Subgroups — #88655

Order 432.

205 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	C₂	1	2	yes	maximal, complemented
2	C₄×#272	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊((C₃⋊D₃)⋊C₃)	C₂	1	2	yes	maximal, complemented
4	#6198	C₂	1	2	yes	maximal, complemented
5	D₃×D₁₂	—	3	2	no	maximal
6	C₂×#272	C₂²	1	1	yes	characteristic
7	C₂×(C₃⋊D₃)⋊C₃	C₂²	1	4	yes	complemented
8	((C₃⋊D₃)⋊C₃)⋊C₂	—	2	4	no	—
9	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
10	#1084	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₃²	—	3	4	no	—
12	C₄⋊(C₃⋊D₃)	D₃	1	2	yes	complemented
13	C₃×D₁₂	—	3	2	no	—
14	C₃×C₄×D₃	—	3	2	no	—
15	C₃×C₄×D₃	—	6	1	no	—
16	#626	—	3	4	no	—
17	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
18	#272	D₄	1	2	yes	complemented
19	(C₃⋊D₃)⋊C₃	—	2	4	no	—
20	D₃×D₄	—	9	2	no	—
21	C₂×D₁₂	—	9	1	no	—
22	C₂×C₃⋊D₃	C₂×D₃	1	4	yes	complemented
23	C₂×C₃×D₃	—	3	2	no	—
24	C₂×C₃×D₃	—	3	4	no	—
25	C₂×C₃×D₃	—	6	1	no	—
26	D₃²	—	6	8	no	—
27	C₃²×C₄	—	2	1	no	—
28	C₃²×C₄	C₂×D₃	1	2	yes	complemented
29	C₃×Dic₃	—	3	2	no	—
30	C₃×Dic₃	—	6	1	no	—
31	H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
32	C₂²×D₃	—	9	2	no	—
33	C₂²×D₃	—	9	4	no	—
34	C₃×D₄	—	9	2	no	—
35	C₂×C₃×C₄	—	9	1	no	—
36	#67	—	9	4	no	—
37	D₁₂	—	3	2	no	—
38	D₁₂	—	9	2	no	—
39	D₁₂	—	18	2	no	—
40	C₄×D₃	—	3	2	no	—
41	C₄×D₃	—	6	1	no	—
42	C₂×C₃²	—	2	1	no	—
43	C₂×C₃²	C₂²×D₃	1	2	yes	—
44	C₃⋊D₃	—	2	4	no	—
45	C₃×D₃	—	3	4	no	—
46	C₃×D₃	—	6	2	no	—
47	C₃×D₃	—	6	4	no	—
48	C₂×D₄	—	27	1	no	Sylow
49	C₂²×C₃	—	9	4	no	—
50	C₂²×C₃	—	9	1	no	—
51	C₂×D₃	—	3	2	no	—
52	C₂×D₃	—	3	4	no	—
53	C₂×D₃	—	6	1	no	—
54	C₂×D₃	—	18	4	no	—
55	C₂×D₃	—	18	8	no	—
56	C₂×D₃	—	18	4	no	—
57	C₃×C₄	—	6	2	no	—
58	C₃×C₄	—	9	1	no	—
59	C₃×C₄	—	12	1	no	—
60	C₃×C₄	D₃²	1	1	yes	characteristic
61	Dic₃	—	3	2	no	—
62	Dic₃	—	6	1	no	—
63	C₃²	—	2	1	no	—
64	C₃²	D₃×D₄	1	2	yes	complemented
65	C₂³	—	27	2	no	—
66	D₄	—	9	2	no	—
67	D₄	—	27	2	no	—
68	C₂×C₄	—	9	1	no	—
69	C₂×C₃	—	6	2	no	—
70	C₂×C₃	—	9	2	no	—
71	C₂×C₃	—	12	1	no	—
72	C₂×C₃	—	18	4	no	—
73	C₂×C₃	C₂×D₃²	1	1	yes	characteristic
74	D₃	—	3	4	no	—
75	D₃	—	6	2	no	—
76	D₃	—	6	4	no	—
77	D₃	—	36	4	no	—
78	C₂²	—	9	1	no	—
79	C₂²	—	9	4	no	—
80	C₂²	—	54	4	no	—
81	C₄	—	9	1	no	—
82	C₄	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic, complemented
83	C₃	—	6	2	no	—
84	C₃	—	12	1	no	—
85	C₃	#3802	1	1	yes	characteristic
86	C₂	—	9	2	no	—
87	C₂	—	18	4	no	—
88	C₂	C₂×((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³
Frattini, socle	C₂×C₃	C₂×D₃²
Fitting	C₄×H(3,𝔽₃)	C₂²
hypercenter	C₄	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₂×((C₃⋊D₃)⋊C₃)⋊C₂	C₂	semidirect
2. C₄×#272	C₂	semidirect
3. C₄⋊((C₃⋊D₃)⋊C₃)	C₂	semidirect
4. #6198	C₂	semidirect
7. C₂×(C₃⋊D₃)⋊C₃	C₂²	semidirect
9. C₄×H(3,𝔽₃)	C₂²	semidirect
10. #1084	C₂²	semidirect
12. C₄⋊(C₃⋊D₃)	D₃	semidirect
18. #272	D₄	semidirect
22. C₂×C₃⋊D₃	C₂×D₃	semidirect
28. C₃²×C₄	C₂×D₃	semidirect
31. H(3,𝔽₃)	C₂×D₄	semidirect
64. C₃²	D₃×D₄	semidirect
82. C₄	((C₃⋊D₃)⋊C₃)⋊C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×((C₃⋊D₃)⋊C₃)⋊C₂	2
C₄×#272	2
C₄⋊((C₃⋊D₃)⋊C₃)	2
#6198	2
D₃×D₁₂	3