Subgroups — C₂²×#874 #85678

Order 384.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₂	C₂	1	6	yes	maximal, complemented
2	C₂²×C₃×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#874	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#127	—	3	1	no	maximal, Sylow
5	C₂⁴×D₃	C₂²	1	4	yes	—
6	C₂³×C₃×C₄	C₂²	1	3	yes	—
7	C₂²×D₁₂	—	2	48	no	—
8	C₂²×D₁₂	C₂²	1	72	yes	complemented
9	C₂×C₃×C₄²	C₂²	1	12	yes	complemented
10	#874	C₂²	1	64	yes	direct product factor
11	C₂³×D₄	—	3	6	no	—
12	C₂×#127	—	3	24	no	—
13	C₂²×C₄²	D₃	1	1	yes	characteristic, complemented
14	C₂⁴×C₃	C₂³	1	1	yes	characteristic
15	C₂³×D₃	—	2	48	no	—
16	C₂³×D₃	C₂³	1	24	yes	—
17	C₂²×C₃×C₄	D₄	1	24	yes	complemented
18	C₂²×C₃×C₄	C₂³	1	18	yes	—
19	C₂×D₁₂	—	2	288	no	—
20	C₂×D₁₂	C₂³	1	96	yes	complemented
21	C₃×C₄²	C₂³	1	16	yes	complemented
22	C₂⁵	—	3	4	no	—
23	C₂²×D₄	—	3	72	no	—
24	C₂²×D₄	—	6	48	no	—
25	C₂³×C₄	C₂×D₃	1	3	yes	—
26	#127	—	3	64	no	—
27	C₂×C₄²	C₂×D₃	1	12	yes	complemented
28	C₂³×C₃	C₂×D₄	1	12	yes	—
29	C₂³×C₃	C₂⁴	1	3	yes	—
30	C₂²×D₃	—	2	144	no	—
31	C₂²×D₃	—	4	64	no	—
32	C₂²×D₃	C₂⁴	1	16	yes	—
33	C₂×C₃×C₄	C₂×D₄	1	72	yes	complemented
34	C₂×C₃×C₄	C₂⁴	1	12	yes	—
35	D₁₂	—	2	192	no	—
36	C₂⁴	—	3	24	no	—
37	C₂⁴	—	6	48	no	—
38	C₂⁴	C₂²×D₃	1	1	yes	characteristic
39	C₂×D₄	—	3	96	no	—
40	C₂×D₄	—	6	288	no	—
41	C₂²×C₄	D₁₂	1	24	yes	complemented
42	C₂²×C₄	C₂²×D₃	1	18	yes	—
43	C₄²	C₂²×D₃	1	16	yes	complemented
44	C₂²×C₃	#127	1	16	yes	complemented
45	C₂²×C₃	C₂²×D₄	1	18	yes	—
46	C₂²×C₃	C₂⁵	1	1	yes	characteristic
47	C₂×D₃	—	2	48	no	—
48	C₂×D₃	—	4	96	no	—
49	C₃×C₄	C₂²×D₄	1	24	yes	complemented
50	C₂³	—	3	16	no	—
51	C₂³	—	6	144	no	—
52	C₂³	—	12	64	no	—
53	C₂³	C₂×D₁₂	1	12	yes	—
54	C₂³	C₂³×D₃	1	3	yes	—
55	D₄	—	6	192	no	—
56	C₂×C₄	C₂×D₁₂	1	72	yes	complemented
57	C₂×C₄	C₂³×D₃	1	12	yes	—
58	C₂×C₃	C₂×#127	1	12	yes	complemented
59	C₂×C₃	C₂³×D₄	1	3	yes	—
60	D₃	—	4	16	no	—
61	C₂²	—	6	48	no	—
62	C₂²	—	12	96	no	—
63	C₂²	#874	1	16	yes	direct product factor
64	C₂²	C₂²×D₁₂	1	18	yes	—
65	C₂²	C₂⁴×D₃	1	1	yes	characteristic
66	C₄	C₂²×D₁₂	1	24	yes	complemented
67	C₃	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	12	16	no	—
69	C₂	C₂×#874	1	12	yes	direct product factor
70	C₂	C₂³×D₁₂	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂²×C₃×C₄²	C₂
socle	C₂⁴×C₃	C₂³
hypercenter	C₂²×C₄²	D₃
hypocenter	C₃	C₂²×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×D₁₂	C₂	semidirect
2. C₂²×C₃×C₄²	C₂	semidirect
3. C₂×#874	C₂	direct (N × H)
8. C₂²×D₁₂	C₂²	semidirect
9. C₂×C₃×C₄²	C₂²	semidirect
10. #874	C₂²	direct (N × H)
13. C₂²×C₄²	D₃	semidirect
17. C₂²×C₃×C₄	D₄	semidirect
20. C₂×D₁₂	C₂³	semidirect
21. C₃×C₄²	C₂³	semidirect
27. C₂×C₄²	C₂×D₃	semidirect
33. C₂×C₃×C₄	C₂×D₄	semidirect
41. C₂²×C₄	D₁₂	semidirect
43. C₄²	C₂²×D₃	semidirect
44. C₂²×C₃	#127	semidirect
49. C₃×C₄	C₂²×D₄	semidirect
56. C₂×C₄	C₂×D₁₂	semidirect
58. C₂×C₃	C₂×#127	semidirect
63. C₂²	#874	direct (N × H)
66. C₄	C₂²×D₁₂	semidirect
67. C₃	C₂²×#127	semidirect
69. C₂	C₂×#874	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₁₂	2
C₂²×C₃×C₄²	2
C₂×#874	2
C₂²×#127	3

Subgroups — C22×#874 #85678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#874 #85678