Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
240 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 37 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 | C3×#578⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10593641 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊#4628 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #73139 | — | 3 | 3 | no | maximal |
| 5 | #72124 | — | 3 | 1 | no | maximal |
| 6 | C3×2−1+4⋊C3 | — | 2 | 1 | no | — |
| 7 | #64267 | — | 2 | 1 | no | — |
| 8 | C3×#131⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | #63653 | — | 4 | 1 | no | maximal |
| 10 | C3×#578 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #578⋊C3 | D3 | 1 | 3 | yes | complemented |
| 12 | #5407 | — | 3 | 3 | no | — |
| 13 | #5193 | — | 3 | 1 | no | — |
| 14 | C2×#828 | — | 3 | 1 | no | — |
| 15 | #5142 | — | 3 | 1 | no | — |
| 16 | #4628 | — | 3 | 3 | no | — |
| 17 | #4567 | — | 3 | 1 | no | — |
| 18 | #4565 | — | 3 | 1 | no | — |
| 19 | #4561 | — | 3 | 1 | no | — |
| 20 | C3×#41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 22 | C3⋊GL(2,𝔽3) | — | 4 | 1 | no | — |
| 23 | C32×D8 | — | 4 | 1 | no | — |
| 24 | #3745 | — | 4 | 1 | no | — |
| 25 | C3⋊(C3⋊C16) | — | 4 | 1 | no | — |
| 26 | #2193 | — | 9 | 1 | no | Sylow |
| 27 | C3×2−1+4 | — | 2 | 1 | no | — |
| 28 | 2−1+4⋊C3 | — | 2 | 3 | no | — |
| 29 | #986 | — | 6 | 3 | no | — |
| 30 | C3×#137 | — | 6 | 1 | no | — |
| 31 | C3×#135 | — | 3 | 1 | no | — |
| 32 | C2×C3×Q16 | — | 3 | 1 | no | — |
| 33 | C3×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | #949 | — | 6 | 1 | no | — |
| 35 | C2×D24 | — | 3 | 1 | no | — |
| 36 | #131⋊C3 | C2×D3 | 1 | 3 | yes | — |
| 37 | #828 | — | 6 | 2 | no | — |
| 38 | #828 | — | 6 | 1 | no | — |
| 39 | #826 | — | 6 | 1 | no | — |
| 40 | #826 | — | 12 | 3 | no | — |
| 41 | #823 | — | 6 | 1 | no | — |
| 42 | #819 | — | 3 | 1 | no | — |
| 43 | #812 | — | 3 | 1 | no | — |
| 44 | C2×C3⋊C16 | — | 3 | 1 | no | — |
| 45 | C32×D4 | — | 8 | 1 | no | — |
| 46 | C4⋊(C3⋊D3) | — | 8 | 1 | no | — |
| 47 | C3×SL(2,𝔽3) | D8 | 1 | 1 | yes | characteristic |
| 48 | C32×C8 | — | 4 | 1 | no | — |
| 49 | #578 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 50 | #509 | — | 9 | 1 | no | — |
| 51 | C2×SD32 | — | 9 | 1 | no | — |
| 52 | #472 | — | 9 | 1 | no | — |
| 53 | #362 | — | 9 | 1 | no | — |
| 54 | #359 | — | 9 | 1 | no | — |
| 55 | #350 | — | 9 | 1 | no | — |
| 56 | C3×#41 | — | 6 | 1 | no | — |
| 57 | C3×#41 | — | 12 | 1 | no | — |
| 58 | C3×#41 | #67 | 1 | 1 | yes | characteristic |
| 59 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 60 | C2×C3×Q8 | — | 6 | 1 | no | — |
| 61 | C2×D12 | — | 6 | 1 | no | — |
| 62 | #41⋊C3 | #67 | 1 | 3 | yes | — |
| 63 | C2×SL(2,𝔽3) | — | 4 | 3 | no | — |
| 64 | GL(2,𝔽3) | — | 12 | 3 | no | — |
| 65 | C3×Q16 | — | 3 | 2 | no | — |
| 66 | C3×Q16 | — | 3 | 1 | no | — |
| 67 | C3×SD16 | — | 6 | 1 | no | — |
| 68 | C3×D8 | — | 4 | 3 | no | — |
| 69 | C3×D8 | S4 | 1 | 1 | yes | characteristic |
| 70 | C3×M16 | — | 3 | 1 | no | — |
| 71 | C2×C3×C8 | — | 3 | 1 | no | — |
| 72 | #215 | — | 12 | 1 | no | — |
| 73 | #213 | — | 12 | 1 | no | — |
| 74 | #208 | — | 6 | 1 | no | — |
| 75 | D24 | — | 6 | 1 | no | — |
| 76 | D24 | — | 6 | 1 | no | — |
| 77 | D24 | — | 12 | 3 | no | — |
| 78 | C3⋊C16 | — | 6 | 1 | no | — |
| 79 | C3⋊C16 | — | 6 | 1 | no | — |
| 80 | C3⋊C16 | — | 12 | 3 | no | — |
| 81 | C22×C32 | — | 16 | 1 | no | — |
| 82 | C2×C3⋊D3 | — | 16 | 1 | no | — |
| 83 | C32×C4 | — | 4 | 1 | no | — |
| 84 | 2−1+4 | — | 2 | 1 | no | — |
| 85 | #137 | — | 6 | 1 | no | — |
| 86 | AGL(1,ℤ/8) | — | 18 | 1 | no | — |
| 87 | #135 | — | 3 | 1 | no | — |
| 88 | C2×Q16 | — | 3 | 1 | no | — |
| 89 | C2×D8 | — | 9 | 1 | no | — |
| 90 | #131 | C2×C3⋊D3 | 1 | 1 | yes | characteristic |
| 91 | SD32 | — | 18 | 2 | no | — |
| 92 | SD32 | — | 18 | 1 | no | — |
| 93 | D16 | — | 18 | 1 | no | — |
| 94 | M32 | — | 9 | 1 | no | — |
| 95 | C2×C16 | — | 9 | 1 | no | — |
| 96 | #108 | — | 9 | 1 | no | — |
| 97 | #100 | — | 18 | 1 | no | — |
| 98 | C22×D3 | — | 12 | 1 | no | — |
| 99 | C3×Q8 | — | 6 | 1 | no | — |
| 100 | C3×Q8 | — | 12 | 1 | no | — |
| 101 | C3×Q8 | #213 | 1 | 1 | yes | characteristic |
| 102 | C3×D4 | — | 2 | 1 | no | — |
| 103 | C3×D4 | — | 3 | 1 | no | — |
| 104 | C3×D4 | — | 8 | 3 | no | — |
| 105 | C3×D4 | — | 12 | 1 | no | — |
| 106 | C2×C3×C4 | — | 3 | 1 | no | — |
| 107 | C2×C3×C4 | — | 12 | 1 | no | — |
| 108 | C2×C3×C4 | — | 12 | 1 | no | — |
| 109 | D12 | — | 12 | 1 | no | — |
| 110 | D12 | — | 12 | 1 | no | — |
| 111 | D12 | — | 24 | 3 | no | — |
| 112 | SL(2,𝔽3) | #213 | 1 | 3 | yes | — |
| 113 | C3×C8 | — | 3 | 1 | no | — |
| 114 | C3×C8 | — | 4 | 3 | no | — |
| 115 | C3×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 116 | C3⋊C8 | — | 12 | 1 | no | — |
| 117 | C2×C32 | — | 4 | 1 | no | — |
| 118 | C2×C32 | — | 32 | 1 | no | — |
| 119 | C3⋊D3 | — | 32 | 1 | no | — |
| 120 | #41 | — | 6 | 1 | no | — |
| 121 | #41 | — | 12 | 1 | no | — |
| 122 | #41 | #638 | 1 | 1 | yes | characteristic |
| 123 | C2×Q8 | — | 4 | 1 | no | — |
| 124 | C2×Q8 | — | 6 | 1 | no | — |
| 125 | C2×D4 | — | 18 | 1 | no | — |
| 126 | Q16 | — | 3 | 2 | no | — |
| 127 | Q16 | — | 3 | 1 | no | — |
| 128 | SD16 | — | 6 | 1 | no | — |
| 129 | SD16 | — | 36 | 1 | no | — |
| 130 | D8 | — | 18 | 1 | no | — |
| 131 | D8 | — | 18 | 1 | no | — |
| 132 | D8 | — | 36 | 1 | no | — |
| 133 | D8 | C3⋊S4 | 1 | 1 | yes | characteristic |
| 134 | M16 | — | 3 | 1 | no | — |
| 135 | M16 | — | 18 | 1 | no | — |
| 136 | C2×C8 | — | 3 | 1 | no | — |
| 137 | C16 | — | 18 | 1 | no | — |
| 138 | C16 | — | 18 | 1 | no | — |
| 139 | C22×C3 | — | 3 | 1 | no | — |
| 140 | C22×C3 | — | 4 | 1 | no | — |
| 141 | C22×C3 | — | 16 | 3 | no | — |
| 142 | C2×D3 | — | 24 | 1 | no | — |
| 143 | C2×D3 | — | 48 | 3 | no | — |
| 144 | C2×D3 | — | 48 | 1 | no | — |
| 145 | C3×C4 | — | 3 | 1 | no | — |
| 146 | C3×C4 | — | 4 | 3 | no | — |
| 147 | C3×C4 | — | 12 | 1 | no | — |
| 148 | C3×C4 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 149 | C32 | — | 4 | 1 | no | Sylow |
| 150 | C23 | — | 36 | 1 | no | — |
| 151 | Q8 | — | 6 | 1 | no | — |
| 152 | Q8 | — | 12 | 1 | no | — |
| 153 | Q8 | #3753 | 1 | 1 | yes | characteristic |
| 154 | D4 | — | 2 | 1 | no | — |
| 155 | D4 | — | 3 | 1 | no | — |
| 156 | D4 | — | 12 | 1 | no | — |
| 157 | D4 | — | 36 | 1 | no | — |
| 158 | D4 | — | 36 | 1 | no | — |
| 159 | C2×C4 | — | 3 | 1 | no | — |
| 160 | C2×C4 | — | 12 | 1 | no | — |
| 161 | C2×C4 | — | 12 | 1 | no | — |
| 162 | C8 | — | 3 | 1 | no | — |
| 163 | C8 | — | 36 | 1 | no | — |
| 164 | C8 | C2×C3⋊S4 | 1 | 1 | yes | characteristic |
| 165 | C2×C3 | — | 4 | 3 | no | — |
| 166 | C2×C3 | — | 6 | 1 | no | — |
| 167 | C2×C3 | — | 8 | 1 | no | — |
| 168 | C2×C3 | — | 32 | 3 | no | — |
| 169 | C2×C3 | #5415 | 1 | 1 | yes | characteristic |
| 170 | D3 | — | 48 | 1 | no | — |
| 171 | D3 | — | 96 | 3 | no | — |
| 172 | C22 | — | 3 | 1 | no | — |
| 173 | C22 | — | 4 | 1 | no | — |
| 174 | C22 | — | 72 | 1 | no | — |
| 175 | C22 | — | 144 | 1 | no | — |
| 176 | C4 | — | 3 | 1 | no | — |
| 177 | C4 | — | 12 | 1 | no | — |
| 178 | C4 | #64269 | 1 | 1 | yes | characteristic |
| 179 | C3 | — | 4 | 3 | no | — |
| 180 | C3 | #73139 | 1 | 1 | yes | characteristic, complemented |
| 181 | C2 | — | 6 | 1 | no | — |
| 182 | C2 | — | 8 | 1 | no | — |
| 183 | C2 | — | 144 | 1 | no | — |
| 184 | C2 | #10593649 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#578⋊C3 | C2 | semidirect |
| 2. #10593641 | C2 | semidirect |
| 3. C3⋊#4628 | C2 | semidirect |
| 10. C3×#578 | D3 | semidirect |
| 11. #578⋊C3 | D3 | semidirect |
| 49. #578 | C3⋊D3 | semidirect |
| 180. C3 | #73139 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#578⋊C3 | 2 |
| #10593641 | 2 |
| C3⋊#4628 | 2 |
| #73139 | 3 |
| #72124 | 3 |
| #63653 | 4 |