Subgroups — #73038

Order 384.

87 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5404	C₂	1	1	yes	maximal, characteristic
2	#504⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#4628	C₂	1	1	yes	maximal, characteristic, complemented
4	#2170	—	3	1	no	maximal, Sylow
5	#985	C₄	1	1	yes	characteristic
6	#131⋊C₃	C₂²	1	1	yes	characteristic
7	GU(2,𝔽₃)	C₄	1	1	yes	characteristic
8	D₃×C₁₆	—	4	1	no	maximal
9	#504	—	3	1	no	—
10	#504	D₃	1	1	yes	characteristic, complemented
11	#443	—	3	1	no	—
12	#364	—	3	1	no	—
13	#350	—	3	1	no	—
14	#350	—	3	1	no	—
15	C₄×C₁₆	—	3	1	no	—
16	#41⋊C₃	C₂×C₄	1	1	yes	characteristic
17	GL(2,𝔽₃)	C₈	1	1	yes	characteristic
18	#226	C₈	1	1	yes	characteristic
19	D₃×C₈	—	4	1	no	—
20	C₃×C₁₆	—	4	1	no	—
21	C₃⋊C₁₆	—	4	1	no	—
22	#135	—	3	1	no	—
23	#131	—	3	1	no	—
24	#131	C₂×D₃	1	1	yes	characteristic
25	M₃₂	—	3	1	no	—
26	M₃₂	—	3	1	no	—
27	M₃₂	—	6	1	no	—
28	C₂×C₁₆	—	3	1	no	—
29	C₂×C₁₆	—	3	1	no	—
30	C₂×C₁₆	—	6	1	no	—
31	#108	—	3	1	no	—
32	#104	—	3	1	no	—
33	#104	—	3	1	no	—
34	C₄×C₈	—	3	1	no	—
35	C₄×D₃	—	4	1	no	—
36	SL(2,𝔽₃)	C₂×C₈	1	1	yes	characteristic
37	C₃×C₈	—	4	1	no	—
38	C₃⋊C₈	—	4	1	no	—
39	#41	—	3	1	no	—
40	#41	C₄×D₃	1	1	yes	characteristic
41	Q₁₆	—	3	1	no	—
42	SD₁₆	—	3	1	no	—
43	SD₁₆	—	3	1	no	—
44	D₈	—	3	1	no	—
45	M₁₆	—	3	1	no	—
46	M₁₆	—	3	1	no	—
47	M₁₆	—	6	1	no	—
48	C₂×C₈	—	3	1	no	—
49	C₂×C₈	—	3	1	no	—
50	C₂×C₈	—	6	1	no	—
51	C₄²	—	3	1	no	—
52	C₁₆	—	3	1	no	—
53	C₁₆	—	6	1	no	—
54	C₁₆	—	6	1	no	—
55	C₁₆	S₄	1	1	yes	characteristic
56	C₂×D₃	—	4	1	no	—
57	C₃×C₄	—	4	1	no	—
58	Dic₃	—	4	1	no	—
59	Q₈	—	3	1	no	—
60	Q₈	D₃×C₈	1	1	yes	characteristic
61	D₄	—	3	1	no	—
62	D₄	—	3	1	no	—
63	D₄	—	6	1	no	—
64	C₂×C₄	—	3	1	no	—
65	C₂×C₄	—	6	1	no	—
66	C₂×C₄	—	6	1	no	—
67	C₈	—	3	1	no	—
68	C₈	—	3	1	no	—
69	C₈	—	3	1	no	—
70	C₈	—	6	1	no	—
71	C₈	C₂×S₄	1	1	yes	characteristic
72	C₂×C₃	—	4	1	no	—
73	D₃	—	4	2	no	—
74	C₂²	—	3	1	no	—
75	C₂²	—	6	1	no	—
76	C₄	—	3	1	no	—
77	C₄	—	6	2	no	—
78	C₄	—	6	1	no	—
79	C₄	C₄×S₄	1	1	yes	characteristic
80	C₃	—	4	1	no	Sylow
81	C₂	—	6	1	no	—
82	C₂	—	12	1	no	—
83	C₂	C₈×S₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₁₆	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₂×C₈
Frattini	C₈	C₂×S₄
Fitting	#504	D₃
socle	C₂	C₈×S₄
prime radical	#985	C₄
cosocle	#131⋊C₃	C₂²
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₁₆

Complements

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

N	H ≅ G/N	type
2. #504⋊C₃	C₂	semidirect
3. #4628	C₂	semidirect
10. #504	D₃	semidirect

Maximal subgroups

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

subgroup	index
#5404	2
#504⋊C₃	2
#4628	2
#2170	3
D₃×C₁₆	4