Subgroups — #4459

Order 192.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊C₈	—	3	1	no	maximal, Sylow
5	C₂×D₁₂	C₄	1	1	yes	characteristic
6	C₂×C₄×D₃	—	2	1	no	—
7	C₂×C₃×C₈	—	2	1	no	—
8	C₃×C₄²	C₂²	1	1	yes	characteristic
9	(C₂×D₃)⋊C₄	—	2	1	no	—
10	C₄⋊Dic₃	C₄	1	1	yes	characteristic
11	C₂×C₃⋊C₈	—	2	1	no	—
12	C₄×D₄	—	3	1	no	—
13	C₄⋊C₈	—	3	1	no	—
14	C₄×C₈	D₃	1	1	yes	characteristic, complemented
15	C₂²×D₃	—	2	1	no	—
16	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄	D₄	1	1	yes	characteristic
18	C₂×C₃×C₄	D₄	1	1	yes	characteristic
19	C₂×Dic₃	—	2	1	no	—
20	D₁₂	—	2	1	no	—
21	D₁₂	C₈	1	2	yes	complemented
22	C₄×D₃	—	4	1	no	—
23	C₄×D₃	—	4	1	no	—
24	C₃×C₈	—	2	2	no	—
25	C₃⋊C₈	—	4	1	no	—
26	C₂×D₄	—	3	1	no	—
27	C₂²×C₄	—	6	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	—	6	1	no	—
30	C₄⋊C₄	—	3	1	no	—
31	C₂²⋊C₄	—	6	1	no	—
32	C₄²	C₂×D₃	1	1	yes	characteristic
33	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×D₃	—	2	2	no	—
35	C₂×D₃	—	4	1	no	—
36	C₂×D₃	—	4	1	no	—
37	C₃×C₄	—	2	1	no	—
38	C₃×C₄	C₂×C₈	1	1	yes	characteristic
39	C₃×C₄	M₁₆	1	1	yes	characteristic
40	C₃×C₄	D₈	1	1	yes	characteristic
41	C₃×C₄	SD₁₆	1	1	yes	characteristic
42	Dic₃	—	4	1	no	—
43	C₂³	—	6	1	no	—
44	D₄	—	3	2	no	—
45	D₄	—	6	1	no	—
46	C₂×C₄	—	6	1	no	—
47	C₂×C₄	—	12	1	no	—
48	C₂×C₄	—	12	1	no	—
49	C₂×C₄	C₄×D₃	1	1	yes	characteristic
50	C₂×C₄	D₁₂	1	1	yes	characteristic
51	C₂×C₄	#67	1	1	yes	characteristic
52	C₈	—	2	2	no	—
53	C₈	—	12	1	no	—
54	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
55	C₂×C₃	#102	1	1	yes	characteristic
56	C₂×C₃	#104	1	1	yes	characteristic
57	D₃	—	4	2	no	—
58	C₂²	—	6	2	no	—
59	C₂²	—	12	1	no	—
60	C₂²	—	12	1	no	—
61	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	—	12	1	no	—
64	C₄	D₃×C₈	1	1	yes	characteristic
65	C₄	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
66	C₄	#204	1	1	yes	characteristic
67	C₄	D₂₄	1	1	yes	characteristic
68	C₃	D₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
69	C₂	—	12	2	no	—
70	C₂	#805	1	1	yes	characteristic
71	C₂	(C₂×D₃)⋊C₈	1	1	yes	characteristic
72	C₂	#821	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	C₂²×C₃	C₂²⋊C₄
prime radical	C₂×D₁₂	C₄
cosocle	C₃×C₄²	C₂²
hypercenter	C₄×C₈	D₃
hypocenter	C₃	D₄⋊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
2. C₃×C₄×C₈	C₂	semidirect
3. C₄⋊(C₃⋊C₈)	C₂	semidirect
14. C₄×C₈	D₃	semidirect
21. D₁₂	C₈	semidirect
68. C₃	D₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₂	2
C₃×C₄×C₈	2
C₄⋊(C₃⋊C₈)	2
D₄⋊C₈	3