Subgroups — #13633

Order 256.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic
2	#2158	C₂	1	2	yes	maximal
3	#1424	C₂	1	2	yes	maximal, complemented
4	#1423	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	#427	—	2	2	no	—
7	C₂×C₃₂	C₄	1	4	yes	complemented
8	C₂×C₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₃₂	C₂²	1	1	yes	characteristic
10	#367	—	2	2	no	—
11	#366	C₂²	1	4	yes	—
12	C₂²×C₈	D₄	1	1	yes	characteristic
13	#117	—	4	2	no	—
14	C₂×C₁₆	C₂×C₄	1	2	yes	—
15	C₂×C₁₆	D₄	1	1	yes	characteristic
16	C₂×C₁₆	Q₈	1	2	yes	—
17	C₂×C₁₆	C₂³	1	1	yes	characteristic
18	#107	—	2	4	no	—
19	#106	—	4	2	no	—
20	C₃₂	C₂×C₄	1	2	yes	—
21	C₃₂	C₂×C₄	1	2	yes	—
22	C₂²×C₄	D₈	1	1	yes	characteristic
23	C₂×C₈	C₄⋊C₄	1	2	yes	—
24	C₂×C₈	D₈	1	1	yes	characteristic
25	C₂×C₈	Q₁₆	1	2	yes	—
26	C₂×C₈	C₂×D₄	1	1	yes	characteristic
27	C₄⋊C₄	—	4	4	no	—
28	C₂²⋊C₄	—	8	2	no	—
29	C₄²	—	8	2	no	—
30	C₁₆	C₄⋊C₄	1	2	yes	—
31	C₁₆	C₂²×C₄	1	1	yes	characteristic
32	C₁₆	C₂×Q₈	1	1	yes	characteristic
33	C₂³	D₁₆	1	1	yes	characteristic
34	C₂×C₄	—	8	4	no	—
35	C₂×C₄	#107	1	2	yes	—
36	C₂×C₄	D₁₆	1	1	yes	characteristic
37	C₂×C₄	Q₃₂	1	2	yes	—
38	C₂×C₄	C₂×D₈	1	1	yes	characteristic
39	C₈	#107	1	2	yes	—
40	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₈	C₂×Q₁₆	1	1	yes	characteristic
42	C₂²	—	2	2	no	—
43	C₂²	#366	1	2	yes	—
44	C₂²	C₂×D₁₆	1	1	yes	characteristic
45	C₄	—	16	4	no	—
46	C₄	#366	1	2	yes	—
47	C₄	C₂×#107	1	1	yes	characteristic
48	C₄	C₂×Q₃₂	1	1	yes	characteristic
49	C₂	—	2	2	no	—
50	C₂	C₂×#366	1	1	yes	characteristic
51	C₂	#2262	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₆
commutator	C₁₆	C₂²×C₄
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂²	C₂×D₁₆
prime radical	C₂³	D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×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
3. #1424	C₂	semidirect
4. #1423	C₂	semidirect
7. C₂×C₃₂	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₂	2
#2158	2
#1424	2
#1423	2