Subgroups — #67595

Order 384.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#4499	C₂	1	1	yes	maximal, characteristic, complemented
4	#1401	—	3	1	no	maximal, Sylow
5	#901	C₄	1	1	yes	characteristic
6	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
7	#812	C₄	1	1	yes	characteristic
8	#798	—	2	1	no	—
9	D₃×C₁₆	—	2	1	no	—
10	C₃×C₃₂	—	2	1	no	—
11	C₃⋊C₃₂	—	2	1	no	—
12	#504	—	3	1	no	—
13	M₆₄	—	3	1	no	—
14	C₂×C₃₂	D₃	1	1	yes	characteristic, complemented
15	#235	C₈	1	1	yes	characteristic
16	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
17	#208	C₈	1	1	yes	characteristic
18	ΓL(1,𝔽₂₅)	—	2	1	no	—
19	D₃×C₈	—	2	1	no	—
20	C₃×C₁₆	D₄	1	1	yes	characteristic
21	C₃×C₁₆	D₄	1	1	yes	characteristic
22	C₃⋊C₁₆	—	2	1	no	—
23	#131	—	3	1	no	—
24	M₃₂	—	3	1	no	—
25	M₃₂	—	6	1	no	—
26	C₂×C₁₆	—	6	1	no	—
27	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
28	C₃₂	—	2	1	no	—
29	C₃₂	—	6	1	no	—
30	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
31	#67	—	2	1	no	—
32	D₁₂	C₁₆	1	1	yes	characteristic
33	C₄×D₃	—	2	1	no	—
34	Dic₆	C₁₆	1	1	yes	characteristic
35	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
36	C₃×C₈	M₁₆	1	1	yes	characteristic
37	C₃⋊C₈	—	2	1	no	—
38	#41	—	3	1	no	—
39	M₁₆	—	3	1	no	—
40	M₁₆	—	6	1	no	—
41	C₂×C₈	—	6	1	no	—
42	C₂×C₈	C₄×D₃	1	1	yes	characteristic
43	C₁₆	—	6	1	no	—
44	C₁₆	D₁₂	1	1	yes	characteristic
45	C₁₆	#67	1	1	yes	characteristic
46	C₂²×C₃	M₃₂	1	1	yes	characteristic
47	C₂×D₃	—	2	1	no	—
48	C₃×C₄	C₂²⋊C₈	1	1	yes	characteristic
49	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
50	Dic₃	—	2	1	no	—
51	Q₈	—	3	1	no	—
52	D₄	—	3	1	no	—
53	D₄	—	6	1	no	—
54	C₂×C₄	—	6	1	no	—
55	C₂×C₄	D₃×C₈	1	1	yes	characteristic
56	C₈	—	6	1	no	—
57	C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
58	C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	C₂²⋊C₁₆	1	1	yes	characteristic
61	D₃	—	4	1	no	—
62	C₂²	—	6	1	no	—
63	C₂²	#798	1	1	yes	characteristic
64	C₄	—	6	1	no	—
65	C₄	D₃×C₁₆	1	1	yes	characteristic
66	C₄	(C₂×D₃)⋊C₈	1	1	yes	characteristic
67	C₃	#1401	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	1	no	—
69	C₂	—	12	1	no	—
70	C₂	(C₂×D₃)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₁₂
commutator	C₃×C₄	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₂×C₃×C₃₂	C₂
socle	C₂×C₃	C₂²⋊C₁₆
prime radical	#235	C₈
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	#1401
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃₂	C₂	semidirect
3. #4499	C₂	semidirect
14. C₂×C₃₂	D₃	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
#4901	2
C₂×C₃×C₃₂	2
#4499	2
#1401	3