Subgroups — #13641

Order 256.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2108	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₃₂	C₂	1	2	yes	maximal
4	#1397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	#405	C₄	1	1	yes	characteristic
9	C₂×C₃₂	C₂²	1	2	yes	—
10	C₂×C₃₂	C₂²	1	2	yes	—
11	#346	C₄	1	1	yes	characteristic
12	#346	C₄	1	1	yes	characteristic
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×M₁₆	C₈	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	#117	C₈	1	1	yes	characteristic
18	M₃₂	—	2	2	no	—
19	C₂×C₁₆	—	2	2	no	—
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	C₂³	1	1	yes	characteristic
24	#97	C₈	1	1	yes	characteristic
25	#97	C₈	1	1	yes	characteristic
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₃₂	—	2	2	no	—
29	C₃₂	—	2	2	no	—
30	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
31	M₁₆	—	2	2	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	—	2	1	no	—
34	C₂×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊C₄	C₁₆	1	2	yes	—
39	C₂²⋊C₄	C₁₆	1	2	yes	—
40	C₄²	C₂×C₈	1	1	yes	characteristic
41	C₄²	C₂×C₈	1	1	yes	characteristic
42	C₁₆	—	2	1	no	—
43	C₁₆	—	2	1	no	—
44	C₁₆	M₁₆	1	2	yes	—
45	C₁₆	#41	1	2	yes	—
46	C₂³	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	C₂×C₁₆	1	2	yes	—
50	C₂×C₄	C₂×C₁₆	1	2	yes	—
51	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	2	1	no	—
55	C₈	M₃₂	1	2	yes	—
56	C₈	#117	1	1	yes	characteristic
57	C₈	C₂×M₁₆	1	1	yes	characteristic
58	C₂²	—	2	1	no	—
59	C₂²	—	2	2	no	—
60	C₂²	C₂²×C₁₆	1	1	yes	characteristic
61	C₄	—	2	1	no	—
62	C₄	—	2	2	no	—
63	C₄	—	2	2	no	—
64	C₄	#431	1	1	yes	characteristic
65	C₄	C₂×M₃₂	1	1	yes	characteristic
66	C₂	—	4	1	no	—
67	C₂	#2162	1	1	yes	characteristic
68	C₂	#2258	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂²×C₁₆
prime radical	C₂³	C₂×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
2. C₄⋊C₃₂	C₂	semidirect
4. #1397	C₂	semidirect
5. C₄×C₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2108	2
C₄⋊C₃₂	2
C₂²⋊C₃₂	2
#1397	2
C₄×C₃₂	2