Subgroups — #5075

Order 192.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×#214	C₂	1	2	yes	maximal, complemented
3	C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#832	C₂	1	2	yes	maximal, complemented
5	C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#460	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	—	2	1	no	—
8	C₂×C₃×D₄	—	2	1	no	—
9	C₂×C₃×D₄	C₂²	1	2	yes	—
10	C₂×Dic₆	C₂²	1	1	yes	characteristic
11	C₃×C₄²	C₂²	1	1	yes	characteristic
12	#214	—	2	4	no	—
13	C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	Dic₃⋊C₄	—	2	1	no	—
15	C₄×Dic₃	—	2	1	no	—
16	C₂×C₃⋊C₈	C₂²	1	2	yes	—
17	C₂×SD₁₆	—	3	2	no	—
18	#127	D₃	1	1	yes	characteristic
19	C₄×Q₈	—	3	1	no	—
20	C₄⋊C₈	—	3	1	no	—
21	#102	—	3	2	no	—
22	C₂³×C₃	—	2	2	no	—
23	C₃×D₄	—	2	2	no	—
24	C₃×D₄	—	2	2	no	—
25	C₃×D₄	—	4	2	no	—
26	C₃×D₄	—	4	2	no	—
27	C₂×C₃×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	C₂×Dic₃	—	2	1	no	—
31	C₂×Dic₃	—	2	1	no	—
32	Dic₆	—	2	1	no	—
33	Dic₆	D₄	1	2	yes	complemented
34	C₃⋊C₈	—	2	2	no	—
35	C₂×Q₈	—	3	1	no	—
36	C₂×D₄	—	2	1	no	—
37	C₂×D₄	—	2	1	no	—
38	C₂×D₄	C₂×D₃	1	2	yes	—
39	SD₁₆	—	6	4	no	—
40	C₂×C₈	—	3	2	no	—
41	C₄⋊C₄	—	3	1	no	—
42	C₄⋊C₄	—	6	1	no	—
43	C₄²	—	6	1	no	—
44	C₄²	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₃	—	4	2	no	—
46	C₂²×C₃	—	4	2	no	—
47	C₂²×C₃	—	4	2	no	—
48	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
49	C₃×C₄	—	2	1	no	—
50	C₃×C₄	SD₁₆	1	2	yes	complemented
51	C₃×C₄	C₂×D₄	1	1	yes	characteristic
52	C₃×C₄	#41	1	1	yes	characteristic
53	Dic₃	—	2	2	no	—
54	Dic₃	—	4	1	no	—
55	C₂³	—	2	2	no	—
56	Q₈	—	3	2	no	—
57	Q₈	—	6	1	no	—
58	D₄	—	2	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	4	2	no	—
61	D₄	—	4	2	no	—
62	C₂×C₄	—	6	1	no	—
63	C₂×C₄	—	6	1	no	—
64	C₂×C₄	#67	1	1	yes	characteristic
65	C₂×C₄	#67	1	1	yes	characteristic
66	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₈	—	6	2	no	—
68	C₂×C₃	—	8	2	no	—
69	C₂×C₃	#121	1	1	yes	characteristic
70	C₂×C₃	C₂×SD₁₆	1	1	yes	characteristic
71	C₂×C₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	C₂×#67	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	6	2	no	—
78	C₄	—	12	1	no	—
79	C₄	#214	1	2	yes	complemented
80	C₄	D₃×D₄	1	1	yes	characteristic
81	C₄	#237	1	1	yes	characteristic
82	C₃	#460	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	8	2	no	—
84	C₂	#932	1	1	yes	characteristic
85	C₂	C₂×#214	1	1	yes	characteristic
86	C₂	#938	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₃×#127	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	#127	D₃
hypocenter	C₃	#460
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
2. C₂×#214	C₂	semidirect
3. C₄×Dic₆	C₂	semidirect
4. #832	C₂	semidirect
5. C₄⋊(C₃⋊C₈)	C₂	semidirect
33. Dic₆	D₄	semidirect
50. C₃×C₄	SD₁₆	semidirect
79. C₄	#214	semidirect
82. C₃	#460	semidirect

Maximal subgroups

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

subgroup	index
C₃×#127	2
C₂×#214	2
C₄×Dic₆	2
#832	2
C₄⋊(C₃⋊C₈)	2
#460	3