Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
307 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 37 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×D9×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×C9×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C9⋊#859 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10593217 | — | 3 | 1 | no | maximal |
| 5 | #10588850 | — | 3 | 1 | no | maximal |
| 6 | C2×C9×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#6194 | — | 4 | 1 | no | maximal |
| 8 | D9×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 9 | C9×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 10 | C2×D3×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 11 | C2×C3×GL(2,𝔽3) | D3 | 1 | 1 | yes | characteristic |
| 12 | C3⋊#859 | — | 3 | 1 | no | — |
| 13 | C2×Q8×D9 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C2×C9×SD16 | — | 3 | 1 | no | — |
| 15 | #63500 | — | 3 | 1 | no | — |
| 16 | C2×#3672 | — | 3 | 1 | no | — |
| 17 | #63421 | — | 6 | 1 | no | — |
| 18 | #63395 | — | 3 | 1 | no | — |
| 19 | (C2×D9)⋊C8 | — | 3 | 1 | no | — |
| 20 | #63376 | — | 3 | 1 | no | — |
| 21 | C22×D3×C9 | — | 4 | 1 | no | — |
| 22 | C22×C3×D9 | — | 4 | 1 | no | — |
| 23 | C2×C3⋊Dic9 | — | 4 | 1 | no | — |
| 24 | C9×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic, complemented |
| 25 | #6194 | — | 4 | 4 | no | — |
| 26 | #5421 | — | 9 | 1 | no | — |
| 27 | #5170 | — | 9 | 1 | no | — |
| 28 | C2×C3×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 29 | C2×#625 | — | 12 | 1 | no | — |
| 30 | D3×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 31 | C3×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 32 | C2×Q8×C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 33 | C2×D4×C9 | — | 3 | 1 | no | — |
| 34 | C2×#611 | — | 6 | 1 | no | — |
| 35 | Q8×D9 | — | 2 | 1 | no | — |
| 36 | Q8×D9 | — | 6 | 1 | no | — |
| 37 | C2×C4×D9 | — | 3 | 1 | no | — |
| 38 | C2×Dic18 | — | 3 | 1 | no | — |
| 39 | C2×Q8⋊C9 | — | 2 | 1 | no | — |
| 40 | C9×SD16 | — | 6 | 1 | no | — |
| 41 | C9×SD16 | — | 6 | 1 | no | — |
| 42 | C2×C8×C9 | — | 3 | 1 | no | — |
| 43 | #3676 | — | 6 | 1 | no | — |
| 44 | #3672 | — | 6 | 2 | no | — |
| 45 | C4⋊Dic9 | — | 3 | 1 | no | — |
| 46 | C2×C9⋊C8 | — | 3 | 1 | no | — |
| 47 | C22×C3×C9 | — | 4 | 1 | no | — |
| 48 | C2×D3×C9 | — | 4 | 2 | no | — |
| 49 | C2×D3×C9 | — | 8 | 1 | no | — |
| 50 | C2×D3×C9 | — | 8 | 1 | no | — |
| 51 | C2×C3×D9 | — | 4 | 2 | no | — |
| 52 | C2×C3×D9 | — | 8 | 1 | no | — |
| 53 | C2×C3×D9 | — | 8 | 1 | no | — |
| 54 | C3⋊Dic9 | — | 4 | 2 | no | — |
| 55 | C2×D3×Q8 | — | 3 | 1 | no | — |
| 56 | C22×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 57 | C2×GL(2,𝔽3) | D9 | 1 | 1 | yes | characteristic, complemented |
| 58 | C2×C3×SD16 | — | 3 | 1 | no | — |
| 59 | #938 | — | 9 | 1 | no | — |
| 60 | C2×#214 | — | 9 | 1 | no | — |
| 61 | #859 | — | 9 | 1 | no | — |
| 62 | #859 | — | 18 | 1 | no | — |
| 63 | #835 | — | 9 | 1 | no | — |
| 64 | (C2×D3)⋊C8 | — | 9 | 1 | no | — |
| 65 | #816 | — | 9 | 1 | no | — |
| 66 | C22×C3×D3 | — | 4 | 1 | no | — |
| 67 | C22×C3×D3 | — | 12 | 1 | no | — |
| 68 | C2×C3⋊Dic3 | — | 12 | 1 | no | — |
| 69 | C3×SL(2,𝔽3) | #67 | 1 | 1 | yes | characteristic |
| 70 | #625 | — | 12 | 4 | no | — |
| 71 | C23×C9 | — | 6 | 1 | no | — |
| 72 | C22×D9 | S4 | 1 | 1 | yes | characteristic |
| 73 | Q8×C9 | — | 3 | 1 | no | — |
| 74 | Q8×C9 | #67 | 1 | 1 | yes | characteristic, complemented |
| 75 | D4×C9 | — | 6 | 1 | no | — |
| 76 | D4×C9 | — | 6 | 1 | no | — |
| 77 | C2×C4×C9 | — | 3 | 1 | no | — |
| 78 | #611 | — | 12 | 2 | no | — |
| 79 | C2×Dic9 | — | 3 | 1 | no | — |
| 80 | C2×Dic9 | — | 6 | 1 | no | — |
| 81 | C2×Dic9 | — | 24 | 1 | no | — |
| 82 | C4×D9 | — | 6 | 1 | no | — |
| 83 | C4×D9 | — | 6 | 1 | no | — |
| 84 | Dic18 | — | 3 | 2 | no | — |
| 85 | Dic18 | — | 6 | 1 | no | — |
| 86 | Q8⋊C9 | — | 2 | 1 | no | — |
| 87 | C8×C9 | — | 6 | 1 | no | — |
| 88 | C9⋊C8 | — | 6 | 1 | no | — |
| 89 | #448 | — | 27 | 1 | no | Sylow |
| 90 | C2×C3×C9 | — | 4 | 1 | no | — |
| 91 | C2×C3×C9 | — | 4 | 1 | no | — |
| 92 | C2×C3×C9 | — | 4 | 1 | no | — |
| 93 | D3×C9 | — | 8 | 2 | no | — |
| 94 | C3×D9 | — | 8 | 2 | no | — |
| 95 | C2×C3×Q8 | D32 | 1 | 1 | yes | characteristic |
| 96 | C2×C3×D4 | — | 3 | 1 | no | — |
| 97 | C2×#67 | — | 18 | 1 | no | — |
| 98 | C2×#67 | — | 36 | 1 | no | — |
| 99 | D3×Q8 | — | 6 | 1 | no | — |
| 100 | D3×Q8 | — | 18 | 1 | no | — |
| 101 | C2×C4×D3 | — | 9 | 1 | no | — |
| 102 | C2×Dic6 | — | 9 | 1 | no | — |
| 103 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 104 | C2×SL(2,𝔽3) | — | 18 | 1 | no | — |
| 105 | C2×SL(2,𝔽3) | C2×D9 | 1 | 1 | yes | characteristic |
| 106 | GL(2,𝔽3) | — | 2 | 1 | no | — |
| 107 | C3×SD16 | — | 6 | 1 | no | — |
| 108 | C3×SD16 | — | 6 | 1 | no | — |
| 109 | C2×C3×C8 | — | 3 | 1 | no | — |
| 110 | #217 | — | 18 | 1 | no | — |
| 111 | #214 | — | 18 | 2 | no | — |
| 112 | C4⋊Dic3 | — | 9 | 1 | no | — |
| 113 | C2×C3⋊C8 | — | 9 | 1 | no | — |
| 114 | C22×C32 | — | 4 | 1 | no | — |
| 115 | C2×C3×D3 | — | 4 | 2 | no | — |
| 116 | C2×C3×D3 | — | 8 | 1 | no | — |
| 117 | C2×C3×D3 | — | 8 | 1 | no | — |
| 118 | C2×C3×D3 | — | 12 | 2 | no | — |
| 119 | C2×C3×D3 | — | 24 | 1 | no | — |
| 120 | C2×C3×D3 | — | 24 | 1 | no | — |
| 121 | C3⋊Dic3 | — | 12 | 2 | no | — |
| 122 | C22×C9 | — | 8 | 1 | no | — |
| 123 | C22×C9 | — | 12 | 1 | no | — |
| 124 | C22×C9 | — | 12 | 1 | no | — |
| 125 | C22×C9 | — | 12 | 1 | no | — |
| 126 | C22×C9 | C2×S4 | 1 | 1 | yes | characteristic |
| 127 | C2×D9 | — | 2 | 1 | no | — |
| 128 | C2×D9 | — | 2 | 1 | no | — |
| 129 | C2×D9 | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 130 | C4×C9 | — | 3 | 1 | no | — |
| 131 | C4×C9 | — | 3 | 1 | no | — |
| 132 | Dic9 | — | 6 | 1 | no | — |
| 133 | Dic9 | — | 12 | 1 | no | — |
| 134 | Dic9 | — | 24 | 2 | no | — |
| 135 | C22×Q8 | — | 9 | 1 | no | — |
| 136 | C2×SD16 | — | 3 | 1 | no | — |
| 137 | C2×SD16 | — | 27 | 1 | no | — |
| 138 | #121 | — | 27 | 1 | no | — |
| 139 | #103 | — | 27 | 1 | no | — |
| 140 | #103 | — | 27 | 1 | no | — |
| 141 | C22⋊C8 | — | 27 | 1 | no | — |
| 142 | C3×C9 | — | 4 | 1 | no | Sylow |
| 143 | C23×C3 | — | 6 | 1 | no | — |
| 144 | C23×C3 | — | 36 | 1 | no | — |
| 145 | C22×D3 | — | 3 | 1 | no | — |
| 146 | C22×D3 | — | 4 | 1 | no | — |
| 147 | C3×Q8 | — | 3 | 1 | no | — |
| 148 | C3×Q8 | #625 | 1 | 1 | yes | characteristic |
| 149 | C3×D4 | — | 6 | 1 | no | — |
| 150 | C3×D4 | — | 6 | 1 | no | — |
| 151 | C2×C3×C4 | — | 3 | 1 | no | — |
| 152 | #67 | — | 36 | 2 | no | — |
| 153 | #67 | — | 36 | 4 | no | — |
| 154 | C2×Dic3 | — | 9 | 1 | no | — |
| 155 | C2×Dic3 | — | 18 | 1 | no | — |
| 156 | C2×Dic3 | — | 36 | 1 | no | — |
| 157 | C2×Dic3 | — | 72 | 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 | 1 | no | — |
| 163 | SL(2,𝔽3) | #611 | 1 | 1 | yes | characteristic, complemented |
| 164 | C3×C8 | — | 6 | 1 | no | — |
| 165 | C3⋊C8 | — | 18 | 1 | no | — |
| 166 | C2×C32 | — | 4 | 1 | no | — |
| 167 | C2×C32 | — | 4 | 1 | no | — |
| 168 | C2×C32 | — | 4 | 1 | no | — |
| 169 | C3×D3 | — | 8 | 2 | no | — |
| 170 | C3×D3 | — | 24 | 2 | no | — |
| 171 | C2×C9 | — | 8 | 1 | no | — |
| 172 | C2×C9 | — | 8 | 1 | no | — |
| 173 | C2×C9 | — | 8 | 1 | no | — |
| 174 | C2×C9 | — | 24 | 1 | no | — |
| 175 | C2×C9 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 176 | C2×C9 | #983 | 1 | 1 | yes | characteristic |
| 177 | C2×C9 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 178 | D9 | — | 2 | 2 | no | — |
| 179 | C2×Q8 | — | 18 | 1 | no | — |
| 180 | C2×Q8 | — | 27 | 1 | no | — |
| 181 | C2×Q8 | — | 54 | 1 | no | — |
| 182 | C2×Q8 | D3×D9 | 1 | 1 | yes | characteristic |
| 183 | C2×D4 | — | 3 | 1 | no | — |
| 184 | C2×D4 | — | 54 | 1 | no | — |
| 185 | C22×C4 | — | 27 | 1 | no | — |
| 186 | SD16 | — | 6 | 1 | no | — |
| 187 | SD16 | — | 6 | 1 | no | — |
| 188 | SD16 | — | 54 | 2 | no | — |
| 189 | C2×C8 | — | 3 | 1 | no | — |
| 190 | C2×C8 | — | 27 | 1 | no | — |
| 191 | C4⋊C4 | — | 27 | 1 | no | — |
| 192 | C22⋊C4 | — | 54 | 1 | no | — |
| 193 | C22×C3 | — | 4 | 1 | no | — |
| 194 | C22×C3 | — | 8 | 1 | no | — |
| 195 | C22×C3 | — | 12 | 1 | no | — |
| 196 | C22×C3 | — | 12 | 1 | no | — |
| 197 | C22×C3 | — | 12 | 1 | no | — |
| 198 | C22×C3 | — | 36 | 2 | no | — |
| 199 | C22×C3 | — | 72 | 1 | no | — |
| 200 | C22×C3 | — | 72 | 1 | no | — |
| 201 | C22×C3 | D3×S4 | 1 | 1 | yes | characteristic |
| 202 | C2×D3 | — | 3 | 2 | no | — |
| 203 | C2×D3 | — | 4 | 2 | no | — |
| 204 | C2×D3 | — | 6 | 1 | no | — |
| 205 | C2×D3 | — | 6 | 1 | no | — |
| 206 | C2×D3 | — | 8 | 1 | no | — |
| 207 | C2×D3 | — | 8 | 1 | no | — |
| 208 | C3×C4 | — | 3 | 1 | no | — |
| 209 | C3×C4 | — | 3 | 1 | no | — |
| 210 | Dic3 | — | 18 | 1 | no | — |
| 211 | Dic3 | — | 36 | 2 | no | — |
| 212 | Dic3 | — | 36 | 1 | no | — |
| 213 | Dic3 | — | 72 | 2 | no | — |
| 214 | C32 | — | 4 | 1 | no | — |
| 215 | C9 | — | 8 | 1 | no | — |
| 216 | C9 | #5421 | 1 | 1 | yes | characteristic, complemented |
| 217 | C23 | — | 6 | 1 | no | — |
| 218 | C23 | — | 9 | 1 | no | — |
| 219 | Q8 | — | 3 | 1 | no | — |
| 220 | Q8 | — | 27 | 2 | no | — |
| 221 | Q8 | — | 54 | 1 | no | — |
| 222 | Q8 | #6194 | 1 | 1 | yes | characteristic, complemented |
| 223 | D4 | — | 6 | 1 | no | — |
| 224 | D4 | — | 6 | 1 | no | — |
| 225 | D4 | — | 108 | 2 | no | — |
| 226 | C2×C4 | — | 3 | 1 | no | — |
| 227 | C2×C4 | — | 27 | 1 | no | — |
| 228 | C2×C4 | — | 54 | 1 | no | — |
| 229 | C2×C4 | — | 54 | 1 | no | — |
| 230 | C2×C4 | — | 54 | 1 | no | — |
| 231 | C8 | — | 6 | 1 | no | — |
| 232 | C8 | — | 54 | 1 | no | — |
| 233 | C2×C3 | — | 4 | 1 | no | — |
| 234 | C2×C3 | — | 4 | 1 | no | — |
| 235 | C2×C3 | — | 4 | 1 | no | — |
| 236 | C2×C3 | — | 8 | 1 | no | — |
| 237 | C2×C3 | — | 8 | 1 | no | — |
| 238 | C2×C3 | — | 8 | 1 | no | — |
| 239 | C2×C3 | — | 24 | 1 | no | — |
| 240 | C2×C3 | — | 72 | 2 | no | — |
| 241 | C2×C3 | #64201 | 1 | 1 | yes | characteristic |
| 242 | C2×C3 | D3×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 243 | C2×C3 | #64209 | 1 | 1 | yes | characteristic |
| 244 | D3 | — | 6 | 2 | no | — |
| 245 | D3 | — | 8 | 2 | no | — |
| 246 | C22 | — | 9 | 2 | no | — |
| 247 | C22 | — | 12 | 1 | no | — |
| 248 | C22 | — | 12 | 1 | no | — |
| 249 | C22 | — | 12 | 1 | no | — |
| 250 | C22 | — | 18 | 1 | no | — |
| 251 | C22 | — | 18 | 1 | no | — |
| 252 | C22 | D9×S4 | 1 | 1 | yes | characteristic |
| 253 | C4 | — | 3 | 1 | no | — |
| 254 | C4 | — | 3 | 1 | no | — |
| 255 | C4 | — | 54 | 1 | no | — |
| 256 | C4 | — | 108 | 1 | no | — |
| 257 | C3 | — | 4 | 1 | no | — |
| 258 | C3 | — | 8 | 1 | no | — |
| 259 | C3 | #10593217 | 1 | 1 | yes | characteristic |
| 260 | C2 | — | 18 | 2 | no | — |
| 261 | C2 | — | 24 | 1 | no | — |
| 262 | C2 | #11722233 | 1 | 1 | yes | characteristic |
| 263 | C2 | D9×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 264 | C2 | #11722241 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D9×S4 |
| commutator, cosocle | C2×C9×SL(2,𝔽3) | C22 |
| Frattini, socle | C22×C3 | D3×S4 |
| Fitting | C2×Q8×C9 | C2×D3 |
| hypocenter | C9×SL(2,𝔽3) | D4 |
| LCS | G ▷ C2×C9×SL(2,𝔽3) ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D9×SL(2,𝔽3) | C2 | semidirect |
| 2. C2×C9×GL(2,𝔽3) | C2 | semidirect |
| 3. C9⋊#859 | C2 | semidirect |
| 13. C2×Q8×D9 | D3 | semidirect |
| 24. C9×SL(2,𝔽3) | D4 | semidirect |
| 57. C2×GL(2,𝔽3) | D9 | semidirect |
| 74. Q8×C9 | #67 | semidirect |
| 129. C2×D9 | GL(2,𝔽3) | semidirect |
| 163. SL(2,𝔽3) | #611 | semidirect |
| 216. C9 | #5421 | semidirect |
| 222. Q8 | #6194 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D9×SL(2,𝔽3) | 2 |
| C2×C9×GL(2,𝔽3) | 2 |
| C9⋊#859 | 2 |
| #10593217 | 3 |
| #10588850 | 3 |
| C2×#6194 | 4 |