Subgroups — #33974

Order 256.

231 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	1	yes	maximal, characteristic
2	#2258	C₂	1	8	yes	maximal, complemented
3	C₂×M₆₄	C₂	1	6	yes	maximal, complemented
4	C₂×#131	C₄	1	1	yes	characteristic
5	#504	C₄	1	4	yes	—
6	#504	C₂²	1	4	yes	—
7	C₂×M₃₂	C₄	1	3	yes	—
8	C₂²×C₁₆	C₂²	1	3	yes	—
9	M₆₄	C₂²	1	4	yes	—
10	M₆₄	C₂²	1	12	yes	complemented
11	C₂×C₃₂	C₂²	1	12	yes	complemented
12	C₂×#41	C₈	1	1	yes	characteristic
13	#131	C₈	1	4	yes	—
14	#131	C₂×C₄	1	4	yes	—
15	C₂×M₁₆	C₈	1	3	yes	—
16	C₂²×C₈	C₂×C₄	1	3	yes	—
17	M₃₂	C₂×C₄	1	12	yes	—
18	C₂×C₁₆	C₂×C₄	1	3	yes	—
19	C₂×C₁₆	C₂×C₄	1	6	yes	—
20	C₂×C₁₆	C₂³	1	6	yes	—
21	C₂×C₁₆	C₂³	1	1	yes	characteristic
22	C₃₂	C₂³	1	8	yes	—
23	#41	C₁₆	1	4	yes	—
24	#41	C₂×C₈	1	4	yes	—
25	C₂×Q₈	C₁₆	1	1	yes	characteristic
26	C₂×D₄	C₁₆	1	3	yes	—
27	C₂²×C₄	C₂×C₈	1	3	yes	—
28	M₁₆	C₂×C₈	1	12	yes	—
29	C₂×C₈	C₂×C₈	1	3	yes	—
30	C₂×C₈	C₂×C₈	1	6	yes	—
31	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈	C₂²×C₄	1	6	yes	—
33	C₁₆	C₂²×C₄	1	1	yes	characteristic
34	C₁₆	C₂²×C₄	1	6	yes	—
35	C₁₆	C₂⁴	1	1	yes	characteristic
36	C₂³	C₂×C₁₆	1	3	yes	—
37	Q₈	C₂×C₁₆	1	4	yes	—
38	D₄	C₂×C₁₆	1	12	yes	—
39	C₂×C₄	C₂×C₁₆	1	3	yes	—
40	C₂×C₄	C₂×C₁₆	1	6	yes	—
41	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
42	C₂×C₄	C₂²×C₈	1	6	yes	—
43	C₈	C₂²×C₈	1	1	yes	characteristic
44	C₈	C₂²×C₈	1	6	yes	—
45	C₈	C₂³×C₄	1	1	yes	characteristic
46	C₂²	—	4	3	no	—
47	C₂²	C₂²×C₁₆	1	1	yes	characteristic
48	C₂²	C₂²×C₁₆	1	6	yes	—
49	C₄	C₂²×C₁₆	1	1	yes	characteristic
50	C₄	C₂²×C₁₆	1	6	yes	—
51	C₄	C₂³×C₈	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	2	6	no	—
54	C₂	C₂³×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₁₆	C₂⁴
commutator, socle	C₂	C₂³×C₁₆
prime radical	C₂×#41	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2258	C₂	semidirect
3. C₂×M₆₄	C₂	semidirect
10. M₆₄	C₂²	semidirect
11. C₂×C₃₂	C₂²	semidirect

Maximal subgroups

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

subgroup	index
C₂×#504	2
#2258	2
C₂×M₆₄	2