Subgroups — C₄×GU(2,𝔽₃) #73066

Order 384.

173 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₂×GU(2,𝔽₃)	C₂	1	2	yes	maximal
3	C₄×#104	—	3	1	no	maximal, Sylow
4	C₂×#41⋊C₃	C₂²	1	1	yes	characteristic
5	C₄×SL(2,𝔽₃)	C₄	1	2	yes	complemented
6	GU(2,𝔽₃)	C₄	1	4	yes	direct product factor
7	C₄×C₃⋊C₈	—	4	1	no	maximal
8	C₄×#41	D₃	1	1	yes	characteristic
9	C₂×#104	—	3	2	no	—
10	C₄×M₁₆	—	3	1	no	—
11	C₄³	—	3	1	no	—
12	#339	—	3	2	no	—
13	#41⋊C₃	C₂×C₄	1	2	yes	—
14	C₂×SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄²	—	4	1	no	—
16	C₂×C₃⋊C₈	—	4	2	no	—
17	C₂×#41	C₂×D₃	1	1	yes	characteristic
18	C₂×M₁₆	—	3	2	no	—
19	C₄×Q₈	Dic₃	1	2	yes	—
20	C₄×D₄	—	3	2	no	—
21	#117	—	3	1	no	—
22	C₂×C₄²	—	3	1	no	—
23	C₂×C₄²	—	3	2	no	—
24	C₂×C₄²	—	6	2	no	—
25	#104	—	3	4	no	—
26	#104	—	3	4	no	—
27	#97	—	6	1	no	—
28	C₄×C₈	—	6	1	no	—
29	C₂×C₃×C₄	—	4	2	no	—
30	C₂×C₃×C₄	—	4	1	no	—
31	SL(2,𝔽₃)	C₄²	1	1	yes	characteristic, complemented
32	C₃⋊C₈	—	4	4	no	—
33	#41	—	3	2	no	—
34	#41	C₄×D₃	1	2	yes	—
35	C₂×Q₈	C₂×Dic₃	1	1	yes	characteristic
36	C₂×D₄	—	3	1	no	—
37	C₂²×C₄	—	3	1	no	—
38	C₂²×C₄	—	3	2	no	—
39	C₂²×C₄	—	6	2	no	—
40	M₁₆	—	3	4	no	—
41	M₁₆	—	6	2	no	—
42	C₂×C₈	—	6	2	no	—
43	C₄⋊C₄	—	3	2	no	—
44	C₂²⋊C₄	—	6	1	no	—
45	C₄²	—	3	2	no	—
46	C₄²	—	3	1	no	—
47	C₄²	—	3	4	no	—
48	C₄²	—	6	2	no	—
49	C₄²	—	6	8	no	—
50	C₄²	S₄	1	1	yes	characteristic
51	C₂²×C₃	—	4	1	no	—
52	C₃×C₄	—	4	4	no	—
53	C₃×C₄	—	4	2	no	—
54	C₂³	—	3	1	no	—
55	Q₈	—	3	1	no	—
56	Q₈	C₄×Dic₃	1	1	yes	characteristic
57	D₄	—	3	2	no	—
58	D₄	—	6	1	no	—
59	C₂×C₄	—	3	1	no	—
60	C₂×C₄	—	3	2	no	—
61	C₂×C₄	—	3	2	no	—
62	C₂×C₄	—	3	2	no	—
63	C₂×C₄	—	6	4	no	—
64	C₂×C₄	—	6	8	no	—
65	C₂×C₄	—	6	4	no	—
66	C₂×C₄	SL(2,ℤ/4)	1	2	yes	—
67	C₂×C₄	C₂×S₄	1	1	yes	characteristic
68	C₈	—	6	4	no	—
69	C₂×C₃	—	4	1	no	—
70	C₂×C₃	—	4	2	no	—
71	C₂²	—	3	2	no	—
72	C₂²	—	6	2	no	—
73	C₂²	C₂×SL(2,ℤ/4)	1	1	yes	characteristic
74	C₄	—	3	1	no	—
75	C₄	—	3	1	no	—
76	C₄	—	6	8	no	—
77	C₄	—	6	2	no	—
78	C₄	GU(2,𝔽₃)	1	4	yes	direct product factor
79	C₄	C₄×S₄	1	2	yes	—
80	C₃	—	4	1	no	Sylow
81	C₂	—	6	2	no	—
82	C₂	C₄×SL(2,ℤ/4)	1	1	yes	characteristic
83	C₂	C₂×GU(2,𝔽₃)	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₄²	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₄²
Frattini	C₂×C₄	C₂×S₄
Fitting	C₄×#41	D₃
socle	C₂²	C₂×SL(2,ℤ/4)
prime radical, cosocle	C₂×#41⋊C₃	C₂²
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
5. C₄×SL(2,𝔽₃)	C₄	semidirect
6. GU(2,𝔽₃)	C₄	direct (N × H)
31. SL(2,𝔽₃)	C₄²	semidirect
78. C₄	GU(2,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#41⋊C₃	2
C₂×GU(2,𝔽₃)	2
C₄×#104	3
C₄×C₃⋊C₈	4

Subgroups — C4×GU(2,𝔽3) #73066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×GU(2,𝔽₃) #73066