Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
334 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 39 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×#393⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(#393⋊C3) | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #76888 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #73210 | — | 3 | 1 | no | maximal |
| 5 | C22×D3×A4 | — | 4 | 1 | no | maximal |
| 6 | C3×C22⋊SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C3×#393 | C2×C3 | 1 | 1 | yes | characteristic, complemented |
| 8 | #5231 | C2×C3 | 1 | 2 | yes | complemented |
| 9 | #4955 | — | 3 | 1 | no | — |
| 10 | #4947 | — | 3 | 1 | no | — |
| 11 | #4673 | — | 3 | 2 | no | — |
| 12 | #393⋊C3 | — | 2 | 1 | no | — |
| 13 | #393⋊C3 | — | 3 | 2 | no | — |
| 14 | #393⋊C3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | C22×C3×A4 | — | 4 | 1 | no | — |
| 16 | C2×D3×A4 | — | 4 | 4 | no | — |
| 17 | C2×D3×A4 | — | 4 | 2 | no | — |
| 18 | #2840 | — | 3 | 1 | no | Sylow |
| 19 | C24×D3 | A4 | 1 | 1 | yes | characteristic |
| 20 | C23×A4 | — | 12 | 1 | no | — |
| 21 | C22×C3×Q8 | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 22 | C22×Dic6 | — | 3 | 1 | no | — |
| 23 | C22⋊SL(2,𝔽3) | — | 2 | 1 | no | — |
| 24 | C22⋊SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 25 | C2×C3×C22⋊C4 | — | 3 | 1 | no | — |
| 26 | C2×#217 | — | 3 | 1 | no | — |
| 27 | C2×(C2×D3)⋊C4 | — | 3 | 2 | no | — |
| 28 | C2×(C2×D3)⋊C4 | — | 3 | 2 | no | — |
| 29 | C3×#95 | — | 3 | 1 | no | — |
| 30 | C3⋊#95 | — | 3 | 1 | no | — |
| 31 | C3⋊#95 | — | 3 | 2 | no | — |
| 32 | C22×C3×D3 | — | 16 | 1 | no | — |
| 33 | C2×C3×A4 | — | 4 | 2 | no | — |
| 34 | C2×C3×A4 | — | 4 | 1 | no | — |
| 35 | D3×A4 | — | 4 | 4 | no | — |
| 36 | C3×SL(2,𝔽3) | — | 16 | 1 | no | — |
| 37 | #393 | — | 3 | 2 | no | — |
| 38 | #393 | — | 9 | 2 | no | — |
| 39 | #393 | — | 9 | 1 | no | — |
| 40 | #393 | C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 41 | #379 | — | 9 | 1 | no | — |
| 42 | C24×C3 | C2×A4 | 1 | 1 | yes | characteristic |
| 43 | C23×D3 | — | 12 | 1 | no | — |
| 44 | C23×D3 | — | 12 | 1 | no | — |
| 45 | C23×D3 | SL(2,𝔽3) | 1 | 4 | yes | complemented |
| 46 | C23×D3 | C2×A4 | 1 | 2 | yes | — |
| 47 | C22×A4 | — | 4 | 1 | no | — |
| 48 | C22×A4 | — | 8 | 1 | no | — |
| 49 | C22×A4 | — | 12 | 2 | no | — |
| 50 | C22×A4 | — | 12 | 4 | no | — |
| 51 | C2×C3×Q8 | — | 12 | 1 | no | — |
| 52 | C22×C3×C4 | — | 3 | 1 | no | — |
| 53 | C22×C3×C4 | — | 3 | 1 | no | — |
| 54 | C22×Dic3 | — | 3 | 2 | no | — |
| 55 | C2×Dic6 | — | 12 | 1 | no | — |
| 56 | C2×Dic6 | — | 12 | 2 | no | — |
| 57 | C3×C22⋊C4 | — | 12 | 1 | no | — |
| 58 | #217 | — | 12 | 1 | no | — |
| 59 | (C2×D3)⋊C4 | — | 12 | 2 | no | — |
| 60 | (C2×D3)⋊C4 | — | 12 | 2 | no | — |
| 61 | C22×C32 | — | 16 | 1 | no | — |
| 62 | C2×C3×D3 | — | 16 | 4 | no | — |
| 63 | C2×C3×D3 | — | 16 | 2 | no | — |
| 64 | C3×A4 | — | 4 | 1 | no | — |
| 65 | C25 | — | 3 | 1 | no | — |
| 66 | C22×Q8 | — | 9 | 1 | no | — |
| 67 | C22×Q8 | C2×C3×D3 | 1 | 1 | yes | characteristic, complemented |
| 68 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 69 | C2×C22⋊C4 | — | 9 | 2 | no | — |
| 70 | C2×C22⋊C4 | — | 9 | 2 | no | — |
| 71 | C2×C22⋊C4 | — | 9 | 1 | no | — |
| 72 | #95 | — | 3 | 1 | no | — |
| 73 | #95 | — | 9 | 2 | no | — |
| 74 | #95 | — | 9 | 1 | no | — |
| 75 | C23×C3 | — | 6 | 1 | no | — |
| 76 | C23×C3 | — | 6 | 1 | no | — |
| 77 | C23×C3 | — | 48 | 1 | no | — |
| 78 | C23×C3 | C2×SL(2,𝔽3) | 1 | 2 | yes | — |
| 79 | C23×C3 | C22×A4 | 1 | 1 | yes | characteristic |
| 80 | C22×D3 | — | 4 | 1 | no | — |
| 81 | C22×D3 | — | 6 | 2 | no | — |
| 82 | C22×D3 | — | 6 | 2 | no | — |
| 83 | C22×D3 | — | 12 | 1 | no | — |
| 84 | C22×D3 | — | 12 | 4 | no | — |
| 85 | C22×D3 | — | 12 | 4 | no | — |
| 86 | C22×D3 | C2×SL(2,𝔽3) | 1 | 4 | yes | — |
| 87 | C2×A4 | — | 4 | 1 | no | — |
| 88 | C2×A4 | — | 4 | 2 | no | — |
| 89 | C2×A4 | — | 8 | 1 | no | — |
| 90 | C2×A4 | — | 8 | 2 | no | — |
| 91 | C2×A4 | — | 12 | 4 | no | — |
| 92 | C3×Q8 | — | 4 | 1 | no | — |
| 93 | C3×Q8 | — | 12 | 1 | no | — |
| 94 | C2×C3×C4 | — | 6 | 2 | no | — |
| 95 | C2×C3×C4 | — | 6 | 2 | no | — |
| 96 | C2×C3×C4 | — | 6 | 1 | no | — |
| 97 | C2×C3×C4 | — | 6 | 1 | no | — |
| 98 | C2×Dic3 | — | 6 | 2 | no | — |
| 99 | C2×Dic3 | — | 6 | 2 | no | — |
| 100 | C2×Dic3 | — | 6 | 2 | no | — |
| 101 | Dic6 | — | 12 | 4 | no | — |
| 102 | SL(2,𝔽3) | — | 16 | 1 | no | — |
| 103 | SL(2,𝔽3) | — | 16 | 2 | no | — |
| 104 | C2×C32 | — | 16 | 2 | no | — |
| 105 | C2×C32 | — | 16 | 1 | no | — |
| 106 | C3×D3 | — | 16 | 4 | no | — |
| 107 | C24 | — | 3 | 2 | no | — |
| 108 | C24 | — | 3 | 4 | no | — |
| 109 | C24 | — | 36 | 1 | no | — |
| 110 | C24 | — | 36 | 1 | no | — |
| 111 | C24 | D3×A4 | 1 | 1 | yes | characteristic |
| 112 | C2×Q8 | — | 12 | 1 | no | — |
| 113 | C2×Q8 | — | 36 | 1 | no | — |
| 114 | C2×Q8 | — | 36 | 2 | no | — |
| 115 | C22×C4 | — | 3 | 1 | no | — |
| 116 | C22×C4 | — | 3 | 1 | no | — |
| 117 | C22×C4 | — | 9 | 2 | no | — |
| 118 | C22⋊C4 | — | 12 | 1 | no | — |
| 119 | C22⋊C4 | — | 36 | 2 | no | — |
| 120 | C22⋊C4 | — | 36 | 2 | no | — |
| 121 | C22⋊C4 | — | 36 | 1 | no | — |
| 122 | C22×C3 | — | 3 | 1 | no | — |
| 123 | C22×C3 | — | 3 | 1 | no | — |
| 124 | C22×C3 | — | 4 | 1 | no | — |
| 125 | C22×C3 | — | 6 | 2 | no | — |
| 126 | C22×C3 | — | 12 | 1 | no | — |
| 127 | C22×C3 | — | 16 | 1 | no | — |
| 128 | C22×C3 | — | 32 | 1 | no | — |
| 129 | C22×C3 | — | 48 | 4 | no | — |
| 130 | C22×C3 | — | 48 | 2 | no | — |
| 131 | C22×C3 | C22×SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 132 | C2×D3 | — | 4 | 4 | no | — |
| 133 | C2×D3 | — | 4 | 2 | no | — |
| 134 | C2×D3 | — | 6 | 4 | no | — |
| 135 | C2×D3 | — | 12 | 2 | no | — |
| 136 | C2×D3 | — | 12 | 4 | no | — |
| 137 | A4 | — | 4 | 1 | no | — |
| 138 | A4 | — | 8 | 1 | no | — |
| 139 | C3×C4 | — | 12 | 1 | no | — |
| 140 | C3×C4 | — | 12 | 1 | no | — |
| 141 | Dic3 | — | 12 | 2 | no | — |
| 142 | C32 | — | 16 | 1 | no | Sylow |
| 143 | C23 | — | 3 | 4 | no | — |
| 144 | C23 | — | 6 | 1 | no | — |
| 145 | C23 | — | 6 | 1 | no | — |
| 146 | C23 | — | 12 | 1 | no | — |
| 147 | C23 | — | 18 | 2 | no | — |
| 148 | C23 | — | 18 | 2 | no | — |
| 149 | C23 | — | 36 | 1 | no | — |
| 150 | C23 | — | 36 | 4 | no | — |
| 151 | C23 | — | 36 | 4 | no | — |
| 152 | C23 | D3×SL(2,𝔽3) | 1 | 2 | yes | — |
| 153 | C23 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 154 | Q8 | — | 4 | 1 | no | — |
| 155 | Q8 | — | 12 | 1 | no | — |
| 156 | Q8 | — | 36 | 4 | no | — |
| 157 | C2×C4 | — | 6 | 1 | no | — |
| 158 | C2×C4 | — | 6 | 1 | no | — |
| 159 | C2×C4 | — | 6 | 2 | no | — |
| 160 | C2×C4 | — | 6 | 2 | no | — |
| 161 | C2×C4 | — | 18 | 2 | no | — |
| 162 | C2×C4 | — | 18 | 2 | no | — |
| 163 | C2×C4 | — | 18 | 2 | no | — |
| 164 | C2×C3 | — | 3 | 1 | no | — |
| 165 | C2×C3 | — | 3 | 1 | no | — |
| 166 | C2×C3 | — | 4 | 2 | no | — |
| 167 | C2×C3 | — | 16 | 2 | no | — |
| 168 | C2×C3 | — | 16 | 1 | no | — |
| 169 | C2×C3 | — | 32 | 2 | no | — |
| 170 | C2×C3 | — | 32 | 1 | no | — |
| 171 | C2×C3 | — | 48 | 4 | no | — |
| 172 | C2×C3 | #5450 | 1 | 1 | yes | characteristic |
| 173 | D3 | — | 4 | 4 | no | — |
| 174 | C22 | — | 3 | 1 | no | — |
| 175 | C22 | — | 3 | 1 | no | — |
| 176 | C22 | — | 4 | 1 | no | — |
| 177 | C22 | — | 6 | 2 | no | — |
| 178 | C22 | — | 12 | 1 | no | — |
| 179 | C22 | — | 12 | 4 | no | — |
| 180 | C22 | — | 12 | 2 | no | — |
| 181 | C22 | — | 18 | 4 | no | — |
| 182 | C22 | — | 36 | 2 | no | — |
| 183 | C22 | — | 36 | 4 | no | — |
| 184 | C22 | C2×D3×SL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 185 | C4 | — | 12 | 1 | no | — |
| 186 | C4 | — | 12 | 1 | no | — |
| 187 | C4 | — | 36 | 2 | no | — |
| 188 | C3 | — | 16 | 1 | no | — |
| 189 | C3 | — | 32 | 1 | no | — |
| 190 | C3 | #73210 | 1 | 1 | yes | characteristic, complemented |
| 191 | C2 | — | 3 | 1 | no | — |
| 192 | C2 | — | 3 | 1 | no | — |
| 193 | C2 | — | 4 | 2 | no | — |
| 194 | C2 | — | 12 | 4 | no | — |
| 195 | C2 | C3⋊#5450 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#5450 |
| commutator, cosocle, hypocenter | C22×C3×Q8 | C22×C3 |
| Frattini | C23 | C2×D3×A4 |
| Fitting | C3×#393 | C2×C3 |
| socle | C23×C3 | C22×A4 |
| LCS | G ▷ C22×C3×Q8 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#393⋊C3 | C2 | semidirect |
| 2. C3⋊(#393⋊C3) | C2 | semidirect |
| 3. #76888 | C3 | semidirect |
| 6. C3×C22⋊SL(2,𝔽3) | C22 | semidirect |
| 7. C3×#393 | C2×C3 | semidirect |
| 8. #5231 | C2×C3 | semidirect |
| 14. #393⋊C3 | D3 | semidirect |
| 21. C22×C3×Q8 | C22×C3 | semidirect |
| 24. C22⋊SL(2,𝔽3) | C2×D3 | semidirect |
| 40. #393 | C3×D3 | semidirect |
| 45. C23×D3 | SL(2,𝔽3) | semidirect |
| 67. C22×Q8 | C2×C3×D3 | semidirect |
| 190. C3 | #73210 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#393⋊C3 | 2 |
| C3⋊(#393⋊C3) | 2 |
| #76888 | 3 |
| #73210 | 3 |
| C22×D3×A4 | 4 |