Subgroups — #68006

Order 384.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504⋊C₃	C₂	1	1	yes	maximal, characteristic
2	#1401	—	3	1	no	maximal, Sylow
3	#131⋊C₃	C₄	1	1	yes	characteristic
4	C₃⋊C₃₂	—	4	1	no	maximal
5	#504	D₃	1	1	yes	characteristic
6	M₆₄	—	3	1	no	—
7	C₂×C₃₂	—	3	1	no	—
8	#41⋊C₃	C₈	1	1	yes	characteristic
9	C₃×C₁₆	—	4	1	no	—
10	#131	Dic₃	1	1	yes	characteristic
11	M₃₂	—	3	1	no	—
12	C₂×C₁₆	—	3	1	no	—
13	C₃₂	—	6	1	no	—
14	C₃₂	—	6	1	no	—
15	SL(2,𝔽₃)	C₁₆	1	1	yes	characteristic
16	C₃×C₈	—	4	1	no	—
17	#41	C₃⋊C₈	1	1	yes	characteristic
18	M₁₆	—	3	1	no	—
19	C₂×C₈	—	3	1	no	—
20	C₁₆	—	3	1	no	—
21	C₁₆	S₄	1	1	yes	characteristic
22	C₃×C₄	—	4	1	no	—
23	Q₈	C₃⋊C₁₆	1	1	yes	characteristic
24	D₄	—	3	1	no	—
25	C₂×C₄	—	3	1	no	—
26	C₈	—	3	1	no	—
27	C₈	SL(2,ℤ/4)	1	1	yes	characteristic
28	C₂×C₃	—	4	1	no	—
29	C₂²	—	3	1	no	—
30	C₄	—	3	1	no	—
31	C₄	A₄⋊C₈	1	1	yes	characteristic
32	C₃	—	4	1	no	Sylow
33	C₂	—	6	1	no	—
34	C₂	A₄⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₁₆	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₁₆
Fitting	#504	D₃
socle	C₂	A₄⋊C₁₆
prime radical	#41⋊C₃	C₈
cosocle	#504⋊C₃	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₁₆

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