Subgroups — #5419

Order 192.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#226	C₂	1	1	yes	maximal, characteristic, complemented
3	#859	C₂	1	1	yes	maximal, characteristic, complemented
4	#451	—	3	1	no	maximal, Sylow
5	C₂×SL(2,𝔽₃)	—	2	1	no	—
6	C₂×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	#226	—	2	1	no	—
8	#217	—	4	1	no	maximal
9	C₂²×Q₈	D₃	1	1	yes	characteristic
10	C₂×Q₁₆	—	3	1	no	—
11	C₂×Q₁₆	—	3	1	no	—
12	C₂²⋊Q₈	—	3	1	no	—
13	#103	—	3	1	no	—
14	#103	—	3	1	no	—
15	C₂²⋊C₈	—	3	1	no	—
16	C₂³×C₃	—	4	1	no	—
17	C₂×Dic₃	—	4	1	no	—
18	C₂×Dic₃	—	4	1	no	—
19	SL(2,𝔽₃)	D₄	1	1	yes	characteristic
20	C₂×Q₈	—	2	1	no	—
21	C₂×Q₈	—	3	1	no	—
22	C₂×Q₈	—	3	1	no	—
23	C₂×Q₈	—	6	1	no	—
24	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
25	C₂²×C₄	—	3	1	no	—
26	Q₁₆	—	6	1	no	—
27	Q₁₆	—	6	2	no	—
28	Q₁₆	—	6	1	no	—
29	C₂×C₈	—	3	1	no	—
30	C₂×C₈	—	3	1	no	—
31	C₄⋊C₄	—	3	1	no	—
32	C₄⋊C₄	—	6	1	no	—
33	C₂²⋊C₄	—	6	1	no	—
34	C₂²×C₃	—	4	2	no	—
35	C₂²×C₃	—	4	1	no	—
36	C₂²×C₃	—	8	1	no	—
37	C₂²×C₃	—	8	1	no	—
38	Dic₃	—	8	1	no	—
39	Dic₃	—	8	1	no	—
40	C₂³	S₄	1	1	yes	characteristic
41	Q₈	—	3	1	no	—
42	Q₈	—	3	2	no	—
43	Q₈	—	6	1	no	—
44	Q₈	—	6	1	no	—
45	Q₈	—	6	1	no	—
46	Q₈	#67	1	1	yes	characteristic
47	C₂×C₄	—	3	1	no	—
48	C₂×C₄	—	3	1	no	—
49	C₂×C₄	—	6	1	no	—
50	C₂×C₄	—	6	1	no	—
51	C₂×C₄	—	6	1	no	—
52	C₂×C₄	—	6	1	no	—
53	C₈	—	6	1	no	—
54	C₈	—	6	1	no	—
55	C₂×C₃	—	4	1	no	—
56	C₂×C₃	—	4	1	no	—
57	C₂×C₃	—	4	1	no	—
58	C₂×C₃	—	8	2	no	—
59	C₂²	—	2	1	no	—
60	C₂²	—	2	1	no	—
61	C₂²	#226	1	2	yes	complemented
62	C₂²	C₂×S₄	1	1	yes	characteristic
63	C₄	—	3	1	no	—
64	C₄	—	3	1	no	—
65	C₄	—	6	1	no	—
66	C₄	—	12	1	no	—
67	C₄	—	12	1	no	—
68	C₃	—	4	1	no	Sylow
69	C₂	—	2	2	no	—
70	C₂	C₂×#226	1	1	yes	characteristic
71	C₂	#983	1	1	yes	characteristic
72	C₂	GL(2,ℤ/4)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂²	C₂×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂²
Fitting	C₂²×Q₈	D₃
prime radical	C₂²×SL(2,𝔽₃)	C₂
hypercenter	C₂³	S₄
hypocenter	SL(2,𝔽₃)	D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×#226	C₂	semidirect
3. #859	C₂	semidirect
61. C₂²	#226	semidirect

Maximal subgroups

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

subgroup	index
C₂²×SL(2,𝔽₃)	2
C₂×#226	2
#859	2
#451	3
#217	4