Subgroups — #67803

Order 384.

82 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×C₃⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₃₂	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
5	C₂×C₃×C₁₆	—	2	2	no	—
6	C₂×C₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₃⋊C₃₂	—	2	2	no	—
9	C₂²×C₁₆	D₃	1	1	yes	characteristic
10	C₂×C₃₂	—	3	2	no	—
11	C₂²×C₃×C₄	C₈	1	1	yes	characteristic
12	C₂×C₃×C₈	—	2	1	no	—
13	C₂×C₃×C₈	—	2	1	no	—
14	C₂×C₃×C₈	C₈	1	1	yes	characteristic
15	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₃×C₁₆	—	2	1	no	—
17	C₃×C₁₆	D₄	1	2	yes	—
18	C₂²×C₈	Dic₃	1	1	yes	characteristic
19	C₂×C₁₆	—	2	2	no	—
20	C₂×C₁₆	Dic₃	1	1	yes	characteristic
21	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
22	C₃₂	—	6	2	no	—
23	C₂³×C₃	C₁₆	1	1	yes	characteristic
24	C₂×C₃×C₄	—	2	1	no	—
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	C₁₆	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
28	C₃×C₈	—	2	1	no	—
29	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
30	C₃×C₈	M₁₆	1	1	yes	characteristic
31	C₂²×C₄	C₃⋊C₈	1	1	yes	characteristic
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	C₃⋊C₈	1	1	yes	characteristic
35	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
36	C₁₆	—	2	1	no	—
37	C₁₆	#67	1	2	yes	—
38	C₂²×C₃	—	2	1	no	—
39	C₂²×C₃	—	2	1	no	—
40	C₂²×C₃	C₃₂	1	2	yes	complemented
41	C₂²×C₃	C₂×C₁₆	1	1	yes	characteristic
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
44	C₃×C₄	M₃₂	1	1	yes	characteristic
45	C₂³	C₃⋊C₁₆	1	1	yes	characteristic
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	C₃⋊C₁₆	1	1	yes	characteristic
49	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
50	C₈	—	2	1	no	—
51	C₈	#208	1	1	yes	characteristic
52	C₈	#217	1	1	yes	characteristic
53	C₂×C₃	—	2	2	no	—
54	C₂×C₃	C₂²⋊C₁₆	1	1	yes	characteristic
55	C₂×C₃	C₂×C₃₂	1	1	yes	characteristic
56	C₂×C₃	M₆₄	1	1	yes	characteristic
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	C₃⋊C₃₂	1	2	yes	complemented
60	C₂²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	#812	1	1	yes	characteristic
63	C₄	#830	1	1	yes	characteristic
64	C₃	C₂²⋊C₃₂	1	1	yes	characteristic, complemented, Sylow
65	C₂	—	2	2	no	—
66	C₂	C₂×C₃⋊C₃₂	1	1	yes	characteristic
67	C₂	#4499	1	1	yes	characteristic
68	C₂	#4549	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊C₃₂	C₂	semidirect
40. C₂²×C₃	C₃₂	semidirect
59. C₂²	C₃⋊C₃₂	semidirect
64. C₃	C₂²⋊C₃₂	semidirect

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×C₃⋊C₃₂	2
C₂²⋊C₃₂	3