Subgroups — #13660

Order 256.

79 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₃₂	C₂	1	2	yes	maximal
4	#1397	C₂	1	2	yes	maximal
5	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₃₂	C₂²	1	2	yes	—
9	C₂×C₃₂	C₂²	1	1	yes	characteristic
10	C₄⋊C₁₆	C₄	1	1	yes	characteristic
11	C₄⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₆	C₂²	1	2	yes	—
14	C₄×Q₈	C₈	1	1	yes	characteristic
15	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₆	C₂³	1	1	yes	characteristic
18	C₄⋊C₈	C₈	1	2	yes	—
19	C₄⋊C₈	C₈	1	1	yes	characteristic
20	C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈	C₂×C₄	1	2	yes	—
22	C₃₂	—	2	1	no	—
23	C₃₂	—	2	2	no	—
24	C₃₂	Q₈	1	2	yes	complemented
25	C₂×Q₈	C₁₆	1	1	yes	characteristic
26	C₂×C₈	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈	C₂×C₈	1	2	yes	—
28	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
29	C₄⋊C₄	C₁₆	1	2	yes	—
30	C₄⋊C₄	C₁₆	1	1	yes	characteristic
31	C₄²	C₂×C₈	1	2	yes	—
32	C₄²	C₂×C₈	1	1	yes	characteristic
33	C₁₆	—	2	1	no	—
34	C₁₆	—	2	2	no	—
35	C₁₆	C₂×Q₈	1	1	yes	characteristic
36	C₁₆	#41	1	1	yes	characteristic
37	Q₈	—	2	2	no	—
38	C₂×C₄	C₂×C₁₆	1	2	yes	—
39	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
40	C₂×C₄	C₂×C₁₆	1	2	yes	—
41	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₈	—	2	2	no	—
45	C₈	C₄×Q₈	1	1	yes	characteristic
46	C₈	#131	1	1	yes	characteristic
47	C₂²	C₂²×C₁₆	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	2	2	no	—
50	C₄	—	2	1	no	—
51	C₄	M₆₄	1	2	yes	—
52	C₄	C₈×Q₈	1	1	yes	characteristic
53	C₄	#504	1	1	yes	characteristic
54	C₂	Q₈×C₁₆	1	1	yes	characteristic
55	C₂	C₂×M₆₄	1	1	yes	characteristic
56	C₂	#2258	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂³
commutator, socle, prime radical	C₂²	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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
24. C₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₁₆	2
C₄⋊C₃₂	2
#1397	2
C₄×C₃₂	2