Subgroups — #88613

Order 432.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×SL(2,ℤ/4)	—	3	3	no	maximal
3	C₃×#217	—	3	1	no	maximal
4	C₂²⋊H(3,𝔽₃)	C₄	1	1	yes	characteristic, complemented
5	#1084	—	4	1	no	maximal
6	C₂³×C₃²	D₃	1	1	yes	characteristic
7	C₂×C₃×A₄	D₃	1	3	yes	—
8	C₂×C₃×Dic₃	—	3	1	no	—
9	C₂×C₃×Dic₃	—	3	1	no	—
10	C₂×H(3,𝔽₃)	—	4	1	no	—
11	SL(2,ℤ/4)	—	3	3	no	—
12	C₃×C₂²⋊C₄	—	9	1	no	—
13	#217	—	3	1	no	—
14	C₂²×C₃²	—	3	1	no	—
15	C₂²×C₃²	—	3	1	no	—
16	C₂²×C₃²	Dic₃	1	1	yes	characteristic, complemented
17	C₃×A₄	Dic₃	1	3	yes	complemented
18	C₃×Dic₃	—	6	1	no	—
19	C₃×Dic₃	—	6	1	no	—
20	C₃×Dic₃	—	12	3	no	—
21	H(3,𝔽₃)	—	4	1	no	Sylow
22	C₂³×C₃	—	3	1	no	—
23	C₂³×C₃	C₃⋊D₃	1	1	yes	characteristic
24	C₂×A₄	—	3	3	no	—
25	C₂×C₃×C₄	—	9	1	no	—
26	C₂×C₃×C₄	—	9	1	no	—
27	C₂×Dic₃	—	3	1	no	—
28	C₂×Dic₃	—	3	1	no	—
29	C₂×C₃²	—	3	1	no	—
30	C₂×C₃²	—	3	1	no	—
31	C₂×C₃²	—	4	3	no	—
32	C₂×C₃²	S₄	1	1	yes	characteristic
33	C₂²⋊C₄	—	9	1	no	Sylow
34	C₂²×C₃	—	3	1	no	—
35	C₂²×C₃	—	3	1	no	—
36	C₂²×C₃	—	3	1	no	—
37	C₂²×C₃	—	3	1	no	—
38	C₂²×C₃	—	3	1	no	—
39	C₂²×C₃	—	6	1	no	—
40	C₂²×C₃	—	6	1	no	—
41	C₂²×C₃	C₃⋊Dic₃	1	1	yes	characteristic
42	A₄	—	3	3	no	—
43	C₃×C₄	—	18	1	no	—
44	C₃×C₄	—	18	1	no	—
45	Dic₃	—	6	1	no	—
46	Dic₃	—	6	1	no	—
47	Dic₃	—	12	3	no	—
48	C₃²	—	4	3	no	—
49	C₃²	SL(2,ℤ/4)	1	1	yes	characteristic, complemented
50	C₂³	#272	1	1	yes	characteristic
51	C₂×C₄	—	9	1	no	—
52	C₂×C₄	—	9	1	no	—
53	C₂×C₃	—	3	1	no	—
54	C₂×C₃	—	3	1	no	—
55	C₂×C₃	—	3	1	no	—
56	C₂×C₃	—	3	1	no	—
57	C₂×C₃	—	3	1	no	—
58	C₂×C₃	—	6	1	no	—
59	C₂×C₃	—	6	1	no	—
60	C₂×C₃	—	12	3	no	—
61	C₂×C₃	C₃⋊S₄	1	1	yes	characteristic
62	C₂²	—	3	1	no	—
63	C₂²	—	3	1	no	—
64	C₂²	#1084	1	1	yes	characteristic, complemented
65	C₄	—	18	1	no	—
66	C₄	—	18	1	no	—
67	C₃	—	3	1	no	—
68	C₃	—	12	3	no	—
69	C₃	C₃⋊SL(2,ℤ/4)	1	1	yes	characteristic
70	C₂	—	3	1	no	—
71	C₂	—	3	1	no	—
72	C₂	#6258	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	C₃⋊S₄
commutator, hypocenter	C₂²⋊H(3,𝔽₃)	C₄
Fitting	C₂³×C₃²	D₃
socle	C₂³×C₃	C₃⋊D₃
prime radical, cosocle	C₂×C₂²⋊H(3,𝔽₃)	C₂
LCS	G ▷ C₂²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
4. C₂²⋊H(3,𝔽₃)	C₄	semidirect
16. C₂²×C₃²	Dic₃	semidirect
17. C₃×A₄	Dic₃	semidirect
49. C₃²	SL(2,ℤ/4)	semidirect
64. C₂²	#1084	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊H(3,𝔽₃)	2
C₃×SL(2,ℤ/4)	3
C₃×#217	3
#1084	4