Subgroups — C₂×#4504 #71402

Order 384.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#818	C₂	1	2	yes	maximal
3	#4504	C₂	1	4	yes	maximal, direct product factor
4	C₂×#366	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₃	—	2	2	no	—
7	C₂×C₃×C₁₆	C₄	1	4	yes	complemented
8	C₂×C₃×C₁₆	C₂²	1	2	yes	—
9	#818	—	2	2	no	—
10	#818	C₂²	1	4	yes	—
11	C₂²×C₁₆	D₃	1	1	yes	characteristic
12	C₂×#107	—	3	2	no	—
13	#366	—	3	4	no	—
14	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
15	C₂²×Dic₃	—	4	2	no	—
16	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₂×C₃×C₈	Q₈	1	2	yes	—
19	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
20	C₄⋊Dic₃	—	2	4	no	—
21	C₄⋊Dic₃	—	4	2	no	—
22	C₃×C₁₆	C₂×C₄	1	4	yes	complemented
23	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
24	C₂×C₄⋊C₄	—	6	2	no	—
25	C₂×C₁₆	Dic₃	1	4	yes	complemented
26	C₂×C₁₆	C₂×D₃	1	2	yes	—
27	#107	—	3	4	no	—
28	#107	—	6	2	no	—
29	C₂³×C₃	D₈	1	1	yes	characteristic
30	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
31	C₂×C₃×C₄	D₈	1	1	yes	characteristic
32	C₂×C₃×C₄	Q₁₆	1	2	yes	—
33	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₃	—	4	4	no	—
35	C₂×Dic₃	—	8	4	no	—
36	C₃×C₈	C₄⋊C₄	1	2	yes	—
37	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄	—	12	2	no	—
40	C₂²×C₄	D₁₂	1	1	yes	characteristic
41	C₂×C₈	Dic₆	1	2	yes	—
42	C₂×C₈	D₁₂	1	1	yes	characteristic
43	C₂×C₈	C₂×Dic₃	1	2	yes	—
44	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
45	C₄⋊C₄	—	6	4	no	—
46	C₄⋊C₄	—	12	2	no	—
47	C₁₆	C₂×Dic₃	1	4	yes	complemented
48	C₂²×C₃	#107	1	2	yes	—
49	C₂²×C₃	D₁₆	1	2	yes	—
50	C₂²×C₃	Q₃₂	1	2	yes	—
51	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
52	C₃×C₄	#107	1	2	yes	—
53	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
55	Dic₃	—	8	4	no	—
56	C₂³	D₂₄	1	1	yes	characteristic
57	C₂×C₄	—	12	4	no	—
58	C₂×C₄	—	24	4	no	—
59	C₂×C₄	D₂₄	1	1	yes	characteristic
60	C₂×C₄	Dic₁₂	1	2	yes	—
61	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
62	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
63	C₈	C₄⋊Dic₃	1	2	yes	—
64	C₈	C₂×Dic₆	1	1	yes	characteristic
65	C₈	C₂²×Dic₃	1	1	yes	characteristic
66	C₂×C₃	#366	1	4	yes	complemented
67	C₂×C₃	C₂×#107	1	1	yes	characteristic
68	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
69	C₂×C₃	C₂×Q₃₂	1	1	yes	characteristic
70	C₂²	D₄₈	1	2	yes	—
71	C₂²	Dic₂₄	1	2	yes	—
72	C₂²	#818	1	2	yes	—
73	C₂²	C₂×D₂₄	1	1	yes	characteristic
74	C₄	—	24	4	no	—
75	C₄	#818	1	2	yes	—
76	C₄	C₂×Dic₁₂	1	1	yes	characteristic
77	C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
78	C₃	C₂×#366	1	1	yes	characteristic, complemented, Sylow
79	C₂	#4504	1	4	yes	direct product factor
80	C₂	C₂×D₄₈	1	1	yes	characteristic
81	C₂	C₂×Dic₂₄	1	1	yes	characteristic
82	C₂	C₂×#818	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₄
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂²×C₃×C₁₆	C₂
socle, prime radical	C₂³×C₃	D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂²×C₁₆	D₃
hypocenter	C₃	C₂×#366
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #4504	C₂	direct (N × H)
7. C₂×C₃×C₁₆	C₄	semidirect
22. C₃×C₁₆	C₂×C₄	semidirect
25. C₂×C₁₆	Dic₃	semidirect
47. C₁₆	C₂×Dic₃	semidirect
66. C₂×C₃	#366	semidirect
78. C₃	C₂×#366	semidirect
79. C₂	#4504	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×C₁₆	2
C₂×#818	2
#4504	2
C₂×#366	3

Subgroups — C2×#4504 #71402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4504 #71402