Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
266 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 33 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⋊#63700 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#63690 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊#63422 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10591664 | — | 3 | 1 | no | maximal |
| 5 | #10589530 | — | 3 | 1 | no | maximal |
| 6 | C3×#41⋊C9 | C22 | 1 | 1 | yes | characteristic |
| 7 | C3⋊(#41⋊C9) | — | 2 | 1 | no | — |
| 8 | C3×#3688 | — | 2 | 1 | no | — |
| 9 | #88427 | — | 4 | 1 | no | maximal |
| 10 | C3×#1009 | D3 | 1 | 1 | yes | characteristic |
| 11 | C3×#951 | — | 3 | 1 | no | — |
| 12 | #63863 | — | 3 | 1 | no | — |
| 13 | #63827 | — | 3 | 1 | no | — |
| 14 | #63700 | — | 3 | 1 | no | — |
| 15 | #63690 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | #63664 | — | 3 | 1 | no | — |
| 17 | #63568 | — | 3 | 1 | no | — |
| 18 | #63558 | — | 3 | 1 | no | — |
| 19 | #63422 | — | 3 | 1 | no | — |
| 20 | #63422 | — | 6 | 1 | no | — |
| 21 | C9×Dic6 | — | 4 | 1 | no | — |
| 22 | C3×Dic18 | — | 4 | 1 | no | — |
| 23 | C3×Q8⋊C9 | D4 | 1 | 1 | yes | characteristic |
| 24 | C3⋊(C9⋊C8) | — | 4 | 1 | no | — |
| 25 | #5242 | — | 9 | 1 | no | — |
| 26 | #5202 | — | 9 | 1 | no | — |
| 27 | C32×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | C3×#239 | — | 2 | 1 | no | — |
| 29 | C3×D3×D4 | — | 6 | 1 | no | — |
| 30 | C3×#235 | — | 3 | 1 | no | — |
| 31 | C2×C3×D12 | — | 3 | 1 | no | — |
| 32 | C2×C3×Dic6 | — | 3 | 1 | no | — |
| 33 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 34 | #3748 | — | 3 | 1 | no | — |
| 35 | C3×#216 | — | 6 | 1 | no | — |
| 36 | C3×#214 | — | 6 | 1 | no | — |
| 37 | C3×#208 | — | 3 | 1 | no | — |
| 38 | (C2×D3)⋊Dic3 | — | 6 | 2 | no | — |
| 39 | #3718 | — | 6 | 1 | no | — |
| 40 | #3715 | — | 6 | 1 | no | — |
| 41 | #41⋊C9 | — | 2 | 1 | no | — |
| 42 | #41⋊C9 | — | 6 | 1 | no | — |
| 43 | #41⋊C9 | C2×D3 | 1 | 1 | yes | characteristic |
| 44 | #3688 | — | 2 | 1 | no | — |
| 45 | #3674 | — | 12 | 1 | no | — |
| 46 | C3×C4×C9 | — | 4 | 1 | no | — |
| 47 | C9×Dic3 | — | 8 | 1 | no | — |
| 48 | C3×Dic9 | — | 8 | 1 | no | — |
| 49 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 50 | #1009 | D9 | 1 | 1 | yes | characteristic |
| 51 | C3×#137 | — | 9 | 1 | no | — |
| 52 | #951 | — | 3 | 1 | no | — |
| 53 | #951 | — | 9 | 1 | no | — |
| 54 | #947 | — | 9 | 1 | no | — |
| 55 | #942 | — | 9 | 1 | no | — |
| 56 | #936 | — | 9 | 1 | no | — |
| 57 | #837 | — | 9 | 1 | no | — |
| 58 | #837 | — | 9 | 1 | no | — |
| 59 | #837 | — | 9 | 1 | no | — |
| 60 | #837 | — | 18 | 1 | no | — |
| 61 | #833 | — | 9 | 1 | no | — |
| 62 | #825 | — | 9 | 1 | no | — |
| 63 | #823 | — | 9 | 1 | no | — |
| 64 | C22×C3×D3 | — | 3 | 2 | no | — |
| 65 | C32×Q8 | #67 | 1 | 1 | yes | characteristic |
| 66 | C32×D4 | — | 3 | 1 | no | — |
| 67 | C2×C32×C4 | — | 3 | 1 | no | — |
| 68 | C2×C3⋊Dic3 | — | 6 | 1 | no | — |
| 69 | C3×#67 | — | 6 | 1 | no | — |
| 70 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 71 | C3×D12 | — | 3 | 1 | no | — |
| 72 | C3×D12 | — | 6 | 1 | no | — |
| 73 | C3×C4×D3 | — | 6 | 1 | no | — |
| 74 | C3×Dic6 | — | 6 | 1 | no | — |
| 75 | C3×Dic6 | — | 6 | 1 | no | — |
| 76 | C3×Dic6 | S4 | 1 | 1 | yes | characteristic |
| 77 | C3⋊(C3⋊C8) | — | 6 | 1 | no | — |
| 78 | C3×C3⋊C8 | — | 6 | 1 | no | — |
| 79 | Q8×C9 | — | 12 | 1 | no | — |
| 80 | Dic18 | — | 4 | 1 | no | — |
| 81 | Q8⋊C9 | — | 2 | 1 | no | — |
| 82 | Q8⋊C9 | #67 | 1 | 1 | yes | characteristic |
| 83 | C9⋊C8 | — | 12 | 1 | no | — |
| 84 | C9⋊C8 | — | 24 | 1 | no | — |
| 85 | #455 | — | 27 | 1 | no | Sylow |
| 86 | C2×C3×C9 | — | 4 | 1 | no | — |
| 87 | C3×#41 | — | 2 | 1 | no | — |
| 88 | C3×#41 | — | 6 | 1 | no | — |
| 89 | C3×#41 | — | 9 | 1 | no | — |
| 90 | C3×#41 | C2×D9 | 1 | 1 | yes | characteristic |
| 91 | C3×#41 | D32 | 1 | 1 | yes | characteristic |
| 92 | C2×C3×Q8 | — | 9 | 1 | no | — |
| 93 | C2×C3×D4 | — | 9 | 1 | no | — |
| 94 | C2×C3×D4 | — | 18 | 1 | no | — |
| 95 | #239 | — | 2 | 1 | no | — |
| 96 | D3×D4 | — | 6 | 1 | no | — |
| 97 | #235 | — | 3 | 1 | no | — |
| 98 | C2×D12 | — | 3 | 1 | no | — |
| 99 | C2×Dic6 | — | 3 | 1 | no | — |
| 100 | C3×Q16 | — | 18 | 1 | no | — |
| 101 | C3×SD16 | — | 18 | 1 | no | — |
| 102 | C3×M16 | — | 9 | 1 | no | — |
| 103 | #217 | — | 18 | 2 | no | — |
| 104 | #216 | — | 6 | 1 | no | — |
| 105 | #216 | — | 18 | 1 | no | — |
| 106 | #216 | — | 18 | 1 | no | — |
| 107 | #215 | — | 18 | 1 | no | — |
| 108 | #214 | — | 6 | 1 | no | — |
| 109 | #214 | — | 18 | 1 | no | — |
| 110 | (C2×D3)⋊C4 | — | 18 | 2 | no | — |
| 111 | C4×Dic3 | — | 9 | 1 | no | — |
| 112 | C4×Dic3 | — | 9 | 1 | no | — |
| 113 | C4×Dic3 | — | 18 | 1 | no | — |
| 114 | #208 | — | 3 | 1 | no | — |
| 115 | #208 | — | 9 | 1 | no | — |
| 116 | #208 | — | 9 | 1 | no | — |
| 117 | #208 | — | 18 | 1 | no | — |
| 118 | C22×C32 | — | 3 | 1 | no | — |
| 119 | C2×C3×D3 | — | 6 | 1 | no | — |
| 120 | C2×C3×D3 | — | 12 | 2 | no | — |
| 121 | C32×C4 | — | 3 | 1 | no | — |
| 122 | C32×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 123 | C3⋊Dic3 | — | 12 | 1 | no | — |
| 124 | C3×Dic3 | — | 2 | 1 | no | — |
| 125 | C3×Dic3 | — | 12 | 1 | no | — |
| 126 | C4×C9 | — | 4 | 1 | no | — |
| 127 | C4×C9 | — | 8 | 1 | no | — |
| 128 | C4×C9 | — | 24 | 1 | no | — |
| 129 | Dic9 | — | 8 | 1 | no | — |
| 130 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 131 | #137 | — | 9 | 1 | no | — |
| 132 | #137 | — | 27 | 1 | no | — |
| 133 | #124 | — | 27 | 1 | no | — |
| 134 | #104 | — | 27 | 1 | no | — |
| 135 | #104 | — | 27 | 1 | no | — |
| 136 | #100 | — | 27 | 1 | no | — |
| 137 | C3×C9 | — | 4 | 1 | no | Sylow |
| 138 | C23×C3 | — | 9 | 2 | no | — |
| 139 | C22×D3 | — | 3 | 2 | no | — |
| 140 | C3×Q8 | — | 2 | 1 | no | — |
| 141 | C3×Q8 | — | 3 | 1 | no | — |
| 142 | C3×Q8 | — | 18 | 1 | no | — |
| 143 | C3×Q8 | — | 18 | 1 | no | — |
| 144 | C3×Q8 | #611 | 1 | 1 | yes | characteristic |
| 145 | C3×Q8 | #625 | 1 | 1 | yes | characteristic |
| 146 | C3×D4 | — | 3 | 1 | no | — |
| 147 | C3×D4 | — | 3 | 1 | no | — |
| 148 | C3×D4 | — | 6 | 1 | no | — |
| 149 | C3×D4 | — | 9 | 1 | no | — |
| 150 | C3×D4 | — | 18 | 1 | no | — |
| 151 | C3×D4 | — | 18 | 1 | no | — |
| 152 | C2×C3×C4 | — | 3 | 1 | no | — |
| 153 | C2×C3×C4 | — | 3 | 1 | no | — |
| 154 | C2×C3×C4 | — | 6 | 1 | no | — |
| 155 | C2×C3×C4 | — | 18 | 1 | no | — |
| 156 | C2×C3×C4 | — | 18 | 1 | no | — |
| 157 | #67 | — | 6 | 1 | no | — |
| 158 | C2×Dic3 | — | 6 | 1 | no | — |
| 159 | C2×Dic3 | — | 18 | 1 | no | — |
| 160 | C2×Dic3 | — | 18 | 1 | no | — |
| 161 | C2×Dic3 | — | 36 | 1 | no | — |
| 162 | D12 | — | 3 | 1 | no | — |
| 163 | D12 | — | 6 | 1 | no | — |
| 164 | C4×D3 | — | 6 | 1 | no | — |
| 165 | Dic6 | — | 6 | 1 | no | — |
| 166 | Dic6 | — | 6 | 1 | no | — |
| 167 | Dic6 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 168 | C3×C8 | — | 18 | 1 | no | — |
| 169 | C3⋊C8 | — | 6 | 1 | no | — |
| 170 | C3⋊C8 | — | 18 | 1 | no | — |
| 171 | C3⋊C8 | — | 18 | 1 | no | — |
| 172 | C3⋊C8 | — | 36 | 1 | no | — |
| 173 | C2×C32 | — | 6 | 1 | no | — |
| 174 | C2×C32 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 175 | C3×D3 | — | 12 | 1 | no | — |
| 176 | C2×C9 | — | 4 | 1 | no | — |
| 177 | C2×C9 | — | 8 | 1 | no | — |
| 178 | #41 | — | 6 | 1 | no | — |
| 179 | #41 | — | 9 | 1 | no | — |
| 180 | #41 | D3×D9 | 1 | 1 | yes | characteristic |
| 181 | C2×Q8 | — | 9 | 1 | no | — |
| 182 | C2×D4 | — | 9 | 1 | no | — |
| 183 | C2×D4 | — | 18 | 1 | no | — |
| 184 | Q16 | — | 18 | 1 | no | — |
| 185 | Q16 | — | 54 | 1 | no | — |
| 186 | SD16 | — | 18 | 1 | no | — |
| 187 | SD16 | — | 54 | 1 | no | — |
| 188 | M16 | — | 9 | 1 | no | — |
| 189 | M16 | — | 27 | 1 | no | — |
| 190 | C22⋊C4 | — | 54 | 2 | no | — |
| 191 | C42 | — | 27 | 1 | no | — |
| 192 | C22×C3 | — | 3 | 1 | no | — |
| 193 | C22×C3 | — | 3 | 1 | no | — |
| 194 | C22×C3 | — | 6 | 1 | no | — |
| 195 | C22×C3 | — | 18 | 1 | no | — |
| 196 | C22×C3 | — | 36 | 2 | no | — |
| 197 | C2×D3 | — | 6 | 1 | no | — |
| 198 | C2×D3 | — | 12 | 2 | no | — |
| 199 | C3×C4 | — | 2 | 1 | no | — |
| 200 | C3×C4 | — | 3 | 1 | no | — |
| 201 | C3×C4 | — | 3 | 1 | no | — |
| 202 | C3×C4 | — | 6 | 1 | no | — |
| 203 | C3×C4 | — | 6 | 1 | no | — |
| 204 | C3×C4 | — | 36 | 1 | no | — |
| 205 | C3×C4 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 206 | C3×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 207 | Dic3 | — | 2 | 1 | no | — |
| 208 | Dic3 | — | 12 | 1 | no | — |
| 209 | Dic3 | — | 36 | 1 | no | — |
| 210 | Dic3 | — | 36 | 1 | no | — |
| 211 | Dic3 | — | 36 | 2 | no | — |
| 212 | C32 | #5428 | 1 | 1 | yes | characteristic |
| 213 | C9 | — | 4 | 1 | no | — |
| 214 | C9 | — | 8 | 1 | no | — |
| 215 | C23 | — | 9 | 2 | no | — |
| 216 | Q8 | — | 3 | 1 | no | — |
| 217 | Q8 | — | 18 | 1 | no | — |
| 218 | Q8 | — | 18 | 1 | no | — |
| 219 | Q8 | #6194 | 1 | 1 | yes | characteristic |
| 220 | D4 | — | 3 | 1 | no | — |
| 221 | D4 | — | 9 | 1 | no | — |
| 222 | D4 | — | 18 | 1 | no | — |
| 223 | D4 | — | 18 | 1 | no | — |
| 224 | C2×C4 | — | 3 | 1 | no | — |
| 225 | C2×C4 | — | 18 | 1 | no | — |
| 226 | C2×C4 | — | 18 | 1 | no | — |
| 227 | C2×C4 | — | 54 | 1 | no | — |
| 228 | C8 | — | 18 | 1 | no | — |
| 229 | C8 | — | 54 | 1 | no | — |
| 230 | C2×C3 | — | 2 | 1 | no | — |
| 231 | C2×C3 | — | 6 | 1 | no | — |
| 232 | C2×C3 | — | 6 | 1 | no | — |
| 233 | C2×C3 | — | 12 | 1 | no | — |
| 234 | C2×C3 | — | 36 | 1 | no | — |
| 235 | C2×C3 | #63694 | 1 | 1 | yes | characteristic |
| 236 | C2×C3 | #64209 | 1 | 1 | yes | characteristic |
| 237 | D3 | — | 12 | 1 | no | — |
| 238 | C22 | — | 3 | 1 | no | — |
| 239 | C22 | — | 18 | 1 | no | — |
| 240 | C22 | — | 36 | 2 | no | — |
| 241 | C4 | — | 3 | 1 | no | — |
| 242 | C4 | — | 6 | 1 | no | — |
| 243 | C4 | — | 36 | 1 | no | — |
| 244 | C4 | — | 108 | 1 | no | — |
| 245 | C4 | D3×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 246 | C3 | — | 2 | 1 | no | — |
| 247 | C3 | #10589530 | 1 | 1 | yes | characteristic, complemented |
| 248 | C3 | #10593235 | 1 | 1 | yes | characteristic |
| 249 | C2 | — | 6 | 1 | no | — |
| 250 | C2 | — | 36 | 1 | no | — |
| 251 | C2 | #11722266 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#63690 | C2 | semidirect |
| 3. C3⋊#63422 | C2 | semidirect |
| 15. #63690 | D3 | semidirect |
| 247. C3 | #10589530 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊#63700 | 2 |
| C3×#63690 | 2 |
| C3⋊#63422 | 2 |
| #10591664 | 3 |
| #10589530 | 3 |
| #88427 | 4 |