Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
304 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 60 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 | #5945 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #5423 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×GU(2,𝔽3) | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #2901 | — | 3 | 1 | no | maximal, Sylow |
| 5 | 2−1+4⋊C3 | C4 | 1 | 3 | yes | complemented |
| 6 | #994 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | C2×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 8 | #948 | — | 4 | 1 | no | maximal |
| 9 | GU(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | GU(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 11 | #585 | D3 | 1 | 1 | yes | characteristic |
| 12 | #568 | — | 3 | 1 | no | — |
| 13 | C4×#41 | — | 3 | 1 | no | — |
| 14 | #421 | — | 3 | 3 | no | — |
| 15 | #421 | — | 3 | 3 | no | — |
| 16 | C2×#104 | — | 3 | 3 | no | — |
| 17 | C2×#104 | — | 3 | 3 | no | — |
| 18 | C3×#41 | — | 4 | 1 | no | — |
| 19 | #41⋊C3 | C2×C4 | 1 | 3 | yes | complemented |
| 20 | #41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 21 | C2×SL(2,𝔽3) | C2×C4 | 1 | 3 | yes | complemented |
| 22 | #208 | — | 4 | 3 | no | — |
| 23 | C2×C3⋊C8 | — | 4 | 3 | no | — |
| 24 | 2−1+4 | — | 3 | 1 | no | — |
| 25 | 2−1+4 | Dic3 | 1 | 3 | yes | — |
| 26 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 27 | H(5,𝔽2) | Dic3 | 1 | 1 | yes | characteristic |
| 28 | C2×#41 | — | 3 | 1 | no | — |
| 29 | C2×#41 | — | 3 | 3 | no | — |
| 30 | C2×#41 | C2×D3 | 1 | 3 | yes | — |
| 31 | #131 | — | 6 | 3 | no | — |
| 32 | #131 | — | 6 | 1 | no | — |
| 33 | C2×M16 | — | 3 | 3 | no | — |
| 34 | C2×M16 | — | 3 | 3 | no | — |
| 35 | C4×Q8 | — | 6 | 1 | no | — |
| 36 | C4×D4 | — | 6 | 3 | no | — |
| 37 | #117 | — | 3 | 3 | no | — |
| 38 | C2×C42 | — | 3 | 3 | no | — |
| 39 | #104 | — | 3 | 3 | no | — |
| 40 | #104 | — | 3 | 6 | no | — |
| 41 | #104 | — | 3 | 3 | no | — |
| 42 | #104 | — | 3 | 1 | no | — |
| 43 | #104 | — | 3 | 3 | no | — |
| 44 | C3×Q8 | — | 4 | 1 | no | — |
| 45 | C3×D4 | — | 4 | 3 | no | — |
| 46 | C2×C3×C4 | — | 4 | 3 | no | — |
| 47 | SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 48 | C3⋊C8 | — | 4 | 1 | no | — |
| 49 | C3⋊C8 | — | 4 | 3 | no | — |
| 50 | #41 | — | 3 | 1 | no | — |
| 51 | #41 | — | 3 | 3 | no | — |
| 52 | #41 | — | 3 | 3 | no | — |
| 53 | #41 | — | 3 | 3 | no | — |
| 54 | #41 | — | 3 | 3 | no | — |
| 55 | #41 | — | 3 | 6 | no | — |
| 56 | #41 | — | 6 | 3 | no | — |
| 57 | #41 | C2×Dic3 | 1 | 3 | yes | — |
| 58 | #41 | S4 | 1 | 1 | yes | characteristic |
| 59 | #41 | C22×D3 | 1 | 1 | yes | characteristic |
| 60 | C2×Q8 | — | 3 | 1 | no | — |
| 61 | C2×Q8 | — | 3 | 3 | no | — |
| 62 | C2×Q8 | C2×Dic3 | 1 | 3 | yes | — |
| 63 | C2×D4 | — | 3 | 3 | no | — |
| 64 | C2×D4 | — | 3 | 3 | no | — |
| 65 | C2×D4 | — | 3 | 3 | no | — |
| 66 | C2×D4 | — | 6 | 3 | no | — |
| 67 | C22×C4 | — | 3 | 3 | no | — |
| 68 | C22×C4 | — | 6 | 1 | no | — |
| 69 | C22×C4 | — | 6 | 3 | no | — |
| 70 | M16 | — | 3 | 3 | no | — |
| 71 | M16 | — | 3 | 1 | no | — |
| 72 | M16 | — | 6 | 3 | no | — |
| 73 | M16 | — | 6 | 3 | no | — |
| 74 | C2×C8 | — | 6 | 3 | no | — |
| 75 | C2×C8 | — | 6 | 3 | no | — |
| 76 | C4⋊C4 | — | 6 | 3 | no | — |
| 77 | C22⋊C4 | — | 6 | 3 | no | — |
| 78 | C42 | — | 3 | 1 | no | — |
| 79 | C42 | — | 3 | 3 | no | — |
| 80 | C42 | — | 6 | 3 | no | — |
| 81 | C22×C3 | — | 4 | 3 | no | — |
| 82 | C3×C4 | — | 4 | 3 | no | — |
| 83 | C3×C4 | — | 4 | 1 | no | — |
| 84 | C23 | — | 3 | 3 | no | — |
| 85 | C23 | — | 6 | 1 | no | — |
| 86 | Q8 | — | 3 | 3 | no | — |
| 87 | Q8 | — | 3 | 3 | no | — |
| 88 | Q8 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 89 | Q8 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 90 | D4 | — | 3 | 3 | no | — |
| 91 | D4 | — | 3 | 1 | no | — |
| 92 | D4 | — | 3 | 6 | no | — |
| 93 | D4 | — | 3 | 3 | no | — |
| 94 | D4 | — | 6 | 3 | no | — |
| 95 | D4 | SL(2,ℤ/4) | 1 | 3 | yes | — |
| 96 | C2×C4 | — | 3 | 3 | no | — |
| 97 | C2×C4 | — | 3 | 3 | no | — |
| 98 | C2×C4 | — | 3 | 3 | no | — |
| 99 | C2×C4 | — | 3 | 1 | no | — |
| 100 | C2×C4 | — | 3 | 3 | no | — |
| 101 | C2×C4 | — | 6 | 3 | no | — |
| 102 | C2×C4 | — | 6 | 1 | no | — |
| 103 | C2×C4 | — | 6 | 3 | no | — |
| 104 | C2×C4 | — | 12 | 6 | no | — |
| 105 | C2×C4 | C2×S4 | 1 | 3 | yes | — |
| 106 | C8 | — | 6 | 3 | no | — |
| 107 | C8 | — | 6 | 1 | no | — |
| 108 | C2×C3 | — | 4 | 1 | no | — |
| 109 | C2×C3 | — | 8 | 3 | no | — |
| 110 | C22 | — | 3 | 1 | no | — |
| 111 | C22 | — | 3 | 3 | no | — |
| 112 | C22 | — | 12 | 3 | no | — |
| 113 | C22 | — | 24 | 1 | no | — |
| 114 | C22 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 115 | C4 | — | 3 | 1 | no | — |
| 116 | C4 | — | 3 | 3 | no | — |
| 117 | C4 | — | 6 | 2 | no | — |
| 118 | C4 | — | 12 | 3 | no | — |
| 119 | C4 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 120 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 121 | C3 | — | 4 | 1 | no | Sylow |
| 122 | C2 | — | 2 | 3 | no | — |
| 123 | C2 | — | 6 | 1 | no | — |
| 124 | C2 | — | 6 | 3 | no | — |
| 125 | C2 | C22×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Fitting | #585 | D3 |
| socle | C2 | C22×SL(2,ℤ/4) |
| prime radical | #5945 | C2 |
| cosocle | #41⋊C3 | C23 |
| hypercenter | #41 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #5423 | C2 | semidirect |
| 3. C2×GU(2,𝔽3) | C2 | semidirect |
| 5. 2−1+4⋊C3 | C4 | semidirect |
| 6. #994 | C4 | semidirect |
| 9. GU(2,𝔽3) | C22 | semidirect |
| 10. GU(2,𝔽3) | C22 | semidirect |
| 19. #41⋊C3 | C2×C4 | semidirect |
| 21. C2×SL(2,𝔽3) | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5945 | 2 |
| #5423 | 2 |
| C2×GU(2,𝔽3) | 2 |
| #2901 | 3 |
| #948 | 4 |