Subgroups — #4830

Order 192.

102 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	Dic₃⋊C₈	C₂	1	2	yes	maximal
4	C₈×Dic₃	C₂	1	2	yes	maximal
5	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈	—	3	1	no	maximal, Sylow
7	C₂×Dic₆	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈	C₂²	1	2	yes	—
9	C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₃	C₄	1	1	yes	characteristic
11	Dic₃⋊C₄	C₄	1	2	yes	—
12	C₄×Dic₃	C₂²	1	2	yes	—
13	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×Q₈	—	3	1	no	—
16	C₄⋊C₈	—	3	2	no	—
17	C₄⋊C₈	D₃	1	1	yes	characteristic
18	C₄×C₈	—	3	2	no	—
19	C₄×C₈	—	3	1	no	—
20	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×Dic₃	C₂×C₄	1	2	yes	—
24	C₂×Dic₃	C₂×C₄	1	2	yes	—
25	Dic₆	C₈	1	4	yes	complemented
26	C₃×C₈	—	2	2	no	—
27	C₃⋊C₈	—	2	1	no	—
28	C₃⋊C₈	Q₈	1	2	yes	complemented
29	C₂×Q₈	—	3	1	no	—
30	C₂×C₈	—	3	1	no	—
31	C₂×C₈	—	3	1	no	—
32	C₂×C₈	C₂×D₃	1	2	yes	—
33	C₄⋊C₄	—	3	2	no	—
34	C₄⋊C₄	—	3	1	no	—
35	C₄²	—	3	2	no	—
36	C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	C₂×C₈	1	2	yes	—
40	C₃×C₄	C₂×Q₈	1	1	yes	characteristic
41	C₃×C₄	#41	1	1	yes	characteristic
42	Dic₃	—	2	2	no	—
43	Dic₃	C₂×C₈	1	4	yes	—
44	Q₈	—	3	4	no	—
45	C₂×C₄	—	3	2	no	—
46	C₂×C₄	—	3	2	no	—
47	C₂×C₄	C₄×D₃	1	1	yes	characteristic
48	C₂×C₄	C₄×D₃	1	1	yes	characteristic
49	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	3	2	no	—
52	C₈	—	6	1	no	—
53	C₂×C₃	C₄×Q₈	1	1	yes	characteristic
54	C₂×C₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₃	#131	1	1	yes	characteristic
56	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
57	C₄	—	2	1	no	—
58	C₄	—	3	4	no	—
59	C₄	—	6	2	no	—
60	C₄	D₃×C₈	1	2	yes	—
61	C₄	#237	1	1	yes	characteristic
62	C₄	D₃×Q₈	1	1	yes	characteristic
63	C₃	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
64	C₂	#887	1	1	yes	characteristic
65	C₂	C₂×D₃×C₈	1	1	yes	characteristic
66	C₂	#907	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂²×C₈
Fitting	C₃×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃	C₂²×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄⋊C₈	D₃
hypocenter	C₃	C₈×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. Dic₆	C₈	semidirect
28. C₃⋊C₈	Q₈	semidirect
63. C₃	C₈×Q₈	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
Dic₃⋊C₈	2
C₈×Dic₃	2
C₄×C₃⋊C₈	2
C₈×Q₈	3