Subgroups — C₂×#4449 #69368

Order 384.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#4449	C₂	1	4	yes	maximal, direct product factor
5	C₂×SD₆₄	—	3	1	no	maximal, Sylow
6	C₂×Dic₁₂	—	2	1	no	—
7	C₂×D₂₄	—	2	1	no	—
8	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	Dic₂₄	—	2	1	no	—
10	Dic₂₄	C₂²	1	2	yes	complemented
11	D₄₈	—	2	1	no	—
12	D₄₈	C₂²	1	2	yes	—
13	C₃×C₃₂	C₂²	1	2	yes	complemented
14	C₂×Q₃₂	—	3	1	no	—
15	C₂×D₁₆	—	3	1	no	—
16	SD₆₄	—	3	4	no	—
17	C₂×C₃₂	D₃	1	1	yes	characteristic, complemented
18	C₂×D₁₂	—	4	1	no	—
19	C₂×Dic₆	—	4	1	no	—
20	C₂×C₃×C₈	D₄	1	1	yes	characteristic
21	Dic₁₂	—	2	2	no	—
22	Dic₁₂	—	4	1	no	—
23	D₂₄	—	2	2	no	—
24	D₂₄	—	4	1	no	—
25	C₃×C₁₆	D₄	1	1	yes	characteristic
26	C₃×C₁₆	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆	—	6	1	no	—
28	C₂×D₈	—	6	1	no	—
29	Q₃₂	—	3	2	no	—
30	Q₃₂	—	6	1	no	—
31	D₁₆	—	3	2	no	—
32	D₁₆	—	6	1	no	—
33	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
34	C₃₂	C₂×D₃	1	2	yes	complemented
35	C₂²×D₃	—	8	1	no	—
36	C₂×C₃×C₄	D₈	1	1	yes	characteristic
37	C₂×Dic₃	—	8	1	no	—
38	D₁₂	—	4	2	no	—
39	D₁₂	—	8	1	no	—
40	Dic₆	—	4	2	no	—
41	Dic₆	—	8	1	no	—
42	C₃×C₈	D₈	1	1	yes	characteristic
43	C₃×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂×Q₈	—	12	1	no	—
45	C₂×D₄	—	12	1	no	—
46	Q₁₆	—	6	2	no	—
47	Q₁₆	—	12	1	no	—
48	D₈	—	6	2	no	—
49	D₈	—	12	1	no	—
50	C₂×C₈	D₁₂	1	1	yes	characteristic
51	C₁₆	D₁₂	1	1	yes	characteristic
52	C₁₆	C₂²×D₃	1	1	yes	characteristic
53	C₂²×C₃	D₁₆	1	1	yes	characteristic
54	C₂×D₃	—	8	2	no	—
55	C₂×D₃	—	16	2	no	—
56	C₃×C₄	D₁₆	1	1	yes	characteristic
57	C₃×C₄	C₂×D₈	1	1	yes	characteristic
58	Dic₃	—	8	2	no	—
59	C₂³	—	24	1	no	—
60	Q₈	—	12	2	no	—
61	Q₈	—	24	1	no	—
62	D₄	—	12	2	no	—
63	D₄	—	24	1	no	—
64	C₂×C₄	—	24	1	no	—
65	C₂×C₄	D₂₄	1	1	yes	characteristic
66	C₈	D₂₄	1	1	yes	characteristic
67	C₈	C₂×D₁₂	1	1	yes	characteristic
68	C₂×C₃	SD₆₄	1	2	yes	complemented
69	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
70	D₃	—	16	2	no	—
71	C₂²	—	24	2	no	—
72	C₂²	—	48	2	no	—
73	C₂²	D₄₈	1	1	yes	characteristic
74	C₄	—	24	2	no	—
75	C₄	D₄₈	1	1	yes	characteristic
76	C₄	C₂×D₂₄	1	1	yes	characteristic
77	C₃	C₂×SD₆₄	1	1	yes	characteristic, complemented, Sylow
78	C₂	—	48	2	no	—
79	C₂	#4449	1	2	yes	direct product factor
80	C₂	C₂×D₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄₈
commutator, cosocle	C₃×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₃
Fitting	C₂×C₃×C₃₂	C₂
socle	C₂²×C₃	D₁₆
prime radical	C₂×D₄₈	C₂
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	C₂×SD₆₄
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₂₄	C₂	semidirect
3. C₂×C₃×C₃₂	C₂	semidirect
4. #4449	C₂	direct (N × H)
10. Dic₂₄	C₂²	semidirect
13. C₃×C₃₂	C₂²	semidirect
17. C₂×C₃₂	D₃	semidirect
34. C₃₂	C₂×D₃	semidirect
68. C₂×C₃	SD₆₄	semidirect
77. C₃	C₂×SD₆₄	semidirect
79. C₂	#4449	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄	2
C₂×D₄₈	2
C₂×C₃×C₃₂	2
#4449	2
C₂×SD₆₄	3

Subgroups — C2×#4449 #69368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4449 #69368