Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1225 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 1225 conjugacy classes of subgroups — too many to draw. Only its 51 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 | C2×#997 | C2 | 1 | 3 | yes | maximal, direct product factor |
| 2 | C22×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C23×A4 | — | 4 | 3 | no | maximal |
| 4 | #997 | C22 | 1 | 1 | yes | characteristic, direct product factor |
| 5 | C22×SL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 6 | C2×H(5,𝔽2) | — | 3 | 6 | no | — |
| 7 | C2×H(5,𝔽2) | — | 3 | 9 | no | — |
| 8 | C2×H(5,𝔽2) | C2×C3 | 1 | 3 | yes | complemented |
| 9 | C22×#41 | — | 3 | 2 | no | — |
| 10 | C23×D4 | — | 3 | 3 | no | — |
| 11 | C22×A4 | — | 4 | 12 | no | — |
| 12 | C22×A4 | — | 4 | 9 | no | — |
| 13 | C2×SL(2,𝔽3) | — | 4 | 6 | no | — |
| 14 | C25 | — | 3 | 1 | no | — |
| 15 | C25 | A4 | 1 | 3 | yes | complemented |
| 16 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 17 | H(5,𝔽2) | — | 3 | 18 | no | — |
| 18 | H(5,𝔽2) | — | 3 | 36 | no | — |
| 19 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 20 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 21 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 22 | H(5,𝔽2) | — | 3 | 9 | no | — |
| 23 | H(5,𝔽2) | C22×C3 | 1 | 1 | yes | characteristic, complemented |
| 24 | C2×#41 | — | 3 | 18 | no | — |
| 25 | C2×#41 | — | 3 | 18 | no | — |
| 26 | C2×#41 | — | 3 | 6 | no | — |
| 27 | C2×#41 | — | 3 | 6 | no | — |
| 28 | C22×Q8 | A4 | 1 | 2 | yes | complemented |
| 29 | C22×D4 | — | 3 | 6 | no | — |
| 30 | C22×D4 | — | 3 | 9 | no | — |
| 31 | C22×D4 | — | 3 | 36 | no | — |
| 32 | C22×D4 | — | 3 | 9 | no | — |
| 33 | C22×D4 | — | 3 | 9 | no | — |
| 34 | C22×D4 | — | 3 | 9 | no | — |
| 35 | C23×C4 | — | 3 | 3 | no | — |
| 36 | C23×C3 | — | 16 | 1 | no | — |
| 37 | C2×A4 | — | 4 | 18 | no | — |
| 38 | C2×A4 | — | 4 | 3 | no | — |
| 39 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 40 | C24 | — | 3 | 9 | no | — |
| 41 | C24 | — | 3 | 3 | no | — |
| 42 | C24 | — | 3 | 3 | no | — |
| 43 | C24 | — | 3 | 9 | no | — |
| 44 | C24 | — | 4 | 12 | no | — |
| 45 | C24 | — | 12 | 4 | no | — |
| 46 | C24 | C2×A4 | 1 | 9 | yes | complemented |
| 47 | #41 | — | 3 | 36 | no | — |
| 48 | #41 | — | 3 | 18 | no | — |
| 49 | #41 | — | 3 | 18 | no | — |
| 50 | #41 | — | 3 | 36 | no | — |
| 51 | #41 | — | 3 | 12 | no | — |
| 52 | #41 | — | 3 | 6 | no | — |
| 53 | #41 | — | 3 | 2 | no | — |
| 54 | C2×Q8 | — | 3 | 6 | no | — |
| 55 | C2×Q8 | C2×A4 | 1 | 6 | yes | complemented |
| 56 | C2×D4 | — | 3 | 36 | no | — |
| 57 | C2×D4 | — | 3 | 18 | no | — |
| 58 | C2×D4 | — | 3 | 18 | no | — |
| 59 | C2×D4 | — | 3 | 36 | no | — |
| 60 | C2×D4 | — | 3 | 36 | no | — |
| 61 | C2×D4 | — | 3 | 36 | no | — |
| 62 | C2×D4 | — | 3 | 18 | no | — |
| 63 | C2×D4 | — | 3 | 9 | no | — |
| 64 | C2×D4 | — | 3 | 36 | no | — |
| 65 | C2×D4 | — | 3 | 9 | no | — |
| 66 | C2×D4 | — | 3 | 9 | no | — |
| 67 | C2×D4 | — | 3 | 3 | no | — |
| 68 | C22×C4 | — | 3 | 9 | no | — |
| 69 | C22×C4 | — | 3 | 18 | no | — |
| 70 | C22×C4 | — | 3 | 9 | no | — |
| 71 | C22×C4 | — | 3 | 2 | no | — |
| 72 | C22×C3 | — | 16 | 4 | no | — |
| 73 | C22×C3 | — | 16 | 3 | no | — |
| 74 | A4 | — | 4 | 3 | no | — |
| 75 | C23 | — | 3 | 6 | no | — |
| 76 | C23 | — | 3 | 6 | no | — |
| 77 | C23 | — | 3 | 9 | no | — |
| 78 | C23 | — | 3 | 1 | no | — |
| 79 | C23 | — | 3 | 9 | no | — |
| 80 | C23 | — | 3 | 9 | no | — |
| 81 | C23 | — | 3 | 9 | no | — |
| 82 | C23 | — | 4 | 18 | no | — |
| 83 | C23 | — | 6 | 12 | no | — |
| 84 | C23 | — | 12 | 6 | no | — |
| 85 | C23 | — | 12 | 18 | no | — |
| 86 | C23 | — | 12 | 18 | no | — |
| 87 | C23 | C22×A4 | 1 | 3 | yes | complemented |
| 88 | C23 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 89 | Q8 | — | 3 | 6 | no | — |
| 90 | Q8 | — | 3 | 4 | no | — |
| 91 | Q8 | C22×A4 | 1 | 2 | yes | complemented |
| 92 | D4 | — | 3 | 36 | no | — |
| 93 | D4 | — | 3 | 36 | no | — |
| 94 | D4 | — | 3 | 18 | no | — |
| 95 | D4 | — | 3 | 6 | no | — |
| 96 | C2×C4 | — | 3 | 18 | no | — |
| 97 | C2×C4 | — | 3 | 9 | no | — |
| 98 | C2×C4 | — | 3 | 18 | no | — |
| 99 | C2×C4 | — | 3 | 3 | no | — |
| 100 | C2×C4 | — | 3 | 6 | no | — |
| 101 | C2×C4 | — | 3 | 6 | no | — |
| 102 | C2×C3 | — | 16 | 6 | no | — |
| 103 | C2×C3 | — | 16 | 1 | no | — |
| 104 | C22 | — | 3 | 9 | no | — |
| 105 | C22 | — | 3 | 3 | no | — |
| 106 | C22 | — | 4 | 3 | no | — |
| 107 | C22 | — | 6 | 18 | no | — |
| 108 | C22 | — | 6 | 18 | no | — |
| 109 | C22 | — | 12 | 9 | no | — |
| 110 | C22 | — | 12 | 1 | no | — |
| 111 | C22 | — | 12 | 9 | no | — |
| 112 | C22 | — | 12 | 6 | no | — |
| 113 | C22 | — | 12 | 6 | no | — |
| 114 | C22 | #997 | 1 | 4 | yes | direct product factor |
| 115 | C22 | C2×C22⋊A4 | 1 | 3 | yes | — |
| 116 | C4 | — | 3 | 6 | no | — |
| 117 | C4 | — | 3 | 2 | no | — |
| 118 | C3 | — | 16 | 1 | no | Sylow |
| 119 | C2 | — | 6 | 3 | no | — |
| 120 | C2 | — | 6 | 9 | no | — |
| 121 | C2 | C2×#997 | 1 | 6 | yes | direct product factor |
| 122 | C2 | C22×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C22×C3 |
| Frattini | C2 | C22×C22⋊A4 |
| Fitting | C22×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#997 | C2 | direct (N × H) |
| 2. C22×H(5,𝔽2) | C3 | semidirect |
| 4. #997 | C22 | direct (N × H) |
| 8. C2×H(5,𝔽2) | C2×C3 | semidirect |
| 15. C25 | A4 | semidirect |
| 23. H(5,𝔽2) | C22×C3 | semidirect |
| 28. C22×Q8 | A4 | semidirect |
| 46. C24 | C2×A4 | semidirect |
| 55. C2×Q8 | C2×A4 | semidirect |
| 87. C23 | C22×A4 | semidirect |
| 91. Q8 | C22×A4 | semidirect |
| 114. C22 | #997 | direct (N × H) |
| 121. C2 | C2×#997 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#997 | 2 |
| C22×H(5,𝔽2) | 3 |
| C23×A4 | 4 |
| C22×SL(2,𝔽3) | 4 |