Subgroups — #4530

Order 192.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	2	yes	maximal
3	#343	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₄	C₄	1	2	yes	—
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₃×C₄⋊C₄	—	2	2	no	—
7	#208	—	2	2	no	—
8	#208	C₄	1	4	yes	complemented
9	C₂×C₃⋊C₈	—	2	2	no	—
10	C₂×M₁₆	—	3	2	no	—
11	C₂×C₄⋊C₄	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	—	4	2	no	—
15	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
16	C₂×C₃×C₄	D₄	1	1	yes	characteristic
17	C₂×C₃×C₄	D₄	1	2	yes	—
18	C₂×C₃×C₄	Q₈	1	1	yes	characteristic
19	C₃⋊C₈	—	2	4	no	—
20	C₂²×C₄	Dic₃	1	2	yes	—
21	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
22	M₁₆	—	3	4	no	—
23	M₁₆	—	6	2	no	—
24	C₂×C₈	—	6	2	no	—
25	C₄⋊C₄	—	2	2	no	—
26	C₂²×C₃	—	2	1	no	—
27	C₂²×C₃	—	2	1	no	—
28	C₂²×C₃	C₄²	1	1	yes	characteristic
29	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₃	C₄⋊C₄	1	1	yes	characteristic
31	C₃×C₄	—	4	2	no	—
32	C₃×C₄	C₂²⋊C₄	1	2	yes	—
33	C₃×C₄	C₄⋊C₄	1	2	yes	—
34	C₂³	C₂×Dic₃	1	1	yes	characteristic
35	C₂×C₄	—	2	2	no	—
36	C₂×C₄	—	4	2	no	—
37	C₂×C₄	Dic₆	1	1	yes	characteristic
38	C₂×C₄	C₄×D₃	1	2	yes	—
39	C₂×C₄	D₁₂	1	1	yes	characteristic
40	C₂×C₄	#67	1	2	yes	—
41	C₈	—	6	4	no	—
42	C₂×C₃	—	2	1	no	—
43	C₂×C₃	—	2	1	no	—
44	C₂×C₃	#95	1	1	yes	characteristic
45	C₂×C₃	#100	1	1	yes	characteristic
46	C₂×C₃	#101	1	1	yes	characteristic
47	C₂²	—	2	1	no	—
48	C₂²	—	2	1	no	—
49	C₂²	C₄×Dic₃	1	1	yes	characteristic
50	C₂²	C₄⋊Dic₃	1	1	yes	characteristic
51	C₂²	#217	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	Dic₃⋊C₄	1	2	yes	—
54	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
55	C₃	#343	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₃⋊#95	1	1	yes	characteristic
59	C₂	#833	1	1	yes	characteristic
60	C₂	#836	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#217
commutator	C₂²×C₃	C₄²
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₂×C₃×C₄⋊C₄	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#343
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
8. #208	C₄	semidirect
55. C₃	#343	semidirect

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×#208	2
#343	3