Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
379 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 379 conjugacy classes of subgroups — too many to draw. Only its 52 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⋊D3×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×C3⋊(#131⋊C3) | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10595443 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5750⋊C3 | — | 3 | 2 | no | maximal |
| 5 | C32×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | C3⋊D3×SL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic |
| 7 | C3⋊(C3⋊(#41⋊C3)) | C4 | 1 | 1 | yes | characteristic |
| 8 | C3×#3712 | — | 4 | 1 | no | maximal |
| 9 | #41×C3⋊D3 | C2×C3 | 1 | 1 | yes | characteristic |
| 10 | D3×#41⋊C3 | — | 3 | 2 | no | — |
| 11 | D3×#41⋊C3 | — | 6 | 1 | no | — |
| 12 | C3×#948 | C2×C3 | 1 | 2 | yes | complemented |
| 13 | C3×#131⋊C3 | — | 3 | 2 | no | — |
| 14 | C2×#3712 | — | 3 | 1 | no | — |
| 15 | #63818 | — | 3 | 1 | no | — |
| 16 | #63817 | — | 3 | 2 | no | — |
| 17 | C3⋊(#131⋊C3) | — | 2 | 2 | no | — |
| 18 | C3⋊(#131⋊C3) | D3 | 1 | 2 | yes | complemented |
| 19 | C3×C4×C3⋊D3 | — | 4 | 1 | no | — |
| 20 | C32×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 21 | C32×C3⋊C8 | — | 4 | 2 | no | — |
| 22 | #5750 | — | 3 | 2 | no | — |
| 23 | #568⋊C3 | — | 9 | 1 | no | — |
| 24 | C32×#41 | C22×C3 | 1 | 1 | yes | characteristic |
| 25 | #3832 | C3×C4 | 1 | 1 | yes | characteristic |
| 26 | Q8×C3⋊D3 | C3×C4 | 1 | 1 | yes | characteristic |
| 27 | #3830 | — | 3 | 1 | no | — |
| 28 | D4×C3⋊D3 | — | 3 | 1 | no | — |
| 29 | #3828 | — | 3 | 1 | no | — |
| 30 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 31 | C3×#41⋊C3 | — | 2 | 2 | no | — |
| 32 | C3×#41⋊C3 | — | 2 | 1 | no | — |
| 33 | C3×#41⋊C3 | — | 4 | 1 | no | — |
| 34 | C3×#41⋊C3 | C2×D3 | 1 | 2 | yes | — |
| 35 | D3×SL(2,𝔽3) | — | 3 | 2 | no | — |
| 36 | D3×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 37 | C3⋊(#41⋊C3) | — | 3 | 2 | no | — |
| 38 | C3⋊(#41⋊C3) | — | 6 | 1 | no | — |
| 39 | C3×#208 | — | 3 | 2 | no | — |
| 40 | C2×C3×C3⋊C8 | — | 3 | 2 | no | — |
| 41 | C3×ΓL(1,𝔽25) | — | 12 | 2 | no | — |
| 42 | #3712 | — | 3 | 1 | no | — |
| 43 | #3712 | — | 3 | 2 | no | — |
| 44 | #3712 | A4 | 1 | 1 | yes | characteristic |
| 45 | #3710 | — | 3 | 2 | no | — |
| 46 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 47 | C33×C4 | — | 4 | 1 | no | — |
| 48 | C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 49 | D3×#41 | — | 3 | 2 | no | — |
| 50 | D3×#41 | — | 6 | 1 | no | — |
| 51 | C2×#41⋊C3 | — | 9 | 1 | no | — |
| 52 | C3×#131 | — | 3 | 2 | no | — |
| 53 | #948 | C3×D3 | 1 | 2 | yes | complemented |
| 54 | #907 | — | 9 | 2 | no | — |
| 55 | D3×M16 | — | 9 | 2 | no | — |
| 56 | #901 | — | 9 | 2 | no | — |
| 57 | C2×ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 58 | #131⋊C3 | — | 3 | 2 | no | — |
| 59 | #131⋊C3 | — | 6 | 2 | no | — |
| 60 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 61 | C32×Q8 | C2×C3×C4 | 1 | 1 | yes | characteristic |
| 62 | C32×D4 | — | 3 | 1 | no | — |
| 63 | C2×C32×C4 | — | 3 | 1 | no | — |
| 64 | #638 | — | 3 | 2 | no | — |
| 65 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 66 | C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 67 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 68 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 69 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 70 | C4×C3⋊D3 | C2×A4 | 1 | 1 | yes | characteristic |
| 71 | #634 | — | 3 | 1 | no | — |
| 72 | C3×C4×D3 | — | 12 | 2 | no | — |
| 73 | C3×C4×D3 | — | 24 | 1 | no | — |
| 74 | C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 75 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 76 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 77 | C3×SL(2,𝔽3) | C4×D3 | 1 | 2 | yes | — |
| 78 | C32×C8 | — | 12 | 2 | no | — |
| 79 | C3×C3⋊C8 | — | 3 | 2 | no | — |
| 80 | C3×C3⋊C8 | — | 4 | 2 | no | — |
| 81 | C3×C3⋊C8 | — | 8 | 2 | no | — |
| 82 | C3×C3⋊C8 | C2×A4 | 1 | 2 | yes | — |
| 83 | #568 | — | 9 | 1 | no | Sylow |
| 84 | C2×C33 | — | 4 | 1 | no | — |
| 85 | C3×C3⋊D3 | — | 8 | 1 | no | — |
| 86 | C3×#41 | — | 2 | 1 | no | — |
| 87 | C3×#41 | C2×C3×D3 | 1 | 2 | yes | — |
| 88 | #239 | — | 3 | 2 | no | — |
| 89 | #239 | — | 6 | 1 | no | — |
| 90 | D3×Q8 | — | 3 | 2 | no | — |
| 91 | D3×Q8 | — | 6 | 1 | no | — |
| 92 | #237 | — | 9 | 2 | no | — |
| 93 | #237 | — | 18 | 1 | no | — |
| 94 | D3×D4 | — | 9 | 2 | no | — |
| 95 | D3×D4 | — | 18 | 1 | no | — |
| 96 | #235 | — | 9 | 2 | no | — |
| 97 | #235 | — | 18 | 1 | no | — |
| 98 | C2×C4×D3 | — | 9 | 2 | no | — |
| 99 | C2×C4×D3 | — | 18 | 1 | no | — |
| 100 | #41⋊C3 | — | 2 | 2 | no | — |
| 101 | #41⋊C3 | — | 4 | 1 | no | — |
| 102 | #41⋊C3 | — | 9 | 1 | no | — |
| 103 | #41⋊C3 | D32 | 1 | 1 | yes | characteristic |
| 104 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 105 | C3×M16 | — | 9 | 2 | no | — |
| 106 | C3×M16 | — | 36 | 1 | no | — |
| 107 | C2×C3×C8 | — | 9 | 2 | no | — |
| 108 | #208 | — | 3 | 2 | no | — |
| 109 | C2×C3⋊C8 | — | 3 | 2 | no | — |
| 110 | ΓL(1,𝔽25) | — | 3 | 2 | no | — |
| 111 | ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 112 | ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 113 | ΓL(1,𝔽25) | — | 9 | 2 | no | — |
| 114 | D3×C8 | — | 9 | 4 | no | — |
| 115 | C22×C32 | — | 3 | 1 | no | — |
| 116 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 117 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 118 | C2×C3⋊D3 | C4×A4 | 1 | 1 | yes | characteristic |
| 119 | C2×C3×D3 | — | 12 | 2 | no | — |
| 120 | C2×C3×D3 | — | 24 | 1 | no | — |
| 121 | C32×C4 | — | 3 | 1 | no | — |
| 122 | C32×C4 | — | 4 | 2 | no | — |
| 123 | C32×C4 | — | 8 | 1 | no | — |
| 124 | C32×C4 | — | 8 | 2 | no | — |
| 125 | C32×C4 | — | 16 | 1 | no | — |
| 126 | C32×C4 | C22×A4 | 1 | 1 | yes | characteristic |
| 127 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 128 | C3⋊Dic3 | C4×A4 | 1 | 1 | yes | characteristic |
| 129 | C3×Dic3 | — | 12 | 2 | no | — |
| 130 | C3×Dic3 | — | 24 | 1 | no | — |
| 131 | C2×#41 | — | 9 | 1 | no | — |
| 132 | #131 | — | 3 | 2 | no | — |
| 133 | #131 | — | 27 | 2 | no | — |
| 134 | C2×M16 | — | 27 | 1 | no | — |
| 135 | C2×M16 | — | 27 | 1 | no | — |
| 136 | C33 | — | 4 | 1 | no | Sylow |
| 137 | C22×D3 | — | 9 | 2 | no | — |
| 138 | C22×D3 | — | 18 | 1 | no | — |
| 139 | C3×Q8 | — | 2 | 1 | no | — |
| 140 | C3×Q8 | C3×C4×D3 | 1 | 2 | yes | — |
| 141 | C3×D4 | — | 3 | 2 | no | — |
| 142 | C3×D4 | — | 6 | 1 | no | — |
| 143 | C2×C3×C4 | — | 3 | 2 | no | — |
| 144 | C2×C3×C4 | — | 6 | 1 | no | — |
| 145 | C2×C3×C4 | — | 36 | 1 | no | — |
| 146 | #67 | — | 9 | 4 | no | — |
| 147 | #67 | — | 18 | 2 | no | — |
| 148 | C2×Dic3 | — | 9 | 2 | no | — |
| 149 | C2×Dic3 | — | 18 | 1 | no | — |
| 150 | D12 | — | 9 | 2 | no | — |
| 151 | D12 | — | 18 | 1 | no | — |
| 152 | C4×D3 | — | 3 | 2 | no | — |
| 153 | C4×D3 | — | 6 | 1 | no | — |
| 154 | C4×D3 | — | 9 | 2 | no | — |
| 155 | C4×D3 | — | 9 | 2 | no | — |
| 156 | C4×D3 | — | 9 | 2 | no | — |
| 157 | C4×D3 | — | 18 | 1 | no | — |
| 158 | C4×D3 | — | 18 | 1 | no | — |
| 159 | C4×D3 | — | 18 | 1 | no | — |
| 160 | Dic6 | — | 9 | 2 | no | — |
| 161 | Dic6 | — | 18 | 1 | no | — |
| 162 | SL(2,𝔽3) | — | 2 | 2 | no | — |
| 163 | SL(2,𝔽3) | — | 4 | 1 | no | — |
| 164 | SL(2,𝔽3) | #624 | 1 | 1 | yes | characteristic |
| 165 | C3×C8 | — | 3 | 2 | no | — |
| 166 | C3×C8 | — | 9 | 2 | no | — |
| 167 | C3×C8 | — | 12 | 2 | no | — |
| 168 | C3×C8 | — | 24 | 2 | no | — |
| 169 | C3⋊C8 | — | 3 | 2 | no | — |
| 170 | C3⋊C8 | D3×A4 | 1 | 2 | yes | — |
| 171 | C2×C32 | — | 4 | 2 | no | — |
| 172 | C2×C32 | — | 6 | 1 | no | — |
| 173 | C2×C32 | — | 8 | 1 | no | — |
| 174 | C2×C32 | — | 8 | 2 | no | — |
| 175 | C2×C32 | — | 16 | 1 | no | — |
| 176 | C2×C32 | C2×C4×A4 | 1 | 1 | yes | characteristic |
| 177 | C3⋊D3 | — | 2 | 1 | no | — |
| 178 | C3⋊D3 | — | 6 | 1 | no | — |
| 179 | C3×D3 | — | 24 | 2 | no | — |
| 180 | C3×D3 | — | 24 | 2 | no | — |
| 181 | #41 | — | 9 | 1 | no | — |
| 182 | #41 | — | 27 | 1 | no | — |
| 183 | #41 | — | 27 | 1 | no | — |
| 184 | #41 | C3×D32 | 1 | 1 | yes | characteristic |
| 185 | C2×Q8 | — | 9 | 1 | no | — |
| 186 | C2×D4 | — | 27 | 1 | no | — |
| 187 | C22×C4 | — | 27 | 1 | no | — |
| 188 | M16 | — | 9 | 1 | no | — |
| 189 | M16 | — | 9 | 2 | no | — |
| 190 | M16 | — | 27 | 2 | no | — |
| 191 | M16 | — | 27 | 1 | no | — |
| 192 | C2×C8 | — | 9 | 2 | no | — |
| 193 | C2×C8 | — | 27 | 2 | no | — |
| 194 | C22×C3 | — | 3 | 2 | no | — |
| 195 | C22×C3 | — | 6 | 1 | no | — |
| 196 | C22×C3 | — | 36 | 1 | no | — |
| 197 | C2×D3 | — | 3 | 2 | no | — |
| 198 | C2×D3 | — | 6 | 1 | no | — |
| 199 | C2×D3 | — | 9 | 2 | no | — |
| 200 | C2×D3 | — | 18 | 1 | no | — |
| 201 | C2×D3 | — | 36 | 2 | no | — |
| 202 | C2×D3 | — | 36 | 2 | no | — |
| 203 | C3×C4 | — | 2 | 1 | no | — |
| 204 | C3×C4 | — | 3 | 2 | no | — |
| 205 | C3×C4 | — | 4 | 1 | no | — |
| 206 | C3×C4 | — | 6 | 1 | no | — |
| 207 | C3×C4 | — | 8 | 2 | no | — |
| 208 | C3×C4 | — | 16 | 1 | no | — |
| 209 | C3×C4 | — | 36 | 1 | no | — |
| 210 | C3×C4 | C2×D3×A4 | 1 | 2 | yes | — |
| 211 | Dic3 | — | 3 | 2 | no | — |
| 212 | Dic3 | — | 6 | 1 | no | — |
| 213 | Dic3 | — | 9 | 2 | no | — |
| 214 | Dic3 | — | 18 | 1 | no | — |
| 215 | C32 | — | 4 | 2 | no | — |
| 216 | C32 | — | 8 | 1 | no | — |
| 217 | C32 | — | 8 | 2 | no | — |
| 218 | C32 | — | 16 | 1 | no | — |
| 219 | C32 | #568⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 220 | C23 | — | 27 | 1 | no | — |
| 221 | Q8 | — | 27 | 1 | no | — |
| 222 | Q8 | C3×#624 | 1 | 1 | yes | characteristic |
| 223 | D4 | — | 3 | 1 | no | — |
| 224 | D4 | — | 27 | 1 | no | — |
| 225 | D4 | — | 27 | 2 | no | — |
| 226 | C2×C4 | — | 3 | 1 | no | — |
| 227 | C2×C4 | — | 9 | 1 | no | — |
| 228 | C2×C4 | — | 27 | 1 | no | — |
| 229 | C2×C4 | — | 27 | 1 | no | — |
| 230 | C2×C4 | — | 27 | 1 | no | — |
| 231 | C2×C4 | — | 27 | 1 | no | — |
| 232 | C8 | — | 3 | 2 | no | — |
| 233 | C8 | — | 9 | 2 | no | — |
| 234 | C2×C3 | — | 2 | 1 | no | — |
| 235 | C2×C3 | — | 4 | 1 | no | — |
| 236 | C2×C3 | — | 6 | 2 | no | — |
| 237 | C2×C3 | — | 8 | 2 | no | — |
| 238 | C2×C3 | — | 12 | 1 | no | — |
| 239 | C2×C3 | — | 16 | 1 | no | — |
| 240 | C2×C3 | — | 72 | 1 | no | — |
| 241 | C2×C3 | C4×D3×A4 | 1 | 2 | yes | — |
| 242 | D3 | — | 6 | 2 | no | — |
| 243 | D3 | — | 6 | 2 | no | — |
| 244 | D3 | — | 18 | 2 | no | — |
| 245 | D3 | — | 36 | 1 | no | — |
| 246 | C22 | — | 3 | 1 | no | — |
| 247 | C22 | — | 9 | 1 | no | — |
| 248 | C22 | — | 27 | 1 | no | — |
| 249 | C22 | — | 108 | 1 | no | — |
| 250 | C4 | — | 3 | 1 | no | — |
| 251 | C4 | — | 9 | 1 | no | — |
| 252 | C4 | — | 27 | 1 | no | — |
| 253 | C4 | D32×A4 | 1 | 1 | yes | characteristic |
| 254 | C3 | — | 2 | 1 | no | — |
| 255 | C3 | — | 4 | 1 | no | — |
| 256 | C3 | — | 8 | 2 | no | — |
| 257 | C3 | — | 16 | 1 | no | — |
| 258 | C3 | #5750⋊C3 | 1 | 2 | yes | complemented |
| 259 | C2 | — | 6 | 1 | no | — |
| 260 | C2 | — | 18 | 1 | no | — |
| 261 | C2 | — | 54 | 1 | no | — |
| 262 | C2 | A4×#624 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D32×A4 |
| commutator, hypocenter | C32×Q8 | C2×C3×C4 |
| Fitting, cosocle | C32×#41 | C22×C3 |
| socle | C2×C32 | C2×C4×A4 |
| prime radical | C3⋊D3×#41⋊C3 | C2 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C3⋊(#131⋊C3) | C2 | semidirect |
| 3. #10595443 | C3 | semidirect |
| 12. C3×#948 | C2×C3 | semidirect |
| 18. C3⋊(#131⋊C3) | D3 | semidirect |
| 53. #948 | C3×D3 | semidirect |
| 219. C32 | #568⋊C3 | semidirect |
| 258. C3 | #5750⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×#41⋊C3 | 2 |
| C3×C3⋊(#131⋊C3) | 2 |
| #10595443 | 3 |
| #5750⋊C3 | 3 |
| C3×#3712 | 4 |