Subgroups — #5155

Order 192.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₈	C₂	1	1	yes	maximal, characteristic
2	#936	C₂	1	2	yes	maximal, complemented
3	C₂×#214	C₂	1	2	yes	maximal, complemented
4	C₂×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#495	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	2	yes	—
8	C₂×Dic₆	C₂²	1	2	yes	—
9	C₃×D₈	—	2	2	no	—
10	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
11	#217	—	2	4	no	—
12	#214	—	2	4	no	—
13	C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	Dic₁₂	—	2	2	no	—
16	C₂×Q₁₆	—	3	1	no	—
17	C₂×SD₁₆	—	3	2	no	—
18	C₂×D₈	D₃	1	1	yes	characteristic
19	#124	—	3	2	no	—
20	C₄×C₈	—	3	1	no	—
21	C₂³×C₃	—	2	2	no	—
22	C₃×D₄	—	2	2	no	—
23	C₃×D₄	—	2	2	no	—
24	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂×Dic₃	—	2	2	no	—
26	C₂×Dic₃	D₄	1	2	yes	—
27	Dic₆	—	2	2	no	—
28	Dic₆	—	2	2	no	—
29	C₃×C₈	D₄	1	2	yes	—
30	C₃⋊C₈	D₄	1	2	yes	—
31	C₂×Q₈	—	3	2	no	—
32	C₂×D₄	C₂×D₃	1	2	yes	—
33	Q₁₆	—	6	2	no	—
34	SD₁₆	—	6	4	no	—
35	D₈	—	2	2	no	—
36	C₂×C₈	—	3	1	no	—
37	C₂×C₈	C₂×D₃	1	1	yes	characteristic
38	C₂²⋊C₄	—	6	4	no	—
39	C₄²	—	3	1	no	—
40	C₂²×C₃	—	4	2	no	—
41	C₂²×C₃	—	4	4	no	—
42	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
43	C₃×C₄	C₂×D₄	1	1	yes	characteristic
44	C₃×C₄	C₂×D₄	1	1	yes	characteristic
45	Dic₃	—	2	2	no	—
46	Dic₃	—	4	2	no	—
47	C₂³	—	2	2	no	—
48	Q₈	—	6	2	no	—
49	Q₈	—	6	2	no	—
50	D₄	—	2	2	no	—
51	D₄	—	2	2	no	—
52	C₂×C₄	—	3	2	no	—
53	C₂×C₄	—	6	2	no	—
54	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
55	C₈	—	3	2	no	—
56	C₈	#67	1	2	yes	—
57	C₂×C₃	—	8	2	no	—
58	C₂×C₃	#127	1	1	yes	characteristic
59	C₂×C₃	#135	1	2	yes	—
60	C₂²	—	4	4	no	—
61	C₂²	—	4	2	no	—
62	C₂²	D₃×D₄	1	1	yes	characteristic
63	C₄	—	6	2	no	—
64	C₄	—	12	2	no	—
65	C₄	D₃×D₄	1	1	yes	characteristic
66	C₄	C₂×#67	1	1	yes	characteristic
67	C₃	#495	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	8	2	no	—
69	C₂	#912	1	2	yes	—
70	C₂	#940	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₂×C₃×D₈	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	C₂×D₈	D₃
hypocenter	C₃	#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
2. #936	C₂	semidirect
3. C₂×#214	C₂	semidirect
4. C₂×Dic₁₂	C₂	semidirect
5. C₈×Dic₃	C₂	semidirect
67. C₃	#495	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₈	2
#936	2
C₂×#214	2
C₂×Dic₁₂	2
C₈×Dic₃	2
#495	3