Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
766 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 53 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×#63703 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×D3×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#63703 | — | 3 | 1 | no | maximal |
| 4 | C2×D3×C22⋊C9 | — | 4 | 3 | no | maximal |
| 5 | D3×Q8⋊C9 | — | 4 | 2 | no | maximal |
| 6 | C32×H(5,𝔽2) | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | C3×#1009 | — | 3 | 2 | no | — |
| 8 | C3×D3×#41 | — | 3 | 2 | no | — |
| 9 | C3×#1004 | — | 3 | 3 | no | — |
| 10 | C2×C3×D3×D4 | — | 3 | 3 | no | — |
| 11 | #63703 | — | 2 | 1 | no | — |
| 12 | #63703 | D3 | 1 | 1 | yes | characteristic, direct product factor |
| 13 | C2×C3×C22⋊C9 | — | 4 | 3 | no | — |
| 14 | D3×C22⋊C9 | — | 4 | 6 | no | — |
| 15 | C3×Q8⋊C9 | — | 4 | 2 | no | — |
| 16 | C2×C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 17 | D3×H(5,𝔽2) | C9 | 1 | 1 | yes | characteristic, complemented |
| 18 | C23×C3×D3 | — | 3 | 1 | no | — |
| 19 | C23×C3×D3 | A4 | 1 | 3 | yes | — |
| 20 | C32×#41 | — | 3 | 2 | no | — |
| 21 | C2×C32×D4 | — | 3 | 3 | no | — |
| 22 | C2×C3×#67 | — | 3 | 3 | no | — |
| 23 | C2×C3×#67 | — | 3 | 3 | no | — |
| 24 | C3×#239 | — | 3 | 2 | no | — |
| 25 | C3×D3×Q8 | A4 | 1 | 2 | yes | — |
| 26 | C3×#237 | — | 3 | 6 | no | — |
| 27 | C3×D3×D4 | — | 3 | 6 | no | — |
| 28 | C3×D3×D4 | — | 3 | 12 | no | — |
| 29 | C3×#235 | — | 3 | 6 | no | — |
| 30 | C2×C3×D12 | — | 3 | 3 | no | — |
| 31 | C2×C3×C4×D3 | — | 3 | 3 | no | — |
| 32 | C22×C22⋊C9 | — | 12 | 3 | no | — |
| 33 | C2×Q8⋊C9 | — | 12 | 2 | no | — |
| 34 | C2×D3×C9 | — | 16 | 1 | no | — |
| 35 | C3×C22⋊C9 | — | 4 | 3 | no | — |
| 36 | C3×H(5,𝔽2) | — | 2 | 1 | no | — |
| 37 | C3×H(5,𝔽2) | — | 9 | 3 | no | — |
| 38 | C3×H(5,𝔽2) | — | 9 | 2 | no | — |
| 39 | C3×H(5,𝔽2) | C2×C9 | 1 | 1 | yes | characteristic, complemented |
| 40 | C3×H(5,𝔽2) | C3×D3 | 1 | 1 | yes | characteristic |
| 41 | C2×C3×#41 | — | 9 | 2 | no | — |
| 42 | C22×C3×D4 | — | 9 | 3 | no | — |
| 43 | #1009 | — | 3 | 2 | no | — |
| 44 | D3×#41 | — | 3 | 2 | no | — |
| 45 | #1004 | — | 3 | 3 | no | — |
| 46 | C2×D3×D4 | — | 3 | 3 | no | — |
| 47 | C23×C32 | — | 3 | 1 | no | — |
| 48 | C23×C32 | C2×A4 | 1 | 3 | yes | — |
| 49 | C22×C3×D3 | — | 3 | 3 | no | — |
| 50 | C22×C3×D3 | — | 3 | 3 | no | — |
| 51 | C22×C3×D3 | — | 3 | 3 | no | — |
| 52 | C22×C3×D3 | — | 4 | 6 | no | — |
| 53 | C22×C3×D3 | — | 12 | 2 | no | — |
| 54 | C32×Q8 | C2×A4 | 1 | 2 | yes | — |
| 55 | C32×D4 | — | 3 | 6 | no | — |
| 56 | C2×C32×C4 | — | 3 | 3 | no | — |
| 57 | C3×#67 | — | 3 | 12 | no | — |
| 58 | C2×C3×Dic3 | — | 3 | 3 | no | — |
| 59 | C3×D12 | — | 3 | 6 | no | — |
| 60 | C3×C4×D3 | — | 3 | 6 | no | — |
| 61 | C3×C4×D3 | — | 3 | 2 | no | — |
| 62 | C3×Dic6 | — | 3 | 2 | no | — |
| 63 | C2×C22⋊C9 | — | 4 | 3 | no | — |
| 64 | C2×C22⋊C9 | — | 8 | 3 | no | — |
| 65 | C2×C22⋊C9 | — | 12 | 6 | no | — |
| 66 | Q8⋊C9 | — | 4 | 2 | no | — |
| 67 | Q8⋊C9 | — | 8 | 2 | no | — |
| 68 | C2×H(5,𝔽2) | — | 3 | 1 | no | Sylow |
| 69 | C2×C3×C9 | — | 16 | 1 | no | — |
| 70 | D3×C9 | — | 16 | 2 | no | — |
| 71 | C24×C3 | — | 3 | 3 | no | — |
| 72 | C24×C3 | — | 9 | 1 | no | — |
| 73 | C23×D3 | — | 3 | 1 | no | — |
| 74 | C23×D3 | C22⋊C9 | 1 | 3 | yes | complemented |
| 75 | C3×#41 | — | 3 | 2 | no | — |
| 76 | C3×#41 | — | 3 | 2 | no | — |
| 77 | C3×#41 | — | 6 | 2 | no | — |
| 78 | C3×#41 | — | 9 | 6 | no | — |
| 79 | C3×#41 | — | 9 | 6 | no | — |
| 80 | C3×#41 | — | 9 | 2 | no | — |
| 81 | C2×C3×Q8 | — | 3 | 2 | no | — |
| 82 | C2×C3×D4 | — | 3 | 3 | no | — |
| 83 | C2×C3×D4 | — | 3 | 3 | no | — |
| 84 | C2×C3×D4 | — | 6 | 3 | no | — |
| 85 | C2×C3×D4 | — | 9 | 3 | no | — |
| 86 | C2×C3×D4 | — | 9 | 6 | no | — |
| 87 | C2×C3×D4 | — | 9 | 12 | no | — |
| 88 | C2×C3×D4 | — | 9 | 3 | no | — |
| 89 | C2×C3×D4 | — | 9 | 3 | no | — |
| 90 | C22×C3×C4 | — | 9 | 3 | no | — |
| 91 | C2×#67 | — | 3 | 3 | no | — |
| 92 | C2×#67 | — | 3 | 3 | no | — |
| 93 | #239 | — | 3 | 2 | no | — |
| 94 | D3×Q8 | C22⋊C9 | 1 | 2 | yes | complemented |
| 95 | #237 | — | 3 | 6 | no | — |
| 96 | D3×D4 | — | 3 | 6 | no | — |
| 97 | D3×D4 | — | 3 | 12 | no | — |
| 98 | #235 | — | 3 | 6 | no | — |
| 99 | C2×D12 | — | 3 | 3 | no | — |
| 100 | C2×C4×D3 | — | 3 | 3 | no | — |
| 101 | C22×C32 | — | 3 | 3 | no | — |
| 102 | C22×C32 | — | 4 | 3 | no | — |
| 103 | C22×C32 | — | 12 | 1 | no | — |
| 104 | C2×C3×D3 | — | 3 | 3 | no | — |
| 105 | C2×C3×D3 | — | 6 | 6 | no | — |
| 106 | C2×C3×D3 | — | 12 | 3 | no | — |
| 107 | C2×C3×D3 | — | 12 | 3 | no | — |
| 108 | C2×C3×D3 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 109 | C32×C4 | — | 3 | 2 | no | — |
| 110 | C3×Dic3 | — | 3 | 2 | no | — |
| 111 | C22×C9 | — | 48 | 1 | no | — |
| 112 | C22⋊C9 | — | 4 | 3 | no | — |
| 113 | C22⋊C9 | — | 8 | 3 | no | — |
| 114 | H(5,𝔽2) | — | 9 | 3 | no | — |
| 115 | H(5,𝔽2) | — | 9 | 2 | no | — |
| 116 | H(5,𝔽2) | D3×C9 | 1 | 1 | yes | characteristic, complemented |
| 117 | C2×#41 | — | 9 | 2 | no | — |
| 118 | C22×D4 | — | 9 | 3 | no | — |
| 119 | C3×C9 | — | 16 | 1 | no | Sylow |
| 120 | C23×C3 | — | 2 | 3 | no | — |
| 121 | C23×C3 | — | 3 | 1 | no | — |
| 122 | C23×C3 | — | 3 | 1 | no | — |
| 123 | C23×C3 | — | 6 | 1 | no | — |
| 124 | C23×C3 | — | 9 | 3 | no | — |
| 125 | C23×C3 | — | 9 | 3 | no | — |
| 126 | C23×C3 | — | 9 | 3 | no | — |
| 127 | C23×C3 | — | 12 | 6 | no | — |
| 128 | C23×C3 | — | 36 | 2 | no | — |
| 129 | C23×C3 | C2×C22⋊C9 | 1 | 3 | yes | complemented |
| 130 | C23×C3 | D3×A4 | 1 | 3 | yes | — |
| 131 | C22×D3 | — | 3 | 3 | no | — |
| 132 | C22×D3 | — | 3 | 3 | no | — |
| 133 | C22×D3 | — | 3 | 3 | no | — |
| 134 | C22×D3 | — | 4 | 6 | no | — |
| 135 | C22×D3 | — | 12 | 2 | no | — |
| 136 | C3×Q8 | — | 2 | 2 | no | — |
| 137 | C3×Q8 | — | 9 | 2 | no | — |
| 138 | C3×Q8 | C2×C22⋊C9 | 1 | 2 | yes | complemented |
| 139 | C3×Q8 | D3×A4 | 1 | 2 | yes | — |
| 140 | C3×D4 | — | 3 | 6 | no | — |
| 141 | C3×D4 | — | 3 | 6 | no | — |
| 142 | C3×D4 | — | 6 | 6 | no | — |
| 143 | C3×D4 | — | 9 | 6 | no | — |
| 144 | C3×D4 | — | 9 | 12 | no | — |
| 145 | C2×C3×C4 | — | 3 | 3 | no | — |
| 146 | C2×C3×C4 | — | 3 | 3 | no | — |
| 147 | C2×C3×C4 | — | 6 | 3 | no | — |
| 148 | C2×C3×C4 | — | 9 | 2 | no | — |
| 149 | C2×C3×C4 | — | 9 | 6 | no | — |
| 150 | C2×C3×C4 | — | 9 | 3 | no | — |
| 151 | #67 | — | 3 | 12 | no | — |
| 152 | C2×Dic3 | — | 3 | 3 | no | — |
| 153 | D12 | — | 3 | 6 | no | — |
| 154 | C4×D3 | — | 3 | 6 | no | — |
| 155 | C4×D3 | — | 3 | 2 | no | — |
| 156 | Dic6 | — | 3 | 2 | no | — |
| 157 | C2×C32 | — | 6 | 3 | no | — |
| 158 | C2×C32 | C2×C22⋊A4 | 1 | 1 | yes | characteristic |
| 159 | C3×D3 | — | 6 | 3 | no | — |
| 160 | C3×D3 | #997 | 1 | 2 | yes | — |
| 161 | C2×C9 | — | 16 | 1 | no | — |
| 162 | C2×C9 | — | 32 | 1 | no | — |
| 163 | C2×C9 | — | 48 | 2 | no | — |
| 164 | C24 | — | 3 | 3 | no | — |
| 165 | C24 | — | 9 | 1 | no | — |
| 166 | #41 | — | 3 | 2 | no | — |
| 167 | #41 | — | 9 | 6 | no | — |
| 168 | #41 | — | 9 | 6 | no | — |
| 169 | #41 | — | 9 | 2 | no | — |
| 170 | C2×Q8 | — | 3 | 2 | no | — |
| 171 | C2×D4 | — | 3 | 3 | no | — |
| 172 | C2×D4 | — | 9 | 3 | no | — |
| 173 | C2×D4 | — | 9 | 6 | no | — |
| 174 | C2×D4 | — | 9 | 12 | no | — |
| 175 | C2×D4 | — | 9 | 3 | no | — |
| 176 | C2×D4 | — | 9 | 3 | no | — |
| 177 | C22×C4 | — | 9 | 3 | no | — |
| 178 | C22×C3 | — | 3 | 1 | no | — |
| 179 | C22×C3 | — | 3 | 3 | no | — |
| 180 | C22×C3 | — | 3 | 3 | no | — |
| 181 | C22×C3 | — | 4 | 3 | no | — |
| 182 | C22×C3 | — | 4 | 3 | no | — |
| 183 | C22×C3 | — | 6 | 3 | no | — |
| 184 | C22×C3 | — | 8 | 3 | no | — |
| 185 | C22×C3 | — | 9 | 3 | no | — |
| 186 | C22×C3 | — | 12 | 1 | no | — |
| 187 | C22×C3 | — | 12 | 1 | no | — |
| 188 | C22×C3 | — | 18 | 6 | no | — |
| 189 | C22×C3 | — | 24 | 1 | no | — |
| 190 | C22×C3 | — | 36 | 3 | no | — |
| 191 | C22×C3 | — | 36 | 3 | no | — |
| 192 | C2×D3 | — | 3 | 3 | no | — |
| 193 | C2×D3 | — | 6 | 6 | no | — |
| 194 | C2×D3 | — | 12 | 3 | no | — |
| 195 | C2×D3 | — | 12 | 3 | no | — |
| 196 | C2×D3 | C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 197 | C3×C4 | — | 3 | 2 | no | — |
| 198 | C3×C4 | — | 3 | 2 | no | — |
| 199 | C3×C4 | — | 6 | 2 | no | — |
| 200 | C3×C4 | — | 9 | 2 | no | — |
| 201 | Dic3 | — | 3 | 2 | no | — |
| 202 | C32 | C2×#997 | 1 | 1 | yes | characteristic |
| 203 | C9 | — | 16 | 1 | no | — |
| 204 | C9 | — | 32 | 1 | no | — |
| 205 | C23 | — | 3 | 1 | no | — |
| 206 | C23 | — | 9 | 3 | no | — |
| 207 | C23 | — | 9 | 3 | no | — |
| 208 | C23 | — | 9 | 3 | no | — |
| 209 | C23 | — | 12 | 6 | no | — |
| 210 | C23 | — | 36 | 2 | no | — |
| 211 | C23 | D3×C22⋊C9 | 1 | 3 | yes | complemented |
| 212 | Q8 | — | 9 | 2 | no | — |
| 213 | Q8 | D3×C22⋊C9 | 1 | 2 | yes | complemented |
| 214 | D4 | — | 3 | 6 | no | — |
| 215 | D4 | — | 9 | 6 | no | — |
| 216 | D4 | — | 9 | 12 | no | — |
| 217 | C2×C4 | — | 3 | 3 | no | — |
| 218 | C2×C4 | — | 9 | 2 | no | — |
| 219 | C2×C4 | — | 9 | 6 | no | — |
| 220 | C2×C4 | — | 9 | 3 | no | — |
| 221 | C2×C3 | — | 2 | 1 | no | — |
| 222 | C2×C3 | — | 3 | 2 | no | — |
| 223 | C2×C3 | — | 6 | 3 | no | — |
| 224 | C2×C3 | — | 6 | 3 | no | — |
| 225 | C2×C3 | — | 12 | 3 | no | — |
| 226 | C2×C3 | — | 18 | 3 | no | — |
| 227 | C2×C3 | C2×C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 228 | C2×C3 | D3×C22⋊A4 | 1 | 1 | yes | characteristic |
| 229 | D3 | — | 6 | 3 | no | — |
| 230 | D3 | #63703 | 1 | 2 | yes | direct product factor |
| 231 | C22 | — | 3 | 1 | no | — |
| 232 | C22 | — | 3 | 3 | no | — |
| 233 | C22 | — | 4 | 3 | no | — |
| 234 | C22 | — | 9 | 3 | no | — |
| 235 | C22 | — | 12 | 1 | no | — |
| 236 | C22 | — | 18 | 6 | no | — |
| 237 | C22 | — | 36 | 3 | no | — |
| 238 | C22 | — | 36 | 3 | no | — |
| 239 | C4 | — | 3 | 2 | no | — |
| 240 | C4 | — | 9 | 2 | no | — |
| 241 | C3 | — | 2 | 1 | no | — |
| 242 | C3 | C2×#63703 | 1 | 1 | yes | characteristic, complemented |
| 243 | C3 | D3×#997 | 1 | 1 | yes | characteristic |
| 244 | C2 | — | 3 | 2 | no | — |
| 245 | C2 | — | 6 | 3 | no | — |
| 246 | C2 | — | 18 | 3 | no | — |
| 247 | C2 | D3×C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#63703 | C2 | semidirect |
| 12. #63703 | D3 | direct (N × H) |
| 17. D3×H(5,𝔽2) | C9 | semidirect |
| 39. C3×H(5,𝔽2) | C2×C9 | semidirect |
| 74. C23×D3 | C22⋊C9 | semidirect |
| 94. D3×Q8 | C22⋊C9 | semidirect |
| 116. H(5,𝔽2) | D3×C9 | semidirect |
| 129. C23×C3 | C2×C22⋊C9 | semidirect |
| 138. C3×Q8 | C2×C22⋊C9 | semidirect |
| 211. C23 | D3×C22⋊C9 | semidirect |
| 213. Q8 | D3×C22⋊C9 | semidirect |
| 230. D3 | #63703 | direct (N × H) |
| 242. C3 | C2×#63703 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#63703 | 2 |
| C3×D3×H(5,𝔽2) | 3 |
| C2×#63703 | 3 |
| C2×D3×C22⋊C9 | 4 |
| D3×Q8⋊C9 | 4 |