Subgroups — #4789

Order 192.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#216	C₂	1	1	yes	maximal, characteristic
3	#887	C₂	1	1	yes	maximal, characteristic
4	C₃×#103	C₂	1	1	yes	maximal, characteristic
5	#816	C₂	1	1	yes	maximal, characteristic
6	#815	C₂	1	1	yes	maximal, characteristic
7	#808	C₂	1	1	yes	maximal, characteristic
8	#441	—	3	1	no	maximal, Sylow
9	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	#216	C₄	1	4	yes	complemented
14	C₄⋊Dic₃	—	2	1	no	—
15	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	Dic₃⋊C₄	—	2	1	no	—
17	C₄×Dic₃	—	2	1	no	—
18	C₄×Dic₃	—	2	1	no	—
19	C₄×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆	—	3	1	no	—
22	C₄×Q₈	—	3	1	no	—
23	C₄×Q₈	—	3	1	no	—
24	#106	—	3	1	no	—
25	#103	—	3	1	no	—
26	#103	D₃	1	1	yes	characteristic
27	#97	—	3	1	no	—
28	C₃×Q₈	—	2	1	no	—
29	C₃×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×Dic₃	—	2	1	no	—
34	C₂×Dic₃	—	2	1	no	—
35	C₂×Dic₃	D₄	1	1	yes	characteristic
36	C₂×Dic₃	D₄	1	1	yes	characteristic
37	Dic₆	—	2	1	no	—
38	Dic₆	C₂×C₄	1	2	yes	—
39	C₃×C₈	—	2	1	no	—
40	C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×Q₈	—	3	1	no	—
42	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
43	Q₁₆	—	3	4	no	—
44	C₂×C₈	—	3	1	no	—
45	C₂×C₈	C₂×D₃	1	1	yes	characteristic
46	C₄⋊C₄	—	3	1	no	—
47	C₄⋊C₄	—	6	1	no	—
48	C₄⋊C₄	—	6	1	no	—
49	C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
50	C₄²	—	3	1	no	—
51	C₄²	—	6	1	no	—
52	C₄²	—	6	1	no	—
53	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
54	C₃×C₄	—	2	2	no	—
55	C₃×C₄	—	4	1	no	—
56	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄	#41	1	1	yes	characteristic
58	Dic₃	—	2	2	no	—
59	Dic₃	—	2	1	no	—
60	Dic₃	—	2	1	no	—
61	Dic₃	—	4	1	no	—
62	Q₈	—	2	1	no	—
63	Q₈	—	3	2	no	—
64	Q₈	—	6	1	no	—
65	Q₈	C₄×D₃	1	2	yes	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	3	1	no	—
69	C₂×C₄	—	3	1	no	—
70	C₂×C₄	—	6	1	no	—
71	C₂×C₄	—	6	1	no	—
72	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	3	2	no	—
75	C₂×C₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₃	#137	1	1	yes	characteristic
77	C₂×C₃	#137	1	1	yes	characteristic
78	C₂²	D₃×D₄	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	4	1	no	—
81	C₄	—	6	2	no	—
82	C₄	—	6	1	no	—
83	C₄	—	6	1	no	—
84	C₄	—	12	1	no	—
85	C₄	C₂×C₄×D₃	1	1	yes	characteristic
86	C₄	#237	1	1	yes	characteristic
87	C₃	#441	1	1	yes	characteristic, complemented, Sylow
88	C₂	#67⋊C₄	1	1	yes	characteristic
89	C₂	#915	1	1	yes	characteristic
90	C₂	#918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#103	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#103	D₃
hypocenter	C₃	#441
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
13. #216	C₄	semidirect
87. C₃	#441	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃	2
C₂×#216	2
#887	2
C₃×#103	2
#816	2
#815	2
#808	2
#441	3