Subgroups — #67823

Order 384.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#504	C₂	1	1	yes	maximal, characteristic
2	#4499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#1401	—	3	1	no	maximal, Sylow
5	C₃×#131	C₄	1	1	yes	characteristic
6	C₃×M₃₂	—	2	1	no	—
7	C₃×M₃₂	C₄	1	1	yes	characteristic
8	C₂×C₃×C₁₆	—	2	1	no	—
9	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₃⋊C₃₂	—	2	1	no	—
11	C₃⋊C₃₂	—	2	1	no	—
12	#504	D₃	1	1	yes	characteristic
13	M₆₄	—	3	1	no	—
14	C₂×C₃₂	—	3	1	no	—
15	C₃×#41	C₈	1	1	yes	characteristic
16	C₃×M₁₆	—	2	1	no	—
17	C₃×M₁₆	C₈	1	1	yes	characteristic
18	C₂×C₃×C₈	—	2	1	no	—
19	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃×C₁₆	—	2	1	no	—
21	C₃×C₁₆	D₄	1	1	yes	characteristic
22	C₃×C₁₆	D₄	1	1	yes	characteristic
23	#131	Dic₃	1	1	yes	characteristic
24	M₃₂	—	2	1	no	—
25	M₃₂	Dic₃	1	1	yes	characteristic
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
28	C₃₂	—	6	1	no	—
29	C₃₂	—	6	1	no	—
30	C₃×Q₈	C₁₆	1	1	yes	characteristic
31	C₃×D₄	—	2	1	no	—
32	C₃×D₄	C₁₆	1	1	yes	characteristic
33	C₂×C₃×C₄	—	2	1	no	—
34	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
35	C₃×C₈	—	2	1	no	—
36	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₃×C₈	M₁₆	1	1	yes	characteristic
38	#41	C₃⋊C₈	1	1	yes	characteristic
39	M₁₆	—	2	1	no	—
40	M₁₆	C₃⋊C₈	1	1	yes	characteristic
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
43	C₁₆	—	2	1	no	—
44	C₁₆	#67	1	1	yes	characteristic
45	C₁₆	#67	1	1	yes	characteristic
46	C₂²×C₃	—	2	1	no	—
47	C₂²×C₃	M₃₂	1	1	yes	characteristic
48	C₃×C₄	—	2	1	no	—
49	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
50	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
51	Q₈	C₃⋊C₁₆	1	1	yes	characteristic
52	D₄	—	2	1	no	—
53	D₄	C₃⋊C₁₆	1	1	yes	characteristic
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	#208	1	1	yes	characteristic
58	C₈	#217	1	1	yes	characteristic
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	4	1	no	—
61	C₂×C₃	C₂²⋊C₁₆	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	#812	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	C₂×C₃⋊C₁₆	1	1	yes	characteristic
66	C₄	#830	1	1	yes	characteristic
67	C₃	#1401	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#4549	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#67
commutator	C₃×C₄	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×#504	C₂
socle	C₂×C₃	C₂²⋊C₁₆
prime radical	C₃×#41	C₈
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	#504	D₃
hypocenter	C₃	#1401
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
2. #4499	C₂	semidirect
3. C₂×C₃⋊C₃₂	C₂	semidirect
67. C₃	#1401	semidirect

Maximal subgroups

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

subgroup	index
C₃×#504	2
#4499	2
C₂×C₃⋊C₃₂	2
#1401	3