Finite Group DatabasebetaFinite Group DatabasebetaOrder 3888.
202 conjugacy classes of subgroups; only the 14 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 14 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 | #14517936 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14517930 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #14517911 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13049804 | — | 3 | 1 | no | maximal |
| 5 | #11757749 | — | 2 | 1 | no | — |
| 6 | #66472⋊C3 | — | 2 | 1 | no | — |
| 7 | #66472⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | #10620211 | D3 | 1 | 1 | yes | characteristic, complemented |
| 9 | #10620206 | — | 3 | 1 | no | — |
| 10 | #10620193 | — | 3 | 1 | no | — |
| 11 | #92301 | D4 | 1 | 1 | yes | characteristic |
| 12 | #92300 | — | 4 | 1 | no | — |
| 13 | #89092 | — | 9 | 1 | no | maximal |
| 14 | C3⋊D32 | — | 6 | 1 | no | — |
| 15 | #66472 | — | 2 | 1 | no | — |
| 16 | #66472 | C2×D3 | 1 | 1 | yes | characteristic |
| 17 | ((C3⋊D3)⋊C3)⋊D3 | — | 12 | 1 | no | — |
| 18 | C32⋊H(3,𝔽3) | SD16 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 19 | C3×PSU(3,𝔽2) | — | 9 | 1 | no | — |
| 20 | #157⋊D3 | — | 9 | 1 | no | — |
| 21 | C3⋊AGL(1,𝔽9) | — | 9 | 1 | no | — |
| 22 | C3⋊(C3⋊(C3⋊D3)) | #67 | 1 | 1 | yes | characteristic |
| 23 | C32×C3⋊D3 | — | 12 | 1 | no | — |
| 24 | C3⋊#272 | — | 4 | 1 | no | — |
| 25 | C3⋊((C3⋊D3)⋊C3) | — | 12 | 1 | no | — |
| 26 | C3×(C3⋊D3)⋊C3 | — | 12 | 1 | no | — |
| 27 | C3×(C3⋊D3)⋊C3 | — | 24 | 1 | no | — |
| 28 | AΓL(1,𝔽9) | — | 27 | 1 | no | — |
| 29 | AΓL(1,𝔽9) | — | 27 | 1 | no | — |
| 30 | #1113 | — | 18 | 1 | no | — |
| 31 | D3×C3⋊D3 | — | 36 | 1 | no | — |
| 32 | D3×C3⋊D3 | — | 36 | 1 | no | — |
| 33 | D3×C3⋊D3 | — | 36 | 2 | no | — |
| 34 | C3×#157 | — | 9 | 1 | no | — |
| 35 | C3×#157 | — | 18 | 1 | no | — |
| 36 | ((C3⋊D3)⋊C3)⋊C2 | — | 36 | 1 | no | — |
| 37 | ((C3⋊D3)⋊C3)⋊C2 | — | 36 | 2 | no | — |
| 38 | C34 | #215 | 1 | 1 | yes | characteristic, complemented |
| 39 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 40 | C3×H(3,𝔽3) | — | 8 | 1 | no | — |
| 41 | PSU(3,𝔽2) | — | 9 | 1 | no | — |
| 42 | PSU(3,𝔽2) | — | 27 | 1 | no | — |
| 43 | GO+(4,𝔽2) | — | 27 | 1 | no | — |
| 44 | GO+(4,𝔽2) | — | 27 | 1 | no | — |
| 45 | GO+(4,𝔽2) | — | 54 | 1 | no | — |
| 46 | AGL(1,𝔽9) | — | 27 | 1 | no | — |
| 47 | AGL(1,𝔽9) | — | 27 | 1 | no | — |
| 48 | C3⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 49 | C3⋊(C3⋊D3) | — | 36 | 1 | no | — |
| 50 | C3⋊(C3⋊D3) | — | 36 | 2 | no | — |
| 51 | C3×C3⋊D3 | — | 9 | 1 | no | — |
| 52 | C3×C3⋊D3 | — | 12 | 1 | no | — |
| 53 | C3×C3⋊D3 | — | 12 | 1 | no | — |
| 54 | C3×C3⋊D3 | — | 12 | 2 | no | — |
| 55 | C3×C3⋊D3 | — | 12 | 1 | no | — |
| 56 | C3×C3⋊D3 | — | 24 | 1 | no | — |
| 57 | C32×D3 | — | 36 | 1 | no | — |
| 58 | C32×D3 | — | 36 | 1 | no | — |
| 59 | C32×D3 | — | 36 | 2 | no | — |
| 60 | #272 | — | 36 | 1 | no | — |
| 61 | #272 | — | 36 | 2 | no | — |
| 62 | (C3⋊D3)⋊C3 | — | 12 | 1 | no | — |
| 63 | (C3⋊D3)⋊C3 | — | 12 | 2 | no | — |
| 64 | (C3⋊D3)⋊C3 | — | 24 | 1 | no | — |
| 65 | (C3⋊D3)⋊C3 | — | 24 | 2 | no | — |
| 66 | (C3⋊D3)⋊C3 | — | 36 | 1 | no | — |
| 67 | (C3⋊D3)⋊C3 | — | 36 | 2 | no | — |
| 68 | #215 | — | 81 | 1 | no | — |
| 69 | C2×C3⋊D3 | — | 108 | 1 | no | — |
| 70 | D32 | — | 54 | 1 | no | — |
| 71 | D32 | — | 54 | 1 | no | — |
| 72 | D32 | — | 54 | 2 | no | — |
| 73 | D32 | — | 108 | 1 | no | — |
| 74 | D32 | — | 108 | 2 | no | — |
| 75 | D32 | — | 108 | 1 | no | — |
| 76 | D32 | — | 108 | 2 | no | — |
| 77 | D32 | — | 108 | 1 | no | — |
| 78 | #157 | — | 9 | 1 | no | — |
| 79 | #157 | — | 18 | 1 | no | — |
| 80 | #157 | — | 27 | 1 | no | — |
| 81 | #157 | — | 54 | 1 | no | — |
| 82 | #157 | — | 54 | 1 | no | — |
| 83 | #157 | — | 108 | 1 | no | — |
| 84 | C33 | — | 4 | 1 | no | — |
| 85 | C33 | — | 8 | 1 | no | — |
| 86 | C33 | — | 12 | 1 | no | — |
| 87 | C33 | — | 12 | 2 | no | — |
| 88 | C33 | AΓL(1,𝔽9) | 1 | 1 | yes | characteristic, complemented |
| 89 | H(3,𝔽3) | — | 12 | 1 | no | — |
| 90 | H(3,𝔽3) | — | 12 | 2 | no | — |
| 91 | H(3,𝔽3) | — | 24 | 1 | no | — |
| 92 | H(3,𝔽3) | — | 24 | 2 | no | — |
| 93 | C3×Q8 | — | 81 | 1 | no | — |
| 94 | D12 | — | 81 | 1 | no | — |
| 95 | C3⋊C8 | — | 81 | 1 | no | — |
| 96 | C2×C32 | — | 108 | 1 | no | — |
| 97 | C3⋊D3 | — | 9 | 1 | no | — |
| 98 | C3⋊D3 | — | 12 | 1 | no | — |
| 99 | C3⋊D3 | — | 12 | 1 | no | — |
| 100 | C3⋊D3 | — | 24 | 1 | no | — |
| 101 | C3⋊D3 | — | 27 | 1 | no | — |
| 102 | C3⋊D3 | — | 36 | 1 | no | — |
| 103 | C3⋊D3 | — | 36 | 2 | no | — |
| 104 | C3⋊D3 | — | 54 | 1 | no | — |
| 105 | C3⋊D3 | — | 108 | 1 | no | — |
| 106 | C3⋊D3 | — | 108 | 1 | no | — |
| 107 | C3⋊D3 | — | 108 | 1 | no | — |
| 108 | C3⋊D3 | — | 108 | 2 | no | — |
| 109 | C3⋊D3 | — | 216 | 1 | no | — |
| 110 | C3⋊D3 | — | 216 | 1 | no | — |
| 111 | C3×D3 | — | 36 | 1 | no | — |
| 112 | C3×D3 | — | 36 | 1 | no | — |
| 113 | C3×D3 | — | 36 | 2 | no | — |
| 114 | C3×D3 | — | 36 | 1 | no | — |
| 115 | C3×D3 | — | 36 | 2 | no | — |
| 116 | C3×D3 | — | 36 | 2 | no | — |
| 117 | C3×D3 | — | 36 | 2 | no | — |
| 118 | C3×D3 | — | 36 | 2 | no | — |
| 119 | C3×D3 | — | 36 | 2 | no | — |
| 120 | C3×D3 | — | 36 | 1 | no | — |
| 121 | C3×D3 | — | 108 | 1 | no | — |
| 122 | C3×D3 | — | 108 | 1 | no | — |
| 123 | C3×D3 | — | 216 | 1 | no | — |
| 124 | SD16 | — | 243 | 1 | no | Sylow |
| 125 | C2×D3 | — | 324 | 1 | no | — |
| 126 | C2×D3 | — | 324 | 2 | no | — |
| 127 | C2×D3 | — | 324 | 1 | no | — |
| 128 | C2×D3 | — | 162 | 1 | no | — |
| 129 | C3×C4 | — | 81 | 1 | no | — |
| 130 | C3×C4 | — | 162 | 1 | no | — |
| 131 | C32 | — | 3 | 1 | no | — |
| 132 | C32 | — | 4 | 1 | no | — |
| 133 | C32 | — | 4 | 2 | no | — |
| 134 | C32 | — | 6 | 1 | no | — |
| 135 | C32 | — | 12 | 1 | no | — |
| 136 | C32 | — | 12 | 1 | no | — |
| 137 | C32 | — | 12 | 2 | no | — |
| 138 | C32 | — | 12 | 1 | no | — |
| 139 | C32 | — | 12 | 1 | no | — |
| 140 | C32 | — | 24 | 1 | no | — |
| 141 | C32 | — | 24 | 1 | no | — |
| 142 | C32 | — | 24 | 1 | no | — |
| 143 | C32 | — | 24 | 1 | no | — |
| 144 | C32 | — | 48 | 1 | no | — |
| 145 | C32 | #89092 | 1 | 1 | yes | characteristic |
| 146 | Q8 | — | 81 | 1 | no | — |
| 147 | D4 | — | 243 | 1 | no | — |
| 148 | C8 | — | 243 | 1 | no | — |
| 149 | C2×C3 | — | 81 | 1 | no | — |
| 150 | C2×C3 | — | 108 | 1 | no | — |
| 151 | C2×C3 | — | 108 | 1 | no | — |
| 152 | C2×C3 | — | 108 | 2 | no | — |
| 153 | D3 | — | 36 | 1 | no | — |
| 154 | D3 | — | 36 | 1 | no | — |
| 155 | D3 | — | 36 | 2 | no | — |
| 156 | D3 | — | 324 | 1 | no | — |
| 157 | D3 | — | 324 | 2 | no | — |
| 158 | D3 | — | 108 | 1 | no | — |
| 159 | D3 | — | 108 | 1 | no | — |
| 160 | D3 | — | 216 | 1 | no | — |
| 161 | C22 | — | 486 | 1 | no | — |
| 162 | C4 | — | 81 | 1 | no | — |
| 163 | C4 | — | 162 | 1 | no | — |
| 164 | C3 | — | 4 | 1 | no | — |
| 165 | C3 | — | 9 | 1 | no | — |
| 166 | C3 | — | 12 | 1 | no | — |
| 167 | C3 | — | 12 | 2 | no | — |
| 168 | C3 | — | 72 | 1 | no | — |
| 169 | C2 | — | 81 | 1 | no | — |
| 170 | C2 | — | 108 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle | #66472⋊C3 | C22 |
| Frattini, socle | C32 | #89092 |
| Fitting, hypocenter | C32⋊H(3,𝔽3) | SD16 |
| prime radical | #14517930 | C2 |
| LCS | G ▷ #66472⋊C3 ▷ #92301 ▷ C32⋊H(3,𝔽3) | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14517936 | C2 | semidirect |
| 3. #14517911 | C2 | semidirect |
| 8. #10620211 | D3 | semidirect |
| 18. C32⋊H(3,𝔽3) | SD16 | semidirect |
| 38. C34 | #215 | semidirect |
| 88. C33 | AΓL(1,𝔽9) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14517936 | 2 |
| #14517930 | 2 |
| #14517911 | 2 |
| #13049804 | 3 |
| #89092 | 9 |