Subgroups — #4566

Order 192.

58 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	#817	C₂	1	1	yes	maximal, characteristic, complemented
3	#812	C₂	1	1	yes	maximal, characteristic, complemented
4	#360	—	3	1	no	maximal, Sylow
5	C₃×#41	—	2	1	no	—
6	C₃×Q₁₆	C₄	1	1	yes	characteristic, complemented
7	C₃×SD₁₆	—	2	1	no	—
8	C₃×D₈	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃	—	2	1	no	—
11	C₃⋊C₁₆	—	2	1	no	—
12	#135	D₃	1	1	yes	characteristic
13	M₃₂	—	3	1	no	—
14	#106	—	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	C₂×Dic₃	—	4	1	no	—
21	C₃×C₈	C₂×C₄	1	1	yes	characteristic
22	C₃×C₈	D₄	1	1	yes	characteristic
23	#41	—	2	1	no	—
24	Q₁₆	Dic₃	1	1	yes	characteristic, complemented
25	SD₁₆	—	2	1	no	—
26	D₈	Dic₃	1	1	yes	characteristic, complemented
27	C₂×C₈	C₂×D₃	1	1	yes	characteristic
28	C₄⋊C₄	—	6	1	no	—
29	C₁₆	—	6	1	no	—
30	C₂²×C₃	—	4	1	no	—
31	C₂²×C₃	D₈	1	1	yes	characteristic
32	C₃×C₄	—	4	1	no	—
33	C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
34	C₃×C₄	SD₁₆	1	1	yes	characteristic
35	Dic₃	—	8	1	no	—
36	Q₈	—	2	1	no	—
37	D₄	—	2	1	no	—
38	D₄	—	4	1	no	—
39	C₂×C₄	—	4	1	no	—
40	C₂×C₄	—	12	1	no	—
41	C₂×C₄	#67	1	1	yes	characteristic
42	C₈	C₂×Dic₃	1	1	yes	characteristic
43	C₈	#67	1	1	yes	characteristic
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	—	8	1	no	—
46	C₂×C₃	#102	1	1	yes	characteristic
47	C₂²	—	4	1	no	—
48	C₂²	#213	1	1	yes	characteristic
49	C₄	—	4	1	no	—
50	C₄	—	24	1	no	—
51	C₄	#214	1	1	yes	characteristic
52	C₄	#217	1	1	yes	characteristic
53	C₃	#360	1	1	yes	characteristic, complemented, Sylow
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	#832	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#832
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	#360
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
2. #817	C₂	semidirect
3. #812	C₂	semidirect
6. C₃×Q₁₆	C₄	semidirect
8. C₃×D₈	C₄	semidirect
24. Q₁₆	Dic₃	semidirect
26. D₈	Dic₃	semidirect
53. C₃	#360	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
#817	2
#812	2
#360	3