Subgroups — #88935

Order 432.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6295	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#625	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃⋊#622	C₂	1	1	yes	maximal, characteristic, complemented
4	#3774	—	3	1	no	maximal
5	C₂×C₃×C₃⋊D₃	—	2	1	no	—
6	C₂×C₃²×D₃	—	2	1	no	—
7	C₃×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
8	C₃×#67	—	6	1	no	—
9	#626	—	6	1	no	—
10	#625	—	3	1	no	—
11	#625	D₃	1	1	yes	characteristic, complemented
12	#622	—	3	1	no	—
13	C₂×C₃³	D₄	1	1	yes	characteristic
14	C₃×C₃⋊D₃	—	4	1	no	—
15	C₃²×D₃	—	4	1	no	—
16	#213	—	9	1	no	maximal
17	C₂²×C₃²	—	6	1	no	—
18	C₂×C₃⋊D₃	—	2	1	no	—
19	C₂×C₃×D₃	—	2	1	no	—
20	C₂×C₃×D₃	—	2	1	no	—
21	C₂×C₃×D₃	—	4	1	no	—
22	C₂×C₃×D₃	—	6	1	no	—
23	C₂×C₃×D₃	—	6	1	no	—
24	C₂×C₃×D₃	—	12	1	no	—
25	C₃⋊Dic₃	C₂×D₃	1	1	yes	characteristic
26	C₃×Dic₃	—	6	1	no	—
27	C₃×Dic₃	—	6	1	no	—
28	C₃³	D₈	1	1	yes	characteristic, complemented, Sylow
29	C₃×D₄	—	9	1	no	—
30	#67	—	6	1	no	—
31	#67	—	18	1	no	—
32	D₁₂	—	9	1	no	—
33	C₃⋊C₈	—	9	1	no	—
34	C₂×C₃²	—	2	1	no	—
35	C₂×C₃²	—	2	1	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₂×C₃²	—	4	1	no	—
38	C₂×C₃²	—	12	1	no	—
39	C₂×C₃²	#67	1	1	yes	characteristic
40	C₃⋊D₃	—	4	1	no	—
41	C₃×D₃	—	4	1	no	—
42	C₃×D₃	—	4	2	no	—
43	C₃×D₃	—	4	1	no	—
44	C₃×D₃	—	12	1	no	—
45	C₃×D₃	—	12	2	no	—
46	C₃×D₃	—	12	1	no	—
47	D₈	—	27	1	no	Sylow
48	C₂²×C₃	—	6	1	no	—
49	C₂²×C₃	—	6	1	no	—
50	C₂²×C₃	—	12	1	no	—
51	C₂²×C₃	—	18	1	no	—
52	C₂×D₃	—	2	1	no	—
53	C₂×D₃	—	6	1	no	—
54	C₂×D₃	—	6	1	no	—
55	C₂×D₃	—	12	1	no	—
56	C₃×C₄	—	9	1	no	—
57	Dic₃	—	6	1	no	—
58	Dic₃	—	6	1	no	—
59	C₃²	—	2	1	no	—
60	C₃²	—	2	1	no	—
61	C₃²	—	4	1	no	—
62	C₃²	—	4	1	no	—
63	C₃²	#213	1	1	yes	characteristic, complemented
64	D₄	—	9	1	no	—
65	D₄	—	27	1	no	—
66	C₈	—	27	1	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	4	1	no	—
70	C₂×C₃	—	4	1	no	—
71	C₂×C₃	—	12	1	no	—
72	C₂×C₃	—	12	2	no	—
73	C₂×C₃	—	12	1	no	—
74	C₂×C₃	—	36	1	no	—
75	C₂×C₃	GO⁺(4,𝔽₂)	1	1	yes	characteristic
76	D₃	—	4	1	no	—
77	D₃	—	12	1	no	—
78	D₃	—	12	2	no	—
79	D₃	—	12	1	no	—
80	C₂²	—	6	1	no	—
81	C₂²	—	18	1	no	—
82	C₄	—	9	1	no	—
83	C₃	—	2	1	no	—
84	C₃	—	2	1	no	—
85	C₃	—	4	1	no	—
86	C₃	—	4	1	no	—
87	C₃	#3774	1	1	yes	characteristic, complemented
88	C₂	—	12	1	no	—
89	C₂	—	36	1	no	—
90	C₂	D₃²⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²⋊D₃
commutator, cosocle	C₃×C₃⋊Dic₃	C₂²
Fitting, socle	C₂×C₃³	D₄
hypocenter	C₃³	D₈
LCS	G ▷ C₃×C₃⋊Dic₃ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. #6295	C₂	semidirect
2. C₃×#625	C₂	semidirect
3. C₃⋊#622	C₂	semidirect
11. #625	D₃	semidirect
28. C₃³	D₈	semidirect
63. C₃²	#213	semidirect
87. C₃	#3774	semidirect

Maximal subgroups

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

subgroup	index
#6295	2
C₃×#625	2
C₃⋊#622	2
#3774	3
#213	9