Subgroups — #4469

Order 192.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₃⋊C₄	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	#324	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
6	C₂²×Dic₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₃	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈	—	2	1	no	—
9	Dic₃⋊C₄	—	2	1	no	—
10	Dic₃⋊C₄	—	2	1	no	—
11	C₂×C₃⋊C₈	—	2	1	no	—
12	C₂×C₄⋊C₄	—	3	1	no	—
13	C₂²⋊C₈	—	3	1	no	—
14	C₂²⋊C₈	D₃	1	1	yes	characteristic
15	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₄	—	4	1	no	—
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	C₂×Dic₃	—	2	1	no	—
21	C₂×Dic₃	—	2	1	no	—
22	C₂×Dic₃	—	4	1	no	—
23	C₂×Dic₃	C₈	1	2	yes	complemented
24	C₃×C₈	—	4	1	no	—
25	C₃⋊C₈	—	4	1	no	—
26	C₂²×C₄	—	3	1	no	—
27	C₂²×C₄	—	3	1	no	—
28	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂×C₈	—	2	1	no	—
30	C₂×C₈	—	6	1	no	—
31	C₄⋊C₄	—	6	1	no	—
32	C₄⋊C₄	—	6	1	no	—
33	C₂²×C₃	—	2	1	no	—
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₃	M₁₆	1	1	yes	characteristic
38	C₃×C₄	—	2	1	no	—
39	C₃×C₄	—	2	1	no	—
40	Dic₃	—	2	2	no	—
41	Dic₃	—	4	1	no	—
42	C₂³	C₄×D₃	1	1	yes	characteristic
43	C₂×C₄	—	3	2	no	—
44	C₂×C₄	—	4	1	no	—
45	C₂×C₄	—	6	1	no	—
46	C₂×C₄	—	6	1	no	—
47	C₂×C₄	—	6	1	no	—
48	C₂×C₄	—	12	1	no	—
49	C₂×C₄	D₁₂	1	1	yes	characteristic
50	C₂×C₄	#67	1	1	yes	characteristic
51	C₈	—	4	1	no	—
52	C₈	—	12	1	no	—
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₃	#99	1	1	yes	characteristic
57	C₂×C₃	#101	1	1	yes	characteristic
58	C₂²	—	2	1	no	—
59	C₂²	—	2	1	no	—
60	C₂²	D₃×C₈	1	1	yes	characteristic
61	C₂²	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
62	C₂²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	2	1	no	—
65	C₄	—	6	2	no	—
66	C₄	—	12	1	no	—
67	C₃	#324	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	1	no	—
69	C₂	—	2	1	no	—
70	C₂	#806	1	1	yes	characteristic
71	C₂	(C₂×D₃)⋊C₈	1	1	yes	characteristic
72	C₂	#824	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂×C₈
Frattini	C₂²×C₄	C₂×D₃
Fitting	C₃×C₂²⋊C₈	C₂
socle	C₂²×C₃	C₂²⋊C₄
prime radical	C₂³×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂²
hypercenter	C₂²⋊C₈	D₃
hypocenter	C₃	#324
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
23. C₂×Dic₃	C₈	semidirect
67. C₃	#324	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₃⋊C₄	2
C₃×C₂²⋊C₈	2
#830	2
#324	3