Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
312 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 21 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 | #11724211 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | SL(2,𝔽3)×#157 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #10593476 | — | 3 | 1 | no | maximal |
| 4 | C2×#157⋊D3 | — | 4 | 1 | no | maximal |
| 5 | C3⋊D3×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 6 | C3×C3⋊GL(2,𝔽3) | — | 2 | 2 | no | — |
| 7 | Q8×#157 | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | #64231 | — | 3 | 1 | no | — |
| 9 | D3×GL(2,𝔽3) | — | 6 | 2 | no | — |
| 10 | #63940 | — | 3 | 2 | no | — |
| 11 | #63732 | — | 3 | 1 | no | — |
| 12 | #63729 | — | 3 | 2 | no | — |
| 13 | C2×#1113 | — | 4 | 2 | no | — |
| 14 | C2×C3×#157 | — | 4 | 1 | no | — |
| 15 | #157⋊D3 | — | 4 | 4 | no | — |
| 16 | C32×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic, complemented |
| 17 | C4⋊GL(2,𝔽3) | — | 9 | 1 | no | maximal |
| 18 | C2×GO+(4,𝔽2) | — | 6 | 1 | no | — |
| 19 | C2×GO+(4,𝔽2) | — | 6 | 1 | no | — |
| 20 | Q8×C3⋊D3 | C2×D3 | 1 | 1 | yes | characteristic |
| 21 | #3802 | — | 3 | 2 | no | — |
| 22 | C4⋊#157 | — | 3 | 1 | no | — |
| 23 | C4×#157 | — | 3 | 1 | no | — |
| 24 | D3×SL(2,𝔽3) | — | 6 | 2 | no | — |
| 25 | C3⋊GL(2,𝔽3) | — | 2 | 2 | no | — |
| 26 | C3×GL(2,𝔽3) | — | 6 | 2 | no | — |
| 27 | C3×GL(2,𝔽3) | — | 12 | 2 | no | — |
| 28 | C3×#215 | — | 6 | 2 | no | — |
| 29 | #3716 | — | 6 | 2 | no | — |
| 30 | #3710 | — | 3 | 2 | no | — |
| 31 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 32 | C2×C3×C3⋊D3 | — | 8 | 2 | no | — |
| 33 | #1113 | — | 4 | 4 | no | — |
| 34 | #1113 | — | 8 | 2 | no | — |
| 35 | C3×#157 | — | 4 | 2 | no | — |
| 36 | C2×GL(2,𝔽3) | — | 9 | 2 | no | — |
| 37 | D3×SD16 | — | 18 | 2 | no | — |
| 38 | C4×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 39 | C2×D32 | — | 6 | 2 | no | — |
| 40 | C2×D32 | — | 24 | 2 | no | — |
| 41 | C2×#157 | — | 3 | 1 | no | — |
| 42 | C2×#157 | S4 | 1 | 1 | yes | characteristic |
| 43 | GO+(4,𝔽2) | — | 12 | 2 | no | — |
| 44 | GO+(4,𝔽2) | — | 12 | 2 | no | — |
| 45 | C32×Q8 | D12 | 1 | 1 | yes | characteristic, complemented |
| 46 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 47 | #634 | — | 3 | 1 | no | — |
| 48 | C3×D12 | — | 6 | 2 | no | — |
| 49 | C3×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 50 | C3×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 51 | #625 | — | 6 | 2 | no | — |
| 52 | C3×C3⋊C8 | — | 6 | 2 | no | — |
| 53 | #460 | — | 27 | 1 | no | Sylow |
| 54 | C2×C33 | — | 4 | 1 | no | — |
| 55 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 56 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 57 | C3×C3⋊D3 | — | 8 | 4 | no | — |
| 58 | D3×Q8 | — | 6 | 2 | no | — |
| 59 | D3×D4 | — | 18 | 2 | no | — |
| 60 | C2×D12 | — | 36 | 1 | no | — |
| 61 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 62 | GL(2,𝔽3) | — | 6 | 2 | no | — |
| 63 | GL(2,𝔽3) | — | 12 | 2 | no | — |
| 64 | C3×SD16 | — | 18 | 2 | no | — |
| 65 | #215 | — | 6 | 2 | no | — |
| 66 | #214 | — | 18 | 2 | no | — |
| 67 | #204 | — | 18 | 2 | no | — |
| 68 | D3×C8 | — | 18 | 2 | no | — |
| 69 | C2×C3⋊D3 | — | 8 | 2 | no | — |
| 70 | C2×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 71 | C2×C3×D3 | — | 12 | 2 | no | — |
| 72 | C2×C3×D3 | — | 24 | 2 | no | — |
| 73 | C2×C3×D3 | — | 24 | 2 | no | — |
| 74 | C2×C3×D3 | — | 48 | 2 | no | — |
| 75 | D32 | — | 12 | 2 | no | — |
| 76 | D32 | — | 12 | 2 | no | — |
| 77 | D32 | — | 24 | 4 | no | — |
| 78 | D32 | — | 24 | 4 | no | — |
| 79 | #157 | — | 6 | 1 | no | — |
| 80 | #157 | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 81 | C32×C4 | — | 3 | 1 | no | — |
| 82 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 83 | C2×SD16 | — | 27 | 2 | no | — |
| 84 | #127 | — | 27 | 1 | no | — |
| 85 | C4×Q8 | — | 9 | 1 | no | — |
| 86 | C4⋊C8 | — | 27 | 1 | no | — |
| 87 | #102 | — | 27 | 2 | no | — |
| 88 | C33 | — | 4 | 1 | no | Sylow |
| 89 | C22×D3 | — | 36 | 2 | no | — |
| 90 | C22×D3 | — | 36 | 2 | no | — |
| 91 | C3×Q8 | — | 2 | 2 | no | — |
| 92 | C3×D4 | — | 18 | 2 | no | — |
| 93 | C2×C3×C4 | — | 36 | 1 | no | — |
| 94 | #67 | — | 36 | 2 | no | — |
| 95 | D12 | — | 6 | 2 | no | — |
| 96 | D12 | — | 36 | 4 | no | — |
| 97 | C4×D3 | — | 18 | 2 | no | — |
| 98 | Dic6 | — | 18 | 2 | no | — |
| 99 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 100 | SL(2,𝔽3) | GO+(4,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 101 | C3×C8 | — | 18 | 2 | no | — |
| 102 | C3⋊C8 | — | 6 | 2 | no | — |
| 103 | C2×C32 | — | 8 | 2 | no | — |
| 104 | C2×C32 | — | 16 | 2 | no | — |
| 105 | C2×C32 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 106 | C3⋊D3 | — | 8 | 4 | no | — |
| 107 | C3⋊D3 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 108 | C3⋊D3 | #986 | 1 | 1 | yes | characteristic |
| 109 | C3×D3 | — | 24 | 2 | no | — |
| 110 | C3×D3 | — | 24 | 2 | no | — |
| 111 | C3×D3 | — | 24 | 4 | no | — |
| 112 | C3×D3 | — | 24 | 2 | no | — |
| 113 | C3×D3 | — | 48 | 4 | no | — |
| 114 | C2×Q8 | — | 9 | 1 | no | — |
| 115 | C2×D4 | — | 27 | 2 | no | — |
| 116 | C2×D4 | — | 54 | 1 | no | — |
| 117 | C2×D4 | — | 54 | 1 | no | — |
| 118 | SD16 | — | 18 | 2 | no | — |
| 119 | SD16 | — | 54 | 2 | no | — |
| 120 | C2×C8 | — | 27 | 2 | no | — |
| 121 | C4⋊C4 | — | 27 | 1 | no | — |
| 122 | C42 | — | 27 | 1 | no | — |
| 123 | C22×C3 | — | 36 | 1 | no | — |
| 124 | C22×C3 | — | 36 | 2 | no | — |
| 125 | C2×D3 | — | 6 | 2 | no | — |
| 126 | C2×D3 | — | 12 | 2 | no | — |
| 127 | C2×D3 | — | 24 | 2 | no | — |
| 128 | C2×D3 | — | 36 | 4 | no | — |
| 129 | C2×D3 | — | 48 | 2 | no | — |
| 130 | C2×D3 | — | 72 | 2 | no | — |
| 131 | C2×D3 | — | 72 | 2 | no | — |
| 132 | C2×D3 | — | 72 | 2 | no | — |
| 133 | C3×C4 | — | 6 | 2 | no | — |
| 134 | C3×C4 | — | 36 | 2 | no | — |
| 135 | Dic3 | — | 18 | 2 | no | — |
| 136 | C32 | — | 8 | 2 | no | — |
| 137 | C32 | — | 16 | 2 | no | — |
| 138 | C32 | C4⋊GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 139 | C23 | — | 54 | 2 | no | — |
| 140 | Q8 | — | 27 | 1 | no | — |
| 141 | Q8 | #157⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 142 | D4 | — | 18 | 2 | no | — |
| 143 | D4 | — | 54 | 2 | no | — |
| 144 | D4 | — | 108 | 2 | no | — |
| 145 | D4 | — | 108 | 2 | no | — |
| 146 | C2×C4 | — | 9 | 1 | no | — |
| 147 | C2×C4 | — | 27 | 1 | no | — |
| 148 | C2×C4 | — | 27 | 1 | no | — |
| 149 | C8 | — | 18 | 2 | no | — |
| 150 | C2×C3 | — | 2 | 2 | no | — |
| 151 | C2×C3 | — | 4 | 1 | no | — |
| 152 | C2×C3 | — | 16 | 2 | no | — |
| 153 | C2×C3 | — | 36 | 1 | no | — |
| 154 | C2×C3 | — | 36 | 1 | no | — |
| 155 | C2×C3 | — | 72 | 2 | no | — |
| 156 | D3 | — | 6 | 2 | no | — |
| 157 | D3 | — | 6 | 2 | no | — |
| 158 | D3 | — | 24 | 2 | no | — |
| 159 | D3 | — | 24 | 4 | no | — |
| 160 | D3 | — | 48 | 4 | no | — |
| 161 | C22 | — | 9 | 1 | no | — |
| 162 | C22 | — | 36 | 2 | no | — |
| 163 | C22 | — | 108 | 2 | no | — |
| 164 | C22 | — | 108 | 2 | no | — |
| 165 | C4 | — | 3 | 1 | no | — |
| 166 | C4 | — | 9 | 2 | no | — |
| 167 | C4 | — | 27 | 1 | no | — |
| 168 | C4 | — | 54 | 1 | no | — |
| 169 | C3 | — | 2 | 2 | no | — |
| 170 | C3 | — | 4 | 1 | no | — |
| 171 | C3 | — | 16 | 2 | no | — |
| 172 | C2 | — | 9 | 1 | no | — |
| 173 | C2 | — | 9 | 1 | no | — |
| 174 | C2 | — | 72 | 2 | no | — |
| 175 | C2 | #157⋊S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #157⋊S4 |
| commutator, cosocle | C3⋊D3×SL(2,𝔽3) | C22 |
| Fitting | C32×Q8 | D12 |
| socle | C2×C32 | C4⋊S4 |
| hypocenter | C32×SL(2,𝔽3) | D4 |
| LCS | G ▷ C3⋊D3×SL(2,𝔽3) ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11724211 | C2 | semidirect |
| 2. SL(2,𝔽3)×#157 | C2 | semidirect |
| 7. Q8×#157 | D3 | semidirect |
| 16. C32×SL(2,𝔽3) | D4 | semidirect |
| 45. C32×Q8 | D12 | semidirect |
| 80. #157 | GL(2,𝔽3) | semidirect |
| 100. SL(2,𝔽3) | GO+(4,𝔽2) | semidirect |
| 138. C32 | C4⋊GL(2,𝔽3) | semidirect |
| 141. Q8 | #157⋊D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11724211 | 2 |
| SL(2,𝔽3)×#157 | 2 |
| #10593476 | 3 |
| C2×#157⋊D3 | 4 |
| C4⋊GL(2,𝔽3) | 9 |