Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
666 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 666 conjugacy classes of subgroups — too many to draw. Only its 65 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 | C2×C3⋊(C3⋊(#41⋊C3)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C3⋊(C3⋊GL(2,𝔽3)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C32×#859 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593206 | — | 3 | 4 | no | maximal |
| 5 | #10592509 | — | 3 | 1 | no | maximal |
| 6 | C2×C32×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊(C3⋊(#41⋊C3)) | — | 2 | 1 | no | — |
| 8 | C3⋊(C3⋊GL(2,𝔽3)) | — | 2 | 1 | no | — |
| 9 | #88804 | — | 4 | 1 | no | maximal |
| 10 | C2×#3832 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×C3⋊(#41⋊C3) | — | 3 | 4 | no | — |
| 12 | C2×C3⋊GL(2,𝔽3) | — | 3 | 4 | no | — |
| 13 | C2×C3⋊GL(2,𝔽3) | — | 6 | 4 | no | — |
| 14 | C2×#3755 | — | 3 | 1 | no | — |
| 15 | C2×#3745 | — | 3 | 1 | no | — |
| 16 | #64104 | — | 3 | 1 | no | — |
| 17 | C3×#859 | D3 | 1 | 4 | yes | complemented |
| 18 | C32×#103 | — | 3 | 1 | no | — |
| 19 | #63647 | — | 3 | 1 | no | — |
| 20 | #63636 | — | 3 | 1 | no | — |
| 21 | C22×C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 22 | C2×C3×C3⋊Dic3 | — | 4 | 1 | no | — |
| 23 | C2×C32×Dic3 | — | 4 | 1 | no | — |
| 24 | C32×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic, complemented |
| 25 | #5427 | — | 9 | 1 | no | — |
| 26 | #4810 | — | 9 | 4 | no | — |
| 27 | C2×C32×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | #3832 | — | 2 | 1 | no | — |
| 29 | #3832 | — | 6 | 1 | no | — |
| 30 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 31 | C2×C4⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 32 | C2×C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 33 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 34 | C2×C3×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 35 | C2×C3×SL(2,𝔽3) | C2×D3 | 1 | 4 | yes | — |
| 36 | C3⋊(#41⋊C3) | — | 6 | 4 | no | — |
| 37 | C3⋊GL(2,𝔽3) | — | 6 | 8 | no | — |
| 38 | C3⋊GL(2,𝔽3) | — | 6 | 4 | no | — |
| 39 | C2×C32×C8 | — | 3 | 1 | no | — |
| 40 | C32×C4⋊C4 | — | 3 | 1 | no | — |
| 41 | #3755 | — | 6 | 1 | no | — |
| 42 | #3755 | — | 6 | 1 | no | — |
| 43 | #3752 | — | 6 | 1 | no | — |
| 44 | C2×C3⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 45 | #3745 | — | 6 | 2 | no | — |
| 46 | #3722 | — | 12 | 4 | no | — |
| 47 | C22×C33 | — | 4 | 1 | no | — |
| 48 | C2×C3⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 49 | C2×C3⋊(C3⋊D3) | — | 8 | 1 | no | — |
| 50 | C2×C3⋊(C3⋊D3) | — | 8 | 1 | no | — |
| 51 | C3×C3⋊Dic3 | — | 8 | 1 | no | — |
| 52 | C32×Dic3 | — | 8 | 1 | no | — |
| 53 | C2×#239 | — | 3 | 4 | no | — |
| 54 | C2×#41⋊C3 | — | 9 | 1 | no | — |
| 55 | C2×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 56 | C2×GL(2,𝔽3) | — | 18 | 4 | no | — |
| 57 | C2×#215 | — | 9 | 4 | no | — |
| 58 | C2×D24 | — | 9 | 4 | no | — |
| 59 | #895 | — | 9 | 4 | no | — |
| 60 | #859 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 61 | C3×#103 | — | 3 | 4 | no | — |
| 62 | (C2×D3)⋊C8 | — | 9 | 4 | no | — |
| 63 | #809 | — | 9 | 4 | no | — |
| 64 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 65 | C22×C3⋊D3 | — | 6 | 1 | no | — |
| 66 | C22×C3⋊D3 | — | 12 | 4 | no | — |
| 67 | C22×C3⋊D3 | — | 24 | 4 | no | — |
| 68 | C32×Q8 | — | 3 | 1 | no | — |
| 69 | C32×Q8 | #67 | 1 | 1 | yes | characteristic |
| 70 | C2×C32×C4 | — | 3 | 1 | no | — |
| 71 | C2×C32×C4 | — | 6 | 1 | no | — |
| 72 | C2×C3⋊Dic3 | S4 | 1 | 1 | yes | characteristic |
| 73 | C4⋊(C3⋊D3) | — | 3 | 2 | no | — |
| 74 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 75 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 76 | C4⋊(C3⋊D3) | — | 6 | 1 | no | — |
| 77 | C4⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 78 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 79 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 80 | C2×C3×Dic3 | — | 4 | 4 | no | — |
| 81 | C2×C3×Dic3 | — | 12 | 4 | no | — |
| 82 | C3×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 83 | C3×SL(2,𝔽3) | D12 | 1 | 4 | yes | complemented |
| 84 | C32×C8 | — | 6 | 1 | no | — |
| 85 | C3⋊(C3⋊C8) | — | 6 | 1 | no | — |
| 86 | #449 | — | 27 | 1 | no | Sylow |
| 87 | C2×C33 | — | 4 | 1 | no | — |
| 88 | C2×C33 | — | 4 | 1 | no | — |
| 89 | C2×C33 | — | 4 | 1 | no | — |
| 90 | C3⋊(C3⋊D3) | — | 8 | 2 | no | — |
| 91 | C2×C3×Q8 | D32 | 1 | 4 | yes | — |
| 92 | #239 | — | 6 | 4 | no | — |
| 93 | #239 | — | 18 | 4 | no | — |
| 94 | C2×D12 | — | 9 | 4 | no | — |
| 95 | C2×D12 | — | 9 | 4 | no | — |
| 96 | C2×D12 | — | 18 | 4 | no | — |
| 97 | C2×C4×D3 | — | 9 | 4 | no | — |
| 98 | #41⋊C3 | — | 18 | 1 | no | — |
| 99 | C2×SL(2,𝔽3) | — | 2 | 4 | no | — |
| 100 | C2×SL(2,𝔽3) | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 101 | GL(2,𝔽3) | — | 18 | 8 | no | — |
| 102 | GL(2,𝔽3) | — | 18 | 1 | no | — |
| 103 | C2×C3×C8 | — | 3 | 4 | no | — |
| 104 | C3×C4⋊C4 | — | 3 | 4 | no | — |
| 105 | #215 | — | 18 | 4 | no | — |
| 106 | #215 | — | 18 | 4 | no | — |
| 107 | (C2×D3)⋊C4 | — | 18 | 4 | no | — |
| 108 | (C2×D3)⋊C4 | — | 36 | 1 | no | — |
| 109 | C2×C3⋊C8 | — | 9 | 4 | no | — |
| 110 | D24 | — | 18 | 8 | no | — |
| 111 | C22×C32 | — | 4 | 4 | no | — |
| 112 | C22×C32 | — | 8 | 4 | no | — |
| 113 | C22×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 114 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 115 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 116 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 117 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 118 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 119 | C2×C3⋊D3 | — | 12 | 1 | no | — |
| 120 | C2×C3⋊D3 | — | 12 | 8 | no | — |
| 121 | C2×C3⋊D3 | — | 24 | 4 | no | — |
| 122 | C2×C3⋊D3 | — | 24 | 4 | no | — |
| 123 | C2×C3⋊D3 | — | 24 | 8 | no | — |
| 124 | C2×C3⋊D3 | — | 24 | 8 | no | — |
| 125 | C2×C3⋊D3 | — | 24 | 8 | no | — |
| 126 | C32×C4 | — | 3 | 1 | no | — |
| 127 | C32×C4 | — | 3 | 1 | no | — |
| 128 | C32×C4 | — | 12 | 1 | no | — |
| 129 | C3⋊Dic3 | — | 2 | 1 | no | — |
| 130 | C3×Dic3 | — | 8 | 4 | no | — |
| 131 | C3×Dic3 | — | 24 | 4 | no | — |
| 132 | C2×#41 | — | 9 | 1 | no | — |
| 133 | C2×SD16 | — | 27 | 1 | no | — |
| 134 | C2×D8 | — | 27 | 1 | no | — |
| 135 | #121 | — | 27 | 1 | no | — |
| 136 | #103 | — | 3 | 1 | no | — |
| 137 | #102 | — | 27 | 1 | no | — |
| 138 | C22⋊C8 | — | 27 | 1 | no | — |
| 139 | C33 | — | 4 | 1 | no | Sylow |
| 140 | C22×D3 | — | 9 | 4 | no | — |
| 141 | C22×D3 | — | 18 | 4 | no | — |
| 142 | C22×D3 | — | 36 | 1 | no | — |
| 143 | C22×D3 | — | 72 | 4 | no | — |
| 144 | C3×Q8 | — | 3 | 4 | no | — |
| 145 | C3×Q8 | #626 | 1 | 4 | yes | — |
| 146 | C2×C3×C4 | — | 3 | 4 | no | — |
| 147 | C2×C3×C4 | — | 6 | 4 | no | — |
| 148 | C2×C3×C4 | — | 36 | 1 | no | — |
| 149 | C2×Dic3 | — | 3 | 4 | no | — |
| 150 | C2×Dic3 | — | 4 | 1 | no | — |
| 151 | D12 | — | 9 | 8 | no | — |
| 152 | D12 | — | 18 | 4 | no | — |
| 153 | D12 | — | 18 | 4 | no | — |
| 154 | D12 | — | 18 | 4 | no | — |
| 155 | D12 | — | 36 | 8 | no | — |
| 156 | C4×D3 | — | 18 | 4 | no | — |
| 157 | C4×D3 | — | 18 | 4 | no | — |
| 158 | SL(2,𝔽3) | — | 2 | 4 | no | — |
| 159 | SL(2,𝔽3) | C4⋊(C3⋊D3) | 1 | 1 | yes | characteristic, complemented |
| 160 | C3×C8 | — | 6 | 4 | no | — |
| 161 | C3⋊C8 | — | 18 | 4 | no | — |
| 162 | C2×C32 | — | 4 | 4 | no | — |
| 163 | C2×C32 | — | 4 | 4 | no | — |
| 164 | C2×C32 | — | 4 | 4 | no | — |
| 165 | C2×C32 | — | 8 | 4 | no | — |
| 166 | C2×C32 | — | 8 | 4 | no | — |
| 167 | C2×C32 | — | 8 | 4 | no | — |
| 168 | C2×C32 | #985 | 1 | 1 | yes | characteristic |
| 169 | C2×C32 | #986 | 1 | 1 | yes | characteristic |
| 170 | C2×C32 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 171 | C3⋊D3 | — | 12 | 1 | no | — |
| 172 | C3⋊D3 | — | 24 | 8 | no | — |
| 173 | C3⋊D3 | — | 24 | 16 | no | — |
| 174 | C3⋊D3 | — | 24 | 1 | no | — |
| 175 | #41 | — | 18 | 1 | no | — |
| 176 | #41 | — | 54 | 1 | no | — |
| 177 | C2×Q8 | D3×C3⋊D3 | 1 | 1 | yes | characteristic |
| 178 | C2×D4 | — | 27 | 1 | no | — |
| 179 | C2×D4 | — | 27 | 1 | no | — |
| 180 | C2×D4 | — | 54 | 1 | no | — |
| 181 | C22×C4 | — | 27 | 1 | no | — |
| 182 | SD16 | — | 54 | 1 | no | — |
| 183 | SD16 | — | 54 | 1 | no | — |
| 184 | D8 | — | 54 | 2 | no | — |
| 185 | C2×C8 | — | 3 | 1 | no | — |
| 186 | C2×C8 | — | 27 | 1 | no | — |
| 187 | C4⋊C4 | — | 3 | 1 | no | — |
| 188 | C22⋊C4 | — | 54 | 1 | no | — |
| 189 | C22×C3 | — | 4 | 1 | no | — |
| 190 | C22×C3 | — | 8 | 4 | no | — |
| 191 | C22×C3 | D3×S4 | 1 | 4 | yes | — |
| 192 | C2×D3 | — | 18 | 4 | no | — |
| 193 | C2×D3 | — | 18 | 4 | no | — |
| 194 | C2×D3 | — | 18 | 4 | no | — |
| 195 | C2×D3 | — | 36 | 2 | no | — |
| 196 | C2×D3 | — | 36 | 4 | no | — |
| 197 | C2×D3 | — | 36 | 4 | no | — |
| 198 | C2×D3 | — | 36 | 4 | no | — |
| 199 | C2×D3 | — | 72 | 1 | no | — |
| 200 | C2×D3 | — | 72 | 1 | no | — |
| 201 | C2×D3 | — | 72 | 8 | no | — |
| 202 | C2×D3 | — | 72 | 8 | no | — |
| 203 | C2×D3 | — | 72 | 8 | no | — |
| 204 | C3×C4 | — | 3 | 4 | no | — |
| 205 | C3×C4 | — | 3 | 4 | no | — |
| 206 | C3×C4 | — | 12 | 4 | no | — |
| 207 | C3×C4 | — | 72 | 1 | no | — |
| 208 | Dic3 | — | 6 | 4 | no | — |
| 209 | Dic3 | — | 8 | 1 | no | — |
| 210 | C32 | — | 4 | 4 | no | — |
| 211 | C32 | — | 8 | 4 | no | — |
| 212 | C32 | #5427 | 1 | 1 | yes | characteristic, complemented |
| 213 | C23 | — | 27 | 1 | no | — |
| 214 | C23 | — | 54 | 1 | no | — |
| 215 | Q8 | — | 3 | 1 | no | — |
| 216 | Q8 | #6292 | 1 | 1 | yes | characteristic |
| 217 | D4 | — | 27 | 2 | no | — |
| 218 | D4 | — | 54 | 1 | no | — |
| 219 | D4 | — | 54 | 1 | no | — |
| 220 | D4 | — | 54 | 1 | no | — |
| 221 | D4 | — | 108 | 2 | no | — |
| 222 | C2×C4 | — | 3 | 1 | no | — |
| 223 | C2×C4 | — | 6 | 1 | no | — |
| 224 | C2×C4 | — | 9 | 1 | no | — |
| 225 | C2×C4 | — | 54 | 1 | no | — |
| 226 | C2×C4 | — | 54 | 1 | no | — |
| 227 | C8 | — | 6 | 1 | no | — |
| 228 | C8 | — | 54 | 1 | no | — |
| 229 | C2×C3 | — | 4 | 1 | no | — |
| 230 | C2×C3 | — | 4 | 1 | no | — |
| 231 | C2×C3 | — | 4 | 1 | no | — |
| 232 | C2×C3 | — | 8 | 4 | no | — |
| 233 | C2×C3 | — | 8 | 4 | no | — |
| 234 | C2×C3 | — | 8 | 4 | no | — |
| 235 | C2×C3 | #64198 | 1 | 4 | yes | — |
| 236 | C2×C3 | #64199 | 1 | 4 | yes | — |
| 237 | C2×C3 | #64210 | 1 | 4 | yes | — |
| 238 | D3 | — | 36 | 4 | no | — |
| 239 | D3 | — | 72 | 2 | no | — |
| 240 | D3 | — | 72 | 16 | no | — |
| 241 | D3 | — | 72 | 4 | no | — |
| 242 | C22 | — | 54 | 1 | no | — |
| 243 | C22 | — | 54 | 1 | no | — |
| 244 | C22 | — | 54 | 1 | no | — |
| 245 | C22 | — | 108 | 1 | no | — |
| 246 | C22 | — | 108 | 1 | no | — |
| 247 | C22 | — | 108 | 1 | no | — |
| 248 | C22 | C3⋊D3×S4 | 1 | 1 | yes | characteristic |
| 249 | C4 | — | 3 | 1 | no | — |
| 250 | C4 | — | 3 | 1 | no | — |
| 251 | C4 | — | 12 | 1 | no | — |
| 252 | C4 | — | 18 | 1 | no | — |
| 253 | C3 | — | 4 | 1 | no | — |
| 254 | C3 | — | 8 | 4 | no | — |
| 255 | C3 | #10593206 | 1 | 4 | yes | complemented |
| 256 | C2 | — | 108 | 1 | no | — |
| 257 | C2 | — | 216 | 1 | no | — |
| 258 | C2 | #11724178 | 1 | 1 | yes | characteristic |
| 259 | C2 | #11724179 | 1 | 1 | yes | characteristic |
| 260 | C2 | #11724190 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C3⋊D3×S4 |
| commutator, cosocle | C2×C32×SL(2,𝔽3) | C22 |
| Fitting | C2×C32×Q8 | C2×D3 |
| socle | C22×C32 | C2×S4 |
| hypocenter | C32×SL(2,𝔽3) | D4 |
| LCS | G ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3⋊(C3⋊(#41⋊C3)) | C2 | semidirect |
| 2. C2×C3⋊(C3⋊GL(2,𝔽3)) | C2 | semidirect |
| 3. C32×#859 | C2 | semidirect |
| 10. C2×#3832 | D3 | semidirect |
| 17. C3×#859 | D3 | semidirect |
| 24. C32×SL(2,𝔽3) | D4 | semidirect |
| 60. #859 | C3⋊D3 | semidirect |
| 83. C3×SL(2,𝔽3) | D12 | semidirect |
| 159. SL(2,𝔽3) | C4⋊(C3⋊D3) | semidirect |
| 212. C32 | #5427 | semidirect |
| 255. C3 | #10593206 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊(C3⋊(#41⋊C3)) | 2 |
| C2×C3⋊(C3⋊GL(2,𝔽3)) | 2 |
| C32×#859 | 2 |
| #10593206 | 3 |
| #10592509 | 3 |
| #88804 | 4 |