Subgroups — #5151

Order 192.

162 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, complemented
2	#940	C₂	1	2	yes	maximal, complemented
3	C₂×#213	C₂	1	2	yes	maximal, complemented
4	C₂×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#493	—	3	1	no	maximal, Sylow
7	C₂×C₃×D₄	C₂²	1	2	yes	—
8	C₂×#67	—	2	4	no	—
9	C₂×D₁₂	C₂²	1	2	yes	—
10	C₃×D₈	—	2	2	no	—
11	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
12	#213	—	2	4	no	—
13	C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	D₂₄	—	2	2	no	—
16	C₂×D₈	—	3	2	no	—
17	C₂×D₈	—	3	1	no	—
18	C₂×D₈	D₃	1	1	yes	characteristic, complemented
19	#127	—	3	2	no	—
20	C₄×C₈	—	3	1	no	—
21	C₂³×C₃	—	2	2	no	—
22	C₂²×D₃	—	2	2	no	—
23	C₃×D₄	—	2	2	no	—
24	C₃×D₄	—	2	2	no	—
25	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
26	#67	—	4	8	no	—
27	C₂×Dic₃	D₄	1	2	yes	—
28	D₁₂	—	2	2	no	—
29	D₁₂	—	2	2	no	—
30	C₃×C₈	D₄	1	2	yes	complemented
31	C₃⋊C₈	D₄	1	2	yes	complemented
32	C₂×D₄	—	3	2	no	—
33	C₂×D₄	—	6	4	no	—
34	C₂×D₄	C₂×D₃	1	2	yes	—
35	D₈	—	2	2	no	—
36	D₈	—	6	4	no	—
37	D₈	—	6	2	no	—
38	C₂×C₈	—	3	1	no	—
39	C₂×C₈	C₂×D₃	1	1	yes	characteristic
40	C₄²	—	3	1	no	—
41	C₂²×C₃	—	4	2	no	—
42	C₂²×C₃	—	4	4	no	—
43	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
44	C₂×D₃	—	4	2	no	—
45	C₂×D₃	—	4	4	no	—
46	C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	Dic₃	D₈	1	4	yes	complemented
49	C₂³	—	2	2	no	—
50	C₂³	—	6	2	no	—
51	D₄	—	2	2	no	—
52	D₄	—	2	2	no	—
53	D₄	—	6	2	no	—
54	D₄	—	6	2	no	—
55	D₄	—	12	8	no	—
56	C₂×C₄	—	3	2	no	—
57	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
58	C₈	—	3	2	no	—
59	C₈	#67	1	2	yes	complemented
60	C₂×C₃	—	8	2	no	—
61	C₂×C₃	#127	1	1	yes	characteristic
62	C₂×C₃	C₂×D₈	1	2	yes	—
63	D₃	—	8	2	no	—
64	C₂²	—	4	4	no	—
65	C₂²	—	4	2	no	—
66	C₂²	—	12	4	no	—
67	C₂²	—	12	2	no	—
68	C₂²	D₃×D₄	1	1	yes	characteristic
69	C₄	—	3	4	no	—
70	C₄	D₃×D₄	1	1	yes	characteristic
71	C₄	C₂×#67	1	1	yes	characteristic
72	C₃	#493	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	8	2	no	—
74	C₂	—	24	2	no	—
75	C₂	D₃×D₈	1	2	yes	—
76	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	C₂×C₃×D₈	C₂
socle	C₂²×C₃	C₂×D₄
hypercenter	C₂×D₈	D₃
hypocenter	C₃	#493
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
1. C₂×C₃×D₈	C₂	semidirect
2. #940	C₂	semidirect
3. C₂×#213	C₂	semidirect
4. C₂×D₂₄	C₂	semidirect
5. C₈×Dic₃	C₂	semidirect
18. C₂×D₈	D₃	semidirect
30. C₃×C₈	D₄	semidirect
31. C₃⋊C₈	D₄	semidirect
48. Dic₃	D₈	semidirect
59. C₈	#67	semidirect
72. C₃	#493	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
#940	2
C₂×#213	2
C₂×D₂₄	2
C₈×Dic₃	2
#493	3