Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
241 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 39 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×#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11722412 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11720753 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5835⋊C3 | — | 3 | 1 | no | maximal |
| 5 | #10589937 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C9×#41⋊C3 | — | 2 | 1 | no | — |
| 7 | C9×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | #88615 | — | 2 | 1 | no | — |
| 9 | C3×#3674 | — | 4 | 1 | no | maximal |
| 10 | C3×#994 | D3 | 1 | 1 | yes | characteristic |
| 11 | C3⋊#994 | — | 3 | 1 | no | — |
| 12 | C3⋊(#131⋊C3) | — | 3 | 1 | no | — |
| 13 | C9×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic |
| 14 | #63715 | C2×C3 | 1 | 1 | yes | characteristic |
| 15 | #63700 | — | 2 | 1 | no | — |
| 16 | #63513 | — | 3 | 1 | no | — |
| 17 | #63512 | — | 3 | 1 | no | — |
| 18 | #63510 | C2×C3 | 1 | 1 | yes | characteristic |
| 19 | #63495 | — | 3 | 1 | no | — |
| 20 | C2×#3673 | — | 3 | 1 | no | — |
| 21 | C3×Q8×C9 | — | 4 | 1 | no | — |
| 22 | C3×Dic18 | — | 4 | 1 | no | — |
| 23 | C9×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 24 | C3×C9⋊C8 | — | 4 | 1 | no | — |
| 25 | #5835 | — | 3 | 1 | no | — |
| 26 | #576⋊C3 | — | 9 | 1 | no | — |
| 27 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 28 | C3×#41⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C3⋊(#41⋊C3) | — | 6 | 1 | no | — |
| 30 | C3×#216 | — | 12 | 1 | no | — |
| 31 | C9×#41 | — | 2 | 1 | no | — |
| 32 | C9×#41 | — | 3 | 1 | no | — |
| 33 | C9×#41 | C22×C3 | 1 | 1 | yes | characteristic |
| 34 | C2×D4×C9 | — | 3 | 1 | no | — |
| 35 | C2×D4×C9 | — | 6 | 1 | no | — |
| 36 | #3701 | — | 2 | 1 | no | — |
| 37 | D4×D9 | — | 6 | 1 | no | — |
| 38 | #3697 | — | 3 | 1 | no | — |
| 39 | C2×D36 | — | 3 | 1 | no | — |
| 40 | #41⋊C9 | — | 2 | 2 | no | — |
| 41 | #41⋊C9 | — | 2 | 1 | no | — |
| 42 | #3675 | — | 3 | 1 | no | — |
| 43 | #3674 | A4 | 1 | 1 | yes | characteristic |
| 44 | #3673 | — | 3 | 2 | no | — |
| 45 | #3673 | — | 3 | 1 | no | — |
| 46 | #3672 | — | 3 | 1 | no | — |
| 47 | #3667 | — | 3 | 1 | no | — |
| 48 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 49 | C3×C4×C9 | — | 4 | 1 | no | — |
| 50 | C3×C4×C9 | — | 8 | 1 | no | — |
| 51 | C3×Dic9 | — | 8 | 1 | no | — |
| 52 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 53 | #1009 | — | 3 | 1 | no | — |
| 54 | #994 | — | 2 | 1 | no | — |
| 55 | #994 | — | 9 | 1 | no | — |
| 56 | #994 | D9 | 1 | 1 | yes | characteristic, complemented |
| 57 | #950 | — | 9 | 1 | no | — |
| 58 | #949 | — | 9 | 1 | no | — |
| 59 | #948 | — | 3 | 1 | no | — |
| 60 | #932 | — | 9 | 1 | no | — |
| 61 | C2×#213 | — | 9 | 1 | no | — |
| 62 | #131⋊C3 | — | 9 | 1 | no | — |
| 63 | C32×Q8 | — | 4 | 1 | no | — |
| 64 | C3×Dic6 | — | 12 | 1 | no | — |
| 65 | C3×SL(2,𝔽3) | #67 | 1 | 1 | yes | characteristic |
| 66 | C23×C9 | — | 6 | 1 | no | — |
| 67 | C22×D9 | — | 6 | 1 | no | — |
| 68 | C3×C3⋊C8 | — | 12 | 1 | no | — |
| 69 | Q8×C9 | — | 8 | 1 | no | — |
| 70 | Q8×C9 | C3×D4 | 1 | 1 | yes | characteristic |
| 71 | Q8×C9 | C2×A4 | 1 | 1 | yes | characteristic |
| 72 | D4×C9 | — | 3 | 1 | no | — |
| 73 | D4×C9 | — | 3 | 1 | no | — |
| 74 | D4×C9 | — | 6 | 1 | no | — |
| 75 | D4×C9 | — | 6 | 1 | no | — |
| 76 | C2×C4×C9 | — | 3 | 1 | no | — |
| 77 | C2×C4×C9 | — | 6 | 1 | no | — |
| 78 | #611 | — | 6 | 1 | no | — |
| 79 | D36 | — | 3 | 1 | no | — |
| 80 | D36 | — | 6 | 1 | no | — |
| 81 | C4×D9 | — | 6 | 1 | no | — |
| 82 | Dic18 | C2×A4 | 1 | 1 | yes | characteristic |
| 83 | Q8⋊C9 | — | 2 | 1 | no | — |
| 84 | C9⋊C8 | — | 3 | 1 | no | — |
| 85 | C9⋊C8 | C2×A4 | 1 | 1 | yes | characteristic |
| 86 | #576 | — | 9 | 1 | no | Sylow |
| 87 | C2×C3×C9 | — | 4 | 1 | no | — |
| 88 | C3×#41 | — | 2 | 1 | no | — |
| 89 | C3×#41 | — | 3 | 1 | no | — |
| 90 | C3×#41 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 91 | C2×C3×D4 | — | 3 | 1 | no | — |
| 92 | C2×C3×D4 | — | 6 | 1 | no | — |
| 93 | #239 | — | 6 | 1 | no | — |
| 94 | D3×D4 | — | 18 | 1 | no | — |
| 95 | #235 | — | 9 | 1 | no | — |
| 96 | C2×D12 | — | 9 | 1 | no | — |
| 97 | #41⋊C3 | — | 2 | 2 | no | — |
| 98 | #41⋊C3 | — | 2 | 1 | no | — |
| 99 | #41⋊C3 | — | 2 | 1 | no | — |
| 100 | #41⋊C3 | — | 18 | 1 | no | — |
| 101 | #41⋊C3 | C2×D9 | 1 | 1 | yes | characteristic |
| 102 | C3×Q16 | — | 36 | 1 | no | — |
| 103 | #216 | — | 3 | 1 | no | — |
| 104 | #215 | — | 9 | 1 | no | — |
| 105 | #214 | — | 9 | 1 | no | — |
| 106 | #213 | — | 9 | 1 | no | — |
| 107 | #213 | — | 9 | 2 | no | — |
| 108 | #208 | — | 9 | 1 | no | — |
| 109 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 110 | C32×C4 | — | 4 | 1 | no | — |
| 111 | C32×C4 | — | 8 | 1 | no | — |
| 112 | C3×Dic3 | — | 24 | 1 | no | — |
| 113 | C22×C9 | — | 3 | 1 | no | — |
| 114 | C22×C9 | — | 6 | 1 | no | — |
| 115 | C22×C9 | — | 24 | 1 | no | — |
| 116 | C2×D9 | — | 6 | 1 | no | — |
| 117 | C2×D9 | — | 24 | 1 | no | — |
| 118 | C4×C9 | — | 2 | 1 | no | — |
| 119 | C4×C9 | — | 3 | 1 | no | — |
| 120 | C4×C9 | — | 8 | 2 | no | — |
| 121 | C4×C9 | — | 8 | 1 | no | — |
| 122 | C4×C9 | C22×A4 | 1 | 1 | yes | characteristic |
| 123 | Dic9 | — | 2 | 1 | no | — |
| 124 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 125 | H(5,𝔽2) | C3×D9 | 1 | 1 | yes | characteristic |
| 126 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 127 | AGL(1,ℤ/8) | — | 27 | 1 | no | — |
| 128 | #135 | — | 27 | 1 | no | — |
| 129 | C2×D8 | — | 27 | 1 | no | — |
| 130 | #131 | — | 9 | 1 | no | — |
| 131 | C3×C9 | — | 4 | 1 | no | Sylow |
| 132 | C23×C3 | — | 6 | 1 | no | — |
| 133 | C22×D3 | — | 18 | 1 | no | — |
| 134 | C3×Q8 | — | 4 | 1 | no | — |
| 135 | C3×Q8 | — | 8 | 1 | no | — |
| 136 | C3×Q8 | — | 36 | 1 | no | — |
| 137 | C3×Q8 | C3×#67 | 1 | 1 | yes | characteristic |
| 138 | C3×Q8 | D3×A4 | 1 | 1 | yes | characteristic |
| 139 | C3×D4 | — | 3 | 1 | no | — |
| 140 | C3×D4 | — | 3 | 1 | no | — |
| 141 | C3×D4 | — | 6 | 1 | no | — |
| 142 | C3×D4 | — | 6 | 1 | no | — |
| 143 | C2×C3×C4 | — | 3 | 1 | no | — |
| 144 | C2×C3×C4 | — | 6 | 1 | no | — |
| 145 | #67 | — | 18 | 1 | no | — |
| 146 | D12 | — | 9 | 1 | no | — |
| 147 | D12 | — | 18 | 1 | no | — |
| 148 | C4×D3 | — | 18 | 1 | no | — |
| 149 | Dic6 | — | 3 | 1 | no | — |
| 150 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 151 | SL(2,𝔽3) | #611 | 1 | 1 | yes | characteristic |
| 152 | C3×C8 | — | 36 | 1 | no | — |
| 153 | C3⋊C8 | — | 3 | 1 | no | — |
| 154 | C3⋊C8 | — | 9 | 1 | no | — |
| 155 | C2×C32 | — | 4 | 1 | no | — |
| 156 | C2×C9 | — | 6 | 1 | no | — |
| 157 | C2×C9 | — | 8 | 1 | no | — |
| 158 | C2×C9 | — | 12 | 1 | no | — |
| 159 | C2×C9 | D4×A4 | 1 | 1 | yes | characteristic |
| 160 | D9 | — | 12 | 1 | no | — |
| 161 | #41 | — | 2 | 1 | no | — |
| 162 | #41 | — | 3 | 1 | no | — |
| 163 | #41 | — | 18 | 1 | no | — |
| 164 | #41 | — | 27 | 1 | no | — |
| 165 | #41 | C2×C3×D9 | 1 | 1 | yes | characteristic |
| 166 | C2×D4 | — | 3 | 1 | no | — |
| 167 | C2×D4 | — | 6 | 1 | no | — |
| 168 | C2×D4 | — | 27 | 1 | no | — |
| 169 | C2×D4 | — | 54 | 1 | no | — |
| 170 | Q16 | — | 9 | 1 | no | — |
| 171 | SD16 | — | 27 | 1 | no | — |
| 172 | SD16 | — | 27 | 1 | no | — |
| 173 | D8 | — | 27 | 1 | no | — |
| 174 | D8 | — | 27 | 2 | no | — |
| 175 | M16 | — | 27 | 1 | no | — |
| 176 | C2×C8 | — | 27 | 1 | no | — |
| 177 | C22×C3 | — | 3 | 1 | no | — |
| 178 | C22×C3 | — | 6 | 1 | no | — |
| 179 | C22×C3 | — | 24 | 1 | no | — |
| 180 | C2×D3 | — | 18 | 1 | no | — |
| 181 | C2×D3 | — | 72 | 1 | no | — |
| 182 | C3×C4 | — | 2 | 1 | no | — |
| 183 | C3×C4 | — | 3 | 1 | no | — |
| 184 | C3×C4 | — | 4 | 1 | no | — |
| 185 | C3×C4 | — | 8 | 1 | no | — |
| 186 | C3×C4 | — | 8 | 2 | no | — |
| 187 | C3×C4 | — | 8 | 1 | no | — |
| 188 | C3×C4 | — | 72 | 1 | no | — |
| 189 | C3×C4 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 190 | Dic3 | — | 6 | 1 | no | — |
| 191 | C32 | — | 4 | 1 | no | — |
| 192 | C9 | — | 8 | 1 | no | — |
| 193 | C9 | #576⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 194 | C23 | — | 6 | 1 | no | — |
| 195 | C23 | — | 54 | 1 | no | — |
| 196 | Q8 | — | 9 | 1 | no | — |
| 197 | Q8 | C3×#611 | 1 | 1 | yes | characteristic |
| 198 | Q8 | A4×D9 | 1 | 1 | yes | characteristic |
| 199 | D4 | — | 3 | 1 | no | — |
| 200 | D4 | — | 3 | 1 | no | — |
| 201 | D4 | — | 6 | 1 | no | — |
| 202 | D4 | — | 6 | 1 | no | — |
| 203 | D4 | — | 27 | 1 | no | — |
| 204 | D4 | — | 54 | 1 | no | — |
| 205 | D4 | — | 54 | 1 | no | — |
| 206 | C2×C4 | — | 3 | 1 | no | — |
| 207 | C2×C4 | — | 6 | 1 | no | — |
| 208 | C2×C4 | — | 54 | 1 | no | — |
| 209 | C8 | — | 9 | 1 | no | — |
| 210 | C8 | — | 27 | 1 | no | — |
| 211 | C2×C3 | — | 4 | 1 | no | — |
| 212 | C2×C3 | — | 6 | 1 | no | — |
| 213 | C2×C3 | — | 8 | 1 | no | — |
| 214 | C2×C3 | — | 12 | 1 | no | — |
| 215 | C2×C3 | A4×#67 | 1 | 1 | yes | characteristic |
| 216 | D3 | — | 36 | 1 | no | — |
| 217 | C22 | — | 3 | 1 | no | — |
| 218 | C22 | — | 6 | 1 | no | — |
| 219 | C22 | — | 24 | 1 | no | — |
| 220 | C22 | — | 54 | 1 | no | — |
| 221 | C22 | — | 216 | 1 | no | — |
| 222 | C4 | — | 2 | 1 | no | — |
| 223 | C4 | — | 3 | 1 | no | — |
| 224 | C4 | — | 18 | 1 | no | — |
| 225 | C4 | C2×A4×D9 | 1 | 1 | yes | characteristic |
| 226 | C3 | — | 4 | 1 | no | — |
| 227 | C3 | — | 8 | 1 | no | — |
| 228 | C3 | #5835⋊C3 | 1 | 1 | yes | characteristic |
| 229 | C2 | — | 6 | 1 | no | — |
| 230 | C2 | — | 12 | 1 | no | — |
| 231 | C2 | — | 108 | 1 | no | — |
| 232 | C2 | A4×#611 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | A4×#611 |
| commutator, cosocle | C9×#41 | C22×C3 |
| Frattini | C3×C4 | C2×D3×A4 |
| Fitting | C9×H(5,𝔽2) | C2×C3 |
| socle | C2×C3 | A4×#67 |
| hypercenter | Q8 | A4×D9 |
| hypocenter | Q8×C9 | C3×D4 |
| LCS | G ▷ C9×#41 ▷ Q8×C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#994 | C2 | semidirect |
| 2. #11722412 | C2 | semidirect |
| 3. #11720753 | C2 | semidirect |
| 5. #10589937 | C3 | semidirect |
| 56. #994 | D9 | semidirect |
| 193. C9 | #576⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#994 | 2 |
| #11722412 | 2 |
| #11720753 | 2 |
| #5835⋊C3 | 3 |
| #10589937 | 3 |
| C3×#3674 | 4 |