Subgroups — #88448

Order 432.

107 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	1	yes	maximal, characteristic
3	C₂×(C₃⋊Dic₃)⋊C₃	C₂	1	1	yes	maximal, characteristic
4	#3724	—	3	1	no	maximal
5	Dic₃⋊Dic₃	—	3	1	no	maximal
6	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	#1084	—	2	1	no	—
8	(C₃⋊Dic₃)⋊C₃	—	2	1	no	—
9	(C₃⋊Dic₃)⋊C₃	C₄	1	2	yes	complemented
10	C₂×C₃⋊Dic₃	D₃	1	1	yes	characteristic
11	C₂×C₃⋊Dic₃	D₃	1	1	yes	characteristic
12	C₂×C₃×Dic₃	—	3	1	no	—
13	C₂×C₃×Dic₃	—	3	1	no	—
14	C₂×C₃×Dic₃	—	3	1	no	—
15	C₂×C₃×Dic₃	—	3	1	no	—
16	C₂×C₃×Dic₃	—	6	1	no	—
17	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
18	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
19	C₂×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
20	C₄⋊Dic₃	—	9	1	no	—
21	Dic₃⋊C₄	—	9	1	no	—
22	Dic₃⋊C₄	—	9	1	no	—
23	C₂²×C₃²	—	2	1	no	—
24	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
25	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
26	C₃⋊Dic₃	—	2	1	no	—
27	C₃⋊Dic₃	Dic₃	1	2	yes	complemented
28	C₃×Dic₃	—	3	2	no	—
29	C₃×Dic₃	—	6	2	no	—
30	C₃×Dic₃	—	6	1	no	—
31	C₃×Dic₃	—	6	1	no	—
32	C₃×Dic₃	—	6	1	no	—
33	H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
34	C₂×C₃×C₄	—	9	1	no	—
35	C₂×C₃×C₄	—	9	1	no	—
36	C₂×C₃×C₄	—	9	1	no	—
37	C₂×Dic₃	—	3	1	no	—
38	C₂×Dic₃	—	3	1	no	—
39	C₂×Dic₃	—	3	1	no	—
40	C₂×Dic₃	—	3	1	no	—
41	C₂×Dic₃	—	6	1	no	—
42	C₂×Dic₃	—	9	1	no	—
43	C₂×Dic₃	—	9	1	no	—
44	C₂×C₃²	—	2	1	no	—
45	C₂×C₃²	—	2	1	no	—
46	C₂×C₃²	—	2	1	no	—
47	C₂×C₃²	Dic₆	1	1	yes	characteristic
48	C₂×C₃²	Dic₆	1	1	yes	characteristic
49	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
50	C₂×C₃²	D₁₂	1	1	yes	characteristic
51	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
52	C₂×C₃²	#67	1	1	yes	characteristic
53	C₄⋊C₄	—	27	1	no	Sylow
54	C₂²×C₃	—	3	1	no	—
55	C₂²×C₃	—	3	1	no	—
56	C₂²×C₃	—	6	1	no	—
57	C₂²×C₃	D₃²	1	1	yes	characteristic
58	C₃×C₄	—	9	2	no	—
59	C₃×C₄	—	18	1	no	—
60	C₃×C₄	—	18	1	no	—
61	Dic₃	—	3	2	no	—
62	Dic₃	—	6	1	no	—
63	Dic₃	—	6	1	no	—
64	Dic₃	—	6	2	no	—
65	Dic₃	—	6	1	no	—
66	Dic₃	—	9	2	no	—
67	Dic₃	—	18	1	no	—
68	C₃²	—	2	1	no	—
69	C₃²	Dic₃⋊C₄	1	1	yes	characteristic, complemented
70	C₃²	C₄⋊Dic₃	1	1	yes	characteristic, complemented
71	C₂×C₄	—	9	1	no	—
72	C₂×C₄	—	9	1	no	—
73	C₂×C₄	—	9	1	no	—
74	C₂×C₃	—	3	1	no	—
75	C₂×C₃	—	3	1	no	—
76	C₂×C₃	—	3	1	no	—
77	C₂×C₃	—	3	1	no	—
78	C₂×C₃	—	3	1	no	—
79	C₂×C₃	—	3	1	no	—
80	C₂×C₃	—	6	1	no	—
81	C₂×C₃	—	6	1	no	—
82	C₂×C₃	—	6	1	no	—
83	C₂×C₃	D₃×Dic₃	1	1	yes	characteristic
84	C₂×C₃	#626	1	1	yes	characteristic
85	C₂×C₃	#627	1	1	yes	characteristic
86	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
87	C₄	—	9	2	no	—
88	C₄	—	18	1	no	—
89	C₄	—	18	1	no	—
90	C₃	—	3	1	no	—
91	C₃	—	3	1	no	—
92	C₃	—	6	1	no	—
93	C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
94	C₂	#6196	1	1	yes	characteristic
95	C₂	((C₃⋊D₃)⋊C₃)⋊C₄	1	1	yes	characteristic
96	C₂	#6200	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²	((C₃⋊D₃)⋊C₃)⋊C₂
commutator	C₂×H(3,𝔽₃)	C₂×C₄
Frattini, socle	C₂²×C₃	D₃²
Fitting, prime radical, cosocle	C₂²×H(3,𝔽₃)	C₂²
hypocenter	H(3,𝔽₃)	C₄⋊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
9. (C₃⋊Dic₃)⋊C₃	C₄	semidirect
27. C₃⋊Dic₃	Dic₃	semidirect
33. H(3,𝔽₃)	C₄⋊C₄	semidirect
69. C₃²	Dic₃⋊C₄	semidirect
70. C₃²	C₄⋊Dic₃	semidirect

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
#3724	3
Dic₃⋊Dic₃	3