Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
571 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 571 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 | #11722577 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #64266⋊C3 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #64265⋊C3 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10596587 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×#5836 | — | 3 | 1 | no | maximal |
| 6 | Q8×(C3⋊D3)⋊C3 | — | 4 | 1 | no | maximal |
| 7 | #41⋊H(3,𝔽3) | C22 | 1 | 3 | yes | — |
| 8 | (C3⋊GL(2,𝔽3))⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 9 | (C3⋊#226)⋊C3 | C22 | 1 | 3 | yes | complemented |
| 10 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | C3×#1010 | — | 3 | 1 | no | — |
| 12 | C3×#994 | — | 2 | 1 | no | — |
| 13 | C3×#994 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #64266 | C2×C3 | 1 | 3 | yes | complemented |
| 15 | #64265 | C2×C3 | 1 | 3 | yes | complemented |
| 16 | C3×#951 | — | 3 | 3 | no | — |
| 17 | C3×#950 | — | 3 | 3 | no | — |
| 18 | C3×#948 | — | 3 | 1 | no | — |
| 19 | C2×C3×#214 | — | 3 | 3 | no | — |
| 20 | C3×#932 | — | 3 | 3 | no | — |
| 21 | Q8×H(3,𝔽3) | — | 4 | 1 | no | — |
| 22 | C4×(C3⋊D3)⋊C3 | — | 4 | 3 | no | — |
| 23 | #634⋊C3 | — | 4 | 3 | no | — |
| 24 | Q8⋊H(3,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 25 | #5924 | — | 9 | 1 | no | — |
| 26 | C3×#577 | — | 9 | 1 | no | — |
| 27 | #5836 | — | 3 | 1 | no | — |
| 28 | C32×#41 | — | 3 | 1 | no | — |
| 29 | C32×#41 | C2×D3 | 1 | 3 | yes | — |
| 30 | C2×C32×D4 | — | 3 | 3 | no | — |
| 31 | Q8×C3⋊D3 | — | 4 | 1 | no | — |
| 32 | C3×D3×Q8 | — | 6 | 1 | no | — |
| 33 | C3×#237 | — | 6 | 3 | no | — |
| 34 | C3×#235 | — | 3 | 3 | no | — |
| 35 | C2×C3×Dic6 | — | 3 | 3 | no | — |
| 36 | C3×#41⋊C3 | — | 2 | 3 | no | — |
| 37 | C3×#41⋊C3 | C22×C3 | 1 | 3 | yes | — |
| 38 | C3⋊GL(2,𝔽3) | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 39 | C3⋊#226 | C22×C3 | 1 | 3 | yes | complemented |
| 40 | C3×#216 | — | 3 | 3 | no | — |
| 41 | C3×#215 | — | 3 | 1 | no | — |
| 42 | C3×#214 | — | 3 | 6 | no | — |
| 43 | C3×#214 | — | 3 | 3 | no | — |
| 44 | C3×#213 | — | 3 | 3 | no | — |
| 45 | C3×#208 | — | 3 | 3 | no | — |
| 46 | C2×C3×C3⋊C8 | — | 3 | 3 | no | — |
| 47 | C2×(C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 48 | C4×H(3,𝔽3) | — | 4 | 3 | no | — |
| 49 | (C3⋊Dic3)⋊C3 | — | 4 | 3 | no | — |
| 50 | C3×2−1+4 | — | 9 | 1 | no | — |
| 51 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 52 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 53 | #1010 | — | 3 | 1 | no | — |
| 54 | #994 | — | 3 | 1 | no | — |
| 55 | #994 | — | 6 | 1 | no | — |
| 56 | #985 | — | 9 | 3 | no | — |
| 57 | #984 | — | 9 | 3 | no | — |
| 58 | C3×#137 | — | 9 | 3 | no | — |
| 59 | C3×AGL(1,ℤ/8) | — | 9 | 3 | no | — |
| 60 | C3×#135 | — | 9 | 3 | no | — |
| 61 | C2×C3×SD16 | — | 9 | 3 | no | — |
| 62 | C3×#131 | — | 9 | 1 | no | — |
| 63 | #951 | — | 3 | 3 | no | — |
| 64 | #950 | — | 3 | 3 | no | — |
| 65 | #948 | — | 3 | 1 | no | — |
| 66 | C2×#214 | — | 3 | 3 | no | — |
| 67 | #932 | — | 3 | 3 | no | — |
| 68 | C23×C32 | — | 6 | 1 | no | — |
| 69 | C32×Q8 | — | 4 | 1 | no | — |
| 70 | C32×Q8 | — | 8 | 1 | no | — |
| 71 | C32×Q8 | S4 | 1 | 1 | yes | characteristic |
| 72 | C32×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 73 | C32×D4 | — | 3 | 3 | no | — |
| 74 | C32×D4 | — | 3 | 3 | no | — |
| 75 | C2×C32×C4 | — | 3 | 3 | no | — |
| 76 | C4×C3⋊D3 | — | 4 | 3 | no | — |
| 77 | #634 | — | 4 | 3 | no | — |
| 78 | C3×#67 | — | 6 | 3 | no | — |
| 79 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 80 | C3×D12 | — | 3 | 1 | no | — |
| 81 | C3×C4×D3 | — | 6 | 3 | no | — |
| 82 | C3×Dic6 | — | 3 | 3 | no | — |
| 83 | C3×Dic6 | — | 6 | 3 | no | — |
| 84 | C3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 85 | C3×SL(2,𝔽3) | C23×C3 | 1 | 1 | yes | characteristic |
| 86 | C3×C3⋊C8 | — | 3 | 1 | no | — |
| 87 | C3×C3⋊C8 | — | 3 | 3 | no | — |
| 88 | #577 | — | 9 | 1 | no | Sylow |
| 89 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 90 | (C3⋊D3)⋊C3 | — | 4 | 2 | no | — |
| 91 | C3×#41 | — | 3 | 1 | no | — |
| 92 | C3×#41 | — | 3 | 1 | no | — |
| 93 | C3×#41 | — | 3 | 3 | no | — |
| 94 | C3×#41 | — | 6 | 1 | no | — |
| 95 | C3×#41 | — | 9 | 3 | no | — |
| 96 | C3×#41 | — | 18 | 3 | no | — |
| 97 | C3×#41 | C2×C3×D3 | 1 | 3 | yes | — |
| 98 | C2×C3×Q8 | — | 9 | 3 | no | — |
| 99 | C2×C3×Q8 | — | 18 | 1 | no | — |
| 100 | C2×C3×D4 | — | 3 | 3 | no | — |
| 101 | C2×C3×D4 | — | 3 | 3 | no | — |
| 102 | C2×C3×D4 | — | 6 | 3 | no | — |
| 103 | D3×Q8 | — | 6 | 1 | no | — |
| 104 | D3×Q8 | — | 36 | 1 | no | — |
| 105 | #237 | — | 6 | 3 | no | — |
| 106 | #235 | — | 3 | 3 | no | — |
| 107 | C2×Dic6 | — | 3 | 3 | no | — |
| 108 | #41⋊C3 | — | 3 | 3 | no | — |
| 109 | #41⋊C3 | — | 6 | 3 | no | — |
| 110 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 111 | #226 | — | 9 | 3 | no | — |
| 112 | C3×Q16 | — | 9 | 3 | no | — |
| 113 | C3×SD16 | — | 9 | 3 | no | — |
| 114 | C3×SD16 | — | 9 | 6 | no | — |
| 115 | C3×SD16 | — | 9 | 1 | no | — |
| 116 | C3×D8 | — | 9 | 3 | no | — |
| 117 | C3×M16 | — | 9 | 3 | no | — |
| 118 | C2×C3×C8 | — | 9 | 3 | no | — |
| 119 | #216 | — | 3 | 3 | no | — |
| 120 | #215 | — | 3 | 1 | no | — |
| 121 | #214 | — | 3 | 6 | no | — |
| 122 | #214 | — | 3 | 3 | no | — |
| 123 | #213 | — | 3 | 3 | no | — |
| 124 | #208 | — | 3 | 3 | no | — |
| 125 | C2×C3⋊C8 | — | 3 | 3 | no | — |
| 126 | C22×C32 | — | 3 | 3 | no | — |
| 127 | C22×C32 | — | 24 | 1 | no | — |
| 128 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 129 | C2×C3×D3 | — | 6 | 1 | no | — |
| 130 | C32×C4 | — | 3 | 1 | no | — |
| 131 | C32×C4 | — | 4 | 3 | no | — |
| 132 | C32×C4 | — | 8 | 3 | no | — |
| 133 | C32×C4 | C2×S4 | 1 | 3 | yes | — |
| 134 | C3⋊Dic3 | — | 4 | 3 | no | — |
| 135 | C3×Dic3 | — | 6 | 3 | no | — |
| 136 | 2−1+4 | — | 9 | 1 | no | — |
| 137 | H(5,𝔽2) | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 138 | #137 | — | 9 | 3 | no | — |
| 139 | AGL(1,ℤ/8) | — | 9 | 3 | no | — |
| 140 | #135 | — | 9 | 3 | no | — |
| 141 | C2×SD16 | — | 9 | 3 | no | — |
| 142 | #131 | — | 9 | 1 | no | — |
| 143 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 144 | C23×C3 | — | 6 | 1 | no | — |
| 145 | C23×C3 | — | 6 | 3 | no | — |
| 146 | C3×Q8 | — | 3 | 1 | no | — |
| 147 | C3×Q8 | — | 3 | 1 | no | — |
| 148 | C3×Q8 | — | 9 | 3 | no | — |
| 149 | C3×Q8 | — | 12 | 1 | no | — |
| 150 | C3×Q8 | — | 18 | 3 | no | — |
| 151 | C3×Q8 | — | 24 | 1 | no | — |
| 152 | C3×Q8 | C3×S4 | 1 | 1 | yes | characteristic |
| 153 | C3×Q8 | C22×C3×D3 | 1 | 1 | yes | characteristic |
| 154 | C3×D4 | — | 3 | 3 | no | — |
| 155 | C3×D4 | — | 3 | 3 | no | — |
| 156 | C3×D4 | — | 3 | 3 | no | — |
| 157 | C3×D4 | — | 3 | 3 | no | — |
| 158 | C3×D4 | — | 6 | 3 | no | — |
| 159 | C3×D4 | — | 6 | 3 | no | — |
| 160 | C3×D4 | — | 9 | 1 | no | — |
| 161 | C3×D4 | — | 18 | 3 | no | — |
| 162 | C2×C3×C4 | — | 3 | 3 | no | — |
| 163 | C2×C3×C4 | — | 3 | 3 | no | — |
| 164 | C2×C3×C4 | — | 6 | 3 | no | — |
| 165 | C2×C3×C4 | — | 18 | 3 | no | — |
| 166 | C2×C3×C4 | — | 18 | 3 | no | — |
| 167 | #67 | — | 6 | 3 | no | — |
| 168 | C2×Dic3 | — | 6 | 3 | no | — |
| 169 | D12 | — | 3 | 1 | no | — |
| 170 | C4×D3 | — | 6 | 3 | no | — |
| 171 | C4×D3 | — | 36 | 3 | no | — |
| 172 | Dic6 | — | 3 | 3 | no | — |
| 173 | Dic6 | — | 6 | 3 | no | — |
| 174 | Dic6 | — | 36 | 3 | no | — |
| 175 | SL(2,𝔽3) | — | 3 | 1 | no | — |
| 176 | SL(2,𝔽3) | — | 6 | 1 | no | — |
| 177 | C3×C8 | — | 9 | 1 | no | — |
| 178 | C3×C8 | — | 9 | 3 | no | — |
| 179 | C3⋊C8 | — | 3 | 1 | no | — |
| 180 | C3⋊C8 | — | 3 | 3 | no | — |
| 181 | C2×C32 | — | 4 | 1 | no | — |
| 182 | C2×C32 | — | 6 | 3 | no | — |
| 183 | C2×C32 | — | 8 | 1 | no | — |
| 184 | C2×C32 | C22×S4 | 1 | 1 | yes | characteristic |
| 185 | C3⋊D3 | — | 4 | 2 | no | — |
| 186 | C3×D3 | — | 12 | 1 | no | — |
| 187 | #41 | — | 3 | 1 | no | — |
| 188 | #41 | — | 9 | 3 | no | — |
| 189 | #41 | — | 18 | 3 | no | — |
| 190 | #41 | C2×(C3⋊D3)⋊C3 | 1 | 3 | yes | — |
| 191 | C2×Q8 | — | 9 | 3 | no | — |
| 192 | C2×Q8 | — | 18 | 1 | no | — |
| 193 | C2×D4 | — | 3 | 3 | no | — |
| 194 | Q16 | — | 9 | 3 | no | — |
| 195 | SD16 | — | 9 | 3 | no | — |
| 196 | SD16 | — | 9 | 6 | no | — |
| 197 | SD16 | — | 9 | 1 | no | — |
| 198 | D8 | — | 9 | 3 | no | — |
| 199 | M16 | — | 9 | 3 | no | — |
| 200 | C2×C8 | — | 9 | 3 | no | — |
| 201 | C22×C3 | — | 3 | 3 | no | — |
| 202 | C22×C3 | — | 3 | 3 | no | — |
| 203 | C22×C3 | — | 6 | 3 | no | — |
| 204 | C22×C3 | — | 18 | 1 | no | — |
| 205 | C22×C3 | — | 24 | 1 | no | — |
| 206 | C22×C3 | — | 24 | 3 | no | — |
| 207 | C2×D3 | — | 6 | 1 | no | — |
| 208 | C2×D3 | — | 36 | 1 | no | — |
| 209 | C3×C4 | — | 3 | 1 | no | — |
| 210 | C3×C4 | — | 3 | 1 | no | — |
| 211 | C3×C4 | — | 3 | 3 | no | — |
| 212 | C3×C4 | — | 6 | 1 | no | — |
| 213 | C3×C4 | — | 12 | 3 | no | — |
| 214 | C3×C4 | — | 18 | 3 | no | — |
| 215 | C3×C4 | — | 24 | 3 | no | — |
| 216 | C3×C4 | C2×C3×S4 | 1 | 3 | yes | — |
| 217 | Dic3 | — | 6 | 3 | no | — |
| 218 | Dic3 | — | 36 | 3 | no | — |
| 219 | C32 | — | 4 | 1 | no | — |
| 220 | C32 | — | 8 | 1 | no | — |
| 221 | C32 | #5924 | 1 | 1 | yes | characteristic, complemented |
| 222 | C23 | — | 6 | 1 | no | — |
| 223 | Q8 | — | 9 | 3 | no | — |
| 224 | Q8 | — | 18 | 3 | no | — |
| 225 | Q8 | (C3⋊S4)⋊C3 | 1 | 1 | yes | characteristic |
| 226 | Q8 | C22×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 227 | D4 | — | 3 | 3 | no | — |
| 228 | D4 | — | 3 | 3 | no | — |
| 229 | D4 | — | 9 | 1 | no | — |
| 230 | D4 | — | 18 | 3 | no | — |
| 231 | C2×C4 | — | 3 | 3 | no | — |
| 232 | C2×C4 | — | 18 | 3 | no | — |
| 233 | C2×C4 | — | 18 | 3 | no | — |
| 234 | C8 | — | 9 | 1 | no | — |
| 235 | C8 | — | 9 | 3 | no | — |
| 236 | C2×C3 | — | 3 | 1 | no | — |
| 237 | C2×C3 | — | 6 | 3 | no | — |
| 238 | C2×C3 | — | 6 | 3 | no | — |
| 239 | C2×C3 | — | 12 | 3 | no | — |
| 240 | C2×C3 | — | 12 | 1 | no | — |
| 241 | C2×C3 | — | 24 | 1 | no | — |
| 242 | C2×C3 | — | 36 | 1 | no | — |
| 243 | C2×C3 | C22×C3×S4 | 1 | 1 | yes | characteristic |
| 244 | D3 | — | 12 | 1 | no | — |
| 245 | D3 | — | 36 | 2 | no | — |
| 246 | C22 | — | 3 | 3 | no | — |
| 247 | C22 | — | 18 | 1 | no | — |
| 248 | C22 | — | 24 | 1 | no | — |
| 249 | C4 | — | 3 | 1 | no | — |
| 250 | C4 | — | 18 | 3 | no | — |
| 251 | C4 | C2×(C3⋊S4)⋊C3 | 1 | 3 | yes | — |
| 252 | C3 | — | 3 | 1 | no | — |
| 253 | C3 | — | 12 | 1 | no | — |
| 254 | C3 | — | 24 | 1 | no | — |
| 255 | C3 | C3×#5924 | 1 | 1 | yes | characteristic |
| 256 | C2 | — | 6 | 3 | no | — |
| 257 | C2 | — | 36 | 1 | no | — |
| 258 | C2 | C22×(C3⋊S4)⋊C3 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×(C3⋊S4)⋊C3 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×S4 |
| Fitting | C32×H(5,𝔽2) | D3 |
| hypercenter | Q8 | (C3⋊S4)⋊C3 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11722577 | C2 | semidirect |
| 2. #64266⋊C3 | C2 | semidirect |
| 3. #64265⋊C3 | C2 | semidirect |
| 4. #10596587 | C3 | semidirect |
| 8. (C3⋊GL(2,𝔽3))⋊C3 | C22 | semidirect |
| 9. (C3⋊#226)⋊C3 | C22 | semidirect |
| 10. C32×H(5,𝔽2) | D3 | semidirect |
| 13. C3×#994 | C2×C3 | semidirect |
| 14. #64266 | C2×C3 | semidirect |
| 15. #64265 | C2×C3 | semidirect |
| 38. C3⋊GL(2,𝔽3) | C22×C3 | semidirect |
| 39. C3⋊#226 | C22×C3 | semidirect |
| 137. H(5,𝔽2) | (C3⋊D3)⋊C3 | semidirect |
| 221. C32 | #5924 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722577 | 2 |
| #64266⋊C3 | 2 |
| #64265⋊C3 | 2 |
| #10596587 | 3 |
| C3×#5836 | 3 |
| Q8×(C3⋊D3)⋊C3 | 4 |