Subgroups — C₂²×#213 #5792

Order 192.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#213	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈	—	3	1	no	maximal, Sylow
6	C₂⁴×C₃	—	2	1	no	—
7	C₂³×D₃	—	2	1	no	—
8	C₂×C₃×D₄	—	2	3	no	—
9	C₂×C₃×D₄	C₂²	1	6	yes	complemented
10	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
11	C₂×D₁₂	—	2	3	no	—
12	C₂×D₁₂	C₂²	1	6	yes	complemented
13	#213	C₂²	1	16	yes	direct product factor
14	C₂×C₃⋊C₈	C₂²	1	6	yes	complemented
15	C₂²×D₄	—	3	1	no	—
16	C₂²×D₄	D₃	1	1	yes	characteristic, complemented
17	C₂×D₈	—	3	12	no	—
18	C₂²×C₈	—	3	1	no	—
19	C₂³×C₃	—	2	6	no	—
20	C₂³×C₃	—	4	4	no	—
21	C₂³×C₃	D₄	1	1	yes	characteristic
22	C₂²×D₃	—	2	6	no	—
23	C₂²×D₃	—	4	4	no	—
24	C₃×D₄	—	2	6	no	—
25	C₃×D₄	C₂³	1	4	yes	complemented
26	C₂×C₃×C₄	D₄	1	3	yes	—
27	C₂×C₃×C₄	C₂³	1	3	yes	—
28	D₁₂	—	2	6	no	—
29	D₁₂	C₂³	1	4	yes	complemented
30	C₃⋊C₈	C₂³	1	4	yes	complemented
31	C₂⁴	—	2	1	no	—
32	C₂⁴	—	6	1	no	—
33	C₂×D₄	—	2	3	no	—
34	C₂×D₄	—	3	6	no	—
35	C₂×D₄	—	6	3	no	—
36	C₂×D₄	C₂×D₃	1	6	yes	complemented
37	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
38	D₈	—	3	16	no	—
39	C₂×C₈	—	3	6	no	—
40	C₂²×C₃	—	2	4	no	—
41	C₂²×C₃	—	4	12	no	—
42	C₂²×C₃	D₈	1	4	yes	complemented
43	C₂²×C₃	C₂×D₄	1	3	yes	—
44	C₂×D₃	—	2	4	no	—
45	C₂×D₃	—	4	12	no	—
46	C₃×C₄	C₂×D₄	1	3	yes	—
47	C₃×C₄	C₂⁴	1	1	yes	characteristic
48	C₂³	—	2	6	no	—
49	C₂³	—	4	4	no	—
50	C₂³	—	6	6	no	—
51	C₂³	—	12	4	no	—
52	C₂³	#67	1	1	yes	characteristic
53	D₄	—	2	6	no	—
54	D₄	—	3	4	no	—
55	D₄	—	6	6	no	—
56	D₄	C₂²×D₃	1	4	yes	complemented
57	C₂×C₄	#67	1	3	yes	—
58	C₂×C₄	C₂²×D₃	1	3	yes	—
59	C₈	—	3	4	no	—
60	C₂×C₃	—	4	4	no	—
61	C₂×C₃	C₂×D₈	1	6	yes	complemented
62	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
63	D₃	—	4	4	no	—
64	C₂²	—	2	4	no	—
65	C₂²	—	4	12	no	—
66	C₂²	—	6	4	no	—
67	C₂²	—	12	12	no	—
68	C₂²	#213	1	4	yes	direct product factor
69	C₂²	C₂×#67	1	3	yes	—
70	C₄	C₂×#67	1	3	yes	—
71	C₄	C₂³×D₃	1	1	yes	characteristic
72	C₃	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
73	C₂	—	4	4	no	—
74	C₂	—	12	4	no	—
75	C₂	C₂×#213	1	6	yes	direct product factor
76	C₂	C₂²×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#67
commutator, cosocle	C₃×C₄	C₂⁴
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×D₄	C₂
socle	C₂³×C₃	D₄
hypercenter	C₂²×D₄	D₃
hypocenter	C₃	C₂²×D₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₄	C₂	semidirect
2. C₂²×D₁₂	C₂	semidirect
3. C₂×#213	C₂	direct (N × H)
4. C₂²×C₃⋊C₈	C₂	semidirect
9. C₂×C₃×D₄	C₂²	semidirect
12. C₂×D₁₂	C₂²	semidirect
13. #213	C₂²	direct (N × H)
14. C₂×C₃⋊C₈	C₂²	semidirect
16. C₂²×D₄	D₃	semidirect
25. C₃×D₄	C₂³	semidirect
29. D₁₂	C₂³	semidirect
30. C₃⋊C₈	C₂³	semidirect
36. C₂×D₄	C₂×D₃	semidirect
42. C₂²×C₃	D₈	semidirect
56. D₄	C₂²×D₃	semidirect
61. C₂×C₃	C₂×D₈	semidirect
68. C₂²	#213	direct (N × H)
72. C₃	C₂²×D₈	semidirect
75. C₂	C₂×#213	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₄	2
C₂²×D₁₂	2
C₂×#213	2
C₂²×C₃⋊C₈	2
C₂²×D₈	3

Subgroups — C22×#213 #5792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#213 #5792