Subgroups — #4720

Order 192.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#926	C₂	1	1	yes	maximal, characteristic
2	C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#830	C₂	1	1	yes	maximal, characteristic
4	#815	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#433	—	3	1	no	maximal, Sylow
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	#217	C₄	1	1	yes	characteristic
13	#217	C₄	1	1	yes	characteristic
14	C₄⋊Dic₃	C₄	1	1	yes	characteristic
15	C₄⋊Dic₃	C₄	1	1	yes	characteristic
16	C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#117	—	3	1	no	—
21	C₄⋊C₈	—	3	1	no	—
22	C₄⋊C₈	—	3	1	no	—
23	C₂²⋊C₈	—	3	1	no	—
24	C₂²⋊C₈	D₃	1	1	yes	characteristic
25	#97	—	3	1	no	—
26	C₄×C₈	—	3	1	no	—
27	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄	—	2	1	no	—
29	C₂×C₃×C₄	—	2	1	no	—
30	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈	—	2	1	no	—
37	C₃×C₈	—	2	1	no	—
38	C₃⋊C₈	—	2	1	no	—
39	C₃⋊C₈	—	2	1	no	—
40	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂×C₈	—	3	1	no	—
42	C₂×C₈	—	3	1	no	—
43	C₂×C₈	C₂×D₃	1	1	yes	characteristic
44	C₂×C₈	C₂×D₃	1	1	yes	characteristic
45	C₄⋊C₄	—	3	1	no	—
46	C₄⋊C₄	—	3	1	no	—
47	C₂²⋊C₄	—	3	1	no	—
48	C₂²⋊C₄	—	3	1	no	—
49	C₄²	—	3	1	no	—
50	C₄²	—	3	1	no	—
51	C₂²×C₃	—	2	1	no	—
52	C₂²×C₃	—	2	1	no	—
53	C₂²×C₃	—	2	1	no	—
54	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
55	C₃×C₄	—	2	1	no	—
56	C₃×C₄	#41	1	1	yes	characteristic
57	C₃×C₄	#41	1	1	yes	characteristic
58	Dic₃	—	2	1	no	—
59	Dic₃	—	2	1	no	—
60	Dic₃	—	2	1	no	—
61	Dic₃	—	2	1	no	—
62	C₂³	C₄×D₃	1	1	yes	characteristic
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	3	1	no	—
67	C₂×C₄	—	3	1	no	—
68	C₂×C₄	—	3	1	no	—
69	C₂×C₄	C₄×D₃	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	2	1	no	—
73	C₈	—	6	1	no	—
74	C₈	—	6	1	no	—
75	C₂×C₃	—	4	1	no	—
76	C₂×C₃	#117	1	1	yes	characteristic
77	C₂×C₃	#131	1	1	yes	characteristic
78	C₂×C₃	#131	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	C₂×C₄×D₃	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	6	1	no	—
85	C₄	—	6	1	no	—
86	C₄	—	6	1	no	—
87	C₄	—	6	1	no	—
88	C₄	#237	1	1	yes	characteristic
89	C₄	#237	1	1	yes	characteristic
90	C₃	#433	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	4	1	no	—
92	C₂	#877	1	1	yes	characteristic
93	C₂	#901	1	1	yes	characteristic
94	C₂	#907	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₃×C₂²⋊C₈	C₂
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²⋊C₈	D₃
hypocenter	C₃	#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
4. #815	C₂	semidirect
5. Dic₃⋊C₈	C₂	semidirect
6. Dic₃⋊C₈	C₂	semidirect
7. C₈×Dic₃	C₂	semidirect
90. C₃	#433	semidirect

Maximal subgroups

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

subgroup	index
#926	2
C₃×C₂²⋊C₈	2
#830	2
#815	2
Dic₃⋊C₈	2
C₈×Dic₃	2
#433	3