Subgroups — #7384

Order 256.

63 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#1371	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂	C₄	1	1	yes	characteristic
5	C₄×Q₁₆	—	2	1	no	—
6	C₂×C₃₂	—	2	1	no	—
7	#366	C₄	1	1	yes	characteristic
8	#358	—	2	1	no	—
9	C₄×C₁₆	C₂²	1	1	yes	characteristic
10	#335	—	2	1	no	—
11	C₂×Q₁₆	—	2	1	no	—
12	C₄×Q₈	—	4	1	no	—
13	Q₃₂	—	2	1	no	—
14	Q₃₂	C₈	1	2	yes	complemented
15	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆	D₄	1	1	yes	characteristic
17	#107	—	2	1	no	—
18	C₄⋊C₈	—	4	1	no	—
19	#103	—	4	1	no	—
20	C₄×C₈	D₄	1	1	yes	characteristic
21	C₃₂	—	2	2	no	—
22	C₂×Q₈	—	4	1	no	—
23	Q₁₆	—	2	2	no	—
24	Q₁₆	—	4	1	no	—
25	C₂×C₈	—	8	1	no	—
26	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈	SD₁₆	1	1	yes	characteristic
28	C₄⋊C₄	—	4	1	no	—
29	C₄⋊C₄	—	8	1	no	—
30	C₄²	—	8	1	no	—
31	C₄²	D₈	1	1	yes	characteristic
32	C₁₆	—	2	1	no	—
33	C₁₆	C₂×C₈	1	1	yes	characteristic
34	C₁₆	M₁₆	1	1	yes	characteristic
35	Q₈	—	4	2	no	—
36	Q₈	—	8	1	no	—
37	C₂×C₄	—	8	1	no	—
38	C₂×C₄	—	8	1	no	—
39	C₂×C₄	#102	1	1	yes	characteristic
40	C₂×C₄	D₁₆	1	1	yes	characteristic
41	C₂×C₄	SD₃₂	1	1	yes	characteristic
42	C₈	—	2	1	no	—
43	C₈	—	16	1	no	—
44	C₈	C₂²⋊C₈	1	1	yes	characteristic
45	C₈	#104	1	1	yes	characteristic
46	C₂²	#357	1	1	yes	characteristic
47	C₄	—	2	1	no	—
48	C₄	—	8	2	no	—
49	C₄	—	16	1	no	—
50	C₄	D₄⋊C₈	1	1	yes	characteristic
51	C₄	#359	1	1	yes	characteristic
52	C₄	SD₆₄	1	1	yes	characteristic
53	C₄	Q₆₄	1	1	yes	characteristic
54	C₂	#1331	1	1	yes	characteristic
55	C₂	#1416	1	1	yes	characteristic
56	C₂	#1417	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₆
commutator	C₁₆	C₂×C₈
Frattini, cosocle	C₄×C₁₆	C₂²
socle, prime radical	C₂²	#357
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ 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
14. Q₃₂	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₃₂	2
C₄×C₃₂	2
#1371	2