Subgroups — #5423

Order 192.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	GU(2,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	#421	—	3	1	no	maximal, Sylow
4	#41⋊C₃	C₄	1	1	yes	characteristic, complemented
5	#41⋊C₃	C₂²	1	1	yes	characteristic
6	C₂×SL(2,𝔽₃)	C₄	1	1	yes	characteristic, complemented
7	#208	—	4	1	no	maximal
8	C₂×#41	D₃	1	1	yes	characteristic
9	C₂×M₁₆	—	3	1	no	—
10	#117	—	3	1	no	—
11	#104	—	3	2	no	—
12	#104	—	3	2	no	—
13	C₂×C₃×C₄	—	4	1	no	—
14	SL(2,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₃⋊C₈	—	4	2	no	—
16	#41	—	3	1	no	—
17	#41	—	3	1	no	—
18	#41	Dic₃	1	1	yes	characteristic
19	#41	C₂×D₃	1	1	yes	characteristic
20	C₂×Q₈	Dic₃	1	1	yes	characteristic
21	C₂×D₄	—	3	1	no	—
22	C₂²×C₄	—	3	1	no	—
23	M₁₆	—	3	2	no	—
24	M₁₆	—	6	1	no	—
25	C₂×C₈	—	6	1	no	—
26	C₄⋊C₄	—	6	1	no	—
27	C₂²⋊C₄	—	6	1	no	—
28	C₄²	—	3	2	no	—
29	C₂²×C₃	—	4	1	no	—
30	C₃×C₄	—	4	1	no	—
31	C₃×C₄	—	4	1	no	—
32	C₂³	—	3	1	no	—
33	Q₈	—	3	1	no	—
34	Q₈	C₂×Dic₃	1	1	yes	characteristic
35	D₄	—	3	1	no	—
36	D₄	—	3	1	no	—
37	D₄	—	6	1	no	—
38	C₂×C₄	—	3	1	no	—
39	C₂×C₄	—	3	1	no	—
40	C₂×C₄	—	3	1	no	—
41	C₂×C₄	—	3	1	no	—
42	C₂×C₄	—	3	1	no	—
43	C₂×C₄	—	6	2	no	—
44	C₂×C₄	S₄	1	1	yes	characteristic
45	C₈	—	6	2	no	—
46	C₂×C₃	—	4	1	no	—
47	C₂×C₃	—	8	1	no	—
48	C₂²	—	3	1	no	—
49	C₂²	—	3	1	no	—
50	C₂²	—	12	1	no	—
51	C₂²	SL(2,ℤ/4)	1	1	yes	characteristic
52	C₄	—	3	1	no	—
53	C₄	—	3	1	no	—
54	C₄	—	12	2	no	—
55	C₄	SL(2,ℤ/4)	1	1	yes	characteristic
56	C₄	C₂×S₄	1	1	yes	characteristic
57	C₃	—	4	1	no	Sylow
58	C₂	—	2	1	no	—
59	C₂	—	6	1	no	—
60	C₂	—	6	1	no	—
61	C₂	C₂×SL(2,ℤ/4)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. GU(2,𝔽₃)	C₂	semidirect
4. #41⋊C₃	C₄	semidirect
6. C₂×SL(2,𝔽₃)	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41⋊C₃	2
GU(2,𝔽₃)	2
#421	3
#208	4