Subgroups — #4524

Order 192.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊C₈	C₂	1	2	yes	maximal
3	#336	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₄	C₄	1	2	yes	—
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₃×C₄²	—	2	2	no	—
7	C₂×C₃⋊C₈	—	2	2	no	—
8	C₂×C₃⋊C₈	—	2	2	no	—
9	C₂×C₃⋊C₈	C₄	1	4	yes	complemented
10	C₂²×C₈	—	3	2	no	—
11	C₂×C₄²	D₃	1	1	yes	characteristic
12	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₄	—	2	2	no	—
14	C₂×C₃×C₄	—	2	2	no	—
15	C₂×C₃×C₄	C₈	1	4	yes	complemented
16	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
17	C₂×C₃×C₄	D₄	1	1	yes	characteristic
18	C₂×C₃×C₄	D₄	1	2	yes	—
19	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
20	C₃⋊C₈	—	2	4	no	—
21	C₂²×C₄	Dic₃	1	2	yes	—
22	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
23	C₂×C₈	—	3	4	no	—
24	C₂×C₈	—	6	2	no	—
25	C₂×C₈	—	6	2	no	—
26	C₄²	—	2	2	no	—
27	C₂²×C₃	C₄²	1	1	yes	characteristic
28	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₃	C₂×C₈	1	2	yes	—
31	C₂²×C₃	M₁₆	1	2	yes	—
32	C₃×C₄	—	2	4	no	—
33	C₃×C₄	C₂²⋊C₄	1	2	yes	—
34	C₃×C₄	C₄⋊C₄	1	2	yes	—
35	C₂³	C₂×Dic₃	1	1	yes	characteristic
36	C₂×C₄	—	2	2	no	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	C₃⋊C₈	1	4	yes	complemented
39	C₂×C₄	Dic₆	1	1	yes	characteristic
40	C₂×C₄	C₄×D₃	1	2	yes	—
41	C₂×C₄	D₁₂	1	1	yes	characteristic
42	C₂×C₄	#67	1	2	yes	—
43	C₈	—	6	4	no	—
44	C₂×C₃	#95	1	1	yes	characteristic
45	C₂×C₃	C₄×C₈	1	1	yes	characteristic
46	C₂×C₃	#97	1	1	yes	characteristic
47	C₂×C₃	C₂²⋊C₈	1	2	yes	—
48	C₂×C₃	C₄⋊C₈	1	2	yes	—
49	C₂²	C₂×C₃⋊C₈	1	2	yes	—
50	C₂²	#208	1	2	yes	—
51	C₂²	C₄×Dic₃	1	1	yes	characteristic
52	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
53	C₂²	#217	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	Dic₃⋊C₄	1	2	yes	—
56	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
57	C₃	#336	1	1	yes	characteristic, complemented, Sylow
58	C₂	C₄×C₃⋊C₈	1	1	yes	characteristic
59	C₂	#803	1	1	yes	characteristic
60	C₂	C₄⋊(C₃⋊C₈)	1	2	yes	—
61	C₂	#830	1	2	yes	—
62	C₂	C₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃
commutator	C₂×C₃	C₄×C₈
Fitting	C₂×C₃×C₄²	C₂
socle, prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×C₄²	D₃
hypocenter	C₃	#336
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
9. C₂×C₃⋊C₈	C₄	semidirect
15. C₂×C₃×C₄	C₈	semidirect
38. C₂×C₄	C₃⋊C₈	semidirect
57. C₃	#336	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²	2
C₂²×C₃⋊C₈	2
#336	3