Finite Group DatabasebetaFinite Group DatabasebetaOrder 3888.
316 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 10 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 | #14519481 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #13049803 | — | 3 | 1 | no | maximal |
| 3 | #11757731 | — | 4 | 1 | no | maximal |
| 4 | #10620211 | D3 | 1 | 1 | yes | characteristic, complemented |
| 5 | #10620206 | — | 3 | 1 | no | — |
| 6 | #10620192 | — | 3 | 1 | no | — |
| 7 | #92338 | — | 4 | 1 | no | — |
| 8 | #92301 | — | 4 | 1 | no | — |
| 9 | #92258 | — | 4 | 1 | no | — |
| 10 | AGL(2,𝔽3) | — | 9 | 4 | no | maximal |
| 11 | C3⋊D32 | — | 6 | 1 | no | — |
| 12 | #66472 | — | 3 | 1 | no | — |
| 13 | #66432 | — | 12 | 4 | no | — |
| 14 | C32⋊H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 15 | PSU(3,𝔽2)⋊C3 | — | 9 | 4 | no | — |
| 16 | C3⋊(C3⋊(C3⋊D3)) | S4 | 1 | 1 | yes | characteristic |
| 17 | C32×C3⋊D3 | — | 12 | 1 | no | — |
| 18 | C3×#272 | — | 12 | 4 | no | — |
| 19 | C3×#272 | — | 24 | 4 | no | — |
| 20 | C3⋊((C3⋊D3)⋊C3) | — | 4 | 4 | no | — |
| 21 | C3⋊((C3⋊D3)⋊C3) | — | 12 | 4 | no | — |
| 22 | AΓL(1,𝔽9) | — | 27 | 4 | no | — |
| 23 | D3×C3⋊D3 | — | 36 | 4 | no | — |
| 24 | D3×C3⋊D3 | — | 36 | 1 | no | — |
| 25 | ((C3⋊D3)⋊C3)⋊C2 | — | 36 | 4 | no | — |
| 26 | C34 | GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 27 | C3×H(3,𝔽3) | — | 4 | 4 | no | — |
| 28 | C3×H(3,𝔽3) | — | 8 | 4 | no | — |
| 29 | PSU(3,𝔽2) | — | 9 | 4 | no | — |
| 30 | GO+(4,𝔽2) | — | 27 | 4 | no | — |
| 31 | AGL(1,𝔽9) | — | 27 | 4 | no | — |
| 32 | AGL(1,𝔽9) | — | 54 | 3 | no | — |
| 33 | C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 34 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 35 | C3⋊(C3⋊D3) | — | 72 | 1 | no | — |
| 36 | C3×C3⋊D3 | — | 12 | 4 | no | — |
| 37 | C3×C3⋊D3 | — | 36 | 1 | no | — |
| 38 | C32×D3 | — | 36 | 1 | no | — |
| 39 | C32×D3 | — | 36 | 4 | no | — |
| 40 | C32×D3 | — | 72 | 4 | no | — |
| 41 | #272 | — | 12 | 4 | no | — |
| 42 | #272 | — | 24 | 4 | no | — |
| 43 | (C3⋊D3)⋊C3 | — | 36 | 4 | no | — |
| 44 | (C3⋊D3)⋊C3 | — | 36 | 4 | no | — |
| 45 | (C3⋊D3)⋊C3 | — | 72 | 4 | no | — |
| 46 | GL(2,𝔽3) | — | 81 | 1 | no | — |
| 47 | C2×C3⋊D3 | — | 108 | 1 | no | — |
| 48 | D32 | — | 54 | 4 | no | — |
| 49 | D32 | — | 108 | 6 | no | — |
| 50 | D32 | — | 108 | 4 | no | — |
| 51 | #157 | — | 27 | 4 | no | — |
| 52 | #157 | — | 54 | 3 | no | — |
| 53 | C33 | — | 4 | 4 | no | — |
| 54 | C33 | — | 4 | 1 | no | — |
| 55 | C33 | — | 8 | 4 | no | — |
| 56 | C33 | — | 24 | 1 | no | — |
| 57 | H(3,𝔽3) | — | 12 | 4 | no | — |
| 58 | H(3,𝔽3) | — | 24 | 4 | no | — |
| 59 | H(3,𝔽3) | — | 24 | 4 | no | — |
| 60 | H(3,𝔽3) | — | 48 | 4 | no | — |
| 61 | SL(2,𝔽3) | — | 81 | 1 | no | — |
| 62 | C2×C32 | — | 108 | 1 | no | — |
| 63 | C3⋊D3 | — | 9 | 4 | no | — |
| 64 | C3⋊D3 | — | 12 | 1 | no | — |
| 65 | C3⋊D3 | — | 36 | 4 | no | — |
| 66 | C3⋊D3 | — | 36 | 1 | no | — |
| 67 | C3⋊D3 | — | 54 | 3 | no | — |
| 68 | C3⋊D3 | — | 72 | 4 | no | — |
| 69 | C3⋊D3 | — | 108 | 6 | no | — |
| 70 | C3×D3 | — | 36 | 4 | no | — |
| 71 | C3×D3 | — | 36 | 12 | no | — |
| 72 | C3×D3 | — | 36 | 4 | no | — |
| 73 | C3×D3 | — | 72 | 4 | no | — |
| 74 | C3×D3 | — | 72 | 12 | no | — |
| 75 | C3×D3 | — | 108 | 4 | no | — |
| 76 | SD16 | — | 243 | 1 | no | Sylow |
| 77 | C2×D3 | — | 324 | 1 | no | — |
| 78 | C2×D3 | — | 324 | 4 | no | — |
| 79 | C32 | — | 4 | 1 | no | — |
| 80 | C32 | — | 6 | 3 | no | — |
| 81 | C32 | — | 8 | 4 | no | — |
| 82 | C32 | — | 12 | 6 | no | — |
| 83 | C32 | — | 12 | 4 | no | — |
| 84 | C32 | — | 24 | 4 | no | — |
| 85 | C32 | — | 24 | 12 | no | — |
| 86 | C32 | AGL(2,𝔽3) | 1 | 4 | yes | complemented |
| 87 | Q8 | — | 81 | 1 | no | — |
| 88 | D4 | — | 243 | 1 | no | — |
| 89 | C8 | — | 243 | 1 | no | — |
| 90 | C2×C3 | — | 324 | 1 | no | — |
| 91 | C2×C3 | — | 108 | 4 | no | — |
| 92 | D3 | — | 36 | 4 | no | — |
| 93 | D3 | — | 36 | 1 | no | — |
| 94 | D3 | — | 324 | 1 | no | — |
| 95 | D3 | — | 72 | 4 | no | — |
| 96 | D3 | — | 108 | 4 | no | — |
| 97 | D3 | — | 648 | 1 | no | — |
| 98 | C22 | — | 486 | 1 | no | — |
| 99 | C4 | — | 243 | 1 | no | — |
| 100 | C3 | — | 4 | 4 | no | — |
| 101 | C3 | — | 24 | 1 | no | — |
| 102 | C3 | — | 36 | 1 | no | — |
| 103 | C3 | — | 72 | 4 | no | — |
| 104 | C2 | — | 81 | 1 | no | — |
| 105 | C2 | — | 108 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #14519481 | C2 |
| Fitting, socle | C34 | GL(2,𝔽3) |
| LCS | G ▷ #14519481 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14519481 | C2 | semidirect |
| 4. #10620211 | D3 | semidirect |
| 26. C34 | GL(2,𝔽3) | semidirect |
| 86. C32 | AGL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14519481 | 2 |
| #13049803 | 3 |
| #11757731 | 4 |
| AGL(2,𝔽3) | 9 |