Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
289 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 52 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⋊D3×D16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10590234 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×#4519 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #10589071 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10589067 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | D3×D32 | — | 3 | 2 | no | maximal |
| 7 | D8×C3⋊D3 | — | 2 | 1 | no | — |
| 8 | #63797 | — | 2 | 1 | no | — |
| 9 | C32×D16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #63653 | — | 2 | 1 | no | — |
| 11 | #63626 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | C16×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 13 | C3×D48 | C22 | 1 | 2 | yes | complemented |
| 14 | #63545 | — | 2 | 1 | no | — |
| 15 | C3×C3⋊C32 | C22 | 1 | 2 | yes | complemented |
| 16 | D3×D16 | — | 3 | 2 | no | — |
| 17 | D3×D16 | — | 3 | 2 | no | — |
| 18 | D3×D16 | — | 6 | 1 | no | — |
| 19 | C3×D32 | — | 3 | 2 | no | — |
| 20 | #4519 | — | 3 | 2 | no | — |
| 21 | #4519 | D3 | 1 | 2 | yes | complemented |
| 22 | D96 | — | 3 | 2 | no | — |
| 23 | D3×C32 | — | 3 | 2 | no | — |
| 24 | D4×C3⋊D3 | — | 4 | 1 | no | — |
| 25 | #3802 | — | 4 | 1 | no | — |
| 26 | C32×D8 | — | 2 | 1 | no | — |
| 27 | #3753 | — | 4 | 1 | no | — |
| 28 | #3745 | — | 2 | 1 | no | — |
| 29 | C8×C3⋊D3 | D4 | 1 | 1 | yes | characteristic |
| 30 | C3×D24 | — | 2 | 2 | no | — |
| 31 | #3713 | — | 4 | 1 | no | — |
| 32 | C32×C16 | C23 | 1 | 1 | yes | characteristic |
| 33 | C3⋊(C3⋊C16) | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×D32 | — | 9 | 1 | no | Sylow |
| 35 | D3×D8 | — | 6 | 2 | no | — |
| 36 | D3×D8 | — | 6 | 2 | no | — |
| 37 | D3×D8 | — | 6 | 2 | no | — |
| 38 | C3×D16 | — | 2 | 1 | no | — |
| 39 | C3×D16 | — | 3 | 2 | no | — |
| 40 | C3×D16 | C2×D3 | 1 | 2 | yes | complemented |
| 41 | #826 | — | 6 | 2 | no | — |
| 42 | #826 | — | 6 | 2 | no | — |
| 43 | #826 | — | 6 | 2 | no | — |
| 44 | D48 | — | 3 | 2 | no | — |
| 45 | D48 | — | 6 | 1 | no | — |
| 46 | D48 | C2×D3 | 1 | 2 | yes | complemented |
| 47 | D3×C16 | — | 3 | 2 | no | — |
| 48 | D3×C16 | — | 6 | 1 | no | — |
| 49 | C3×C32 | — | 3 | 2 | no | — |
| 50 | C3⋊C32 | C2×D3 | 1 | 2 | yes | complemented |
| 51 | C22×C3⋊D3 | — | 8 | 1 | no | — |
| 52 | C2×D32 | — | 8 | 1 | no | — |
| 53 | C32×D4 | — | 4 | 1 | no | — |
| 54 | #638 | — | 8 | 1 | no | — |
| 55 | C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 56 | C4×C3⋊D3 | D8 | 1 | 1 | yes | characteristic |
| 57 | C3×D12 | — | 4 | 2 | no | — |
| 58 | #625 | — | 8 | 1 | no | — |
| 59 | C32×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 60 | C3⋊(C3⋊C8) | D8 | 1 | 1 | yes | characteristic |
| 61 | C2×D16 | — | 9 | 1 | no | — |
| 62 | C2×D16 | — | 9 | 1 | no | — |
| 63 | D32 | — | 3 | 2 | no | — |
| 64 | D32 | — | 9 | 2 | no | — |
| 65 | C2×C32 | — | 9 | 1 | no | — |
| 66 | D3×D4 | — | 12 | 2 | no | — |
| 67 | D3×D4 | — | 12 | 2 | no | — |
| 68 | D3×D4 | — | 12 | 2 | no | — |
| 69 | C3×D8 | — | 2 | 2 | no | — |
| 70 | C3×D8 | — | 2 | 2 | no | — |
| 71 | C3×D8 | — | 6 | 2 | no | — |
| 72 | #213 | — | 12 | 2 | no | — |
| 73 | #213 | — | 12 | 2 | no | — |
| 74 | #213 | — | 12 | 2 | no | — |
| 75 | D24 | — | 2 | 2 | no | — |
| 76 | D24 | — | 6 | 2 | no | — |
| 77 | D24 | — | 6 | 2 | no | — |
| 78 | D3×C8 | — | 3 | 2 | no | — |
| 79 | D3×C8 | — | 6 | 1 | no | — |
| 80 | C3×C16 | — | 2 | 1 | no | — |
| 81 | C3×C16 | C22×D3 | 1 | 2 | yes | — |
| 82 | C3⋊C16 | — | 3 | 2 | no | — |
| 83 | C3⋊C16 | — | 6 | 1 | no | — |
| 84 | C22×C32 | — | 8 | 1 | no | — |
| 85 | C2×C3⋊D3 | — | 8 | 1 | no | — |
| 86 | C2×C3⋊D3 | — | 16 | 2 | no | — |
| 87 | C2×C3⋊D3 | D16 | 1 | 1 | yes | characteristic |
| 88 | C2×C3×D3 | — | 8 | 2 | no | — |
| 89 | D32 | — | 16 | 2 | no | — |
| 90 | C32×C4 | C2×D8 | 1 | 1 | yes | characteristic |
| 91 | C3⋊Dic3 | D16 | 1 | 1 | yes | characteristic |
| 92 | C2×D8 | — | 18 | 1 | no | — |
| 93 | C2×D8 | — | 18 | 1 | no | — |
| 94 | D16 | — | 3 | 2 | no | — |
| 95 | D16 | — | 9 | 1 | no | — |
| 96 | D16 | — | 18 | 1 | no | — |
| 97 | D16 | — | 18 | 1 | no | — |
| 98 | D16 | D32 | 1 | 1 | yes | characteristic, complemented |
| 99 | C2×C16 | — | 9 | 1 | no | — |
| 100 | C32 | — | 3 | 2 | no | — |
| 101 | C22×D3 | — | 24 | 2 | no | — |
| 102 | C22×D3 | — | 24 | 2 | no | — |
| 103 | C22×D3 | — | 24 | 2 | no | — |
| 104 | C3×D4 | — | 4 | 2 | no | — |
| 105 | C3×D4 | — | 4 | 2 | no | — |
| 106 | C3×D4 | — | 12 | 2 | no | — |
| 107 | #67 | — | 24 | 2 | no | — |
| 108 | #67 | — | 24 | 2 | no | — |
| 109 | #67 | — | 24 | 2 | no | — |
| 110 | D12 | — | 4 | 2 | no | — |
| 111 | D12 | — | 12 | 2 | no | — |
| 112 | D12 | — | 12 | 2 | no | — |
| 113 | C4×D3 | — | 3 | 2 | no | — |
| 114 | C4×D3 | — | 6 | 1 | no | — |
| 115 | C3×C8 | — | 2 | 1 | no | — |
| 116 | C3×C8 | D3×D4 | 1 | 2 | yes | — |
| 117 | C3⋊C8 | — | 3 | 2 | no | — |
| 118 | C3⋊C8 | — | 6 | 1 | no | — |
| 119 | C2×C32 | — | 16 | 1 | no | — |
| 120 | C2×C32 | C2×D16 | 1 | 1 | yes | characteristic |
| 121 | C3⋊D3 | — | 16 | 1 | no | — |
| 122 | C3⋊D3 | D32 | 1 | 2 | yes | complemented |
| 123 | C3×D3 | — | 16 | 2 | no | — |
| 124 | C2×D4 | — | 36 | 1 | no | — |
| 125 | C2×D4 | — | 36 | 1 | no | — |
| 126 | D8 | — | 2 | 1 | no | — |
| 127 | D8 | — | 6 | 2 | no | — |
| 128 | D8 | — | 18 | 1 | no | — |
| 129 | D8 | — | 36 | 1 | no | — |
| 130 | D8 | — | 36 | 1 | no | — |
| 131 | C2×C8 | — | 9 | 1 | no | — |
| 132 | C16 | — | 9 | 1 | no | — |
| 133 | C16 | C2×D32 | 1 | 1 | yes | characteristic |
| 134 | C22×C3 | — | 8 | 2 | no | — |
| 135 | C22×C3 | — | 8 | 2 | no | — |
| 136 | C22×C3 | — | 24 | 2 | no | — |
| 137 | C2×D3 | — | 3 | 2 | no | — |
| 138 | C2×D3 | — | 6 | 1 | no | — |
| 139 | C2×D3 | — | 8 | 2 | no | — |
| 140 | C2×D3 | — | 24 | 2 | no | — |
| 141 | C2×D3 | — | 24 | 2 | no | — |
| 142 | C2×D3 | — | 48 | 4 | no | — |
| 143 | C2×D3 | — | 48 | 4 | no | — |
| 144 | C2×D3 | — | 48 | 4 | no | — |
| 145 | C3×C4 | — | 2 | 1 | no | — |
| 146 | C3×C4 | D3×D8 | 1 | 2 | yes | — |
| 147 | Dic3 | — | 3 | 2 | no | — |
| 148 | Dic3 | — | 6 | 1 | no | — |
| 149 | C32 | C2×D32 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 150 | C23 | — | 72 | 1 | no | — |
| 151 | C23 | — | 72 | 1 | no | — |
| 152 | D4 | — | 4 | 1 | no | — |
| 153 | D4 | — | 12 | 2 | no | — |
| 154 | D4 | — | 36 | 1 | no | — |
| 155 | D4 | — | 72 | 1 | no | — |
| 156 | D4 | — | 72 | 1 | no | — |
| 157 | C2×C4 | — | 9 | 1 | no | — |
| 158 | C8 | — | 9 | 1 | no | — |
| 159 | C8 | #3811 | 1 | 1 | yes | characteristic |
| 160 | C2×C3 | — | 2 | 1 | no | — |
| 161 | C2×C3 | — | 16 | 2 | no | — |
| 162 | C2×C3 | — | 16 | 2 | no | — |
| 163 | C2×C3 | — | 48 | 2 | no | — |
| 164 | C2×C3 | D3×D16 | 1 | 2 | yes | — |
| 165 | D3 | — | 3 | 4 | no | — |
| 166 | D3 | — | 6 | 2 | no | — |
| 167 | D3 | — | 16 | 2 | no | — |
| 168 | D3 | — | 48 | 2 | no | — |
| 169 | D3 | — | 48 | 2 | no | — |
| 170 | C22 | — | 8 | 1 | no | — |
| 171 | C22 | — | 9 | 1 | no | — |
| 172 | C22 | — | 24 | 2 | no | — |
| 173 | C22 | — | 72 | 1 | no | — |
| 174 | C22 | — | 144 | 2 | no | — |
| 175 | C22 | — | 144 | 2 | no | — |
| 176 | C4 | — | 9 | 1 | no | — |
| 177 | C4 | #63926 | 1 | 1 | yes | characteristic |
| 178 | C3 | — | 2 | 1 | no | — |
| 179 | C3 | D3×D32 | 1 | 2 | yes | complemented |
| 180 | C2 | — | 9 | 2 | no | — |
| 181 | C2 | — | 16 | 1 | no | — |
| 182 | C2 | — | 48 | 2 | no | — |
| 183 | C2 | — | 144 | 1 | no | — |
| 184 | C2 | #10591025 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10591025 |
| commutator, cosocle | C32×C16 | C23 |
| Frattini | C16 | C2×D32 |
| Fitting | C32×D16 | C22 |
| socle | C2×C32 | C2×D16 |
| hypercenter | D16 | D32 |
| hypocenter | C32 | C2×D32 |
| LCS | G ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3⋊D3×D16 | C2 | semidirect |
| 2. #10590234 | C2 | semidirect |
| 3. C3×#4519 | C2 | semidirect |
| 4. #10589071 | C2 | semidirect |
| 5. #10589067 | C2 | semidirect |
| 9. C32×D16 | C22 | semidirect |
| 11. #63626 | C22 | semidirect |
| 13. C3×D48 | C22 | semidirect |
| 15. C3×C3⋊C32 | C22 | semidirect |
| 21. #4519 | D3 | semidirect |
| 40. C3×D16 | C2×D3 | semidirect |
| 46. D48 | C2×D3 | semidirect |
| 50. C3⋊C32 | C2×D3 | semidirect |
| 98. D16 | D32 | semidirect |
| 122. C3⋊D3 | D32 | semidirect |
| 149. C32 | C2×D32 | semidirect |
| 179. C3 | D3×D32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3⋊D3×D16 | 2 |
| #10590234 | 2 |
| C3×#4519 | 2 |
| #10589071 | 2 |
| #10589067 | 2 |
| D3×D32 | 3 |