Subgroups — #67825

Order 384.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₆	C₂	1	1	yes	maximal, characteristic
2	#4506	C₂	1	1	yes	maximal, characteristic, complemented
3	#4499	C₂	1	1	yes	maximal, characteristic, complemented
4	#1419	—	3	1	no	maximal, Sylow
5	C₂×C₃×D₈	—	2	1	no	—
6	C₃×D₁₆	—	2	1	no	—
7	C₃×D₁₆	C₄	1	2	yes	—
8	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	#819	—	2	1	no	—
10	C₃⋊C₃₂	—	2	1	no	—
11	C₂×D₁₆	D₃	1	1	yes	characteristic
12	M₆₄	—	3	1	no	—
13	#368	—	3	1	no	—
14	C₂×C₃×D₄	—	4	1	no	—
15	C₃×D₈	—	2	2	no	—
16	C₃×D₈	—	4	1	no	—
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	#208	—	4	1	no	—
19	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×D₈	—	2	1	no	—
22	D₁₆	—	2	1	no	—
23	D₁₆	Dic₃	1	2	yes	—
24	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
25	#108	—	6	1	no	—
26	C₃₂	—	6	1	no	—
27	C₂³×C₃	—	8	1	no	—
28	C₃×D₄	—	4	2	no	—
29	C₃×D₄	—	8	1	no	—
30	C₂×C₃×C₄	D₈	1	1	yes	characteristic
31	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
32	C₃×C₈	SD₁₆	1	1	yes	characteristic
33	C₃⋊C₈	—	8	1	no	—
34	C₂×D₄	—	4	1	no	—
35	D₈	—	2	2	no	—
36	D₈	—	4	1	no	—
37	M₁₆	—	12	1	no	—
38	C₂×C₈	#67	1	1	yes	characteristic
39	C₁₆	C₂×Dic₃	1	1	yes	characteristic
40	C₁₆	#67	1	1	yes	characteristic
41	C₂²×C₃	—	8	2	no	—
42	C₂²×C₃	—	16	2	no	—
43	C₂²×C₃	SD₃₂	1	1	yes	characteristic
44	C₃×C₄	#102	1	1	yes	characteristic
45	C₃×C₄	D₁₆	1	1	yes	characteristic
46	C₂³	—	8	1	no	—
47	D₄	—	4	2	no	—
48	D₄	—	8	1	no	—
49	C₂×C₄	#213	1	1	yes	characteristic
50	C₈	—	24	1	no	—
51	C₈	#214	1	1	yes	characteristic
52	C₈	#217	1	1	yes	characteristic
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	—	16	2	no	—
55	C₂×C₃	#357	1	1	yes	characteristic
56	C₂²	—	8	2	no	—
57	C₂²	—	16	2	no	—
58	C₂²	#827	1	1	yes	characteristic
59	C₄	#826	1	1	yes	characteristic
60	C₄	#832	1	1	yes	characteristic
61	C₃	#1419	1	1	yes	characteristic, complemented, Sylow
62	C₂	—	2	1	no	—
63	C₂	—	16	2	no	—
64	C₂	#4562	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4562
commutator	C₃×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₃
Fitting, prime radical	C₂×C₃×D₁₆	C₂
socle	C₂×C₃	#357
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₂×D₁₆	D₃
hypocenter	C₃	#1419
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

Complements

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

N	H ≅ G/N	type
2. #4506	C₂	semidirect
3. #4499	C₂	semidirect
61. C₃	#1419	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₁₆	2
#4506	2
#4499	2
#1419	3