Subgroups — C₃×#396 #5270

Order 192.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₂×C₃×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	#396	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁴×C₃	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₄	C₂²	1	3	yes	—
7	C₂²×C₃×C₄	C₂²	1	3	yes	—
8	C₃×C₄⋊C₄	—	2	6	no	—
9	C₃×C₂²⋊C₄	—	2	6	no	—
10	C₂×C₄⋊C₄	C₂×C₃	1	3	yes	complemented
11	C₂×C₂²⋊C₄	C₂×C₃	1	3	yes	—
12	#95	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂³×C₃	—	2	3	no	—
14	C₂³×C₃	—	2	3	no	—
15	C₂³×C₃	Q₈	1	2	yes	—
16	C₂³×C₃	C₂³	1	1	yes	characteristic
17	C₂×C₃×C₄	—	2	3	no	—
18	C₂×C₃×C₄	—	2	3	no	—
19	C₂×C₃×C₄	—	2	3	no	—
20	C₂×C₃×C₄	—	2	3	no	—
21	C₂×C₃×C₄	—	2	3	no	—
22	C₂×C₃×C₄	D₄	1	6	yes	—
23	C₂⁴	C₂²×C₃	1	1	yes	characteristic
24	C₂²×C₄	C₂²×C₃	1	3	yes	—
25	C₂²×C₄	C₂²×C₃	1	3	yes	—
26	C₄⋊C₄	—	2	6	no	—
27	C₂²⋊C₄	—	2	6	no	—
28	C₂²×C₃	—	2	6	no	—
29	C₂²×C₃	—	4	3	no	—
30	C₂²×C₃	—	4	1	no	—
31	C₂²×C₃	C₂×D₄	1	3	yes	—
32	C₂²×C₃	C₂×Q₈	1	1	yes	characteristic
33	C₂²×C₃	#41	1	3	yes	—
34	C₃×C₄	—	2	6	no	—
35	C₃×C₄	—	4	3	no	—
36	C₂³	—	2	3	no	—
37	C₂³	—	2	3	no	—
38	C₂³	C₃×Q₈	1	2	yes	—
39	C₂³	C₂³×C₃	1	1	yes	characteristic
40	C₂×C₄	—	2	3	no	—
41	C₂×C₄	—	2	3	no	—
42	C₂×C₄	—	2	3	no	—
43	C₂×C₄	—	2	3	no	—
44	C₂×C₄	—	2	3	no	—
45	C₂×C₄	C₃×D₄	1	6	yes	—
46	C₂×C₃	—	4	2	no	—
47	C₂×C₃	#121	1	3	yes	—
48	C₂×C₃	C₂²⋊Q₈	1	3	yes	—
49	C₂×C₃	#126	1	1	yes	characteristic
50	C₂²	—	2	6	no	—
51	C₂²	—	4	3	no	—
52	C₂²	—	4	1	no	—
53	C₂²	C₂×C₃×D₄	1	3	yes	—
54	C₂²	C₂×C₃×Q₈	1	1	yes	characteristic
55	C₂²	C₃×#41	1	3	yes	—
56	C₄	—	2	6	no	—
57	C₄	—	4	3	no	—
58	C₃	#396	1	1	yes	characteristic, direct product factor, Sylow
59	C₂	—	4	2	no	—
60	C₂	C₃×#121	1	3	yes	—
61	C₂	C₃×C₂²⋊Q₈	1	3	yes	—
62	C₂	C₃×#126	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂³×C₃
prime radical	C₂⁴×C₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄⋊C₄	C₂	semidirect
3. C₃×#95	C₂	semidirect
4. #396	C₃	direct (N × H)
10. C₂×C₄⋊C₄	C₂×C₃	semidirect
12. #95	C₂×C₃	semidirect
58. C₃	#396	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄⋊C₄	2
C₂×C₃×C₂²⋊C₄	2
C₃×#95	2
#396	3

Subgroups — C3×#396 #5270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#396 #5270