Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
485 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 485 conjugacy classes of subgroups — too many to draw. Only its 23 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, complemented |
| 2 | C3×#5934 | — | 3 | 3 | no | maximal |
| 3 | C3×#5244 | — | 3 | 1 | no | maximal |
| 4 | C2×#6258 | — | 4 | 3 | no | maximal |
| 5 | C32×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C3×#997 | D3 | 1 | 3 | yes | complemented |
| 7 | C3×#937 | — | 3 | 3 | no | — |
| 8 | C3×#834 | — | 3 | 3 | no | — |
| 9 | C2×C22⋊H(3,𝔽3) | — | 4 | 3 | no | — |
| 10 | #6258 | — | 4 | 6 | no | — |
| 11 | Q8⋊H(3,𝔽3) | — | 8 | 1 | no | — |
| 12 | #5934 | — | 3 | 3 | no | — |
| 13 | C3×B(4,𝔽2) | — | 9 | 1 | no | — |
| 14 | #5244 | — | 3 | 1 | no | — |
| 15 | C23×C3×D3 | — | 3 | 1 | no | — |
| 16 | C2×C3×S4 | — | 12 | 9 | no | — |
| 17 | C32×#41 | — | 6 | 1 | no | — |
| 18 | C2×C32×D4 | — | 3 | 3 | no | — |
| 19 | C2×C3×#67 | — | 6 | 3 | no | — |
| 20 | C3×#217 | — | 3 | 3 | no | — |
| 21 | C3×(C2×D3)⋊C4 | — | 6 | 3 | no | — |
| 22 | C2×#272 | — | 16 | 1 | no | — |
| 23 | C22⋊H(3,𝔽3) | — | 4 | 3 | no | — |
| 24 | C3×H(5,𝔽2) | — | 3 | 1 | no | — |
| 25 | C3×H(5,𝔽2) | C3⋊D3 | 1 | 1 | yes | characteristic |
| 26 | #997 | — | 3 | 3 | no | — |
| 27 | C3×#120 | — | 9 | 3 | no | — |
| 28 | #937 | — | 3 | 3 | no | — |
| 29 | C3×#99 | — | 9 | 3 | no | — |
| 30 | #834 | — | 3 | 3 | no | — |
| 31 | C23×C32 | — | 3 | 1 | no | — |
| 32 | C23×C32 | S4 | 1 | 3 | yes | complemented |
| 33 | C22×C3×D3 | — | 6 | 3 | no | — |
| 34 | C22×C3×D3 | — | 12 | 2 | no | — |
| 35 | C2×C3×A4 | — | 4 | 9 | no | — |
| 36 | C3×S4 | — | 12 | 18 | no | — |
| 37 | C32×Q8 | — | 2 | 1 | no | — |
| 38 | C32×D4 | — | 6 | 3 | no | — |
| 39 | C2×C32×C4 | — | 3 | 3 | no | — |
| 40 | C3×#67 | — | 12 | 6 | no | — |
| 41 | C2×C3×Dic3 | — | 6 | 3 | no | — |
| 42 | C3×SL(2,𝔽3) | — | 8 | 3 | no | — |
| 43 | B(4,𝔽2) | — | 9 | 1 | no | Sylow |
| 44 | C2×H(3,𝔽3) | — | 16 | 1 | no | — |
| 45 | #272 | — | 16 | 2 | no | — |
| 46 | C24×C3 | — | 9 | 1 | no | — |
| 47 | C23×D3 | — | 3 | 1 | no | — |
| 48 | C2×S4 | — | 12 | 9 | no | — |
| 49 | C3×#41 | — | 6 | 1 | no | — |
| 50 | C3×#41 | — | 6 | 3 | no | — |
| 51 | C2×C3×D4 | — | 3 | 3 | no | — |
| 52 | C2×C3×D4 | — | 3 | 3 | no | — |
| 53 | C2×C3×D4 | — | 6 | 3 | no | — |
| 54 | C2×C3×D4 | — | 18 | 3 | no | — |
| 55 | C2×#67 | — | 6 | 3 | no | — |
| 56 | C3×C22⋊C4 | — | 9 | 3 | no | — |
| 57 | C3×C22⋊C4 | — | 18 | 3 | no | — |
| 58 | #217 | — | 3 | 3 | no | — |
| 59 | (C2×D3)⋊C4 | — | 6 | 3 | no | — |
| 60 | C22×C32 | — | 3 | 3 | no | — |
| 61 | C22×C32 | — | 4 | 3 | no | — |
| 62 | C22×C32 | — | 12 | 1 | no | — |
| 63 | C2×C3×D3 | — | 6 | 6 | no | — |
| 64 | C2×C3×D3 | — | 12 | 3 | no | — |
| 65 | C2×C3×D3 | — | 12 | 1 | no | — |
| 66 | C2×C3×D3 | — | 48 | 3 | no | — |
| 67 | C3×A4 | — | 4 | 9 | no | — |
| 68 | C32×C4 | — | 6 | 1 | no | — |
| 69 | C3×Dic3 | — | 12 | 3 | no | — |
| 70 | H(5,𝔽2) | #272 | 1 | 1 | yes | characteristic, complemented |
| 71 | #120 | — | 9 | 3 | no | — |
| 72 | #99 | — | 9 | 3 | no | — |
| 73 | H(3,𝔽3) | — | 16 | 1 | no | Sylow |
| 74 | C23×C3 | — | 3 | 1 | no | — |
| 75 | C23×C3 | — | 3 | 1 | no | — |
| 76 | C23×C3 | — | 3 | 3 | no | — |
| 77 | C23×C3 | — | 6 | 1 | no | — |
| 78 | C23×C3 | — | 18 | 3 | no | — |
| 79 | C23×C3 | — | 36 | 2 | no | — |
| 80 | C23×C3 | C3⋊S4 | 1 | 3 | yes | — |
| 81 | C22×D3 | — | 6 | 3 | no | — |
| 82 | C22×D3 | — | 12 | 2 | no | — |
| 83 | C2×A4 | — | 12 | 9 | no | — |
| 84 | S4 | — | 12 | 18 | no | — |
| 85 | C3×Q8 | — | 2 | 1 | no | — |
| 86 | C3×Q8 | — | 6 | 1 | no | — |
| 87 | C3×D4 | — | 6 | 3 | no | — |
| 88 | C3×D4 | — | 6 | 6 | no | — |
| 89 | C3×D4 | — | 6 | 3 | no | — |
| 90 | C3×D4 | — | 36 | 6 | no | — |
| 91 | C2×C3×C4 | — | 3 | 3 | no | — |
| 92 | C2×C3×C4 | — | 3 | 3 | no | — |
| 93 | C2×C3×C4 | — | 6 | 3 | no | — |
| 94 | C2×C3×C4 | — | 18 | 3 | no | — |
| 95 | #67 | — | 12 | 6 | no | — |
| 96 | C2×Dic3 | — | 6 | 3 | no | — |
| 97 | SL(2,𝔽3) | — | 24 | 3 | no | — |
| 98 | C2×C32 | — | 6 | 3 | no | — |
| 99 | C2×C32 | — | 16 | 3 | no | — |
| 100 | C2×C32 | #1020 | 1 | 1 | yes | characteristic |
| 101 | C3×D3 | — | 12 | 2 | no | — |
| 102 | C3×D3 | — | 48 | 6 | no | — |
| 103 | C24 | — | 9 | 1 | no | — |
| 104 | #41 | — | 6 | 1 | no | — |
| 105 | C2×D4 | — | 3 | 3 | no | — |
| 106 | C2×D4 | — | 18 | 3 | no | — |
| 107 | C22⋊C4 | — | 9 | 3 | no | — |
| 108 | C22⋊C4 | — | 18 | 3 | no | — |
| 109 | C22×C3 | — | 3 | 3 | no | — |
| 110 | C22×C3 | — | 3 | 3 | no | — |
| 111 | C22×C3 | — | 4 | 3 | no | — |
| 112 | C22×C3 | — | 6 | 3 | no | — |
| 113 | C22×C3 | — | 12 | 3 | no | — |
| 114 | C22×C3 | — | 12 | 1 | no | — |
| 115 | C22×C3 | — | 12 | 1 | no | — |
| 116 | C22×C3 | — | 18 | 6 | no | — |
| 117 | C22×C3 | — | 24 | 1 | no | — |
| 118 | C22×C3 | — | 36 | 3 | no | — |
| 119 | C22×C3 | — | 36 | 1 | no | — |
| 120 | C2×D3 | — | 6 | 6 | no | — |
| 121 | C2×D3 | — | 12 | 1 | no | — |
| 122 | C2×D3 | — | 12 | 3 | no | — |
| 123 | C2×D3 | — | 48 | 3 | no | — |
| 124 | A4 | — | 12 | 9 | no | — |
| 125 | C3×C4 | — | 6 | 3 | no | — |
| 126 | C3×C4 | — | 6 | 1 | no | — |
| 127 | C3×C4 | — | 36 | 3 | no | — |
| 128 | Dic3 | — | 12 | 3 | no | — |
| 129 | C32 | — | 16 | 3 | no | — |
| 130 | C32 | #5934 | 1 | 1 | yes | characteristic, complemented |
| 131 | C23 | — | 3 | 1 | no | — |
| 132 | C23 | — | 18 | 3 | no | — |
| 133 | C23 | — | 36 | 2 | no | — |
| 134 | C23 | #6258 | 1 | 3 | yes | complemented |
| 135 | Q8 | — | 2 | 1 | no | — |
| 136 | D4 | — | 6 | 3 | no | — |
| 137 | D4 | — | 36 | 6 | no | — |
| 138 | C2×C4 | — | 3 | 3 | no | — |
| 139 | C2×C4 | — | 18 | 3 | no | — |
| 140 | C2×C3 | — | 3 | 1 | no | — |
| 141 | C2×C3 | — | 6 | 3 | no | — |
| 142 | C2×C3 | — | 6 | 3 | no | — |
| 143 | C2×C3 | — | 12 | 3 | no | — |
| 144 | C2×C3 | — | 36 | 2 | no | — |
| 145 | C2×C3 | — | 48 | 3 | no | — |
| 146 | C2×C3 | C3⋊#1020 | 1 | 1 | yes | characteristic |
| 147 | D3 | — | 12 | 2 | no | — |
| 148 | D3 | — | 48 | 6 | no | — |
| 149 | C22 | — | 3 | 3 | no | — |
| 150 | C22 | — | 4 | 3 | no | — |
| 151 | C22 | — | 12 | 1 | no | — |
| 152 | C22 | — | 18 | 6 | no | — |
| 153 | C22 | — | 36 | 3 | no | — |
| 154 | C22 | — | 36 | 1 | no | — |
| 155 | C4 | — | 6 | 1 | no | — |
| 156 | C4 | — | 36 | 3 | no | — |
| 157 | C3 | — | 3 | 1 | no | — |
| 158 | C3 | — | 48 | 3 | no | — |
| 159 | C3 | C3⋊#5934 | 1 | 1 | yes | characteristic |
| 160 | C2 | — | 6 | 3 | no | — |
| 161 | C2 | — | 36 | 2 | no | — |
| 162 | C2 | #11724055 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | C3⋊#1020 |
| commutator, cosocle, hypocenter | C32⋊#997 | C2 |
| Fitting | C32×H(5,𝔽2) | D3 |
| LCS | G ▷ C32⋊#997 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32⋊#997 | C2 | semidirect |
| 5. C32×H(5,𝔽2) | D3 | semidirect |
| 6. C3×#997 | D3 | semidirect |
| 32. C23×C32 | S4 | semidirect |
| 70. H(5,𝔽2) | #272 | semidirect |
| 130. C32 | #5934 | semidirect |
| 134. C23 | #6258 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32⋊#997 | 2 |
| C3×#5934 | 3 |
| C3×#5244 | 3 |
| C2×#6258 | 4 |