Subgroups — #4909

Order 192.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#904	C₂	1	1	yes	maximal, characteristic
3	C₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₂₄	C₂	1	2	yes	maximal, complemented
5	#800	C₂	1	2	yes	maximal, complemented
6	#510	—	3	1	no	maximal, Sylow
7	#235	—	2	1	no	—
8	C₂×Dic₆	—	2	1	no	—
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	Dic₁₂	—	2	1	no	—
11	Dic₁₂	C₂²	1	2	yes	—
12	Dic₁₂	C₂²	1	1	yes	characteristic
13	D₂₄	C₂²	1	1	yes	characteristic
14	#204	—	2	1	no	—
15	C₃×C₁₆	C₂²	1	2	yes	—
16	#135	—	3	1	no	—
17	C₂×Q₁₆	—	3	1	no	—
18	Q₃₂	—	3	2	no	—
19	SD₃₂	—	3	2	no	—
20	M₃₂	D₃	1	1	yes	characteristic, complemented
21	C₂×C₃×C₄	D₄	1	1	yes	characteristic
22	#67	—	4	1	no	—
23	C₂×Dic₃	—	4	1	no	—
24	D₁₂	—	2	1	no	—
25	C₄×D₃	—	4	1	no	—
26	Dic₆	—	2	1	no	—
27	Dic₆	—	2	2	no	—
28	Dic₆	—	4	1	no	—
29	C₃×C₈	D₄	1	1	yes	characteristic
30	C₃×C₈	C₂³	1	1	yes	characteristic
31	#41	—	6	1	no	—
32	C₂×Q₈	—	6	1	no	—
33	Q₁₆	—	3	2	no	—
34	Q₁₆	—	3	1	no	—
35	Q₁₆	—	6	1	no	—
36	SD₁₆	—	6	1	no	—
37	D₈	—	3	1	no	—
38	C₂×C₈	C₂×D₃	1	1	yes	characteristic
39	C₁₆	C₂×D₃	1	2	yes	—
40	C₂²×C₃	D₈	1	1	yes	characteristic
41	C₂×D₃	—	4	1	no	—
42	C₃×C₄	D₈	1	1	yes	characteristic
43	C₃×C₄	C₂×D₄	1	1	yes	characteristic
44	Dic₃	—	4	2	no	—
45	Dic₃	—	4	1	no	—
46	Q₈	—	6	1	no	—
47	Q₈	—	6	2	no	—
48	Q₈	—	12	1	no	—
49	D₄	—	6	1	no	—
50	D₄	—	12	1	no	—
51	C₂×C₄	—	12	1	no	—
52	C₂×C₄	—	12	1	no	—
53	C₂×C₄	D₁₂	1	1	yes	characteristic
54	C₈	D₁₂	1	1	yes	characteristic
55	C₈	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	C₂×D₈	1	1	yes	characteristic
58	D₃	—	8	1	no	—
59	C₂²	—	12	1	no	—
60	C₂²	D₂₄	1	1	yes	characteristic
61	C₄	—	12	1	no	—
62	C₄	—	12	2	no	—
63	C₄	D₂₄	1	1	yes	characteristic
64	C₄	C₂×D₁₂	1	1	yes	characteristic
65	C₃	#510	1	1	yes	characteristic, complemented, Sylow
66	C₂	—	2	1	no	—
67	C₂	—	24	1	no	—
68	C₂	C₂×D₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₄
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting	C₃×M₃₂	C₂
socle	C₂×C₃	C₂×D₈
prime radical	#904	C₂
hypercenter	M₃₂	D₃
hypocenter	C₃	#510
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₂	C₂	semidirect
3. C₃×M₃₂	C₂	semidirect
4. Dic₂₄	C₂	semidirect
5. #800	C₂	semidirect
20. M₃₂	D₃	semidirect
65. C₃	#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₂	2
#904	2
C₃×M₃₂	2
Dic₂₄	2
#800	2
#510	3