Subgroups — #5125

Order 192.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	2	yes	maximal
3	#819	C₂	1	4	yes	maximal, complemented
4	#430	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₃×M₁₆	C₄	1	4	yes	—
7	C₂×C₃×C₈	C₂²	1	2	yes	—
8	#208	—	2	2	no	—
9	#208	C₂²	1	4	yes	—
10	C₂×C₃⋊C₈	—	2	2	no	—
11	C₂×M₁₆	—	3	2	no	—
12	C₂×M₁₆	D₃	1	1	yes	characteristic
13	#108	—	3	4	no	—
14	C₂³×C₃	Q₈	1	1	yes	characteristic
15	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
16	C₂×C₃×C₄	D₄	1	2	yes	—
17	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
18	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
19	C₃×C₈	C₂×C₄	1	4	yes	—
20	C₃⋊C₈	—	2	4	no	—
21	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
22	M₁₆	—	3	4	no	—
23	M₁₆	—	6	2	no	—
24	M₁₆	Dic₃	1	4	yes	—
25	C₂×C₈	—	6	2	no	—
26	C₂×C₈	C₂×D₃	1	2	yes	—
27	C₂²×C₃	—	4	1	no	—
28	C₂²×C₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
30	C₃×C₄	C₄⋊C₄	1	2	yes	—
31	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
32	C₃×C₄	C₂×D₄	1	1	yes	characteristic
33	C₂³	Dic₆	1	1	yes	characteristic
34	C₂×C₄	Dic₆	1	1	yes	characteristic
35	C₂×C₄	D₁₂	1	2	yes	—
36	C₂×C₄	C₂×Dic₃	1	2	yes	—
37	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
38	C₈	—	6	4	no	—
39	C₈	C₂×Dic₃	1	4	yes	—
40	C₂×C₃	—	2	2	no	—
41	C₂×C₃	—	2	1	no	—
42	C₂×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²	—	4	1	no	—
44	C₂²	C₄⋊Dic₃	1	2	yes	—
45	C₂²	C₂×Dic₆	1	1	yes	characteristic
46	C₄	C₄⋊Dic₃	1	2	yes	—
47	C₄	C₂×D₁₂	1	1	yes	characteristic
48	C₄	C₂²×Dic₃	1	1	yes	characteristic
49	C₃	#430	1	1	yes	characteristic, complemented, Sylow
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	C₂×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂×C₃×M₁₆	C₂
socle	C₂×C₃	C₂×C₄⋊C₄
prime radical	C₂³×C₃	Q₈
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×M₁₆	D₃
hypocenter	C₃	#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #819	C₂	semidirect
49. C₃	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×M₁₆	2
C₂×#208	2
#819	2
#430	3