Subgroups — #13654

Order 256.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₆	C₂	1	2	yes	maximal, complemented
3	#1424	C₂	1	1	yes	maximal, characteristic, complemented
4	#1415	C₂	1	2	yes	maximal, complemented
5	#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆	C₂²	1	2	yes	—
7	C₄×D₈	—	2	2	no	—
8	D₃₂	C₄	1	4	yes	complemented
9	C₂×C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₃₂	C₂²	1	1	yes	characteristic
11	#366	C₂²	1	2	yes	—
12	#357	—	2	2	no	—
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×D₈	—	2	2	no	—
15	C₄×D₄	—	4	2	no	—
16	D₁₆	—	2	2	no	—
17	D₁₆	C₂×C₄	1	4	yes	complemented
18	C₂×C₁₆	D₄	1	1	yes	characteristic
19	C₂×C₁₆	C₂³	1	1	yes	characteristic
20	#107	—	2	2	no	—
21	#102	—	4	2	no	—
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₃₂	—	2	1	no	—
24	C₃₂	C₂×C₄	1	2	yes	complemented
25	C₂×D₄	—	4	2	no	—
26	C₂²×C₄	—	8	2	no	—
27	D₈	—	2	4	no	—
28	D₈	—	4	2	no	—
29	C₂×C₈	D₈	1	1	yes	characteristic
30	C₂×C₈	C₂×D₄	1	1	yes	characteristic
31	C₄⋊C₄	—	4	2	no	—
32	C₂²⋊C₄	—	8	2	no	—
33	C₄²	D₈	1	1	yes	characteristic
34	C₁₆	—	2	1	no	—
35	C₁₆	C₂²×C₄	1	1	yes	characteristic
36	C₁₆	#41	1	1	yes	characteristic
37	C₂³	—	8	2	no	—
38	D₄	—	4	4	no	—
39	D₄	—	8	2	no	—
40	C₂×C₄	—	8	2	no	—
41	C₂×C₄	—	16	4	no	—
42	C₂×C₄	D₁₆	1	2	yes	—
43	C₂×C₄	C₂×D₈	1	1	yes	characteristic
44	C₈	—	2	1	no	—
45	C₈	C₄×D₄	1	1	yes	characteristic
46	C₈	#135	1	1	yes	characteristic
47	C₂²	—	8	4	no	—
48	C₂²	—	16	2	no	—
49	C₂²	—	16	2	no	—
50	C₂²	C₂×D₁₆	1	1	yes	characteristic
51	C₄	—	2	2	no	—
52	C₄	—	16	2	no	—
53	C₄	C₄×D₈	1	1	yes	characteristic
54	C₄	#508	1	1	yes	characteristic
55	C₂	—	16	4	no	—
56	C₂	C₄×D₁₆	1	1	yes	characteristic
57	C₂	#2263	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂×D₃₂	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₆	C₂	semidirect
3. #1424	C₂	semidirect
4. #1415	C₂	semidirect
5. #1397	C₂	semidirect
8. D₃₂	C₄	semidirect
17. D₁₆	C₂×C₄	semidirect
24. C₃₂	C₂×C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₂	2
C₄×D₁₆	2
#1424	2
#1415	2
#1397	2