Subgroups — C₂×#6196 #88669

Order 432.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1084	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal
3	#6196	C₂	1	4	yes	maximal, direct product factor
4	C₂×#627	—	3	2	no	maximal
5	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	#1084	C₂²	1	2	yes	—
7	(C₃⋊Dic₃)⋊C₃	C₂²	1	4	yes	—
8	C₂×C₃⋊Dic₃	D₃	1	2	yes	—
9	C₂×C₃×Dic₃	—	3	2	no	—
10	C₂×C₃×Dic₃	—	3	2	no	—
11	C₂×C₃×Dic₃	—	6	1	no	—
12	#627	—	3	8	no	—
13	C₂×H(3,𝔽₃)	Q₈	1	2	yes	complemented
14	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
15	C₂×Dic₆	—	9	1	no	—
16	C₂×Dic₆	—	9	2	no	—
17	C₂²×C₃²	—	2	1	no	—
18	C₂²×C₃²	C₂×D₃	1	2	yes	—
19	C₃⋊Dic₃	C₂×D₃	1	4	yes	—
20	C₃×Dic₃	—	3	4	no	—
21	C₃×Dic₃	—	3	4	no	—
22	C₃×Dic₃	—	6	2	no	—
23	H(3,𝔽₃)	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₃×C₄	—	9	2	no	—
25	C₂×C₃×C₄	—	9	1	no	—
26	C₂×Dic₃	—	3	2	no	—
27	C₂×Dic₃	—	3	2	no	—
28	C₂×Dic₃	—	6	1	no	—
29	C₂×Dic₃	—	9	2	no	—
30	Dic₆	—	9	4	no	—
31	Dic₆	—	9	8	no	—
32	C₂×C₃²	—	2	2	no	—
33	C₂×C₃²	—	2	1	no	—
34	C₂×C₃²	Dic₆	1	4	yes	complemented
35	C₂×C₃²	C₂²×D₃	1	2	yes	—
36	C₂×Q₈	—	27	1	no	Sylow
37	C₂²×C₃	—	3	2	no	—
38	C₂²×C₃	—	6	1	no	—
39	C₂²×C₃	D₃²	1	1	yes	characteristic
40	C₃×C₄	—	9	4	no	—
41	C₃×C₄	—	9	2	no	—
42	Dic₃	—	3	4	no	—
43	Dic₃	—	3	4	no	—
44	Dic₃	—	6	2	no	—
45	Dic₃	—	9	4	no	—
46	C₃²	—	2	1	no	—
47	C₃²	C₂×Dic₆	1	2	yes	complemented
48	Q₈	—	27	4	no	—
49	C₂×C₄	—	9	1	no	—
50	C₂×C₄	—	9	2	no	—
51	C₂×C₃	—	3	4	no	—
52	C₂×C₃	—	3	2	no	—
53	C₂×C₃	—	6	2	no	—
54	C₂×C₃	—	6	1	no	—
55	C₂×C₃	#627	1	2	yes	—
56	C₂×C₃	C₂×D₃²	1	1	yes	characteristic
57	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
58	C₄	—	9	2	no	—
59	C₄	—	9	4	no	—
60	C₃	—	3	2	no	—
61	C₃	—	6	1	no	—
62	C₃	C₂×#627	1	1	yes	characteristic
63	C₂	#6196	1	2	yes	direct product factor
64	C₂	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Frattini	C₂×C₃	C₂×D₃²
Fitting, prime radical	C₂²×H(3,𝔽₃)	C₂²
socle	C₂²×C₃	D₃²
hypocenter	H(3,𝔽₃)	C₂×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
3. #6196	C₂	direct (N × H)
13. C₂×H(3,𝔽₃)	Q₈	semidirect
23. H(3,𝔽₃)	C₂×Q₈	semidirect
34. C₂×C₃²	Dic₆	semidirect
47. C₃²	C₂×Dic₆	semidirect
63. C₂	#6196	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#1084	2
C₂×(C₃⋊Dic₃)⋊C₃	2
#6196	2
C₂×#627	3

Subgroups — C2×#6196 #88669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6196 #88669