Subgroups — #5395

Order 192.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#464	—	3	1	no	maximal, Sylow
5	C₂×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	GL(2,𝔽₃)	—	2	1	no	—
7	#226	—	2	1	no	—
8	(C₂×D₃)⋊C₄	—	4	1	no	maximal
9	C₂×Q₁₆	—	3	1	no	—
10	C₂×SD₁₆	—	3	1	no	—
11	#124	—	3	1	no	—
12	C₄×Q₈	D₃	1	1	yes	characteristic, complemented
13	C₄⋊C₈	—	3	1	no	—
14	#103	—	3	1	no	—
15	#102	—	3	1	no	—
16	C₂²×D₃	—	4	1	no	—
17	C₂×C₃×C₄	—	4	1	no	—
18	C₂×Dic₃	—	4	1	no	—
19	SL(2,𝔽₃)	D₄	1	1	yes	characteristic
20	C₂×Q₈	—	3	1	no	—
21	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
22	C₂×D₄	—	3	1	no	—
23	Q₁₆	—	6	1	no	—
24	Q₁₆	—	6	1	no	—
25	SD₁₆	—	6	1	no	—
26	SD₁₆	—	6	1	no	—
27	C₂×C₈	—	3	1	no	—
28	C₂×C₈	—	3	1	no	—
29	C₄⋊C₄	—	3	1	no	—
30	C₂²⋊C₄	—	6	1	no	—
31	C₂²⋊C₄	—	6	1	no	—
32	C₄²	—	3	1	no	—
33	C₂²×C₃	—	4	1	no	—
34	C₂×D₃	—	4	2	no	—
35	C₂×D₃	—	8	1	no	—
36	C₂×D₃	—	8	1	no	—
37	C₃×C₄	—	8	1	no	—
38	Dic₃	—	8	1	no	—
39	C₂³	—	6	1	no	—
40	Q₈	—	3	1	no	—
41	Q₈	—	6	1	no	—
42	Q₈	—	6	1	no	—
43	Q₈	D₁₂	1	1	yes	characteristic
44	D₄	—	6	1	no	—
45	D₄	—	6	1	no	—
46	C₂×C₄	—	3	1	no	—
47	C₂×C₄	—	3	1	no	—
48	C₂×C₄	—	6	1	no	—
49	C₂×C₄	S₄	1	1	yes	characteristic
50	C₈	—	6	1	no	—
51	C₈	—	6	1	no	—
52	C₂×C₃	—	4	1	no	—
53	C₂×C₃	—	4	1	no	—
54	C₂×C₃	—	4	1	no	—
55	D₃	—	8	2	no	—
56	C₂²	—	12	1	no	—
57	C₂²	—	12	1	no	—
58	C₂²	—	12	1	no	—
59	C₂²	C₂×S₄	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	3	1	no	—
62	C₄	—	3	1	no	—
63	C₄	—	6	1	no	—
64	C₄	—	12	1	no	—
65	C₃	—	4	1	no	Sylow
66	C₂	—	24	1	no	—
67	C₂	C₄⋊S₄	1	1	yes	characteristic
68	C₂	#983	1	1	yes	characteristic
69	C₂	#985	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂²	C₂×S₄
commutator, cosocle	C₂×SL(2,𝔽₃)	C₂²
Fitting	C₄×Q₈	D₃
prime radical	C₂×GL(2,𝔽₃)	C₂
hypercenter	C₂×C₄	S₄
hypocenter	SL(2,𝔽₃)	D₄
LCS	G ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#226	C₂	semidirect
3. C₄×SL(2,𝔽₃)	C₂	semidirect
12. C₄×Q₈	D₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×GL(2,𝔽₃)	2
C₂×#226	2
C₄×SL(2,𝔽₃)	2
#464	3
(C₂×D₃)⋊C₄	4