Subgroups — C₃×#456 #5330

Order 192.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₃×#137	C₂	1	2	yes	maximal, complemented
3	C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#104	C₂	1	2	yes	maximal, complemented
5	C₃×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	#456	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#41	—	2	2	no	—
8	C₃×#41	—	2	2	no	—
9	C₃×#41	C₂²	1	2	yes	—
10	C₂×C₃×Q₈	—	2	2	no	—
11	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
13	C₃×Q₁₆	—	2	2	no	—
14	C₃×SD₁₆	—	2	2	no	—
15	C₃×M₁₆	C₂²	1	2	yes	—
16	C₃×C₄⋊C₄	—	2	2	no	—
17	C₃×C₄²	C₂²	1	1	yes	characteristic
18	2₋¹⁺⁴	C₂×C₃	1	1	yes	characteristic
19	#137	C₂×C₃	1	2	yes	complemented
20	C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
21	#104	C₂×C₃	1	2	yes	complemented
22	#101	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×Q₈	—	2	2	no	—
24	C₃×Q₈	—	2	2	no	—
25	C₃×Q₈	—	2	2	no	—
26	C₃×Q₈	D₄	1	2	yes	—
27	C₃×D₄	—	2	2	no	—
28	C₃×D₄	—	2	2	no	—
29	C₃×D₄	D₄	1	2	yes	—
30	C₂×C₃×C₄	—	2	4	no	—
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	—	2	2	no	—
33	C₂×C₃×C₄	—	2	1	no	—
34	C₂×C₃×C₄	D₄	1	2	yes	—
35	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₃×C₈	—	2	2	no	—
37	#41	—	2	2	no	—
38	#41	—	2	2	no	—
39	#41	C₂²×C₃	1	2	yes	—
40	C₂×Q₈	—	2	2	no	—
41	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×Q₈	C₂²×C₃	1	1	yes	characteristic
43	Q₁₆	—	2	2	no	—
44	SD₁₆	—	2	2	no	—
45	M₁₆	C₂²×C₃	1	2	yes	—
46	C₄⋊C₄	—	2	2	no	—
47	C₄²	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃	—	2	2	no	—
49	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
50	C₃×C₄	—	2	2	no	—
51	C₃×C₄	—	2	2	no	—
52	C₃×C₄	—	2	2	no	—
53	C₃×C₄	—	4	1	no	—
54	C₃×C₄	C₂×D₄	1	2	yes	—
55	Q₈	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	Q₈	—	2	2	no	—
58	Q₈	C₃×D₄	1	2	yes	—
59	D₄	—	2	2	no	—
60	D₄	—	2	2	no	—
61	D₄	C₃×D₄	1	2	yes	—
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	C₃×D₄	1	2	yes	—
67	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂×C₃	—	2	1	no	—
70	C₂×C₃	—	4	2	no	—
71	C₂×C₃	#120	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₃×D₄	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	C₂×C₃×D₄	1	2	yes	—
79	C₃	#456	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	C₃×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₃
prime radical	C₃×2₋¹⁺⁴	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#137	C₂	semidirect
3. C₃×C₄⋊Q₈	C₂	semidirect
4. C₃×#104	C₂	semidirect
5. C₃×#101	C₂	semidirect
6. #456	C₃	direct (N × H)
19. #137	C₂×C₃	semidirect
20. C₄⋊Q₈	C₂×C₃	semidirect
21. #104	C₂×C₃	semidirect
22. #101	C₂×C₃	semidirect
79. C₃	#456	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×2₋¹⁺⁴	2
C₃×#137	2
C₃×C₄⋊Q₈	2
C₃×#104	2
C₃×#101	2
#456	3

Subgroups — C3×#456 #5330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#456 #5330