Subgroups — #7447

Order 256.

65 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2249	C₂	1	1	yes	maximal, characteristic
2	#1422	C₂	1	1	yes	maximal, characteristic, complemented
3	#1373	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Q₃₂	C₄	1	1	yes	characteristic
5	C₂×SD₃₂	—	2	1	no	—
6	C₂×D₁₆	C₄	1	1	yes	characteristic
7	#495	—	2	1	no	—
8	M₆₄	—	2	1	no	—
9	#364	—	2	1	no	—
10	C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆	—	2	1	no	—
12	C₂×SD₁₆	—	4	1	no	—
13	C₂×D₈	—	2	1	no	—
14	#124	—	4	1	no	—
15	Q₃₂	—	4	1	no	—
16	SD₃₂	—	4	1	no	—
17	SD₃₂	—	4	1	no	—
18	D₁₆	—	4	1	no	—
19	M₃₂	—	4	1	no	—
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	D₄	1	1	yes	characteristic
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₃₂	—	4	1	no	—
24	C₂×Q₈	—	4	1	no	—
25	C₂×D₄	—	4	1	no	—
26	Q₁₆	—	4	1	no	—
27	Q₁₆	—	4	1	no	—
28	SD₁₆	—	8	1	no	—
29	SD₁₆	—	8	1	no	—
30	D₈	—	4	1	no	—
31	D₈	—	4	1	no	—
32	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈	D₈	1	1	yes	characteristic
34	C₂²⋊C₄	—	8	2	no	—
35	C₄²	SD₁₆	1	1	yes	characteristic
36	C₁₆	—	2	1	no	—
37	C₁₆	—	8	1	no	—
38	C₁₆	D₈	1	1	yes	characteristic
39	C₁₆	SD₁₆	1	1	yes	characteristic
40	C₂³	—	8	1	no	—
41	Q₈	—	8	1	no	—
42	Q₈	—	8	1	no	—
43	D₄	—	8	1	no	—
44	D₄	—	8	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	8	1	no	—
47	C₂×C₄	#102	1	1	yes	characteristic
48	C₈	#100	1	1	yes	characteristic
49	C₈	#102	1	1	yes	characteristic
50	C₈	D₁₆	1	1	yes	characteristic
51	C₈	SD₃₂	1	1	yes	characteristic
52	C₂²	—	16	2	no	—
53	C₂²	—	16	1	no	—
54	C₂²	#361	1	1	yes	characteristic
55	C₄	—	4	1	no	—
56	C₄	—	16	1	no	—
57	C₄	#331	1	1	yes	characteristic
58	C₄	#357	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	—	32	1	no	—
61	C₂	#1361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1361
commutator	C₂×C₁₆	C₂×C₄
Frattini, cosocle	C₄×C₁₆	C₂²
prime radical	C₂×D₁₆	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 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. #1422	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
#2249	2
#1422	2
#1373	2