Subgroups — #12002

Order 256.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#364	C₂	1	2	yes	maximal
2	C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	#1369	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂	—	2	2	no	—
5	C₂×M₃₂	C₄	1	1	yes	characteristic
6	C₂×M₃₂	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
8	#364	—	2	2	no	—
9	#364	C₂²	1	4	yes	—
10	#346	C₂²	1	2	yes	—
11	C₄×C₁₆	C₄	1	2	yes	—
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈	D₄	1	1	yes	characteristic
14	C₂×C₄²	Q₈	1	1	yes	characteristic
15	M₃₂	—	2	2	no	—
16	M₃₂	—	2	4	no	—
17	M₃₂	—	4	2	no	—
18	M₃₂	C₈	1	4	yes	—
19	C₂×C₁₆	—	4	2	no	—
20	C₂×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₆	C₂×C₄	1	2	yes	—
22	C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₄×C₈	Q₈	1	1	yes	characteristic
25	C₄×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₄	—	2	1	no	—
27	C₂²×C₄	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈	C₂×C₈	1	2	yes	—
31	C₂×C₈	M₁₆	1	2	yes	—
32	C₂×C₈	SD₁₆	1	2	yes	—
33	C₂×C₈	SD₁₆	1	2	yes	—
34	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈	C₂×D₄	1	1	yes	characteristic
36	C₄²	—	2	1	no	—
37	C₄²	C₄⋊C₄	1	1	yes	characteristic
38	C₄²	C₂×Q₈	1	1	yes	characteristic
39	C₁₆	—	2	2	no	—
40	C₁₆	—	4	4	no	—
41	C₁₆	C₂×C₈	1	4	yes	—
42	C₂³	#108	1	1	yes	characteristic
43	C₂×C₄	—	2	1	no	—
44	C₂×C₄	—	2	1	no	—
45	C₂×C₄	—	4	2	no	—
46	C₂×C₄	C₄⋊C₈	1	2	yes	—
47	C₂×C₄	#106	1	2	yes	—
48	C₂×C₄	#108	1	1	yes	characteristic
49	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈	C₄⋊C₈	1	2	yes	—
51	C₈	#106	1	1	yes	characteristic
52	C₈	#106	1	1	yes	characteristic
53	C₈	C₂²×C₈	1	1	yes	characteristic
54	C₈	C₂×M₁₆	1	1	yes	characteristic
55	C₈	C₂×SD₁₆	1	2	yes	—
56	C₂²	—	4	1	no	—
57	C₂²	#334	1	2	yes	—
58	C₂²	C₂×#108	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	4	1	no	—
61	C₄	#334	1	2	yes	—
62	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
63	C₄	C₂×#106	1	1	yes	characteristic
64	C₂	—	2	1	no	—
65	C₂	—	2	2	no	—
66	C₂	C₂×#334	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#106
commutator	C₈	C₂²×C₈
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂	C₂×#334
prime radical	C₂³	#108
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ 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. #1369	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#364	2
C₄×M₃₂	2
#1369	2