Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
265 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 265 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 | C2×A4×#157 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×A4⋊#157 | C2 | 1 | 2 | yes | maximal |
| 3 | #10593568 | — | 3 | 1 | no | maximal |
| 4 | C2×A4×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 5 | A4×#157 | C4 | 1 | 2 | yes | complemented |
| 6 | A4⋊#157 | — | 2 | 2 | no | — |
| 7 | #88946 | — | 4 | 1 | no | maximal |
| 8 | C23×#157 | D3 | 1 | 1 | yes | characteristic |
| 9 | C2×C22⋊#157 | — | 3 | 2 | no | — |
| 10 | C2×#3777 | — | 3 | 1 | no | — |
| 11 | C2×#3777 | — | 3 | 1 | no | — |
| 12 | #63747 | — | 3 | 2 | no | — |
| 13 | C2×C32×A4 | Q8 | 1 | 1 | yes | characteristic |
| 14 | C2×C3⋊#157 | — | 4 | 2 | no | — |
| 15 | C2×C3×#157 | — | 4 | 1 | no | — |
| 16 | A4×C3⋊D3 | C2×C4 | 1 | 1 | yes | characteristic |
| 17 | A4×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 18 | C4⋊SL(2,ℤ/4) | — | 9 | 1 | no | maximal |
| 19 | C23×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 20 | C22×#157 | — | 3 | 1 | no | — |
| 21 | C22×#157 | — | 3 | 1 | no | — |
| 22 | C22×#157 | — | 3 | 2 | no | — |
| 23 | C22×#157 | — | 3 | 2 | no | — |
| 24 | C22×#157 | — | 3 | 2 | no | — |
| 25 | C22×#157 | Dic3 | 1 | 2 | yes | complemented |
| 26 | C2×D3×A4 | — | 12 | 1 | no | — |
| 27 | C22⋊#157 | — | 6 | 2 | no | — |
| 28 | C22⋊#157 | — | 6 | 2 | no | — |
| 29 | #3777 | — | 6 | 1 | no | — |
| 30 | #3777 | — | 6 | 2 | no | — |
| 31 | #3777 | — | 6 | 1 | no | — |
| 32 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 33 | C32×A4 | C4⋊C4 | 1 | 1 | yes | characteristic, complemented |
| 34 | C3⋊#157 | — | 8 | 2 | no | — |
| 35 | C3×#157 | — | 4 | 2 | no | — |
| 36 | C2×C4×A4 | — | 9 | 1 | no | — |
| 37 | C2×SL(2,ℤ/4) | — | 9 | 2 | no | — |
| 38 | C23×C32 | Dic6 | 1 | 1 | yes | characteristic |
| 39 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 40 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 41 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 42 | C22×C3⋊D3 | — | 3 | 1 | no | — |
| 43 | C22×C3⋊D3 | D12 | 1 | 1 | yes | characteristic |
| 44 | C22×C3⋊D3 | C2×Dic3 | 1 | 1 | yes | characteristic |
| 45 | C2×C3×A4 | — | 4 | 1 | no | — |
| 46 | C2×C3×A4 | — | 8 | 1 | no | — |
| 47 | C2×#157 | — | 3 | 1 | no | — |
| 48 | C2×#157 | — | 3 | 2 | no | — |
| 49 | C2×#157 | — | 3 | 2 | no | — |
| 50 | C2×#157 | — | 3 | 2 | no | — |
| 51 | C2×#157 | — | 6 | 1 | no | — |
| 52 | C2×#157 | — | 6 | 2 | no | — |
| 53 | C2×#157 | — | 6 | 2 | no | — |
| 54 | C2×#157 | — | 6 | 2 | no | — |
| 55 | C2×#157 | — | 6 | 2 | no | — |
| 56 | C2×#157 | — | 6 | 2 | no | — |
| 57 | C2×#157 | — | 6 | 2 | no | — |
| 58 | C2×#157 | S4 | 1 | 1 | yes | characteristic |
| 59 | D3×A4 | — | 12 | 1 | no | — |
| 60 | D3×A4 | — | 12 | 1 | no | — |
| 61 | #380 | — | 27 | 1 | no | Sylow |
| 62 | C2×C33 | — | 4 | 1 | no | — |
| 63 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 64 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 65 | C23×D3 | — | 12 | 1 | no | — |
| 66 | C22×A4 | — | 9 | 1 | no | — |
| 67 | C4×A4 | — | 9 | 2 | no | — |
| 68 | SL(2,ℤ/4) | — | 18 | 2 | no | — |
| 69 | C4⋊Dic3 | — | 36 | 1 | no | — |
| 70 | C22×C32 | — | 3 | 1 | no | — |
| 71 | C22×C32 | — | 3 | 1 | no | — |
| 72 | C22×C32 | C4⋊Dic3 | 1 | 1 | yes | characteristic, complemented |
| 73 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 74 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 75 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 76 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 77 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 78 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 79 | C2×C3⋊D3 | — | 3 | 1 | no | — |
| 80 | C2×C3⋊D3 | — | 6 | 1 | no | — |
| 81 | C2×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 82 | C2×C3×D3 | — | 48 | 1 | no | — |
| 83 | C3×A4 | — | 4 | 1 | no | — |
| 84 | C3×A4 | — | 8 | 1 | no | — |
| 85 | #157 | — | 3 | 2 | no | — |
| 86 | #157 | — | 12 | 2 | no | — |
| 87 | #157 | — | 12 | 2 | no | — |
| 88 | #157 | SL(2,ℤ/4) | 1 | 2 | yes | complemented |
| 89 | C23×C4 | — | 9 | 1 | no | — |
| 90 | C2×C4⋊C4 | — | 27 | 1 | no | — |
| 91 | C2×C4⋊C4 | — | 27 | 1 | no | — |
| 92 | C2×C22⋊C4 | — | 27 | 2 | no | — |
| 93 | #95 | — | 27 | 2 | no | — |
| 94 | C33 | — | 4 | 1 | no | Sylow |
| 95 | C23×C3 | — | 4 | 1 | no | — |
| 96 | C22×D3 | — | 12 | 1 | no | — |
| 97 | C22×D3 | — | 12 | 1 | no | — |
| 98 | C22×D3 | — | 36 | 1 | no | — |
| 99 | C22×D3 | — | 36 | 1 | no | — |
| 100 | C22×D3 | — | 36 | 1 | no | — |
| 101 | C22×D3 | — | 36 | 1 | no | — |
| 102 | C2×A4 | — | 8 | 1 | no | — |
| 103 | C2×A4 | — | 9 | 1 | no | — |
| 104 | C2×A4 | — | 9 | 1 | no | — |
| 105 | C2×A4 | PSU(3,𝔽2) | 1 | 1 | yes | characteristic |
| 106 | C2×C3×C4 | — | 36 | 1 | no | — |
| 107 | C2×Dic3 | — | 36 | 2 | no | — |
| 108 | C2×C32 | — | 3 | 1 | no | — |
| 109 | C2×C32 | — | 3 | 1 | no | — |
| 110 | C2×C32 | — | 16 | 1 | no | — |
| 111 | C2×C32 | — | 32 | 1 | no | — |
| 112 | C2×C32 | A4⋊Q8 | 1 | 1 | yes | characteristic |
| 113 | C3⋊D3 | — | 3 | 1 | no | — |
| 114 | C3⋊D3 | — | 3 | 1 | no | — |
| 115 | C3⋊D3 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 116 | C3⋊D3 | C2×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 117 | C3×D3 | — | 48 | 1 | no | — |
| 118 | C3×D3 | — | 48 | 1 | no | — |
| 119 | C24 | — | 9 | 1 | no | — |
| 120 | C22×C4 | — | 9 | 2 | no | — |
| 121 | C22×C4 | — | 27 | 1 | no | — |
| 122 | C22×C4 | — | 27 | 1 | no | — |
| 123 | C22×C4 | — | 27 | 2 | no | — |
| 124 | C22×C4 | — | 27 | 2 | no | — |
| 125 | C22×C4 | — | 27 | 2 | no | — |
| 126 | C4⋊C4 | — | 54 | 1 | no | — |
| 127 | C4⋊C4 | — | 54 | 2 | no | — |
| 128 | C4⋊C4 | — | 54 | 1 | no | — |
| 129 | C22⋊C4 | — | 54 | 2 | no | — |
| 130 | C22⋊C4 | — | 54 | 2 | no | — |
| 131 | C22×C3 | — | 4 | 1 | no | — |
| 132 | C22×C3 | — | 12 | 1 | no | — |
| 133 | C22×C3 | — | 12 | 1 | no | — |
| 134 | C22×C3 | — | 36 | 1 | no | — |
| 135 | C2×D3 | — | 12 | 1 | no | — |
| 136 | C2×D3 | — | 36 | 1 | no | — |
| 137 | C2×D3 | — | 36 | 1 | no | — |
| 138 | C2×D3 | — | 36 | 1 | no | — |
| 139 | C2×D3 | — | 36 | 1 | no | — |
| 140 | C2×D3 | — | 36 | 1 | no | — |
| 141 | C2×D3 | — | 36 | 1 | no | — |
| 142 | C2×D3 | — | 36 | 1 | no | — |
| 143 | C2×D3 | — | 36 | 2 | no | — |
| 144 | A4 | — | 8 | 1 | no | — |
| 145 | A4 | #3777 | 1 | 1 | yes | characteristic, complemented |
| 146 | C3×C4 | — | 36 | 2 | no | — |
| 147 | Dic3 | — | 72 | 2 | no | — |
| 148 | C32 | — | 16 | 1 | no | — |
| 149 | C32 | — | 32 | 1 | no | — |
| 150 | C32 | C4⋊SL(2,ℤ/4) | 1 | 1 | yes | characteristic, complemented |
| 151 | C23 | — | 9 | 1 | no | — |
| 152 | C23 | — | 9 | 1 | no | — |
| 153 | C23 | — | 27 | 1 | no | — |
| 154 | C23 | — | 27 | 1 | no | — |
| 155 | C23 | — | 27 | 1 | no | — |
| 156 | C23 | — | 27 | 1 | no | — |
| 157 | C23 | C3⋊PSU(3,𝔽2) | 1 | 1 | yes | characteristic |
| 158 | C2×C4 | — | 9 | 1 | no | — |
| 159 | C2×C4 | — | 27 | 2 | no | — |
| 160 | C2×C4 | — | 27 | 2 | no | — |
| 161 | C2×C4 | — | 27 | 1 | no | — |
| 162 | C2×C4 | — | 27 | 2 | no | — |
| 163 | C2×C4 | — | 54 | 1 | no | — |
| 164 | C2×C4 | — | 54 | 2 | no | — |
| 165 | C2×C4 | — | 54 | 2 | no | — |
| 166 | C2×C4 | — | 54 | 2 | no | — |
| 167 | C2×C4 | — | 54 | 2 | no | — |
| 168 | C2×C4 | — | 54 | 2 | no | — |
| 169 | C2×C4 | — | 54 | 2 | no | — |
| 170 | C2×C3 | — | 4 | 1 | no | — |
| 171 | C2×C3 | — | 4 | 1 | no | — |
| 172 | C2×C3 | — | 12 | 1 | no | — |
| 173 | C2×C3 | — | 12 | 1 | no | — |
| 174 | C2×C3 | — | 32 | 1 | no | — |
| 175 | C2×C3 | — | 36 | 1 | no | — |
| 176 | C2×C3 | — | 36 | 1 | no | — |
| 177 | D3 | — | 12 | 1 | no | — |
| 178 | D3 | — | 12 | 1 | no | — |
| 179 | D3 | — | 36 | 1 | no | — |
| 180 | D3 | — | 36 | 1 | no | — |
| 181 | C22 | — | 3 | 1 | no | — |
| 182 | C22 | — | 3 | 1 | no | — |
| 183 | C22 | — | 9 | 1 | no | — |
| 184 | C22 | — | 27 | 1 | no | — |
| 185 | C22 | — | 27 | 1 | no | — |
| 186 | C22 | — | 27 | 1 | no | — |
| 187 | C22 | — | 27 | 1 | no | — |
| 188 | C22 | — | 27 | 1 | no | — |
| 189 | C22 | — | 27 | 1 | no | — |
| 190 | C22 | — | 27 | 1 | no | — |
| 191 | C22 | — | 54 | 1 | no | — |
| 192 | C22 | #88946 | 1 | 1 | yes | characteristic, complemented |
| 193 | C4 | — | 9 | 2 | no | — |
| 194 | C4 | — | 27 | 2 | no | — |
| 195 | C4 | — | 108 | 2 | no | — |
| 196 | C4 | — | 108 | 2 | no | — |
| 197 | C3 | — | 4 | 1 | no | — |
| 198 | C3 | — | 4 | 1 | no | — |
| 199 | C3 | — | 32 | 1 | no | — |
| 200 | C2 | — | 3 | 1 | no | — |
| 201 | C2 | — | 3 | 1 | no | — |
| 202 | C2 | — | 9 | 1 | no | — |
| 203 | C2 | — | 9 | 1 | no | — |
| 204 | C2 | — | 27 | 1 | no | — |
| 205 | C2 | — | 27 | 1 | no | — |
| 206 | C2 | A4⋊PSU(3,𝔽2) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4⋊PSU(3,𝔽2) |
| commutator | A4×C3⋊D3 | C2×C4 |
| Fitting, socle | C23×C32 | Dic6 |
| prime radical, cosocle | C2×A4×C3⋊D3 | C22 |
| hypocenter | C32×A4 | C4⋊C4 |
| LCS | G ▷ A4×C3⋊D3 ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. A4×#157 | C4 | semidirect |
| 25. C22×#157 | Dic3 | semidirect |
| 33. C32×A4 | C4⋊C4 | semidirect |
| 72. C22×C32 | C4⋊Dic3 | semidirect |
| 88. #157 | SL(2,ℤ/4) | semidirect |
| 145. A4 | #3777 | semidirect |
| 150. C32 | C4⋊SL(2,ℤ/4) | semidirect |
| 192. C22 | #88946 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×A4×#157 | 2 |
| C2×A4⋊#157 | 2 |
| #10593568 | 3 |
| #88946 | 4 |
| C4⋊SL(2,ℤ/4) | 9 |