Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
330 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 30 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×#127⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#63419 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C3×#5077 | — | 3 | 1 | no | maximal |
| 4 | #63419⋊C2 | C3 | 1 | 1 | yes | maximal, characteristic, direct product factor |
| 5 | C2×C3×(C22⋊C9)⋊C2 | — | 4 | 1 | no | maximal |
| 6 | C3×C42⋊C9 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3×#1004 | — | 3 | 1 | no | — |
| 8 | C32×#127 | D3 | 1 | 1 | yes | characteristic |
| 9 | C3×#932 | — | 3 | 2 | no | — |
| 10 | C3×#833 | — | 3 | 1 | no | — |
| 11 | C3×#805 | — | 3 | 2 | no | — |
| 12 | #127⋊C9 | — | 2 | 1 | no | — |
| 13 | #127⋊C9 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 14 | #63419 | C2×C3 | 1 | 2 | yes | complemented |
| 15 | C2×C3×C22⋊C9 | — | 4 | 1 | no | — |
| 16 | C3×(C22⋊C9)⋊C2 | — | 4 | 2 | no | — |
| 17 | C3×#453 | — | 9 | 1 | no | — |
| 18 | #5077 | — | 3 | 1 | no | — |
| 19 | C2×C32×D4 | — | 3 | 1 | no | — |
| 20 | C2×C32×D4 | — | 3 | 1 | no | — |
| 21 | C2×C3×#67 | — | 6 | 1 | no | — |
| 22 | C2×C3×#67 | — | 6 | 1 | no | — |
| 23 | C3×#237 | — | 6 | 2 | no | — |
| 24 | C3×D3×D4 | — | 6 | 2 | no | — |
| 25 | C3×#235 | — | 3 | 2 | no | — |
| 26 | C2×(C22⋊C9)⋊C2 | — | 4 | 1 | no | — |
| 27 | C32×C42 | C2×D3 | 1 | 1 | yes | characteristic |
| 28 | C3×#214 | — | 6 | 2 | no | — |
| 29 | C3×#213 | — | 6 | 2 | no | — |
| 30 | C3×#208 | — | 3 | 2 | no | — |
| 31 | C42⋊C9 | — | 2 | 1 | no | — |
| 32 | C42⋊C9 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 33 | C2×C3×D9 | — | 16 | 1 | no | — |
| 34 | C3×C22⋊C9 | — | 4 | 1 | no | — |
| 35 | C3×H(5,𝔽2) | — | 9 | 1 | no | — |
| 36 | #1004 | — | 3 | 1 | no | — |
| 37 | C3×AGL(1,ℤ/8) | — | 9 | 2 | no | — |
| 38 | C3×#127 | — | 2 | 1 | no | — |
| 39 | C3×#127 | D9 | 1 | 1 | yes | characteristic, complemented |
| 40 | C3×#127 | C3×D3 | 1 | 1 | yes | characteristic |
| 41 | #932 | — | 3 | 2 | no | — |
| 42 | C3×#104 | — | 9 | 2 | no | — |
| 43 | C3×#100 | — | 9 | 1 | no | — |
| 44 | #833 | — | 3 | 1 | no | — |
| 45 | #805 | — | 3 | 2 | no | — |
| 46 | C23×C32 | — | 3 | 1 | no | — |
| 47 | C23×C32 | S4 | 1 | 1 | yes | characteristic |
| 48 | C22×C3×D3 | — | 6 | 1 | no | — |
| 49 | C22×C3×D3 | — | 6 | 1 | no | — |
| 50 | C32×D4 | — | 6 | 2 | no | — |
| 51 | C32×D4 | — | 6 | 2 | no | — |
| 52 | C2×C32×C4 | — | 3 | 1 | no | — |
| 53 | C3×#67 | — | 6 | 2 | no | — |
| 54 | C3×#67 | — | 6 | 2 | no | — |
| 55 | C3×#67 | — | 6 | 2 | no | — |
| 56 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 57 | C2×C3×Dic3 | — | 6 | 1 | no | — |
| 58 | C3×D12 | — | 3 | 2 | no | — |
| 59 | C3×C4×D3 | — | 6 | 2 | no | — |
| 60 | C3×Dic6 | — | 3 | 2 | no | — |
| 61 | C2×C22⋊C9 | — | 4 | 1 | no | — |
| 62 | C2×C22⋊C9 | — | 8 | 1 | no | — |
| 63 | (C22⋊C9)⋊C2 | — | 4 | 2 | no | — |
| 64 | C3×C3⋊C8 | — | 6 | 2 | no | — |
| 65 | #453 | — | 9 | 1 | no | Sylow |
| 66 | C2×C3×C9 | — | 16 | 1 | no | — |
| 67 | C3×D9 | — | 16 | 2 | no | — |
| 68 | C3×#41 | — | 9 | 2 | no | — |
| 69 | C3×#41 | — | 18 | 2 | no | — |
| 70 | C2×C3×D4 | — | 3 | 1 | no | — |
| 71 | C2×C3×D4 | — | 3 | 1 | no | — |
| 72 | C2×C3×D4 | — | 3 | 1 | no | — |
| 73 | C2×C3×D4 | — | 3 | 1 | no | — |
| 74 | C2×C3×D4 | — | 6 | 1 | no | — |
| 75 | C2×C3×D4 | — | 6 | 1 | no | — |
| 76 | C2×C3×D4 | — | 18 | 2 | no | — |
| 77 | C2×C3×D4 | — | 18 | 1 | no | — |
| 78 | C2×C3×D4 | — | 18 | 1 | no | — |
| 79 | C2×#67 | — | 6 | 1 | no | — |
| 80 | C2×#67 | — | 6 | 1 | no | — |
| 81 | #237 | — | 6 | 2 | no | — |
| 82 | D3×D4 | — | 6 | 2 | no | — |
| 83 | #235 | — | 3 | 2 | no | — |
| 84 | C3×SD16 | — | 18 | 2 | no | — |
| 85 | C3×D8 | — | 18 | 2 | no | — |
| 86 | C3×M16 | — | 9 | 2 | no | — |
| 87 | C3×C42 | — | 2 | 1 | no | — |
| 88 | C3×C42 | C2×D9 | 1 | 1 | yes | characteristic, complemented |
| 89 | C3×C42 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 90 | #214 | — | 6 | 2 | no | — |
| 91 | #213 | — | 6 | 2 | no | — |
| 92 | #208 | — | 3 | 2 | no | — |
| 93 | C22×C32 | — | 6 | 1 | no | — |
| 94 | C22×C32 | — | 6 | 1 | no | — |
| 95 | C22×C32 | — | 12 | 1 | no | — |
| 96 | C22×C32 | C2×S4 | 1 | 1 | yes | characteristic |
| 97 | C2×C3×D3 | — | 6 | 2 | no | — |
| 98 | C2×C3×D3 | — | 24 | 1 | no | — |
| 99 | C2×C3×D3 | — | 24 | 1 | no | — |
| 100 | C32×C4 | — | 3 | 2 | no | — |
| 101 | C3×Dic3 | — | 6 | 2 | no | — |
| 102 | C2×D9 | — | 16 | 1 | no | — |
| 103 | C22⋊C9 | — | 4 | 1 | no | — |
| 104 | C22⋊C9 | — | 8 | 1 | no | — |
| 105 | H(5,𝔽2) | — | 9 | 1 | no | — |
| 106 | AGL(1,ℤ/8) | — | 9 | 2 | no | — |
| 107 | #127 | C3×D9 | 1 | 1 | yes | characteristic, complemented |
| 108 | #104 | — | 9 | 2 | no | — |
| 109 | #100 | — | 9 | 1 | no | — |
| 110 | C3×C9 | — | 16 | 1 | no | Sylow |
| 111 | C23×C3 | — | 2 | 1 | no | — |
| 112 | C23×C3 | — | 3 | 1 | no | — |
| 113 | C23×C3 | — | 3 | 1 | no | — |
| 114 | C23×C3 | — | 6 | 1 | no | — |
| 115 | C23×C3 | — | 18 | 1 | no | — |
| 116 | C23×C3 | — | 18 | 1 | no | — |
| 117 | C23×C3 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 118 | C23×C3 | C3×S4 | 1 | 1 | yes | characteristic |
| 119 | C22×D3 | — | 6 | 1 | no | — |
| 120 | C22×D3 | — | 6 | 1 | no | — |
| 121 | C3×Q8 | — | 9 | 2 | no | — |
| 122 | C3×D4 | — | 6 | 2 | no | — |
| 123 | C3×D4 | — | 6 | 2 | no | — |
| 124 | C3×D4 | — | 6 | 2 | no | — |
| 125 | C3×D4 | — | 6 | 2 | no | — |
| 126 | C3×D4 | — | 9 | 2 | no | — |
| 127 | C3×D4 | — | 12 | 2 | no | — |
| 128 | C3×D4 | — | 12 | 2 | no | — |
| 129 | C3×D4 | — | 18 | 2 | no | — |
| 130 | C3×D4 | — | 18 | 2 | no | — |
| 131 | C3×D4 | — | 18 | 2 | no | — |
| 132 | C2×C3×C4 | — | 3 | 1 | no | — |
| 133 | C2×C3×C4 | — | 3 | 1 | no | — |
| 134 | C2×C3×C4 | — | 6 | 1 | no | — |
| 135 | C2×C3×C4 | — | 18 | 2 | no | — |
| 136 | C2×C3×C4 | — | 18 | 1 | no | — |
| 137 | C2×C3×C4 | — | 18 | 1 | no | — |
| 138 | #67 | — | 6 | 2 | no | — |
| 139 | #67 | — | 6 | 2 | no | — |
| 140 | #67 | — | 6 | 2 | no | — |
| 141 | C2×Dic3 | — | 6 | 1 | no | — |
| 142 | C2×Dic3 | — | 6 | 1 | no | — |
| 143 | D12 | — | 3 | 2 | no | — |
| 144 | C4×D3 | — | 6 | 2 | no | — |
| 145 | Dic6 | — | 3 | 2 | no | — |
| 146 | C3×C8 | — | 18 | 2 | no | — |
| 147 | C3⋊C8 | — | 6 | 2 | no | — |
| 148 | C2×C32 | — | 3 | 1 | no | — |
| 149 | C2×C32 | — | 4 | 1 | no | — |
| 150 | C2×C32 | — | 12 | 1 | no | — |
| 151 | C3×D3 | — | 12 | 2 | no | — |
| 152 | C2×C9 | — | 16 | 1 | no | — |
| 153 | C2×C9 | — | 32 | 1 | no | — |
| 154 | D9 | — | 16 | 2 | no | — |
| 155 | #41 | — | 9 | 2 | no | — |
| 156 | #41 | — | 18 | 2 | no | — |
| 157 | C2×D4 | — | 3 | 1 | no | — |
| 158 | C2×D4 | — | 3 | 1 | no | — |
| 159 | C2×D4 | — | 18 | 2 | no | — |
| 160 | C2×D4 | — | 18 | 1 | no | — |
| 161 | C2×D4 | — | 18 | 1 | no | — |
| 162 | SD16 | — | 18 | 2 | no | — |
| 163 | D8 | — | 18 | 2 | no | — |
| 164 | M16 | — | 9 | 2 | no | — |
| 165 | C42 | C2×C3×D9 | 1 | 1 | yes | characteristic, complemented |
| 166 | C22×C3 | — | 2 | 1 | no | — |
| 167 | C22×C3 | — | 6 | 1 | no | — |
| 168 | C22×C3 | — | 6 | 1 | no | — |
| 169 | C22×C3 | — | 6 | 1 | no | — |
| 170 | C22×C3 | — | 6 | 1 | no | — |
| 171 | C22×C3 | — | 12 | 1 | no | — |
| 172 | C22×C3 | — | 12 | 1 | no | — |
| 173 | C22×C3 | — | 12 | 1 | no | — |
| 174 | C22×C3 | — | 12 | 1 | no | — |
| 175 | C22×C3 | — | 12 | 2 | no | — |
| 176 | C22×C3 | — | 18 | 2 | no | — |
| 177 | C22×C3 | — | 72 | 1 | no | — |
| 178 | C22×C3 | — | 72 | 1 | no | — |
| 179 | C22×C3 | C2×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 180 | C22×C3 | C2×C3×S4 | 1 | 1 | yes | characteristic |
| 181 | C2×D3 | — | 6 | 2 | no | — |
| 182 | C2×D3 | — | 24 | 1 | no | — |
| 183 | C2×D3 | — | 24 | 1 | no | — |
| 184 | C3×C4 | — | 3 | 2 | no | — |
| 185 | C3×C4 | — | 3 | 2 | no | — |
| 186 | C3×C4 | — | 6 | 2 | no | — |
| 187 | C3×C4 | — | 18 | 2 | no | — |
| 188 | Dic3 | — | 6 | 2 | no | — |
| 189 | C32 | #857⋊C2 | 1 | 1 | yes | characteristic |
| 190 | C9 | — | 16 | 1 | no | — |
| 191 | C9 | — | 32 | 1 | no | — |
| 192 | C23 | — | 3 | 1 | no | — |
| 193 | C23 | — | 18 | 1 | no | — |
| 194 | C23 | — | 18 | 1 | no | — |
| 195 | C23 | C3×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 196 | Q8 | — | 9 | 2 | no | — |
| 197 | D4 | — | 6 | 2 | no | — |
| 198 | D4 | — | 6 | 2 | no | — |
| 199 | D4 | — | 9 | 2 | no | — |
| 200 | D4 | — | 18 | 2 | no | — |
| 201 | D4 | — | 18 | 2 | no | — |
| 202 | D4 | — | 18 | 2 | no | — |
| 203 | C2×C4 | — | 3 | 1 | no | — |
| 204 | C2×C4 | — | 18 | 2 | no | — |
| 205 | C2×C4 | — | 18 | 1 | no | — |
| 206 | C2×C4 | — | 18 | 1 | no | — |
| 207 | C8 | — | 18 | 2 | no | — |
| 208 | C2×C3 | — | 3 | 1 | no | — |
| 209 | C2×C3 | — | 3 | 1 | no | — |
| 210 | C2×C3 | — | 4 | 1 | no | — |
| 211 | C2×C3 | — | 4 | 1 | no | — |
| 212 | C2×C3 | — | 6 | 1 | no | — |
| 213 | C2×C3 | — | 8 | 1 | no | — |
| 214 | C2×C3 | — | 12 | 1 | no | — |
| 215 | C2×C3 | — | 12 | 1 | no | — |
| 216 | C2×C3 | — | 24 | 1 | no | — |
| 217 | C2×C3 | — | 36 | 2 | no | — |
| 218 | D3 | — | 12 | 2 | no | — |
| 219 | C22 | — | 6 | 1 | no | — |
| 220 | C22 | — | 6 | 1 | no | — |
| 221 | C22 | — | 12 | 1 | no | — |
| 222 | C22 | — | 18 | 2 | no | — |
| 223 | C22 | — | 72 | 1 | no | — |
| 224 | C22 | — | 72 | 1 | no | — |
| 225 | C22 | C2×C3×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 226 | C4 | — | 3 | 2 | no | — |
| 227 | C4 | — | 18 | 2 | no | — |
| 228 | C3 | — | 2 | 1 | no | — |
| 229 | C3 | #63419⋊C2 | 1 | 1 | yes | characteristic, direct product factor |
| 230 | C3 | C3×#857⋊C2 | 1 | 1 | yes | characteristic |
| 231 | C2 | — | 3 | 1 | no | — |
| 232 | C2 | — | 4 | 1 | no | — |
| 233 | C2 | — | 12 | 1 | no | — |
| 234 | C2 | — | 36 | 2 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #63419⋊C2 |
| commutator, cosocle, hypocenter | C42⋊C9 | C22×C3 |
| Frattini | C22×C3 | C2×C3×S4 |
| Fitting | C32×#127 | D3 |
| socle | C22×C32 | C2×S4 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#127⋊C9 | C2 | semidirect |
| 2. C3×#63419 | C2 | semidirect |
| 4. #63419⋊C2 | C3 | direct (N × H) |
| 6. C3×C42⋊C9 | C22 | semidirect |
| 13. #127⋊C9 | C2×C3 | semidirect |
| 14. #63419 | C2×C3 | semidirect |
| 32. C42⋊C9 | C22×C3 | semidirect |
| 39. C3×#127 | D9 | semidirect |
| 88. C3×C42 | C2×D9 | semidirect |
| 107. #127 | C3×D9 | semidirect |
| 165. C42 | C2×C3×D9 | semidirect |
| 229. C3 | #63419⋊C2 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#127⋊C9 | 2 |
| C3×#63419 | 2 |
| C3×#5077 | 3 |
| #63419⋊C2 | 3 |
| C2×C3×(C22⋊C9)⋊C2 | 4 |