Subgroups — #7387

Order 256.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2178	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#1373	C₂	1	1	yes	maximal, characteristic, complemented
4	#508	C₄	1	1	yes	characteristic
5	#443	—	2	1	no	—
6	C₂×C₃₂	—	2	1	no	—
7	#368	C₄	1	1	yes	characteristic
8	#364	—	2	1	no	—
9	#359	—	2	1	no	—
10	C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#135	—	2	1	no	—
12	#131	—	4	1	no	—
13	Q₃₂	C₈	1	1	yes	characteristic
14	SD₃₂	—	2	1	no	—
15	D₁₆	C₈	1	1	yes	characteristic
16	M₃₂	—	4	1	no	—
17	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆	D₄	1	1	yes	characteristic
19	#108	—	2	1	no	—
20	#104	—	4	1	no	—
21	C₄×C₈	D₄	1	1	yes	characteristic
22	C₃₂	—	2	2	no	—
23	#41	—	4	1	no	—
24	Q₁₆	—	2	1	no	—
25	SD₁₆	—	4	1	no	—
26	D₈	—	2	1	no	—
27	M₁₆	—	4	1	no	—
28	M₁₆	—	8	1	no	—
29	C₂×C₈	—	8	1	no	—
30	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈	D₈	1	1	yes	characteristic
32	C₄²	SD₁₆	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	C₁₆	—	8	1	no	—
35	C₁₆	C₂×C₈	1	1	yes	characteristic
36	C₁₆	M₁₆	1	1	yes	characteristic
37	Q₈	—	4	1	no	—
38	D₄	—	4	1	no	—
39	D₄	—	8	1	no	—
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	8	1	no	—
42	C₂×C₄	#102	1	1	yes	characteristic
43	C₈	—	8	1	no	—
44	C₈	C₂²⋊C₈	1	1	yes	characteristic
45	C₈	#104	1	1	yes	characteristic
46	C₈	D₁₆	1	1	yes	characteristic
47	C₈	SD₃₂	1	1	yes	characteristic
48	C₂²	—	8	1	no	—
49	C₂²	#359	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	—	8	1	no	—
52	C₄	D₄⋊C₈	1	1	yes	characteristic
53	C₄	#357	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	16	1	no	—
56	C₂	#1331	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₆
commutator	C₁₆	C₂×C₈
Frattini, cosocle	C₄×C₁₆	C₂²
socle	C₂	#1331
prime radical	#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ 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. C₄×C₃₂	C₂	semidirect
3. #1373	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2178	2
C₄×C₃₂	2
#1373	2