Subgroups — #5195

Order 192.

82 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	C₂×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#829	C₂	1	2	yes	maximal, complemented
4	#827	C₂	1	2	yes	maximal, complemented
5	#812	C₂	1	1	yes	maximal, characteristic, complemented
6	#510	—	3	1	no	maximal, Sylow
7	C₃×#41	—	2	1	no	—
8	C₂×Dic₆	—	2	1	no	—
9	C₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₃×SD₁₆	—	2	1	no	—
11	C₃×D₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
13	Dic₁₂	—	2	1	no	—
14	Dic₁₂	C₂²	1	2	yes	—
15	C₃⋊C₁₆	C₂²	1	2	yes	—
16	#135	D₃	1	1	yes	characteristic
17	C₂×Q₁₆	—	3	1	no	—
18	Q₃₂	—	3	2	no	—
19	SD₃₂	—	3	2	no	—
20	M₃₂	—	3	1	no	—
21	C₃×Q₈	—	2	1	no	—
22	C₃×D₄	—	2	1	no	—
23	C₃×D₄	—	4	1	no	—
24	C₂×C₃×C₄	—	4	1	no	—
25	C₂×C₃×C₄	D₄	1	1	yes	characteristic
26	C₂×Dic₃	—	4	1	no	—
27	Dic₆	—	2	2	no	—
28	Dic₆	—	4	1	no	—
29	C₃×C₈	D₄	1	1	yes	characteristic
30	C₃×C₈	C₂³	1	1	yes	characteristic
31	#41	—	2	1	no	—
32	C₂×Q₈	—	6	1	no	—
33	Q₁₆	—	3	2	no	—
34	Q₁₆	—	6	1	no	—
35	Q₁₆	C₂×D₃	1	1	yes	characteristic
36	SD₁₆	—	2	1	no	—
37	D₈	C₂×D₃	1	1	yes	characteristic
38	C₂×C₈	C₂×D₃	1	1	yes	characteristic
39	C₁₆	—	3	2	no	—
40	C₂²×C₃	—	4	1	no	—
41	C₂²×C₃	D₈	1	1	yes	characteristic
42	C₃×C₄	—	4	1	no	—
43	C₃×C₄	D₈	1	1	yes	characteristic
44	C₃×C₄	C₂×D₄	1	1	yes	characteristic
45	Dic₃	—	4	2	no	—
46	Q₈	—	2	1	no	—
47	Q₈	—	6	2	no	—
48	Q₈	—	12	1	no	—
49	D₄	—	2	1	no	—
50	D₄	—	4	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	—	12	1	no	—
53	C₂×C₄	#67	1	1	yes	characteristic
54	C₈	#67	1	1	yes	characteristic
55	C₈	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	8	1	no	—
58	C₂×C₃	C₂×D₈	1	1	yes	characteristic
59	C₂²	—	4	1	no	—
60	C₂²	#213	1	1	yes	characteristic
61	C₄	—	4	1	no	—
62	C₄	—	12	2	no	—
63	C₄	#213	1	1	yes	characteristic
64	C₄	C₂×#67	1	1	yes	characteristic
65	C₃	#510	1	1	yes	characteristic, complemented, Sylow
66	C₂	—	2	1	no	—
67	C₂	—	8	1	no	—
68	C₂	C₂×#213	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#213
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting, prime radical	C₃×#135	C₂
socle	C₂×C₃	C₂×D₈
hypercenter	#135	D₃
hypocenter	C₃	#510
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. C₂×Dic₁₂	C₂	semidirect
3. #829	C₂	semidirect
4. #827	C₂	semidirect
5. #812	C₂	semidirect
65. C₃	#510	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
C₂×Dic₁₂	2
#829	2
#827	2
#812	2
#510	3