Finite Group DatabasebetaFinite Group DatabasebetaOrder 4374.
10867 conjugacy classes of subgroups; only the 2674 normal subgroups are shown.
This group has 10867 conjugacy classes of subgroups — too many to draw. Only its 2674 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 | #14555252 | C2 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 2 | #13091539 | — | 3 | 27 | no | maximal |
| 3 | #13091502 | — | 3 | 54 | no | maximal |
| 4 | #13091466 | — | 3 | 36 | no | maximal |
| 5 | C3⋊#92338 | — | 3 | 4 | no | maximal |
| 6 | H(3,𝔽9) | D3 | 1 | 27 | yes | complemented |
| 7 | H(3,𝔽3)2 | D3 | 1 | 54 | yes | complemented |
| 8 | #10624880 | D3 | 1 | 36 | yes | complemented |
| 9 | C3×C32⋊H(3,𝔽3) | D3 | 1 | 4 | yes | complemented |
| 10 | C32×C3⋊(C3⋊D3) | — | 9 | 1 | no | — |
| 11 | C3×C3⋊#272 | — | 9 | 12 | no | — |
| 12 | C3×C3⋊#272 | — | 9 | 36 | no | — |
| 13 | C3×C3⋊#272 | — | 9 | 108 | no | — |
| 14 | #92338 | — | 9 | 36 | no | — |
| 15 | #92338 | — | 9 | 36 | no | — |
| 16 | #92338 | — | 9 | 36 | no | — |
| 17 | #92338 | — | 9 | 486 | no | — |
| 18 | #92338 | — | 9 | 216 | no | — |
| 19 | #92338 | — | 9 | 243 | no | — |
| 20 | C35 | C3⋊D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 12 | yes | complemented |
| 22 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 36 | yes | complemented |
| 23 | C32×H(3,𝔽3) | C3⋊D3 | 1 | 108 | yes | complemented |
| 24 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 36 | yes | complemented |
| 25 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 36 | yes | complemented |
| 26 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 36 | yes | complemented |
| 27 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 486 | yes | complemented |
| 28 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 243 | yes | complemented |
| 29 | C32⋊H(3,𝔽3) | C3⋊D3 | 1 | 216 | yes | complemented |
| 30 | C3×C3⋊(C3⋊D3) | — | 9 | 4 | no | — |
| 31 | C32×C3⋊D3 | — | 27 | 3 | no | — |
| 32 | C32×C3⋊D3 | — | 27 | 4 | no | — |
| 33 | C32×C3⋊D3 | — | 27 | 36 | no | — |
| 34 | C32×C3⋊D3 | — | 27 | 6 | no | — |
| 35 | C32×C3⋊D3 | — | 27 | 81 | no | — |
| 36 | C3⋊#272 | — | 9 | 108 | no | — |
| 37 | C3⋊#272 | — | 9 | 36 | no | — |
| 38 | C3⋊#272 | — | 9 | 12 | no | — |
| 39 | C3×#272 | — | 27 | 108 | no | — |
| 40 | C3×#272 | — | 27 | 108 | no | — |
| 41 | C3×#272 | — | 27 | 108 | no | — |
| 42 | C3×#272 | — | 27 | 108 | no | — |
| 43 | C3×#272 | — | 27 | 648 | no | — |
| 44 | C34 | — | 9 | 4 | no | — |
| 45 | C34 | — | 18 | 4 | no | — |
| 46 | C34 | C3⋊(C3⋊D3) | 1 | 6 | yes | — |
| 47 | C34 | C3⋊(C3⋊D3) | 1 | 36 | yes | — |
| 48 | C34 | C3⋊(C3⋊D3) | 1 | 4 | yes | — |
| 49 | C34 | C3⋊(C3⋊D3) | 1 | 3 | yes | — |
| 50 | C34 | C3⋊(C3⋊D3) | 1 | 81 | yes | complemented |
| 51 | C3×H(3,𝔽3) | — | 9 | 36 | no | — |
| 52 | C3×H(3,𝔽3) | — | 9 | 108 | no | — |
| 53 | C3×H(3,𝔽3) | — | 9 | 12 | no | — |
| 54 | C3×H(3,𝔽3) | — | 18 | 36 | no | — |
| 55 | C3×H(3,𝔽3) | — | 18 | 108 | no | — |
| 56 | C3×H(3,𝔽3) | — | 18 | 12 | no | — |
| 57 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 108 | yes | — |
| 58 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 108 | yes | — |
| 59 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 108 | yes | — |
| 60 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 108 | yes | — |
| 61 | C3×H(3,𝔽3) | C3⋊(C3⋊D3) | 1 | 648 | yes | complemented |
| 62 | C3⋊(C3⋊D3) | — | 9 | 1 | no | — |
| 63 | C3×C3⋊D3 | — | 27 | 36 | no | — |
| 64 | C3×C3⋊D3 | — | 27 | 4 | no | — |
| 65 | C3×C3⋊D3 | — | 27 | 12 | no | — |
| 66 | C3×C3⋊D3 | — | 27 | 324 | no | — |
| 67 | C3×C3⋊D3 | — | 27 | 108 | no | — |
| 68 | C3×C3⋊D3 | — | 27 | 12 | no | — |
| 69 | C3×C3⋊D3 | — | 27 | 12 | no | — |
| 70 | C3×C3⋊D3 | — | 27 | 12 | no | — |
| 71 | C32×D3 | — | 81 | 3 | no | — |
| 72 | C32×D3 | — | 81 | 6 | no | — |
| 73 | C32×D3 | — | 81 | 108 | no | — |
| 74 | C32×D3 | — | 81 | 4 | no | — |
| 75 | #272 | — | 27 | 108 | no | — |
| 76 | #272 | — | 27 | 108 | no | — |
| 77 | #272 | — | 27 | 108 | no | — |
| 78 | #272 | — | 27 | 108 | no | — |
| 79 | #272 | — | 27 | 648 | no | — |
| 80 | C33 | — | 3 | 12 | no | — |
| 81 | C33 | — | 3 | 4 | no | — |
| 82 | C33 | — | 3 | 36 | no | — |
| 83 | C33 | — | 6 | 12 | no | — |
| 84 | C33 | — | 6 | 4 | no | — |
| 85 | C33 | — | 6 | 36 | no | — |
| 86 | C33 | — | 9 | 12 | no | — |
| 87 | C33 | — | 9 | 12 | no | — |
| 88 | C33 | — | 9 | 12 | no | — |
| 89 | C33 | — | 9 | 108 | no | — |
| 90 | C33 | — | 9 | 324 | no | — |
| 91 | C33 | — | 9 | 1 | no | — |
| 92 | C33 | — | 18 | 4 | no | — |
| 93 | C33 | — | 18 | 12 | no | — |
| 94 | C33 | — | 18 | 4 | no | — |
| 95 | C33 | — | 18 | 12 | no | — |
| 96 | C33 | — | 18 | 8 | no | — |
| 97 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 4 | yes | — |
| 98 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 3 | yes | — |
| 99 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 6 | yes | — |
| 100 | C33 | C3⋊(C3⋊(C3⋊D3)) | 1 | 108 | yes | — |
| 101 | H(3,𝔽3) | — | 3 | 108 | no | — |
| 102 | H(3,𝔽3) | — | 6 | 108 | no | — |
| 103 | H(3,𝔽3) | — | 9 | 108 | no | — |
| 104 | H(3,𝔽3) | — | 9 | 108 | no | — |
| 105 | H(3,𝔽3) | — | 9 | 108 | no | — |
| 106 | H(3,𝔽3) | — | 9 | 648 | no | — |
| 107 | C2×C32 | — | 243 | 1 | no | — |
| 108 | C3⋊D3 | — | 27 | 6 | no | — |
| 109 | C3⋊D3 | — | 27 | 4 | no | — |
| 110 | C3⋊D3 | — | 27 | 3 | no | — |
| 111 | C3⋊D3 | — | 27 | 36 | no | — |
| 112 | C3⋊D3 | — | 27 | 81 | no | — |
| 113 | C3×D3 | — | 81 | 324 | no | — |
| 114 | C3×D3 | — | 81 | 108 | no | — |
| 115 | C3×D3 | — | 81 | 12 | no | — |
| 116 | C3×D3 | — | 81 | 12 | no | — |
| 117 | C3×D3 | — | 81 | 12 | no | — |
| 118 | C3×D3 | — | 81 | 12 | no | — |
| 119 | C3×D3 | — | 81 | 4 | no | — |
| 120 | C32 | — | 2 | 4 | no | — |
| 121 | C32 | — | 3 | 12 | no | — |
| 122 | C32 | — | 3 | 324 | no | — |
| 123 | C32 | — | 3 | 12 | no | — |
| 124 | C32 | — | 3 | 12 | no | — |
| 125 | C32 | — | 3 | 12 | no | — |
| 126 | C32 | — | 3 | 108 | no | — |
| 127 | C32 | — | 9 | 6 | no | — |
| 128 | C32 | — | 9 | 4 | no | — |
| 129 | C32 | — | 9 | 3 | no | — |
| 130 | C32 | — | 18 | 12 | no | — |
| 131 | C32 | — | 18 | 4 | no | — |
| 132 | C32 | — | 18 | 12 | no | — |
| 133 | C32 | — | 18 | 4 | no | — |
| 134 | C32 | — | 18 | 8 | no | — |
| 135 | C32 | — | 18 | 12 | no | — |
| 136 | C32 | — | 27 | 36 | no | — |
| 137 | C32 | — | 27 | 81 | no | — |
| 138 | C32 | — | 54 | 36 | no | — |
| 139 | C32 | — | 54 | 81 | no | — |
| 140 | C32 | #92404 | 1 | 4 | yes | — |
| 141 | C32 | C3⋊(C3⋊(C3⋊(C3⋊D3))) | 1 | 1 | yes | characteristic |
| 142 | C2×C3 | — | 243 | 4 | no | — |
| 143 | D3 | — | 81 | 4 | no | — |
| 144 | D3 | — | 81 | 3 | no | — |
| 145 | D3 | — | 81 | 6 | no | — |
| 146 | D3 | — | 81 | 108 | no | — |
| 147 | C3 | — | 3 | 4 | no | — |
| 148 | C3 | — | 6 | 4 | no | — |
| 149 | C3 | — | 9 | 3 | no | — |
| 150 | C3 | — | 9 | 6 | no | — |
| 151 | C3 | — | 9 | 108 | no | — |
| 152 | C3 | C3⋊#92404 | 1 | 4 | yes | — |
| 153 | C2 | — | 243 | 1 | no | Sylow |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C32 | C3⋊(C3⋊(C3⋊(C3⋊D3))) |
| commutator, Fitting, cosocle, hypocenter | #14555252 | C2 |
| LCS | G ▷ #14555252 | |
| UCS | C1 ◁ C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14555252 | C2 | semidirect |
| 6. H(3,𝔽9) | D3 | semidirect |
| 7. H(3,𝔽3)2 | D3 | semidirect |
| 8. #10624880 | D3 | semidirect |
| 9. C3×C32⋊H(3,𝔽3) | D3 | semidirect |
| 20. C35 | C3⋊D3 | semidirect |
| 21. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 22. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 23. C32×H(3,𝔽3) | C3⋊D3 | semidirect |
| 24. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 25. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 26. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 27. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 28. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 29. C32⋊H(3,𝔽3) | C3⋊D3 | semidirect |
| 50. C34 | C3⋊(C3⋊D3) | semidirect |
| 61. C3×H(3,𝔽3) | C3⋊(C3⋊D3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14555252 | 2 |
| #13091539 | 3 |
| #13091502 | 3 |
| #13091466 | 3 |
| C3⋊#92338 | 3 |