Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
270 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 270 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 | C3×#5948 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #72136 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #1780⋊C3 | — | 3 | 1 | no | maximal |
| 4 | C3×C22⋊SL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic |
| 5 | A4×C3⋊C8 | — | 4 | 1 | no | maximal |
| 6 | C22×C3×#41 | C2×C3 | 1 | 1 | yes | characteristic |
| 7 | #5948 | — | 2 | 1 | no | — |
| 8 | #5948 | D3 | 1 | 1 | yes | characteristic |
| 9 | C2×#830 | — | 3 | 1 | no | — |
| 10 | C2×#830 | — | 3 | 1 | no | — |
| 11 | C3×#41⋊C3 | — | 8 | 2 | no | — |
| 12 | C3×C4×A4 | — | 4 | 1 | no | — |
| 13 | #1780 | — | 3 | 1 | no | Sylow |
| 14 | C2×C3×#41 | — | 6 | 2 | no | — |
| 15 | C2×C3×#41 | — | 6 | 2 | no | — |
| 16 | C22×C3×Q8 | C3×C4 | 1 | 1 | yes | characteristic |
| 17 | C22×C3×D4 | — | 3 | 1 | no | — |
| 18 | C23×C3×C4 | — | 3 | 1 | no | — |
| 19 | C22⋊SL(2,𝔽3) | — | 2 | 1 | no | — |
| 20 | C22⋊SL(2,𝔽3) | Dic3 | 1 | 1 | yes | characteristic |
| 21 | C22×C3⋊C8 | — | 3 | 1 | no | — |
| 22 | C22×C3⋊C8 | A4 | 1 | 1 | yes | characteristic |
| 23 | C8×A4 | — | 12 | 1 | no | — |
| 24 | #830 | — | 6 | 2 | no | — |
| 25 | #830 | — | 6 | 2 | no | — |
| 26 | C2×C3×A4 | — | 4 | 1 | no | — |
| 27 | C3×SL(2,𝔽3) | — | 8 | 2 | no | — |
| 28 | C3×C3⋊C8 | — | 16 | 1 | no | — |
| 29 | C22×#41 | C3×D3 | 1 | 1 | yes | characteristic |
| 30 | C2×C22⋊C8 | — | 9 | 1 | no | — |
| 31 | C2×C22⋊C8 | — | 9 | 1 | no | — |
| 32 | C24×C3 | — | 3 | 1 | no | — |
| 33 | C3×#41 | — | 2 | 2 | no | — |
| 34 | C3×#41 | — | 6 | 2 | no | — |
| 35 | C3×#41 | — | 6 | 4 | no | — |
| 36 | C3×#41 | — | 6 | 2 | no | — |
| 37 | C3×#41 | — | 6 | 2 | no | — |
| 38 | C2×C3×Q8 | — | 6 | 2 | no | — |
| 39 | C2×C3×D4 | — | 6 | 2 | no | — |
| 40 | C2×C3×D4 | — | 6 | 4 | no | — |
| 41 | C22×C3×C4 | — | 3 | 2 | no | — |
| 42 | C22×C3×C4 | — | 3 | 2 | no | — |
| 43 | C22×C3×C4 | — | 3 | 1 | no | — |
| 44 | C22×C3×C4 | — | 6 | 2 | no | — |
| 45 | C22×C3×C4 | — | 6 | 2 | no | — |
| 46 | C22×C3×C4 | C2×A4 | 1 | 1 | yes | characteristic |
| 47 | #41⋊C3 | — | 8 | 2 | no | — |
| 48 | #41⋊C3 | — | 8 | 4 | no | — |
| 49 | C4×A4 | — | 4 | 1 | no | — |
| 50 | C4×A4 | — | 8 | 1 | no | — |
| 51 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 52 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 53 | C2×C3⋊C8 | — | 6 | 2 | no | — |
| 54 | C3×A4 | — | 4 | 1 | no | — |
| 55 | C32×C4 | — | 16 | 1 | no | — |
| 56 | C2×#41 | — | 6 | 2 | no | — |
| 57 | C2×#41 | — | 6 | 2 | no | — |
| 58 | C22×Q8 | C3×Dic3 | 1 | 1 | yes | characteristic |
| 59 | C22×D4 | — | 3 | 1 | no | — |
| 60 | C23×C4 | — | 3 | 1 | no | — |
| 61 | C22×C8 | — | 3 | 1 | no | — |
| 62 | C22×C8 | — | 9 | 1 | no | — |
| 63 | C22⋊C8 | — | 18 | 2 | no | — |
| 64 | C22⋊C8 | — | 18 | 2 | no | — |
| 65 | C23×C3 | — | 3 | 2 | no | — |
| 66 | C23×C3 | — | 6 | 2 | no | — |
| 67 | C23×C3 | — | 6 | 2 | no | — |
| 68 | C23×C3 | — | 6 | 1 | no | — |
| 69 | C23×C3 | — | 6 | 1 | no | — |
| 70 | C23×C3 | C4×A4 | 1 | 1 | yes | characteristic |
| 71 | C2×A4 | — | 4 | 1 | no | — |
| 72 | C2×A4 | — | 8 | 1 | no | — |
| 73 | C3×Q8 | — | 2 | 2 | no | — |
| 74 | C3×Q8 | — | 6 | 2 | no | — |
| 75 | C3×D4 | — | 6 | 2 | no | — |
| 76 | C3×D4 | — | 6 | 2 | no | — |
| 77 | C3×D4 | — | 6 | 4 | no | — |
| 78 | C2×C3×C4 | — | 3 | 1 | no | — |
| 79 | C2×C3×C4 | — | 3 | 1 | no | — |
| 80 | C2×C3×C4 | — | 3 | 2 | no | — |
| 81 | C2×C3×C4 | — | 6 | 2 | no | — |
| 82 | C2×C3×C4 | — | 6 | 4 | no | — |
| 83 | C2×C3×C4 | — | 6 | 2 | no | — |
| 84 | C2×C3×C4 | — | 6 | 2 | no | — |
| 85 | SL(2,𝔽3) | — | 8 | 2 | no | — |
| 86 | SL(2,𝔽3) | — | 8 | 4 | no | — |
| 87 | C3×C8 | — | 48 | 1 | no | — |
| 88 | C3⋊C8 | — | 4 | 1 | no | — |
| 89 | C3⋊C8 | — | 12 | 1 | no | — |
| 90 | C2×C32 | — | 16 | 1 | no | — |
| 91 | C24 | — | 3 | 1 | no | — |
| 92 | #41 | — | 2 | 2 | no | — |
| 93 | #41 | — | 6 | 2 | no | — |
| 94 | #41 | — | 6 | 4 | no | — |
| 95 | #41 | — | 6 | 2 | no | — |
| 96 | #41 | — | 6 | 2 | no | — |
| 97 | C2×Q8 | — | 6 | 2 | no | — |
| 98 | C2×D4 | — | 6 | 2 | no | — |
| 99 | C2×D4 | — | 6 | 4 | no | — |
| 100 | C22×C4 | — | 3 | 2 | no | — |
| 101 | C22×C4 | — | 3 | 2 | no | — |
| 102 | C22×C4 | — | 3 | 1 | no | — |
| 103 | C22×C4 | — | 6 | 2 | no | — |
| 104 | C22×C4 | — | 6 | 2 | no | — |
| 105 | C22×C4 | D3×A4 | 1 | 1 | yes | characteristic |
| 106 | C2×C8 | — | 18 | 1 | no | — |
| 107 | C2×C8 | — | 18 | 1 | no | — |
| 108 | C2×C8 | — | 18 | 2 | no | — |
| 109 | C22×C3 | — | 3 | 1 | no | — |
| 110 | C22×C3 | — | 3 | 1 | no | — |
| 111 | C22×C3 | — | 6 | 2 | no | — |
| 112 | C22×C3 | — | 6 | 2 | no | — |
| 113 | C22×C3 | — | 6 | 2 | no | — |
| 114 | C22×C3 | — | 12 | 2 | no | — |
| 115 | C22×C3 | — | 12 | 2 | no | — |
| 116 | C22×C3 | #131⋊C3 | 1 | 1 | yes | characteristic |
| 117 | A4 | — | 4 | 1 | no | — |
| 118 | A4 | — | 8 | 1 | no | — |
| 119 | C3×C4 | — | 3 | 1 | no | — |
| 120 | C3×C4 | — | 6 | 2 | no | — |
| 121 | C3×C4 | — | 16 | 1 | no | — |
| 122 | C3×C4 | — | 32 | 1 | no | — |
| 123 | C3×C4 | #120⋊C3 | 1 | 1 | yes | characteristic |
| 124 | C32 | — | 16 | 1 | no | Sylow |
| 125 | C23 | — | 3 | 2 | no | — |
| 126 | C23 | — | 6 | 2 | no | — |
| 127 | C23 | — | 6 | 1 | no | — |
| 128 | C23 | — | 6 | 2 | no | — |
| 129 | C23 | — | 6 | 1 | no | — |
| 130 | C23 | Dic3×A4 | 1 | 1 | yes | characteristic |
| 131 | Q8 | — | 2 | 2 | no | — |
| 132 | Q8 | — | 6 | 2 | no | — |
| 133 | D4 | — | 6 | 2 | no | — |
| 134 | D4 | — | 6 | 2 | no | — |
| 135 | D4 | — | 6 | 4 | no | — |
| 136 | C2×C4 | — | 3 | 1 | no | — |
| 137 | C2×C4 | — | 3 | 1 | no | — |
| 138 | C2×C4 | — | 3 | 2 | no | — |
| 139 | C2×C4 | — | 6 | 2 | no | — |
| 140 | C2×C4 | — | 6 | 4 | no | — |
| 141 | C2×C4 | — | 6 | 2 | no | — |
| 142 | C2×C4 | — | 6 | 2 | no | — |
| 143 | C8 | — | 12 | 1 | no | — |
| 144 | C8 | — | 36 | 1 | no | — |
| 145 | C2×C3 | — | 3 | 1 | no | — |
| 146 | C2×C3 | — | 3 | 1 | no | — |
| 147 | C2×C3 | — | 12 | 2 | no | — |
| 148 | C2×C3 | — | 16 | 1 | no | — |
| 149 | C2×C3 | — | 32 | 1 | no | — |
| 150 | C2×C3 | #379⋊C3 | 1 | 1 | yes | characteristic |
| 151 | C22 | — | 3 | 1 | no | — |
| 152 | C22 | — | 3 | 1 | no | — |
| 153 | C22 | — | 6 | 2 | no | — |
| 154 | C22 | — | 6 | 2 | no | — |
| 155 | C22 | — | 6 | 2 | no | — |
| 156 | C22 | — | 12 | 2 | no | — |
| 157 | C22 | — | 12 | 2 | no | — |
| 158 | C22 | C3⋊(#131⋊C3) | 1 | 1 | yes | characteristic |
| 159 | C4 | — | 3 | 1 | no | — |
| 160 | C4 | — | 6 | 2 | no | — |
| 161 | C4 | C3⋊(#120⋊C3) | 1 | 1 | yes | characteristic |
| 162 | C3 | — | 16 | 1 | no | — |
| 163 | C3 | — | 32 | 1 | no | — |
| 164 | C3 | #1780⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 165 | C2 | — | 3 | 1 | no | — |
| 166 | C2 | — | 3 | 1 | no | — |
| 167 | C2 | — | 12 | 2 | no | — |
| 168 | C2 | C3⋊(#379⋊C3) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊(#120⋊C3) |
| commutator, hypocenter | C22×C3×Q8 | C3×C4 |
| Frattini | C22×C4 | D3×A4 |
| Fitting, cosocle | C22×C3×#41 | C2×C3 |
| socle | C23×C3 | C4×A4 |
| prime radical | C3×#5948 | C2 |
| LCS | G ▷ C22×C3×Q8 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #72136 | C3 | semidirect |
| 164. C3 | #1780⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#5948 | 2 |
| #72136 | 3 |
| #1780⋊C3 | 3 |
| A4×C3⋊C8 | 4 |