Subgroups — #7362

Order 256.

69 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₃₂	C₂	1	2	yes	maximal
3	C₂²×C₁₆	C₄	1	1	yes	characteristic
4	C₂²×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂×C₃₂	—	2	2	no	—
7	C₄⋊C₁₆	—	2	2	no	—
8	C₂²×C₈	C₈	1	1	yes	characteristic
9	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
10	C₂×C₄²	C₈	1	1	yes	characteristic
11	C₂×C₁₆	—	2	1	no	—
12	C₂×C₁₆	—	4	1	no	—
13	C₂×C₁₆	—	4	1	no	—
14	C₂×C₁₆	D₄	1	2	yes	—
15	C₄×C₈	—	2	1	no	—
16	C₄×C₈	—	2	1	no	—
17	C₃₂	—	4	2	no	—
18	C₂²×C₄	C₁₆	1	1	yes	characteristic
19	C₂²×C₄	C₁₆	1	1	yes	characteristic
20	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
21	C₂×C₈	—	2	1	no	—
22	C₂×C₈	—	2	1	no	—
23	C₂×C₈	—	2	1	no	—
24	C₂×C₈	—	4	1	no	—
25	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈	M₁₆	1	1	yes	characteristic
27	C₄²	—	2	1	no	—
28	C₄²	—	2	1	no	—
29	C₁₆	—	2	2	no	—
30	C₁₆	—	4	1	no	—
31	C₂³	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	2	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	4	1	no	—
38	C₂×C₄	C₃₂	1	2	yes	complemented
39	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
40	C₂×C₄	M₃₂	1	1	yes	characteristic
41	C₈	—	2	1	no	—
42	C₈	—	4	1	no	—
43	C₈	#99	1	1	yes	characteristic
44	C₈	#101	1	1	yes	characteristic
45	C₂²	—	2	1	no	—
46	C₂²	—	2	1	no	—
47	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
48	C₂²	C₂×C₃₂	1	1	yes	characteristic
49	C₂²	M₆₄	1	1	yes	characteristic
50	C₄	—	2	1	no	—
51	C₄	—	2	2	no	—
52	C₄	—	4	1	no	—
53	C₄	#324	1	1	yes	characteristic
54	C₄	#349	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	—	2	1	no	—
57	C₂	#1322	1	1	yes	characteristic
58	C₂	C₂²⋊C₃₂	1	1	yes	characteristic
59	C₂	#1400	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊C₄
commutator	C₂²	C₂×C₃₂
Frattini, cosocle	C₂²×C₁₆	C₂²
socle	C₂²	C₂²⋊C₁₆
prime radical	C₂³	C₂×C₁₆
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
38. C₂×C₄	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₁₆	2
C₂²⋊C₃₂	2