Subgroups — #12021

Order 256.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#1568	C₂	1	2	yes	maximal
3	#1368	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂	C₄	1	2	yes	—
5	#432	—	2	2	no	—
6	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
7	#346	C₂²	1	2	yes	—
8	C₄×C₁₆	C₄	1	1	yes	characteristic
9	C₄×C₁₆	C₄	1	1	yes	characteristic
10	#335	—	2	2	no	—
11	#334	C₂²	1	4	yes	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	Q₈	1	1	yes	characteristic
14	C₂×C₄²	D₄	1	1	yes	characteristic
15	M₃₂	—	2	4	no	—
16	C₂×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₆	C₂×C₄	1	2	yes	—
18	C₄⋊C₈	—	2	4	no	—
19	C₂²⋊C₈	—	4	2	no	—
20	#97	—	4	2	no	—
21	C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈	D₄	1	1	yes	characteristic
23	C₄×C₈	Q₈	1	1	yes	characteristic
24	C₄×C₈	C₂³	1	1	yes	characteristic
25	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
26	C₂²×C₄	SD₁₆	1	2	yes	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	—	2	2	no	—
29	C₂×C₈	—	4	4	no	—
30	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₄²	C₄⋊C₄	1	1	yes	characteristic
35	C₄²	SD₁₆	1	2	yes	—
36	C₄²	C₂×D₄	1	1	yes	characteristic
37	C₁₆	—	2	2	no	—
38	C₁₆	M₁₆	1	4	yes	—
39	C₂³	#106	1	1	yes	characteristic
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	4	2	no	—
43	C₂×C₄	#106	1	2	yes	—
44	C₂×C₄	#106	1	1	yes	characteristic
45	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄	C₂×SD₁₆	1	2	yes	—
47	C₈	—	2	1	no	—
48	C₈	—	2	1	no	—
49	C₈	—	8	4	no	—
50	C₈	#108	1	2	yes	—
51	C₈	C₂×M₁₆	1	2	yes	—
52	C₂²	—	2	1	no	—
53	C₂²	—	2	2	no	—
54	C₂²	C₂×#106	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	—	2	2	no	—
57	C₄	—	2	2	no	—
58	C₄	#423	1	1	yes	characteristic
59	C₄	C₂×#108	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	#1566	1	1	yes	characteristic
62	C₂	#2155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×#106
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂³	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #1368	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×M₃₂	2
#1568	2
#1368	2