Subgroups — #5386

Order 192.

65 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#859	C₂	1	2	yes	maximal
3	#475	—	3	1	no	maximal, Sylow
4	C₂×Dic₆	—	4	1	no	maximal
5	C₂×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄⋊Q₈	—	3	1	no	—
7	C₄×Q₈	D₃	1	1	yes	characteristic
8	#106	—	3	2	no	—
9	C₄⋊C₈	—	3	1	no	—
10	#103	—	3	2	no	—
11	C₂×C₃×C₄	—	4	1	no	—
12	C₂×Dic₃	—	4	2	no	—
13	Dic₆	—	4	4	no	—
14	SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, complemented
15	C₂×Q₈	—	6	1	no	—
16	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
17	C₂×C₈	—	3	2	no	—
18	C₄⋊C₄	—	3	1	no	—
19	C₄⋊C₄	—	3	2	no	—
20	C₄⋊C₄	—	6	1	no	—
21	C₄²	—	3	1	no	—
22	C₂²×C₃	—	4	1	no	—
23	C₃×C₄	—	4	2	no	—
24	Dic₃	—	4	4	no	—
25	Q₈	—	3	1	no	—
26	Q₈	—	12	2	no	—
27	Q₈	Dic₆	1	1	yes	characteristic, complemented
28	C₂×C₄	—	3	1	no	—
29	C₂×C₄	—	3	1	no	—
30	C₂×C₄	—	6	2	no	—
31	C₂×C₄	S₄	1	1	yes	characteristic
32	C₈	—	6	2	no	—
33	C₂×C₃	—	4	1	no	—
34	C₂×C₃	—	4	1	no	—
35	C₂×C₃	—	4	1	no	—
36	C₂²	C₂×S₄	1	1	yes	characteristic
37	C₄	—	3	1	no	—
38	C₄	—	3	1	no	—
39	C₄	—	6	1	no	—
40	C₄	—	12	2	no	—
41	C₄	GL(2,𝔽₃)	1	2	yes	—
42	C₃	—	4	1	no	Sylow
43	C₂	A₄⋊Q₈	1	1	yes	characteristic
44	C₂	C₂×GL(2,𝔽₃)	1	1	yes	characteristic
45	C₂	#984	1	1	yes	characteristic

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

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

N	H ≅ G/N	type
14. SL(2,𝔽₃)	Q₈	semidirect
27. Q₈	Dic₆	semidirect

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