Subgroups — #73014

Order 384.

87 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5391	C₂	1	2	yes	maximal
2	C₈×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	#1676	—	3	1	no	maximal, Sylow
4	C₄×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
5	#859	—	2	2	no	—
6	#818	—	4	1	no	maximal
7	#500	—	3	1	no	—
8	#477	—	3	2	no	—
9	C₈×Q₈	D₃	1	1	yes	characteristic
10	#335	—	3	1	no	—
11	#333	—	3	2	no	—
12	C₂×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
13	C₂×C₃×C₈	—	4	1	no	—
14	C₄⋊Dic₃	—	4	2	no	—
15	C₄⋊Q₈	—	3	2	no	—
16	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
17	#107	—	6	2	no	—
18	#107	—	6	1	no	—
19	#107	—	6	1	no	—
20	C₄⋊C₈	—	3	1	no	—
21	C₄⋊C₈	—	3	2	no	—
22	#103	—	6	2	no	—
23	C₄×C₈	—	3	1	no	—
24	C₂×C₃×C₄	—	4	1	no	—
25	C₂×Dic₃	—	8	2	no	—
26	SL(2,𝔽₃)	Q₁₆	1	1	yes	characteristic
27	C₃×C₈	—	4	2	no	—
28	C₂×Q₈	—	6	2	no	—
29	C₂×Q₈	D₁₂	1	1	yes	characteristic
30	C₂×C₈	—	3	1	no	—
31	C₂×C₈	—	6	2	no	—
32	C₂×C₈	S₄	1	1	yes	characteristic
33	C₄⋊C₄	—	3	1	no	—
34	C₄⋊C₄	—	6	2	no	—
35	C₄⋊C₄	—	6	2	no	—
36	C₄²	—	3	1	no	—
37	C₂²×C₃	—	4	1	no	—
38	C₃×C₄	—	4	1	no	—
39	C₃×C₄	—	4	1	no	—
40	Dic₃	—	16	2	no	—
41	Q₈	—	3	1	no	—
42	Q₈	—	24	2	no	—
43	Q₈	Dic₁₂	1	1	yes	characteristic
44	C₂×C₄	—	3	1	no	—
45	C₂×C₄	—	3	1	no	—
46	C₂×C₄	—	12	2	no	—
47	C₂×C₄	C₂×S₄	1	1	yes	characteristic
48	C₈	—	6	1	no	—
49	C₈	—	12	2	no	—
50	C₈	#226	1	2	yes	—
51	C₂×C₃	—	4	1	no	—
52	C₂×C₃	—	4	1	no	—
53	C₂×C₃	—	4	1	no	—
54	C₂²	C₄⋊S₄	1	1	yes	characteristic
55	C₄	—	3	1	no	—
56	C₄	—	3	1	no	—
57	C₄	—	6	1	no	—
58	C₄	—	24	2	no	—
59	C₄	C₂×#226	1	1	yes	characteristic
60	C₄	#984	1	1	yes	characteristic
61	C₃	—	4	1	no	Sylow
62	C₂	C₄⋊#226	1	1	yes	characteristic
63	C₂	#5398	1	1	yes	characteristic
64	C₂	#5407	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₄⋊S₄
commutator, cosocle	C₄×SL(2,𝔽₃)	C₂²
Frattini	C₂×C₄	C₂×S₄
Fitting	C₈×Q₈	D₃
prime radical	C₂×SL(2,𝔽₃)	D₄
hypercenter	C₂×C₈	S₄
hypocenter	SL(2,𝔽₃)	Q₁₆
LCS	G ▷ C₄×SL(2,𝔽₃) ▷ C₂×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Maximal subgroups

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

subgroup	index
#5391	2
C₈×SL(2,𝔽₃)	2
#1676	3
#818	4