Subgroups — #88542

Order 432.

187 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×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#1084	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×(C₂×D₃)⋊C₄	—	3	4	no	maximal
5	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₂×#272	—	2	1	no	—
7	C₂×#272	—	2	1	no	—
8	C₂×#272	C₄	1	2	yes	complemented
9	C₄×H(3,𝔽₃)	—	2	1	no	—
10	#1084	—	2	1	no	—
11	C₂²×C₃×D₃	—	3	4	no	—
12	C₂×C₃²×C₄	D₃	1	4	yes	complemented
13	C₂×C₃×Dic₃	—	3	4	no	—
14	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
16	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
17	#272	—	2	2	no	—
18	C₃×C₂²⋊C₄	—	9	1	no	—
19	(C₂×D₃)⋊C₄	—	3	4	no	—
20	C₂²×C₃²	C₂×D₃	1	4	yes	—
21	C₂×C₃×D₃	—	3	8	no	—
22	C₂×C₃×D₃	—	6	4	no	—
23	C₂×C₃×D₃	—	6	4	no	—
24	C₃²×C₄	—	2	4	no	—
25	C₃×Dic₃	—	6	4	no	—
26	H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
27	C₂³×C₃	—	9	1	no	—
28	C₂²×D₃	—	3	4	no	—
29	C₂×C₃×C₄	—	3	4	no	—
30	C₂×C₃×C₄	—	9	1	no	—
31	C₂×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
32	C₂×Dic₃	—	3	4	no	—
33	C₂×C₃²	C₄×D₃	1	4	yes	—
34	C₂×C₃²	D₁₂	1	4	yes	—
35	C₂×C₃²	#67	1	4	yes	—
36	C₃×D₃	—	6	8	no	—
37	C₂²⋊C₄	—	9	1	no	Sylow
38	C₂²×C₃	—	3	4	no	—
39	C₂²×C₃	—	9	2	no	—
40	C₂²×C₃	—	18	1	no	—
41	C₂²×C₃	—	18	1	no	—
42	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
43	C₂×D₃	—	3	8	no	—
44	C₂×D₃	—	6	4	no	—
45	C₂×D₃	—	6	4	no	—
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	—	6	4	no	—
48	C₃×C₄	—	18	1	no	—
49	Dic₃	—	6	4	no	—
50	C₃²	(C₂×D₃)⋊C₄	1	4	yes	complemented
51	C₂³	—	9	1	no	—
52	C₂×C₄	—	9	1	no	—
53	C₂×C₄	#272	1	1	yes	characteristic, complemented
54	C₂×C₃	—	3	4	no	—
55	C₂×C₃	—	3	4	no	—
56	C₂×C₃	—	3	4	no	—
57	C₂×C₃	—	18	2	no	—
58	C₂×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
59	C₂×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
60	C₂×C₃	#638	1	1	yes	characteristic
61	D₃	—	6	8	no	—
62	C₂²	—	9	2	no	—
63	C₂²	—	18	1	no	—
64	C₂²	—	18	1	no	—
65	C₂²	C₂×#272	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	18	1	no	—
68	C₃	—	3	4	no	—
69	C₃	#3752	1	1	yes	characteristic
70	C₂	—	18	2	no	—
71	C₂	C₄×#272	1	1	yes	characteristic
72	C₂	C₄⋊#272	1	1	yes	characteristic
73	C₂	#6235	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₂×C₃⋊D₃
commutator	C₂×H(3,𝔽₃)	C₂×C₄
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
prime radical	C₂²×#272	C₂
cosocle	C₂²×H(3,𝔽₃)	C₂²
hypercenter	C₂×C₃×C₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂²⋊C₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×H(3,𝔽₃)	C₂	semidirect
3. C₂×#1084	C₂	semidirect
8. C₂×#272	C₄	semidirect
12. C₂×C₃²×C₄	D₃	semidirect
26. H(3,𝔽₃)	C₂²⋊C₄	semidirect
50. C₃²	(C₂×D₃)⋊C₄	semidirect
53. C₂×C₄	#272	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×C₄×H(3,𝔽₃)	2
C₂×#1084	2
C₃×(C₂×D₃)⋊C₄	3