Subgroups — #4567

Order 192.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#135	C₂	1	1	yes	maximal, characteristic
2	#819	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#359	—	3	1	no	maximal, Sylow
5	C₃×#41	—	2	1	no	—
6	C₃×Q₁₆	C₄	1	1	yes	characteristic
7	C₃×SD₁₆	—	2	1	no	—
8	C₃×D₈	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	#208	—	2	1	no	—
11	C₃⋊C₁₆	—	2	1	no	—
12	#135	D₃	1	1	yes	characteristic
13	C₂×C₁₆	—	3	1	no	—
14	#108	—	3	1	no	—
15	C₃×Q₈	—	2	1	no	—
16	C₃×D₄	—	2	1	no	—
17	C₃×D₄	—	4	1	no	—
18	C₂×C₃×C₄	—	4	1	no	—
19	C₂×C₃×C₄	D₄	1	1	yes	characteristic
20	C₃×C₈	C₂×C₄	1	1	yes	characteristic
21	C₃×C₈	D₄	1	1	yes	characteristic
22	C₃⋊C₈	—	4	1	no	—
23	#41	—	2	1	no	—
24	Q₁₆	Dic₃	1	1	yes	characteristic
25	SD₁₆	—	2	1	no	—
26	D₈	Dic₃	1	1	yes	characteristic
27	M₁₆	—	6	1	no	—
28	C₂×C₈	C₂×D₃	1	1	yes	characteristic
29	C₁₆	—	6	1	no	—
30	C₂²×C₃	—	4	1	no	—
31	C₂²×C₃	SD₁₆	1	1	yes	characteristic
32	C₃×C₄	—	4	1	no	—
33	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₃×C₄	D₈	1	1	yes	characteristic
35	Q₈	—	2	1	no	—
36	D₄	—	2	1	no	—
37	D₄	—	4	1	no	—
38	C₂×C₄	—	4	1	no	—
39	C₂×C₄	#67	1	1	yes	characteristic
40	C₈	—	12	1	no	—
41	C₈	C₂×Dic₃	1	1	yes	characteristic
42	C₈	#67	1	1	yes	characteristic
43	C₂×C₃	—	2	1	no	—
44	C₂×C₃	—	8	1	no	—
45	C₂×C₃	#102	1	1	yes	characteristic
46	C₂²	—	4	1	no	—
47	C₂²	#214	1	1	yes	characteristic
48	C₄	—	4	1	no	—
49	C₄	#213	1	1	yes	characteristic
50	C₄	#217	1	1	yes	characteristic
51	C₃	#359	1	1	yes	characteristic, complemented, Sylow
52	C₂	—	2	1	no	—
53	C₂	—	8	1	no	—
54	C₂	#832	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#213
commutator	C₃×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₃
Fitting, prime radical	C₃×#135	C₂
socle	C₂×C₃	#102
cosocle	C₂×C₃×C₈	C₂²
hypercenter	#135	D₃
hypocenter	C₃	#359
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #819	C₂	semidirect
3. C₂×C₃⋊C₁₆	C₂	semidirect
51. C₃	#359	semidirect

Maximal subgroups

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

subgroup	index
C₃×#135	2
#819	2
C₂×C₃⋊C₁₆	2
#359	3