Subgroups — #4456

Order 192.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₈	—	3	1	no	maximal, Sylow
5	C₂×Dic₆	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈	—	2	1	no	—
7	C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₃	C₄	1	1	yes	characteristic
9	Dic₃⋊C₄	—	2	1	no	—
10	C₄×Dic₃	—	2	1	no	—
11	C₂×C₃⋊C₈	—	2	1	no	—
12	C₄×Q₈	—	3	1	no	—
13	C₄⋊C₈	—	3	1	no	—
14	C₄×C₈	D₃	1	1	yes	characteristic
15	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₄	D₄	1	1	yes	characteristic
17	C₂×C₃×C₄	D₄	1	1	yes	characteristic
18	C₂×Dic₃	—	2	1	no	—
19	C₂×Dic₃	—	2	1	no	—
20	Dic₆	—	2	1	no	—
21	Dic₆	C₈	1	2	yes	complemented
22	C₃×C₈	—	2	2	no	—
23	C₃⋊C₈	—	4	1	no	—
24	C₂×Q₈	—	3	1	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	6	1	no	—
27	C₄⋊C₄	—	3	1	no	—
28	C₄⋊C₄	—	6	1	no	—
29	C₄²	—	6	1	no	—
30	C₄²	C₂×D₃	1	1	yes	characteristic
31	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₃×C₄	—	2	1	no	—
33	C₃×C₄	C₂×C₈	1	1	yes	characteristic
34	C₃×C₄	M₁₆	1	1	yes	characteristic
35	C₃×C₄	SD₁₆	1	1	yes	characteristic
36	C₃×C₄	Q₁₆	1	1	yes	characteristic
37	Dic₃	—	2	2	no	—
38	Dic₃	—	4	1	no	—
39	Q₈	—	3	2	no	—
40	Q₈	—	6	1	no	—
41	C₂×C₄	—	6	1	no	—
42	C₂×C₄	—	6	1	no	—
43	C₂×C₄	C₄×D₃	1	1	yes	characteristic
44	C₂×C₄	D₁₂	1	1	yes	characteristic
45	C₂×C₄	#67	1	1	yes	characteristic
46	C₈	—	2	2	no	—
47	C₈	—	12	1	no	—
48	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
49	C₂×C₃	#103	1	1	yes	characteristic
50	C₂×C₃	#104	1	1	yes	characteristic
51	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
52	C₄	—	2	1	no	—
53	C₄	—	6	2	no	—
54	C₄	—	12	1	no	—
55	C₄	D₃×C₈	1	1	yes	characteristic
56	C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
57	C₄	#204	1	1	yes	characteristic
58	C₄	Dic₁₂	1	1	yes	characteristic
59	C₃	Q₈⋊C₈	1	1	yes	characteristic, complemented, Sylow
60	C₂	#805	1	1	yes	characteristic
61	C₂	#816	1	1	yes	characteristic
62	C₂	(C₂×D₃)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂
commutator	C₃×C₄	C₂×C₈
Frattini	C₄²	C₂×D₃
Fitting	C₃×C₄×C₈	C₂
socle, prime radical	C₂²×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂²
hypercenter	C₄×C₈	D₃
hypocenter	C₃	Q₈⋊C₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
21. Dic₆	C₈	semidirect
59. C₃	Q₈⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆	2
C₃×C₄×C₈	2
C₄⋊(C₃⋊C₈)	2
Q₈⋊C₈	3