Subgroups — #13628

Order 256.

103 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#2108	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂×M₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	#405	C₄	1	1	yes	characteristic
8	M₆₄	—	2	2	no	—
9	C₂×C₃₂	—	2	2	no	—
10	C₂×C₃₂	C₂²	1	2	yes	—
11	C₂×C₃₂	C₂²	1	2	yes	—
12	#346	C₄	1	2	yes	—
13	C₄×C₁₆	C₂²	1	2	yes	—
14	C₂×M₁₆	C₈	1	1	yes	characteristic
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	#117	C₈	1	1	yes	characteristic
17	M₃₂	—	2	2	no	—
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆	D₄	1	2	yes	—
21	C₂×C₁₆	Q₈	1	2	yes	—
22	C₂×C₁₆	C₂³	1	1	yes	characteristic
23	#97	C₈	1	2	yes	—
24	C₄×C₈	C₂×C₄	1	2	yes	—
25	C₃₂	—	2	2	no	—
26	C₃₂	—	2	2	no	—
27	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
28	M₁₆	—	2	2	no	—
29	C₂×C₈	C₄⋊C₄	1	2	yes	—
30	C₂×C₈	C₂×C₈	1	2	yes	—
31	C₂×C₈	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈	M₁₆	1	2	yes	—
33	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₄⋊C₄	C₁₆	1	2	yes	—
35	C₂²⋊C₄	C₁₆	1	2	yes	—
36	C₄²	C₂×C₈	1	2	yes	—
37	C₁₆	—	2	2	no	—
38	C₁₆	C₄⋊C₄	1	2	yes	—
39	C₁₆	C₂×D₄	1	1	yes	characteristic
40	C₁₆	C₂×Q₈	1	1	yes	characteristic
41	C₂³	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄	C₄⋊C₈	1	2	yes	—
43	C₂×C₄	C₂×C₁₆	1	4	yes	—
44	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄	M₃₂	1	2	yes	—
46	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
47	C₈	—	2	2	no	—
48	C₈	C₄⋊C₈	1	2	yes	—
49	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈	C₂×M₁₆	1	1	yes	characteristic
51	C₂²	—	2	2	no	—
52	C₂²	C₄⋊C₁₆	1	2	yes	—
53	C₂²	C₂²×C₁₆	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	C₄⋊C₁₆	1	2	yes	—
56	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄	C₂×M₃₂	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄⋊C₁₆	1	1	yes	characteristic
60	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
4. 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
C₂×M₆₄	2
C₂²×C₃₂	2
#2108	2
C₄⋊C₃₂	2