Subgroups — #88451

Order 432.

151 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#272	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal, complemented
3	(C₂×D₃)⋊Dic₃	—	3	2	no	maximal
4	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₂×#272	—	2	2	no	—
6	C₂×#272	C₄	1	2	yes	complemented
7	(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
8	C₂²×C₃×D₃	—	3	2	no	—
9	C₂²×C₃×D₃	—	6	1	no	—
10	C₂×C₃⋊Dic₃	D₃	1	2	yes	complemented
11	C₂×C₃×Dic₃	—	3	2	no	—
12	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
13	C₂×H(3,𝔽₃)	D₄	1	2	yes	—
14	#272	—	2	2	no	—
15	#217	—	9	1	no	—
16	(C₂×D₃)⋊C₄	—	9	2	no	—
17	C₂²×C₃²	—	2	1	no	—
18	C₂²×C₃²	C₂×D₃	1	2	yes	—
19	C₂×C₃×D₃	—	3	4	no	—
20	C₂×C₃×D₃	—	6	2	no	—
21	C₂×C₃×D₃	—	6	4	no	—
22	C₂×C₃×D₃	—	6	2	no	—
23	C₂×C₃×D₃	—	6	2	no	—
24	C₃⋊Dic₃	—	2	2	no	—
25	C₃×Dic₃	—	6	2	no	—
26	H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₃	—	9	1	no	—
28	C₂²×D₃	—	3	2	no	—
29	C₂²×D₃	—	6	1	no	—
30	C₂×C₃×C₄	—	9	2	no	—
31	C₂×Dic₃	—	3	2	no	—
32	C₂×Dic₃	—	9	2	no	—
33	C₂×C₃²	—	2	1	no	—
34	C₂×C₃²	—	2	2	no	—
35	C₂×C₃²	C₄×D₃	1	2	yes	—
36	C₂×C₃²	D₁₂	1	2	yes	—
37	C₂×C₃²	#67	1	2	yes	—
38	C₃×D₃	—	6	4	no	—
39	C₃×D₃	—	6	4	no	—
40	C₂²⋊C₄	—	27	1	no	Sylow
41	C₂²×C₃	—	3	2	no	—
42	C₂²×C₃	—	6	1	no	—
43	C₂²×C₃	—	9	2	no	—
44	C₂²×C₃	—	18	2	no	—
45	C₂²×C₃	D₃²	1	1	yes	characteristic
46	C₂×D₃	—	3	4	no	—
47	C₂×D₃	—	6	2	no	—
48	C₂×D₃	—	6	2	no	—
49	C₂×D₃	—	6	2	no	—
50	C₂×D₃	—	6	4	no	—
51	C₃×C₄	—	18	2	no	—
52	Dic₃	—	6	2	no	—
53	Dic₃	—	18	2	no	—
54	C₃²	—	2	1	no	—
55	C₃²	(C₂×D₃)⋊C₄	1	2	yes	complemented
56	C₂³	—	9	1	no	—
57	C₂×C₄	—	9	2	no	—
58	C₂×C₃	—	3	2	no	—
59	C₂×C₃	—	3	2	no	—
60	C₂×C₃	—	3	2	no	—
61	C₂×C₃	—	6	1	no	—
62	C₂×C₃	—	6	2	no	—
63	C₂×C₃	—	18	2	no	—
64	C₂×C₃	#624	1	1	yes	characteristic
65	C₂×C₃	#626	1	2	yes	—
66	D₃	—	6	4	no	—
67	D₃	—	6	4	no	—
68	C₂²	—	9	2	no	—
69	C₂²	—	18	2	no	—
70	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
71	C₄	—	18	2	no	—
72	C₃	—	3	2	no	—
73	C₃	—	6	1	no	—
74	C₃	#3722	1	1	yes	characteristic
75	C₂	—	18	2	no	—
76	C₂	#6199	1	1	yes	characteristic
77	C₂	#6200	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂
commutator	C₂×H(3,𝔽₃)	C₂×C₄
Frattini, socle	C₂²×C₃	D₃²
Fitting, cosocle	C₂²×H(3,𝔽₃)	C₂²
prime radical	C₂²×#272	C₂
hypocenter	H(3,𝔽₃)	C₂²⋊C₄
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
2. C₂×(C₃⋊Dic₃)⋊C₃	C₂	semidirect
6. C₂×#272	C₄	semidirect
10. C₂×C₃⋊Dic₃	D₃	semidirect
26. H(3,𝔽₃)	C₂²⋊C₄	semidirect
55. C₃²	(C₂×D₃)⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#272	2
C₂×(C₃⋊Dic₃)⋊C₃	2
(C₂×D₃)⋊Dic₃	3