Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
322 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 35 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⋊D3×C42⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×C3⋊(#126⋊C3) | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10594020 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D3×#126⋊C3 | — | 3 | 2 | no | maximal |
| 5 | D32×A4 | — | 4 | 1 | no | maximal |
| 6 | C32×C42⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C42×C3⋊D3 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C3×#876 | C2×C3 | 1 | 2 | yes | complemented |
| 9 | C3×#126⋊C3 | — | 3 | 2 | no | — |
| 10 | #63921 | — | 3 | 1 | no | — |
| 11 | #63883 | — | 3 | 1 | no | — |
| 12 | #63862 | — | 3 | 2 | no | — |
| 13 | D3×C42⋊C3 | — | 3 | 2 | no | — |
| 14 | D3×C42⋊C3 | — | 6 | 1 | no | — |
| 15 | C3⋊(#126⋊C3) | — | 2 | 2 | no | — |
| 16 | C3⋊(#126⋊C3) | D3 | 1 | 2 | yes | complemented |
| 17 | A4×C3⋊D3 | — | 4 | 1 | no | — |
| 18 | C3×D3×A4 | — | 4 | 2 | no | — |
| 19 | D3×#126 | — | 3 | 2 | no | — |
| 20 | C2×#126⋊C3 | — | 9 | 1 | no | — |
| 21 | C22×D32 | A4 | 1 | 1 | yes | characteristic |
| 22 | C2×D3×A4 | — | 12 | 2 | no | — |
| 23 | C2×C4×C3⋊D3 | — | 3 | 1 | no | — |
| 24 | C2×#624 | — | 3 | 1 | no | — |
| 25 | C32×C42 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 26 | C4×C3⋊Dic3 | — | 3 | 1 | no | — |
| 27 | C3×(C2×D3)⋊C4 | — | 3 | 2 | no | — |
| 28 | C3×Dic3⋊C4 | — | 3 | 2 | no | — |
| 29 | C3×C42⋊C3 | — | 2 | 2 | no | — |
| 30 | C3×C42⋊C3 | — | 2 | 1 | no | — |
| 31 | C3×C42⋊C3 | — | 4 | 1 | no | — |
| 32 | C3×C42⋊C3 | C2×D3 | 1 | 2 | yes | complemented |
| 33 | Dic3⋊Dic3 | — | 3 | 2 | no | — |
| 34 | (C2×D3)⋊Dic3 | — | 3 | 2 | no | — |
| 35 | C32×A4 | — | 4 | 1 | no | — |
| 36 | C3×D32 | — | 16 | 1 | no | — |
| 37 | C3×#126 | — | 3 | 2 | no | — |
| 38 | #898 | — | 9 | 2 | no | — |
| 39 | D3×C4⋊C4 | — | 9 | 2 | no | — |
| 40 | D3×C22⋊C4 | — | 9 | 2 | no | — |
| 41 | #879 | — | 9 | 2 | no | — |
| 42 | #876 | C3×D3 | 1 | 2 | yes | complemented |
| 43 | C42×D3 | — | 3 | 2 | no | — |
| 44 | C42×D3 | — | 6 | 1 | no | — |
| 45 | #126⋊C3 | — | 3 | 2 | no | — |
| 46 | #126⋊C3 | — | 6 | 2 | no | — |
| 47 | C2×C42⋊C3 | — | 9 | 1 | no | — |
| 48 | C22×C3⋊D3 | C2×A4 | 1 | 1 | yes | characteristic |
| 49 | C22×C3×D3 | C2×A4 | 1 | 2 | yes | — |
| 50 | C2×C3×A4 | — | 12 | 2 | no | — |
| 51 | C2×D32 | — | 6 | 1 | no | — |
| 52 | C2×D32 | — | 6 | 1 | no | — |
| 53 | D3×A4 | — | 4 | 2 | no | — |
| 54 | D3×A4 | — | 8 | 2 | no | — |
| 55 | D3×A4 | — | 12 | 2 | no | — |
| 56 | D3×A4 | — | 24 | 1 | no | — |
| 57 | C2×C32×C4 | — | 3 | 1 | no | — |
| 58 | C2×C3⋊Dic3 | — | 3 | 1 | no | — |
| 59 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 60 | C4×C3⋊D3 | — | 6 | 1 | no | — |
| 61 | C2×C3×Dic3 | — | 3 | 2 | no | — |
| 62 | #624 | — | 6 | 1 | no | — |
| 63 | #624 | — | 6 | 1 | no | — |
| 64 | C2×#126 | — | 9 | 1 | no | Sylow |
| 65 | C3×C3⋊D3 | — | 16 | 1 | no | — |
| 66 | C32×D3 | — | 16 | 2 | no | — |
| 67 | C23×D3 | — | 3 | 2 | no | — |
| 68 | C22×A4 | — | 36 | 1 | no | — |
| 69 | C2×C4×D3 | — | 9 | 2 | no | — |
| 70 | C2×C4×D3 | — | 9 | 2 | no | — |
| 71 | C2×C4×D3 | — | 18 | 1 | no | — |
| 72 | C3×C4⋊C4 | — | 9 | 2 | no | — |
| 73 | C3×C22⋊C4 | — | 9 | 2 | no | — |
| 74 | C3×C42 | — | 2 | 1 | no | — |
| 75 | C3×C42 | C2×C3×D3 | 1 | 2 | yes | complemented |
| 76 | #217 | — | 9 | 2 | no | — |
| 77 | (C2×D3)⋊C4 | — | 3 | 2 | no | — |
| 78 | (C2×D3)⋊C4 | — | 9 | 2 | no | — |
| 79 | C4⋊Dic3 | — | 9 | 2 | no | — |
| 80 | Dic3⋊C4 | — | 3 | 2 | no | — |
| 81 | Dic3⋊C4 | — | 9 | 2 | no | — |
| 82 | C4×Dic3 | — | 9 | 2 | no | — |
| 83 | C4×Dic3 | — | 18 | 1 | no | — |
| 84 | C42⋊C3 | — | 2 | 2 | no | — |
| 85 | C42⋊C3 | — | 4 | 1 | no | — |
| 86 | C42⋊C3 | D32 | 1 | 1 | yes | characteristic, complemented |
| 87 | C22×C32 | C22×A4 | 1 | 1 | yes | characteristic |
| 88 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 89 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 90 | C2×C3×D3 | — | 6 | 2 | no | — |
| 91 | C2×C3×D3 | — | 48 | 2 | no | — |
| 92 | C3×A4 | — | 4 | 2 | no | — |
| 93 | C3×A4 | — | 8 | 1 | no | — |
| 94 | C3×A4 | — | 8 | 2 | no | — |
| 95 | C3×A4 | — | 16 | 1 | no | — |
| 96 | D32 | — | 4 | 1 | no | — |
| 97 | D32 | — | 12 | 1 | no | — |
| 98 | C32×C4 | — | 6 | 1 | no | — |
| 99 | C3⋊Dic3 | — | 6 | 1 | no | — |
| 100 | C3×Dic3 | — | 6 | 2 | no | — |
| 101 | #126 | — | 3 | 2 | no | — |
| 102 | #126 | — | 27 | 2 | no | — |
| 103 | C2×C4⋊C4 | — | 27 | 1 | no | — |
| 104 | C2×C22⋊C4 | — | 27 | 1 | no | — |
| 105 | C2×C42 | — | 9 | 1 | no | — |
| 106 | C33 | — | 16 | 1 | no | Sylow |
| 107 | C23×C3 | — | 3 | 2 | no | — |
| 108 | C22×D3 | — | 3 | 2 | no | — |
| 109 | C22×D3 | — | 6 | 1 | no | — |
| 110 | C22×D3 | — | 18 | 2 | no | — |
| 111 | C22×D3 | — | 18 | 2 | no | — |
| 112 | C22×D3 | D3×A4 | 1 | 2 | yes | — |
| 113 | C2×A4 | — | 12 | 2 | no | — |
| 114 | C2×A4 | — | 24 | 2 | no | — |
| 115 | C2×A4 | — | 36 | 1 | no | — |
| 116 | C2×C3×C4 | — | 3 | 2 | no | — |
| 117 | C2×C3×C4 | — | 6 | 1 | no | — |
| 118 | C2×C3×C4 | — | 9 | 2 | no | — |
| 119 | C2×Dic3 | — | 3 | 2 | no | — |
| 120 | C2×Dic3 | — | 9 | 2 | no | — |
| 121 | C2×Dic3 | — | 18 | 1 | no | — |
| 122 | C4×D3 | — | 18 | 2 | no | — |
| 123 | C4×D3 | — | 18 | 2 | no | — |
| 124 | C4×D3 | — | 18 | 2 | no | — |
| 125 | C4×D3 | — | 18 | 2 | no | — |
| 126 | C4×D3 | — | 18 | 2 | no | — |
| 127 | C4×D3 | — | 18 | 2 | no | — |
| 128 | C2×C32 | — | 3 | 1 | no | — |
| 129 | C2×C32 | — | 48 | 2 | no | — |
| 130 | C3⋊D3 | — | 3 | 1 | no | — |
| 131 | C3⋊D3 | #126⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 132 | C3×D3 | — | 4 | 2 | no | — |
| 133 | C3×D3 | — | 16 | 2 | no | — |
| 134 | C3×D3 | — | 32 | 2 | no | — |
| 135 | C3×D3 | — | 48 | 2 | no | — |
| 136 | C3×D3 | — | 96 | 1 | no | — |
| 137 | C24 | — | 9 | 1 | no | — |
| 138 | C22×C4 | — | 27 | 1 | no | — |
| 139 | C22×C4 | — | 27 | 1 | no | — |
| 140 | C4⋊C4 | — | 9 | 2 | no | — |
| 141 | C4⋊C4 | — | 27 | 2 | no | — |
| 142 | C22⋊C4 | — | 9 | 2 | no | — |
| 143 | C22⋊C4 | — | 27 | 2 | no | — |
| 144 | C42 | — | 27 | 1 | no | — |
| 145 | C42 | C3×D32 | 1 | 1 | yes | characteristic, complemented |
| 146 | C22×C3 | — | 2 | 1 | no | — |
| 147 | C22×C3 | — | 18 | 2 | no | — |
| 148 | C22×C3 | — | 144 | 1 | no | — |
| 149 | C22×C3 | C2×D3×A4 | 1 | 2 | yes | — |
| 150 | C2×D3 | — | 6 | 2 | no | — |
| 151 | C2×D3 | — | 9 | 2 | no | — |
| 152 | C2×D3 | — | 9 | 2 | no | — |
| 153 | C2×D3 | — | 12 | 2 | no | — |
| 154 | C2×D3 | — | 18 | 1 | no | — |
| 155 | C2×D3 | — | 18 | 1 | no | — |
| 156 | C2×D3 | — | 36 | 2 | no | — |
| 157 | A4 | — | 4 | 1 | no | — |
| 158 | A4 | — | 8 | 2 | no | — |
| 159 | A4 | — | 16 | 1 | no | — |
| 160 | C3×C4 | — | 6 | 2 | no | — |
| 161 | C3×C4 | — | 6 | 2 | no | — |
| 162 | C3×C4 | — | 18 | 2 | no | — |
| 163 | Dic3 | — | 6 | 2 | no | — |
| 164 | Dic3 | — | 18 | 2 | no | — |
| 165 | Dic3 | — | 18 | 2 | no | — |
| 166 | C32 | — | 16 | 2 | no | — |
| 167 | C32 | — | 32 | 1 | no | — |
| 168 | C32 | — | 32 | 2 | no | — |
| 169 | C32 | — | 64 | 1 | no | — |
| 170 | C32 | C2×#126⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 171 | C23 | — | 3 | 2 | no | — |
| 172 | C23 | — | 9 | 1 | no | — |
| 173 | C23 | — | 54 | 1 | no | — |
| 174 | C23 | — | 54 | 1 | no | — |
| 175 | C2×C4 | — | 3 | 1 | no | — |
| 176 | C2×C4 | — | 9 | 2 | no | — |
| 177 | C2×C4 | — | 27 | 1 | no | — |
| 178 | C2×C4 | — | 54 | 1 | no | — |
| 179 | C2×C4 | — | 54 | 1 | no | — |
| 180 | C2×C4 | — | 54 | 1 | no | — |
| 181 | C2×C4 | — | 54 | 1 | no | — |
| 182 | C2×C3 | — | 3 | 2 | no | — |
| 183 | C2×C3 | — | 6 | 1 | no | — |
| 184 | C2×C3 | — | 12 | 2 | no | — |
| 185 | C2×C3 | — | 48 | 2 | no | — |
| 186 | C2×C3 | — | 96 | 2 | no | — |
| 187 | C2×C3 | — | 144 | 1 | no | — |
| 188 | D3 | — | 3 | 2 | no | — |
| 189 | D3 | — | 4 | 2 | no | — |
| 190 | D3 | — | 6 | 1 | no | — |
| 191 | D3 | — | 9 | 2 | no | — |
| 192 | D3 | — | 18 | 1 | no | — |
| 193 | C22 | — | 18 | 2 | no | — |
| 194 | C22 | — | 27 | 1 | no | — |
| 195 | C22 | — | 27 | 1 | no | — |
| 196 | C22 | — | 36 | 1 | no | — |
| 197 | C22 | — | 108 | 1 | no | — |
| 198 | C22 | D32×A4 | 1 | 1 | yes | characteristic |
| 199 | C4 | — | 6 | 1 | no | — |
| 200 | C4 | — | 18 | 2 | no | — |
| 201 | C4 | — | 54 | 1 | no | — |
| 202 | C3 | — | 2 | 1 | no | — |
| 203 | C3 | — | 16 | 1 | no | — |
| 204 | C3 | — | 32 | 2 | no | — |
| 205 | C3 | — | 64 | 1 | no | — |
| 206 | C3 | D3×#126⋊C3 | 1 | 2 | yes | complemented |
| 207 | C2 | — | 3 | 1 | no | — |
| 208 | C2 | — | 9 | 1 | no | — |
| 209 | C2 | — | 12 | 2 | no | — |
| 210 | C2 | — | 27 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C32×C42 | C22×C3 |
| Frattini | C22 | D32×A4 |
| socle | C22×C32 | C22×A4 |
| LCS | G ▷ C32×C42 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊D3×C42⋊C3 | C2 | semidirect |
| 2. C3×C3⋊(#126⋊C3) | C2 | semidirect |
| 3. #10594020 | C3 | semidirect |
| 6. C32×C42⋊C3 | C22 | semidirect |
| 7. C42×C3⋊D3 | C2×C3 | semidirect |
| 8. C3×#876 | C2×C3 | semidirect |
| 16. C3⋊(#126⋊C3) | D3 | semidirect |
| 25. C32×C42 | C22×C3 | semidirect |
| 32. C3×C42⋊C3 | C2×D3 | semidirect |
| 42. #876 | C3×D3 | semidirect |
| 75. C3×C42 | C2×C3×D3 | semidirect |
| 86. C42⋊C3 | D32 | semidirect |
| 131. C3⋊D3 | #126⋊C3 | semidirect |
| 145. C42 | C3×D32 | semidirect |
| 170. C32 | C2×#126⋊C3 | semidirect |
| 206. C3 | D3×#126⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×C42⋊C3 | 2 |
| C3×C3⋊(#126⋊C3) | 2 |
| #10594020 | 3 |
| D3×#126⋊C3 | 3 |
| D32×A4 | 4 |