Subgroups — C₃×#562 #5892

Order 192.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#124	C₂	1	2	yes	maximal, complemented
3	C₃×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
5	C₃×#121	C₂	1	4	yes	maximal, complemented
6	#562	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×D₄	C₂²	1	4	yes	—
9	C₂×C₃×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄	C₂²	1	4	yes	—
11	C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₄⋊C₄	C₂²	1	2	yes	—
14	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₃×C₄²	C₂²	1	1	yes	characteristic
18	#125	C₂×C₃	1	1	yes	characteristic, complemented
19	#124	C₂×C₃	1	2	yes	complemented
20	#123	C₂×C₃	1	4	yes	complemented
21	C₂²⋊Q₈	C₂×C₃	1	4	yes	complemented
22	#121	C₂×C₃	1	4	yes	complemented
23	C₂³×C₃	C₂³	1	4	yes	—
24	C₃×Q₈	—	4	2	no	—
25	C₃×D₄	—	4	4	no	—
26	C₃×D₄	—	4	2	no	—
27	C₂×C₃×C₄	—	4	4	no	—
28	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₄	C₂³	1	2	yes	—
30	C₂×C₃×C₄	C₂³	1	4	yes	—
31	C₂×C₃×C₄	C₂³	1	4	yes	—
32	C₂×Q₈	C₂²×C₃	1	2	yes	—
33	C₂×D₄	C₂²×C₃	1	4	yes	—
34	C₂×D₄	C₂²×C₃	1	2	yes	—
35	C₂²×C₄	C₂²×C₃	1	4	yes	—
36	C₄⋊C₄	C₂²×C₃	1	4	yes	—
37	C₄⋊C₄	C₂²×C₃	1	4	yes	—
38	C₄⋊C₄	C₂²×C₃	1	2	yes	—
39	C₂²⋊C₄	C₂²×C₃	1	4	yes	—
40	C₂²⋊C₄	C₂²×C₃	1	4	yes	—
41	C₂²⋊C₄	C₂²×C₃	1	4	yes	—
42	C₄²	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃	—	2	4	no	—
44	C₂²×C₃	—	2	4	no	—
45	C₂²×C₃	—	2	4	no	—
46	C₂²×C₃	C₂⁴	1	1	yes	characteristic
47	C₃×C₄	—	2	4	no	—
48	C₃×C₄	—	2	4	no	—
49	C₃×C₄	—	2	2	no	—
50	C₃×C₄	—	2	1	no	—
51	C₂³	C₂³×C₃	1	4	yes	—
52	Q₈	—	4	2	no	—
53	D₄	—	4	4	no	—
54	D₄	—	4	2	no	—
55	C₂×C₄	—	4	4	no	—
56	C₂×C₄	C₂³×C₃	1	1	yes	characteristic
57	C₂×C₄	C₂³×C₃	1	2	yes	—
58	C₂×C₄	C₂³×C₃	1	4	yes	—
59	C₂×C₄	C₂³×C₃	1	4	yes	—
60	C₂×C₃	—	4	4	no	—
61	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
62	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
63	C₂²	—	2	4	no	—
64	C₂²	—	2	4	no	—
65	C₂²	—	2	4	no	—
66	C₂²	C₂⁴×C₃	1	1	yes	characteristic
67	C₄	—	2	4	no	—
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₃	#562	1	1	yes	characteristic, direct product factor, Sylow
72	C₂	—	4	4	no	—
73	C₂	C₃×H(5,𝔽₂)	1	2	yes	—
74	C₂	C₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#125	C₂	semidirect
2. C₃×#124	C₂	semidirect
3. C₃×#123	C₂	semidirect
4. C₃×C₂²⋊Q₈	C₂	semidirect
5. C₃×#121	C₂	semidirect
6. #562	C₃	direct (N × H)
18. #125	C₂×C₃	semidirect
19. #124	C₂×C₃	semidirect
20. #123	C₂×C₃	semidirect
21. C₂²⋊Q₈	C₂×C₃	semidirect
22. #121	C₂×C₃	semidirect
71. C₃	#562	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×#125	2
C₃×#124	2
C₃×#123	2
C₃×C₂²⋊Q₈	2
C₃×#121	2
#562	3

Subgroups — C3×#562 #5892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#562 #5892