Subgroups — #4915

Order 192.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#912	C₂	1	1	yes	maximal, characteristic
3	C₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#829	C₂	1	1	yes	maximal, characteristic, complemented
5	#827	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#798	C₂	1	1	yes	maximal, characteristic, complemented
8	#510	—	3	1	no	maximal, Sylow
9	D₃×Q₈	—	2	1	no	—
10	#237	—	2	1	no	—
11	C₃×Q₁₆	C₂²	1	1	yes	characteristic
12	C₃×D₈	C₂²	1	1	yes	characteristic
13	#216	—	2	1	no	—
14	#214	—	2	1	no	—
15	Dic₁₂	C₂²	1	1	yes	characteristic
16	Dic₁₂	C₂²	1	1	yes	characteristic
17	D₃×C₈	C₂²	1	1	yes	characteristic
18	C₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	#135	—	3	1	no	—
21	C₂×Q₁₆	—	3	1	no	—
22	Q₃₂	—	3	1	no	—
23	Q₃₂	—	3	1	no	—
24	SD₃₂	—	3	1	no	—
25	SD₃₂	D₃	1	1	yes	characteristic, complemented
26	M₃₂	—	3	1	no	—
27	C₃×Q₈	—	2	1	no	—
28	C₃×D₄	—	2	1	no	—
29	#67	—	4	1	no	—
30	C₂×Dic₃	—	4	1	no	—
31	C₄×D₃	—	4	1	no	—
32	C₄×D₃	D₄	1	1	yes	characteristic
33	Dic₆	—	2	1	no	—
34	Dic₆	—	2	1	no	—
35	Dic₆	—	4	1	no	—
36	C₃×C₈	C₂³	1	1	yes	characteristic
37	C₃⋊C₈	D₄	1	1	yes	characteristic
38	#41	—	6	1	no	—
39	C₂×Q₈	—	6	1	no	—
40	Q₁₆	—	3	1	no	—
41	Q₁₆	—	3	1	no	—
42	Q₁₆	—	6	1	no	—
43	Q₁₆	C₂×D₃	1	1	yes	characteristic
44	SD₁₆	—	6	1	no	—
45	D₈	C₂×D₃	1	1	yes	characteristic
46	C₂×C₈	—	3	1	no	—
47	C₁₆	—	3	1	no	—
48	C₁₆	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃	—	4	1	no	—
50	C₂×D₃	D₈	1	1	yes	characteristic
51	C₃×C₄	—	4	1	no	—
52	C₃×C₄	C₂×D₄	1	1	yes	characteristic
53	Dic₃	—	4	1	no	—
54	Dic₃	—	4	1	no	—
55	Dic₃	D₈	1	1	yes	characteristic
56	Q₈	—	2	1	no	—
57	Q₈	—	6	1	no	—
58	Q₈	—	6	1	no	—
59	Q₈	—	12	1	no	—
60	D₄	—	2	1	no	—
61	D₄	—	12	1	no	—
62	C₂×C₄	—	3	1	no	—
63	C₂×C₄	—	12	1	no	—
64	C₂×C₄	—	12	1	no	—
65	C₈	—	3	1	no	—
66	C₈	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₃	—	8	1	no	—
68	C₂×C₃	C₂×D₈	1	1	yes	characteristic
69	D₃	—	2	1	no	—
70	C₂²	—	3	1	no	—
71	C₂²	—	4	1	no	—
72	C₄	—	3	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	12	1	no	—
75	C₄	—	12	1	no	—
76	C₄	D₃×D₄	1	1	yes	characteristic
77	C₃	#510	1	1	yes	characteristic, complemented, Sylow
78	C₂	—	6	1	no	—
79	C₂	—	8	1	no	—
80	C₂	D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting	C₃×SD₃₂	C₂
socle	C₂×C₃	C₂×D₈
prime radical	#912	C₂
hypercenter	SD₃₂	D₃
hypocenter	C₃	#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. D₃×Q₁₆	C₂	semidirect
3. C₃×SD₃₂	C₂	semidirect
4. #829	C₂	semidirect
5. #827	C₂	semidirect
6. Dic₂₄	C₂	semidirect
7. #798	C₂	semidirect
25. SD₃₂	D₃	semidirect
77. C₃	#510	semidirect

Maximal subgroups

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

subgroup	index
D₃×Q₁₆	2
#912	2
C₃×SD₃₂	2
#829	2
#827	2
Dic₂₄	2
#798	2
#510	3