Subgroups — #67809

Order 384.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#4499	C₂	1	2	yes	maximal, complemented
3	#1400	—	3	1	no	maximal, Sylow
4	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
5	C₃×M₃₂	—	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	C₂×M₃₂	D₃	1	1	yes	characteristic
10	M₆₄	—	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	M₃₂	—	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	—	4	1	no	—
40	C₂²×C₃	C₂×C₁₆	1	1	yes	characteristic
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
43	C₃×C₄	M₃₂	1	1	yes	characteristic
44	C₂³	C₃⋊C₁₆	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	C₃⋊C₁₆	1	1	yes	characteristic
48	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
49	C₈	—	2	1	no	—
50	C₈	#208	1	1	yes	characteristic
51	C₈	#217	1	1	yes	characteristic
52	C₂×C₃	—	2	1	no	—
53	C₂×C₃	—	4	1	no	—
54	C₂×C₃	C₂²⋊C₁₆	1	1	yes	characteristic
55	C₂²	—	2	1	no	—
56	C₂²	—	4	1	no	—
57	C₂²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	#812	1	1	yes	characteristic
60	C₄	#830	1	1	yes	characteristic
61	C₃	#1400	1	1	yes	characteristic, complemented, Sylow
62	C₂	—	2	1	no	—
63	C₂	—	4	1	no	—
64	C₂	#4549	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#217
commutator	C₂²×C₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₂×C₃×M₃₂	C₂
socle	C₂×C₃	C₂²⋊C₁₆
prime radical	C₂³×C₃	C₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	#1400
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. #4499	C₂	semidirect
61. C₃	#1400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×M₃₂	2
#4499	2
#1400	3