Subgroups — #7381

Order 256.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2170	C₂	1	1	yes	maximal, characteristic
2	#1422	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#504	—	2	1	no	—
5	#443	C₄	1	1	yes	characteristic
6	M₆₄	—	2	1	no	—
7	C₂×C₃₂	—	2	1	no	—
8	#364	C₄	1	1	yes	characteristic
9	#350	—	2	1	no	—
10	C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#135	C₈	1	1	yes	characteristic
12	#131	—	2	1	no	—
13	M₃₂	—	2	1	no	—
14	M₃₂	—	4	1	no	—
15	C₂×C₁₆	—	4	1	no	—
16	C₂×C₁₆	D₄	1	1	yes	characteristic
17	C₂×C₁₆	D₄	1	1	yes	characteristic
18	#108	C₈	1	1	yes	characteristic
19	#104	—	2	1	no	—
20	C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₃₂	—	2	2	no	—
22	C₃₂	—	4	1	no	—
23	#41	—	2	1	no	—
24	Q₁₆	C₁₆	1	1	yes	characteristic
25	SD₁₆	—	2	1	no	—
26	D₈	C₁₆	1	1	yes	characteristic
27	M₁₆	—	2	1	no	—
28	M₁₆	—	4	1	no	—
29	C₂×C₈	—	4	1	no	—
30	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈	C₂×C₈	1	1	yes	characteristic
32	C₄²	M₁₆	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	C₁₆	—	4	1	no	—
35	C₁₆	D₈	1	1	yes	characteristic
36	C₁₆	SD₁₆	1	1	yes	characteristic
37	Q₈	—	2	1	no	—
38	D₄	—	2	1	no	—
39	D₄	—	4	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	4	1	no	—
42	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
43	C₈	—	4	1	no	—
44	C₈	#102	1	1	yes	characteristic
45	C₈	#104	1	1	yes	characteristic
46	C₈	C₂×C₁₆	1	1	yes	characteristic
47	C₈	M₃₂	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	#350	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	4	1	no	—
52	C₄	D₄⋊C₈	1	1	yes	characteristic
53	C₄	C₂²⋊C₁₆	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	8	1	no	—
56	C₂	D₄⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₈
commutator	C₈	C₂×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂²
socle	C₂	D₄⋊C₁₆
prime radical	#135	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #1422	C₂	semidirect
3. C₄×C₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2170	2
#1422	2
C₄×C₃₂	2