Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
484 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 48 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 | C32×#997 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#5933 | C3 | 1 | 4 | yes | maximal, direct product factor |
| 3 | C32×#455 | — | 3 | 1 | no | maximal |
| 4 | C32×SL(2,ℤ/4) | — | 4 | 1 | no | maximal |
| 5 | C32×#226 | — | 4 | 2 | no | maximal |
| 6 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic |
| 7 | C3×#997 | — | 2 | 4 | no | — |
| 8 | C3×#997 | C2×C3 | 1 | 4 | yes | — |
| 9 | C32×#137 | — | 3 | 2 | no | — |
| 10 | C32×#124 | — | 3 | 1 | no | — |
| 11 | C32×#104 | — | 3 | 2 | no | — |
| 12 | C32×#100 | — | 3 | 1 | no | — |
| 13 | C2×C32×A4 | — | 4 | 1 | no | — |
| 14 | C2×C32×A4 | — | 8 | 1 | no | — |
| 15 | C32×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 16 | #5933 | C32 | 1 | 1 | yes | characteristic, direct product factor |
| 17 | C3×#455 | — | 3 | 4 | no | — |
| 18 | C32×#41 | — | 3 | 2 | no | — |
| 19 | C2×C32×Q8 | — | 3 | 1 | no | — |
| 20 | C2×C32×D4 | — | 3 | 1 | no | — |
| 21 | C2×C32×D4 | — | 6 | 1 | no | — |
| 22 | C3×SL(2,ℤ/4) | — | 4 | 4 | no | — |
| 23 | C3×#226 | — | 4 | 8 | no | — |
| 24 | C32×Q16 | — | 6 | 2 | no | — |
| 25 | C32×SD16 | — | 6 | 2 | no | — |
| 26 | C32×M16 | — | 3 | 2 | no | — |
| 27 | C32×C22⋊C4 | — | 6 | 1 | no | — |
| 28 | C32×C22⋊C4 | — | 6 | 1 | no | — |
| 29 | C32×C42 | — | 3 | 1 | no | — |
| 30 | C32×A4 | — | 4 | 1 | no | — |
| 31 | C32×A4 | — | 8 | 1 | no | — |
| 32 | C32×Dic3 | — | 16 | 1 | no | — |
| 33 | C3×H(5,𝔽2) | C3×D3 | 1 | 4 | yes | — |
| 34 | #997 | — | 2 | 4 | no | — |
| 35 | #997 | C2×C32 | 1 | 1 | yes | characteristic |
| 36 | C3×#137 | — | 3 | 8 | no | — |
| 37 | C3×#124 | — | 3 | 4 | no | — |
| 38 | C3×#104 | — | 3 | 8 | no | — |
| 39 | C3×#100 | — | 3 | 4 | no | — |
| 40 | C23×C32 | — | 2 | 1 | no | — |
| 41 | C23×C32 | — | 3 | 1 | no | — |
| 42 | C23×C32 | S4 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×A4 | — | 4 | 4 | no | — |
| 44 | C2×C3×A4 | — | 8 | 4 | no | — |
| 45 | C2×C3×A4 | — | 8 | 8 | no | — |
| 46 | C2×C3×A4 | — | 8 | 4 | no | — |
| 47 | C32×Q8 | — | 6 | 2 | no | — |
| 48 | C32×Q8 | S4 | 1 | 2 | yes | — |
| 49 | C32×D4 | — | 3 | 2 | no | — |
| 50 | C32×D4 | — | 6 | 2 | no | — |
| 51 | C2×C32×C4 | — | 3 | 1 | no | — |
| 52 | C2×C32×C4 | — | 6 | 1 | no | — |
| 53 | C2×C32×C4 | — | 6 | 1 | no | — |
| 54 | C2×C32×C4 | — | 6 | 1 | no | — |
| 55 | C3×SL(2,𝔽3) | — | 4 | 8 | no | — |
| 56 | C3×SL(2,𝔽3) | — | 8 | 8 | no | — |
| 57 | C32×C8 | — | 6 | 2 | no | — |
| 58 | #455 | — | 3 | 1 | no | Sylow |
| 59 | C2×C33 | — | 16 | 1 | no | — |
| 60 | C3×#41 | — | 3 | 8 | no | — |
| 61 | C2×C3×Q8 | — | 3 | 4 | no | — |
| 62 | C2×C3×D4 | — | 3 | 4 | no | — |
| 63 | C2×C3×D4 | — | 6 | 4 | no | — |
| 64 | SL(2,ℤ/4) | — | 4 | 1 | no | — |
| 65 | #226 | — | 4 | 2 | no | — |
| 66 | C3×Q16 | — | 6 | 8 | no | — |
| 67 | C3×SD16 | — | 6 | 8 | no | — |
| 68 | C3×M16 | — | 3 | 8 | no | — |
| 69 | C3×C22⋊C4 | — | 6 | 4 | no | — |
| 70 | C3×C22⋊C4 | — | 6 | 4 | no | — |
| 71 | C3×C42 | — | 3 | 4 | no | — |
| 72 | C22×C32 | — | 3 | 1 | no | — |
| 73 | C22×C32 | — | 4 | 1 | no | — |
| 74 | C22×C32 | — | 6 | 1 | no | — |
| 75 | C22×C32 | — | 8 | 1 | no | — |
| 76 | C22×C32 | — | 12 | 1 | no | — |
| 77 | C3×A4 | — | 4 | 4 | no | — |
| 78 | C3×A4 | — | 8 | 4 | no | — |
| 79 | C3×A4 | — | 8 | 4 | no | — |
| 80 | C3×A4 | — | 8 | 8 | no | — |
| 81 | C32×C4 | — | 3 | 2 | no | — |
| 82 | C32×C4 | — | 12 | 1 | no | — |
| 83 | C32×C4 | — | 12 | 1 | no | — |
| 84 | C3×Dic3 | — | 16 | 4 | no | — |
| 85 | H(5,𝔽2) | C32×D3 | 1 | 1 | yes | characteristic |
| 86 | #137 | — | 3 | 2 | no | — |
| 87 | #124 | — | 3 | 1 | no | — |
| 88 | #104 | — | 3 | 2 | no | — |
| 89 | #100 | — | 3 | 1 | no | — |
| 90 | C33 | — | 16 | 1 | no | Sylow |
| 91 | C23×C3 | — | 2 | 4 | no | — |
| 92 | C23×C3 | — | 3 | 4 | no | — |
| 93 | C23×C3 | C3×S4 | 1 | 4 | yes | — |
| 94 | C2×A4 | — | 4 | 1 | no | — |
| 95 | C2×A4 | — | 8 | 1 | no | — |
| 96 | C2×A4 | — | 8 | 8 | no | — |
| 97 | C2×A4 | — | 8 | 4 | no | — |
| 98 | C3×Q8 | — | 6 | 8 | no | — |
| 99 | C3×Q8 | C3×S4 | 1 | 8 | yes | — |
| 100 | C3×D4 | — | 3 | 8 | no | — |
| 101 | C3×D4 | — | 6 | 8 | no | — |
| 102 | C2×C3×C4 | — | 3 | 4 | no | — |
| 103 | C2×C3×C4 | — | 6 | 4 | no | — |
| 104 | C2×C3×C4 | — | 6 | 4 | no | — |
| 105 | C2×C3×C4 | — | 6 | 4 | no | — |
| 106 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 107 | SL(2,𝔽3) | — | 8 | 8 | no | — |
| 108 | C3×C8 | — | 6 | 8 | no | — |
| 109 | C2×C32 | — | 6 | 1 | no | — |
| 110 | C2×C32 | — | 12 | 1 | no | — |
| 111 | C2×C32 | — | 16 | 4 | no | — |
| 112 | C2×C32 | — | 32 | 4 | no | — |
| 113 | C2×C32 | #1020 | 1 | 1 | yes | characteristic |
| 114 | #41 | — | 3 | 2 | no | — |
| 115 | C2×Q8 | — | 3 | 1 | no | — |
| 116 | C2×D4 | — | 3 | 1 | no | — |
| 117 | C2×D4 | — | 6 | 1 | no | — |
| 118 | Q16 | — | 6 | 2 | no | — |
| 119 | SD16 | — | 6 | 2 | no | — |
| 120 | M16 | — | 3 | 2 | no | — |
| 121 | C22⋊C4 | — | 6 | 1 | no | — |
| 122 | C22⋊C4 | — | 6 | 1 | no | — |
| 123 | C42 | — | 3 | 1 | no | — |
| 124 | C22×C3 | — | 3 | 4 | no | — |
| 125 | C22×C3 | — | 4 | 4 | no | — |
| 126 | C22×C3 | — | 6 | 4 | no | — |
| 127 | C22×C3 | — | 8 | 4 | no | — |
| 128 | C22×C3 | — | 12 | 4 | no | — |
| 129 | A4 | — | 4 | 1 | no | — |
| 130 | A4 | — | 8 | 1 | no | — |
| 131 | A4 | — | 8 | 4 | no | — |
| 132 | A4 | — | 8 | 8 | no | — |
| 133 | C3×C4 | — | 3 | 8 | no | — |
| 134 | C3×C4 | — | 12 | 4 | no | — |
| 135 | C3×C4 | — | 12 | 4 | no | — |
| 136 | Dic3 | — | 16 | 1 | no | — |
| 137 | C32 | — | 16 | 4 | no | — |
| 138 | C32 | — | 32 | 4 | no | — |
| 139 | C32 | #5933 | 1 | 1 | yes | characteristic, direct product factor |
| 140 | C23 | — | 2 | 1 | no | — |
| 141 | C23 | — | 3 | 1 | no | — |
| 142 | C23 | C32×S4 | 1 | 1 | yes | characteristic |
| 143 | Q8 | — | 6 | 2 | no | — |
| 144 | Q8 | C32×S4 | 1 | 2 | yes | — |
| 145 | D4 | — | 3 | 2 | no | — |
| 146 | D4 | — | 6 | 2 | no | — |
| 147 | C2×C4 | — | 3 | 1 | no | — |
| 148 | C2×C4 | — | 6 | 1 | no | — |
| 149 | C2×C4 | — | 6 | 1 | no | — |
| 150 | C2×C4 | — | 6 | 1 | no | — |
| 151 | C8 | — | 6 | 2 | no | — |
| 152 | C2×C3 | — | 6 | 4 | no | — |
| 153 | C2×C3 | — | 12 | 4 | no | — |
| 154 | C2×C3 | — | 16 | 1 | no | — |
| 155 | C2×C3 | — | 32 | 4 | no | — |
| 156 | C2×C3 | C3×#1020 | 1 | 4 | yes | — |
| 157 | C22 | — | 3 | 1 | no | — |
| 158 | C22 | — | 4 | 1 | no | — |
| 159 | C22 | — | 6 | 1 | no | — |
| 160 | C22 | — | 8 | 1 | no | — |
| 161 | C22 | — | 12 | 1 | no | — |
| 162 | C4 | — | 3 | 2 | no | — |
| 163 | C4 | — | 12 | 1 | no | — |
| 164 | C4 | — | 12 | 1 | no | — |
| 165 | C3 | — | 16 | 1 | no | — |
| 166 | C3 | — | 32 | 4 | no | — |
| 167 | C3 | C3×#5933 | 1 | 4 | yes | direct product factor |
| 168 | C2 | — | 6 | 1 | no | — |
| 169 | C2 | — | 12 | 1 | no | — |
| 170 | C2 | C32×#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C32 | #1020 |
| commutator, cosocle, hypocenter | #997 | C2×C32 |
| Frattini | C2 | C32×#1020 |
| Fitting | C32×H(5,𝔽2) | D3 |
| prime radical | C32×#997 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#5933 | C3 | direct (N × H) |
| 16. #5933 | C32 | direct (N × H) |
| 139. C32 | #5933 | direct (N × H) |
| 167. C3 | C3×#5933 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#997 | 2 |
| C3×#5933 | 3 |
| C32×#455 | 3 |
| C32×SL(2,ℤ/4) | 4 |
| C32×#226 | 4 |