Finite Group Databasebeta

← C3×(#41⋊C3)⋊A4 #13180921

Subgroups — C3×(#41⋊C3)⋊A4 #13180921

Order 1728.

468 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Ω+(4,𝔽3)C211yesmaximal, characteristic, complemented
2C3×#5950C312yesmaximal, complemented
3C3×#5945C312yesmaximal, complemented
4(#41⋊C3)⋊A4C319yesmaximal, direct product factor
5C32×#41⋊C3—42nomaximal
6C3×#997C2×C312yescomplemented
7C3×2−1+4⋊C3—32no—
8C3×#994C2×C312yescomplemented
9C2×C3×#41⋊C3—32no—
10Ω+(4,𝔽3)C2×C319yescomplemented
11C32×SL(2,𝔽3)—42no—
12C3×#585C3211yescharacteristic, complemented
13#5950C3216yescomplemented
14#5945C3216yescomplemented
15C32×#41—42no—
16C3×#41⋊C3—32no—
17C3×#41⋊C3—44no—
18C3×#41⋊C3—418no—
19C3×#41⋊C3A412yescomplemented
20C2×C3×SL(2,𝔽3)—32no—
21C3×C4×A4—122no—
22C33×C4—161no—
23C3×2−1+4—32no—
24C3×H(5,𝔽2)—91no—
25C3×H(5,𝔽2)C2×C3211yescharacteristic
26C2×C3×#41—32no—
27C2×C3×#41—91no—
28#997C2×C3216yescomplemented
292−1+4⋊C3—36no—
30#994C2×C3216yescomplemented
31C2×#41⋊C3—36no—
32C2×C3×A4—122no—
33C32×Q8—42no—
34C32×D4—122no—
35C2×C32×C4—122no—
36C3×SL(2,𝔽3)—44no—
37C3×SL(2,𝔽3)—418no—
38C3×SL(2,𝔽3)C2×A412yes—
39#585C3311yescharacteristic, complemented, Sylow
40C2×C33—161no—
41C3×#41—32no—
42C3×#41—46no—
43C3×#41—92no—
44C3×#41—94no—
45C3×#41—92no—
46C3×#41C3×A412yes—
47C2×C3×Q8—32no—
48C2×C3×Q8—91no—
49C2×C3×D4—92no—
50C2×C3×D4—92no—
51C2×C3×D4—91no—
52C22×C3×C4—32no—
53C22×C3×C4—91no—
54#41⋊C3—36no—
55#41⋊C3—412no—
56#41⋊C3C3×A416yescomplemented
57C2×SL(2,𝔽3)—36no—
58C4×A4—126no—
59C22×C32—122no—
60C3×A4—122no—
61C32×C4—42no—
62C32×C4—122no—
63C32×C4—169no—
64C32×C4—162no—
652−1+4—32no—
66H(5,𝔽2)—91no—
67H(5,𝔽2)C2×C3311yescharacteristic
68C2×#41—32no—
69C2×#41—91no—
70C33—161noSylow
71C23×C3—32no—
72C23×C3—91no—
73C2×A4—126no—
74C3×Q8—46no—
75C3×Q8—92no—
76C3×Q8C2×C3×A412yes—
77C3×D4—32no—
78C3×D4—92no—
79C3×D4—94no—
80C3×D4—126no—
81C2×C3×C4—32no—
82C2×C3×C4—91no—
83C2×C3×C4—92no—
84C2×C3×C4—91no—
85C2×C3×C4—92no—
86C2×C3×C4—126no—
87SL(2,𝔽3)—412no—
88SL(2,𝔽3)C2×C3×A416yes—
89C2×C32—42no—
90C2×C32—169no—
91C2×C32—162no—
92C2×C32—242no—
93#41—32no—
94#41—92no—
95#41—94no—
96#41—92no—
97#41C32×A412yes—
98C2×Q8—32no—
99C2×Q8—91no—
100C2×D4—92no—
101C2×D4—92no—
102C2×D4—91no—
103C22×C4—32no—
104C22×C4—91no—
105C22×C3—32no—
106C22×C3—91no—
107C22×C3—126no—
108C22×C3—122no—
109C22×C3—361no—
110A4—126no—
111C3×C4—32no—
112C3×C4—46no—
113C3×C4—91no—
114C3×C4—126no—
115C3×C4—166no—
116C3×C4A4211yescharacteristic
117C32—42no—
118C32—169no—
119C32—162no—
120C23—32no—
121C23—91no—
122Q8—92no—
123Q8C2×C32×A412yes—
124D4—32no—
125D4—92no—
126D4—94no—
127C2×C4—32no—
128C2×C4—91no—
129C2×C4—92no—
130C2×C4—91no—
131C2×C4—92no—
132C2×C3—46no—
133C2×C3—62no—
134C2×C3—166no—
135C2×C3—181no—
136C2×C3—246no—
137C2×C3C2×A4211yescharacteristic
138C22—32no—
139C22—91no—
140C22—122no—
141C22—361no—
142C4—32no—
143C4—91no—
144C4C3×A4211yescharacteristic
145C3—46no—
146C3—166no—
147C3(#41⋊C3)⋊A411yescharacteristic, direct product factor
148C2—62no—
149C2—181no—
150C2C2×C3×A4211yescharacteristic
subgroupG/H
center, hypercenterC3×C4A42
commutator, cosocle, hypocenterH(5,𝔽2)C2×C33
FrattiniC2C2×C3×A42
FittingC3×#585C32
socleC2×C3C2×A42
LCSG ▷ H(5,𝔽2)
UCSC1 ◁ C3×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×Ω+(4,𝔽3)C2semidirect
2. C3×#5950C3semidirect
3. C3×#5945C3semidirect
4. (#41⋊C3)⋊A4C3direct (N × H)
6. C3×#997C2×C3semidirect
8. C3×#994C2×C3semidirect
10. Ω+(4,𝔽3)C2×C3semidirect
12. C3×#585C32semidirect
13. #5950C32semidirect
14. #5945C32semidirect
19. C3×#41⋊C3A4semidirect
28. #997C2×C32semidirect
30. #994C2×C32semidirect
39. #585C33semidirect
56. #41⋊C3C3×A4semidirect
147. C3(#41⋊C3)⋊A4direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×Ω+(4,𝔽3)2
C3×#59503
C3×#59453
(#41⋊C3)⋊A43
C32×#41⋊C34