Subgroups — #5012

Order 192.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#208	C₂	1	2	yes	maximal, complemented
3	#830	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#436	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄	C₂²	1	2	yes	—
9	C₃×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₃×C₂²⋊C₄	C₄	1	2	yes	—
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	#208	—	2	4	no	—
13	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×M₁₆	—	3	2	no	—
17	C₄×D₄	D₃	1	1	yes	characteristic
18	C₄⋊C₈	—	3	1	no	—
19	C₂²⋊C₈	—	3	2	no	—
20	C₄×C₈	—	3	1	no	—
21	C₂³×C₃	C₂×C₄	1	2	yes	—
22	C₃×D₄	—	2	2	no	—
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	—	2	2	no	—
25	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₃⋊C₈	—	2	2	no	—
30	C₃⋊C₈	—	2	1	no	—
31	C₃⋊C₈	D₄	1	2	yes	complemented
32	C₂×D₄	Dic₃	1	1	yes	characteristic
33	C₂²×C₄	C₂×D₃	1	2	yes	—
34	M₁₆	—	6	4	no	—
35	C₂×C₈	—	3	1	no	—
36	C₂×C₈	—	3	1	no	—
37	C₂×C₈	—	3	2	no	—
38	C₄⋊C₄	Dic₃	1	1	yes	characteristic
39	C₂²⋊C₄	Dic₃	1	2	yes	—
40	C₄²	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃	—	2	2	no	—
42	C₂²×C₃	—	2	2	no	—
43	C₂²×C₃	—	2	2	no	—
44	C₂²×C₃	C₂²×C₄	1	1	yes	characteristic
45	C₃×C₄	—	2	2	no	—
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	M₁₆	1	2	yes	complemented
48	C₃×C₄	C₂×D₄	1	1	yes	characteristic
49	C₃×C₄	#41	1	1	yes	characteristic
50	C₂³	C₂×Dic₃	1	2	yes	—
51	D₄	—	2	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
55	C₂×C₄	C₂×Dic₃	1	2	yes	—
56	C₂×C₄	C₂×Dic₃	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₈	—	3	2	no	—
59	C₈	—	6	2	no	—
60	C₈	—	6	1	no	—
61	C₂×C₃	—	4	2	no	—
62	C₂×C₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₃	#131	1	1	yes	characteristic
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂²×Dic₃	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	#208	1	2	yes	complemented
72	C₄	D₃×D₄	1	1	yes	characteristic
73	C₄	#237	1	1	yes	characteristic
74	C₃	#436	1	1	yes	characteristic, complemented, Sylow
75	C₂	—	4	2	no	—
76	C₂	C₂×#208	1	1	yes	characteristic
77	C₂	D₄×Dic₃	1	1	yes	characteristic
78	C₂	#948	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₄
Fitting	C₃×C₄×D₄	C₂
prime radical	C₂×C₃×D₄	C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₄×D₄	D₃
hypocenter	C₃	#436
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#208	C₂	semidirect
3. #830	C₂	semidirect
4. C₄⋊(C₃⋊C₈)	C₂	semidirect
5. C₄×C₃⋊C₈	C₂	semidirect
31. C₃⋊C₈	D₄	semidirect
47. C₃×C₄	M₁₆	semidirect
71. C₄	#208	semidirect
74. C₃	#436	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₄	2
C₂×#208	2
#830	2
C₄⋊(C₃⋊C₈)	2
C₄×C₃⋊C₈	2
#436	3