Subgroups — #4851

Order 192.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#869	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#817	C₂	1	2	yes	maximal
5	#816	C₂	1	2	yes	maximal
6	#475	—	3	1	no	maximal, Sylow
7	C₂×Dic₆	—	2	1	no	—
8	C₂×Dic₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₈	C₂²	1	2	yes	—
10	C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃	—	2	1	no	—
12	C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	Dic₃⋊C₄	—	2	1	no	—
15	C₄×Dic₃	—	2	1	no	—
16	C₄⋊Q₈	—	3	1	no	—
17	C₄×Q₈	—	3	1	no	—
18	#106	—	3	2	no	—
19	C₄⋊C₈	D₃	1	1	yes	characteristic
20	#103	—	3	2	no	—
21	C₂×C₃×C₄	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂×Dic₃	—	2	1	no	—
25	C₂×Dic₃	—	2	1	no	—
26	C₂×Dic₃	—	2	2	no	—
27	Dic₆	—	2	1	no	—
28	Dic₆	—	4	2	no	—
29	Dic₆	Q₈	1	2	yes	complemented
30	C₃×C₈	—	2	2	no	—
31	C₂×Q₈	—	3	1	no	—
32	C₂×Q₈	—	6	1	no	—
33	C₂×C₈	C₂×D₃	1	2	yes	—
34	C₄⋊C₄	—	3	2	no	—
35	C₄⋊C₄	—	3	1	no	—
36	C₄⋊C₄	—	6	1	no	—
37	C₄⋊C₄	—	6	1	no	—
38	C₄²	—	6	1	no	—
39	C₄²	C₂×D₃	1	1	yes	characteristic
40	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	SD₁₆	1	2	yes	—
43	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
44	C₃×C₄	#41	1	1	yes	characteristic
45	Dic₃	—	2	2	no	—
46	Dic₃	—	4	2	no	—
47	Dic₃	—	4	1	no	—
48	Q₈	—	3	2	no	—
49	Q₈	—	6	1	no	—
50	Q₈	—	12	2	no	—
51	C₂×C₄	—	6	1	no	—
52	C₂×C₄	—	6	1	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	D₁₂	1	1	yes	characteristic
55	C₂×C₄	D₁₂	1	1	yes	characteristic
56	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
57	C₈	—	2	2	no	—
58	C₂×C₃	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₃	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₃	#137	1	1	yes	characteristic
61	C₂²	C₂×D₁₂	1	1	yes	characteristic
62	C₄	—	2	1	no	—
63	C₄	—	6	2	no	—
64	C₄	—	12	2	no	—
65	C₄	—	12	1	no	—
66	C₄	#204	1	2	yes	—
67	C₄	#237	1	1	yes	characteristic
68	C₄	D₃×Q₈	1	1	yes	characteristic
69	C₃	#475	1	1	yes	characteristic, complemented, Sylow
70	C₂	#897	1	1	yes	characteristic
71	C₂	C₂×#204	1	1	yes	characteristic
72	C₂	#909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	#475
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. Dic₆	Q₈	semidirect
69. C₃	#475	semidirect

Maximal subgroups

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

subgroup	index
#869	2
C₄×Dic₆	2
C₃×C₄⋊C₈	2
#817	2
#816	2
#475	3