Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
281 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 27 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×C22⋊#1088 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22⋊#6249 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C23×#272 | C4 | 1 | 1 | yes | characteristic |
| 4 | C22×#1088 | C4 | 1 | 1 | yes | characteristic |
| 5 | C22×#1088 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×#6249 | — | 2 | 2 | no | — |
| 7 | C22⋊#1088 | — | 2 | 2 | no | — |
| 8 | C23×H(3,𝔽3) | C8 | 1 | 1 | yes | characteristic, complemented |
| 9 | C22×#272 | — | 2 | 1 | no | — |
| 10 | C22×#272 | — | 2 | 1 | no | — |
| 11 | C22×#272 | — | 4 | 1 | no | — |
| 12 | C22×#272 | — | 4 | 1 | no | — |
| 13 | C22×#272 | C8 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×#272 | C2×C4 | 1 | 1 | yes | characteristic |
| 15 | C2×#1088 | — | 2 | 1 | no | — |
| 16 | C2×#1088 | — | 4 | 1 | no | — |
| 17 | C2×#1088 | — | 4 | 1 | no | — |
| 18 | C2×#1088 | D4 | 1 | 2 | yes | — |
| 19 | #6249 | — | 4 | 2 | no | — |
| 20 | #4525 | — | 9 | 1 | no | maximal |
| 21 | C23×C3×D3 | — | 12 | 1 | no | — |
| 22 | C22×H(3,𝔽3) | — | 2 | 1 | no | — |
| 23 | C22×H(3,𝔽3) | — | 4 | 1 | no | — |
| 24 | C22×H(3,𝔽3) | C2×C8 | 1 | 1 | yes | characteristic |
| 25 | C2×#272 | — | 2 | 1 | no | — |
| 26 | C2×#272 | — | 2 | 1 | no | — |
| 27 | C2×#272 | — | 2 | 1 | no | — |
| 28 | C2×#272 | — | 4 | 2 | no | — |
| 29 | C2×#272 | — | 4 | 1 | no | — |
| 30 | C2×#272 | — | 4 | 1 | no | — |
| 31 | C2×#272 | — | 4 | 1 | no | — |
| 32 | C2×#272 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 33 | C2×#272 | M16 | 1 | 1 | yes | characteristic |
| 34 | #1088 | — | 2 | 2 | no | — |
| 35 | #1088 | — | 4 | 1 | no | — |
| 36 | C2×C3×C22⋊C4 | — | 9 | 1 | no | — |
| 37 | #830 | — | 9 | 2 | no | — |
| 38 | C23×C32 | — | 4 | 1 | no | — |
| 39 | C22×C3×D3 | — | 12 | 1 | no | — |
| 40 | C22×C3×D3 | — | 12 | 1 | no | — |
| 41 | C22×C3×D3 | — | 12 | 2 | no | — |
| 42 | C22×C3×D3 | — | 12 | 2 | no | — |
| 43 | C22×C3×D3 | — | 12 | 4 | no | — |
| 44 | C22×C3×D3 | — | 12 | 4 | no | — |
| 45 | #323 | — | 27 | 1 | no | Sylow |
| 46 | C2×H(3,𝔽3) | — | 2 | 1 | no | — |
| 47 | C2×H(3,𝔽3) | — | 4 | 1 | no | — |
| 48 | C2×H(3,𝔽3) | C22⋊C8 | 1 | 1 | yes | characteristic |
| 49 | #272 | — | 2 | 1 | no | — |
| 50 | #272 | — | 4 | 1 | no | — |
| 51 | #272 | #99 | 1 | 1 | yes | characteristic |
| 52 | #272 | #100 | 1 | 1 | yes | characteristic |
| 53 | C24×C3 | — | 9 | 1 | no | — |
| 54 | C23×D3 | — | 12 | 1 | no | — |
| 55 | C22×C3×C4 | — | 9 | 1 | no | — |
| 56 | C22×C3×C4 | — | 9 | 1 | no | — |
| 57 | C3×C22⋊C4 | — | 18 | 2 | no | — |
| 58 | C2×C3⋊C8 | — | 18 | 2 | no | — |
| 59 | C22×C32 | — | 4 | 2 | no | — |
| 60 | C22×C32 | — | 4 | 4 | no | — |
| 61 | C22×C32 | — | 4 | 1 | no | — |
| 62 | C2×C3×D3 | — | 12 | 1 | no | — |
| 63 | C2×C3×D3 | — | 12 | 1 | no | — |
| 64 | C2×C3×D3 | — | 12 | 2 | no | — |
| 65 | C2×C3×D3 | — | 12 | 2 | no | — |
| 66 | C2×C3×D3 | — | 12 | 2 | no | — |
| 67 | C2×C3×D3 | — | 12 | 4 | no | — |
| 68 | C2×C3×D3 | — | 12 | 4 | no | — |
| 69 | C2×C3×D3 | — | 12 | 4 | no | — |
| 70 | C2×C3×D3 | — | 12 | 8 | no | — |
| 71 | C2×C22⋊C4 | — | 9 | 1 | no | — |
| 72 | C22⋊C8 | — | 27 | 2 | no | — |
| 73 | H(3,𝔽3) | #323 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 74 | C23×C3 | — | 9 | 1 | no | — |
| 75 | C23×C3 | — | 9 | 1 | no | — |
| 76 | C23×C3 | — | 12 | 1 | no | — |
| 77 | C23×C3 | — | 18 | 1 | no | — |
| 78 | C23×C3 | — | 18 | 1 | no | — |
| 79 | C23×C3 | — | 36 | 1 | no | — |
| 80 | C23×C3 | — | 36 | 1 | no | — |
| 81 | C23×C3 | AGL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 82 | C22×D3 | — | 12 | 1 | no | — |
| 83 | C22×D3 | — | 12 | 1 | no | — |
| 84 | C22×D3 | — | 12 | 2 | no | — |
| 85 | C22×D3 | — | 12 | 2 | no | — |
| 86 | C22×D3 | — | 12 | 4 | no | — |
| 87 | C22×D3 | — | 12 | 4 | no | — |
| 88 | C2×C3×C4 | — | 9 | 2 | no | — |
| 89 | C2×C3×C4 | — | 18 | 1 | no | — |
| 90 | C2×C3×C4 | — | 36 | 1 | no | — |
| 91 | C2×C3×C4 | — | 36 | 1 | no | — |
| 92 | C3⋊C8 | — | 36 | 2 | no | — |
| 93 | C2×C32 | — | 4 | 2 | no | — |
| 94 | C2×C32 | — | 4 | 4 | no | — |
| 95 | C2×C32 | — | 4 | 1 | no | — |
| 96 | C3×D3 | — | 12 | 1 | no | — |
| 97 | C3×D3 | — | 12 | 1 | no | — |
| 98 | C3×D3 | — | 12 | 2 | no | — |
| 99 | C3×D3 | — | 12 | 4 | no | — |
| 100 | C24 | — | 9 | 1 | no | — |
| 101 | C22×C4 | — | 9 | 1 | no | — |
| 102 | C22×C4 | — | 9 | 1 | no | — |
| 103 | C2×C8 | — | 54 | 2 | no | — |
| 104 | C22⋊C4 | — | 18 | 2 | no | — |
| 105 | C22×C3 | — | 2 | 1 | no | — |
| 106 | C22×C3 | — | 4 | 1 | no | — |
| 107 | C22×C3 | — | 9 | 1 | no | — |
| 108 | C22×C3 | — | 9 | 1 | no | — |
| 109 | C22×C3 | — | 12 | 2 | no | — |
| 110 | C22×C3 | — | 12 | 4 | no | — |
| 111 | C22×C3 | — | 12 | 1 | no | — |
| 112 | C22×C3 | — | 18 | 1 | no | — |
| 113 | C22×C3 | — | 18 | 1 | no | — |
| 114 | C22×C3 | — | 18 | 1 | no | — |
| 115 | C22×C3 | — | 36 | 1 | no | — |
| 116 | C22×C3 | — | 36 | 1 | no | — |
| 117 | C22×C3 | — | 36 | 1 | no | — |
| 118 | C22×C3 | — | 36 | 2 | no | — |
| 119 | C22×C3 | C2×AGL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 120 | C2×D3 | — | 12 | 1 | no | — |
| 121 | C2×D3 | — | 12 | 1 | no | — |
| 122 | C2×D3 | — | 12 | 2 | no | — |
| 123 | C2×D3 | — | 12 | 2 | no | — |
| 124 | C2×D3 | — | 12 | 2 | no | — |
| 125 | C2×D3 | — | 12 | 4 | no | — |
| 126 | C2×D3 | — | 12 | 4 | no | — |
| 127 | C2×D3 | — | 12 | 4 | no | — |
| 128 | C2×D3 | — | 12 | 8 | no | — |
| 129 | C3×C4 | — | 18 | 2 | no | — |
| 130 | C3×C4 | — | 36 | 1 | no | — |
| 131 | C32 | — | 4 | 1 | no | — |
| 132 | C23 | — | 9 | 1 | no | — |
| 133 | C23 | — | 9 | 1 | no | — |
| 134 | C23 | — | 18 | 1 | no | — |
| 135 | C23 | — | 18 | 1 | no | — |
| 136 | C23 | — | 36 | 1 | no | — |
| 137 | C23 | — | 36 | 1 | no | — |
| 138 | C23 | #6249 | 1 | 1 | yes | characteristic, complemented |
| 139 | C2×C4 | — | 9 | 2 | no | — |
| 140 | C2×C4 | — | 18 | 1 | no | — |
| 141 | C2×C4 | — | 36 | 1 | no | — |
| 142 | C2×C4 | — | 36 | 1 | no | — |
| 143 | C8 | — | 108 | 2 | no | — |
| 144 | C2×C3 | — | 2 | 1 | no | — |
| 145 | C2×C3 | — | 4 | 1 | no | — |
| 146 | C2×C3 | — | 9 | 1 | no | — |
| 147 | C2×C3 | — | 9 | 1 | no | — |
| 148 | C2×C3 | — | 12 | 2 | no | — |
| 149 | C2×C3 | — | 12 | 4 | no | — |
| 150 | C2×C3 | — | 12 | 1 | no | — |
| 151 | C2×C3 | — | 18 | 1 | no | — |
| 152 | C2×C3 | — | 36 | 1 | no | — |
| 153 | C2×C3 | C22⋊AGL(1,𝔽9) | 1 | 1 | yes | characteristic |
| 154 | D3 | — | 12 | 1 | no | — |
| 155 | D3 | — | 12 | 1 | no | — |
| 156 | D3 | — | 12 | 2 | no | — |
| 157 | D3 | — | 12 | 4 | no | — |
| 158 | C22 | — | 2 | 1 | no | — |
| 159 | C22 | — | 4 | 1 | no | — |
| 160 | C22 | — | 9 | 1 | no | — |
| 161 | C22 | — | 9 | 1 | no | — |
| 162 | C22 | — | 18 | 1 | no | — |
| 163 | C22 | — | 18 | 1 | no | — |
| 164 | C22 | — | 18 | 1 | no | — |
| 165 | C22 | — | 36 | 1 | no | — |
| 166 | C22 | — | 36 | 1 | no | — |
| 167 | C22 | — | 36 | 1 | no | — |
| 168 | C22 | — | 36 | 2 | no | — |
| 169 | C22 | C2×#6249 | 1 | 1 | yes | characteristic |
| 170 | C4 | — | 18 | 2 | no | — |
| 171 | C4 | — | 36 | 1 | no | — |
| 172 | C3 | — | 12 | 1 | no | — |
| 173 | C3 | #10593272 | 1 | 1 | yes | characteristic |
| 174 | C2 | — | 2 | 1 | no | — |
| 175 | C2 | — | 4 | 1 | no | — |
| 176 | C2 | — | 9 | 1 | no | — |
| 177 | C2 | — | 9 | 1 | no | — |
| 178 | C2 | — | 18 | 1 | no | — |
| 179 | C2 | — | 36 | 1 | no | — |
| 180 | C2 | C22⋊#6249 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22⋊#6249 | C2 | semidirect |
| 8. C23×H(3,𝔽3) | C8 | semidirect |
| 13. C22×#272 | C8 | semidirect |
| 73. H(3,𝔽3) | #323 | semidirect |
| 138. C23 | #6249 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊#1088 | 2 |
| C22⋊#6249 | 2 |
| #4525 | 9 |