1 | no | — |
| 199 | C8 | — | 54 | 1 | no | — |
| 200 | C2×C3 | — | 4 | 1 | no | — |
| 201 | C2×C3 | — | 6 | 1 | no | — |
| 202 | C2×C3 | — | 8 | 1 | no | — |
| 203 | C2×C3 | — | 24 | 1 | no | — |
| 204 | C2×C3 | #64209 | 1 | 1 | yes | characteristic |
| 205 | D3 | — | 16 | 1 | no | — |
| 206 | D3 | — | 36 | 1 | no | — |
| 207 | C22 | — | 3 | 1 | no | — |
| 208 | C22 | — | 12 | 2 | no | — |
| 209 | C22 | — | 12 | 1 | no | — |
| 210 | C22 | — | 54 | 1 | no | — |
| 211 | C22 | — | 108 | 2 | no | — |
| 212 | C4 | — | 3 | 1 | no | — |
| 213 | C4 | — | 18 | 1 | no | — |
| 214 | C4 | — | 54 | 2 | no | — |
| 215 | C4 | D9×S4 | 1 | 1 | yes | characteristic |
| 216 | C3 | — | 4 | 1 | no | — |
| 217 | C3 | — | 8 | 1 | no | — |
| 218 | C3 | #10593233 | 1 | 1 | yes | characteristic |
| 219 | C2 | — | 6 | 1 | no | — |
| 220 | C2 | — | 24 | 1 | no | — |
| 221 | C2 | — | 108 | 1 | no | — |
| 222 | C2 | #11722241 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11722241 |
| commutator, cosocle | C9×#41⋊C3 | C22 |
| Frattini | C3×C4 | D3×S4 |
| Fitting | C9×#41 | C2×D3 |
| socle | C2×C3 | #64209 |
| hypercenter | C4 | D9×S4 |
| hypocenter | C9×SL(2,𝔽3) | D4 |
| LCS | G ▷ C9×#41⋊C3 ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722412 | C2 | semidirect |
| 2. C9×#986 | C2 | semidirect |
| 3. C9⋊GU(2,𝔽3) | C2 | semidirect |
| 13. #63715 | D3 | semidirect |
| 23. C9×SL(2,𝔽3) | D4 | semidirect |
| 51. #986 | D9 | semidirect |
| 142. SL(2,𝔽3) | #611 | semidirect |
| 183. C9 | #5429 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722412 | 2 |
| C9×#986 | 2 |
| C9⋊GU(2,𝔽3) | 2 |
| #10593233 | 3 |
| #10588879 | 3 |
| #88423 | 4 |