Subgroups — #33414

Order 256.

311 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	#2916	C₂	1	2	yes	maximal
3	#2161	C₂	1	6	yes	maximal, complemented
4	C₂×#368	C₂	1	6	yes	maximal, complemented
5	#568	—	2	2	no	—
6	C₂×#131	C₂²	1	1	yes	characteristic
7	#504	C₄	1	8	yes	—
8	C₂×M₃₂	C₂²	1	3	yes	—
9	C₂²×C₁₆	C₂²	1	3	yes	—
10	#430	—	2	6	no	—
11	C₂×#108	C₂²	1	12	yes	—
12	#368	C₂²	1	4	yes	—
13	#368	C₂²	1	12	yes	—
14	C₂×#41	D₄	1	1	yes	characteristic
15	#131	—	4	8	no	—
16	#131	C₂×C₄	1	4	yes	—
17	#131	Q₈	1	4	yes	—
18	C₂×M₁₆	—	2	12	no	—
19	C₂×M₁₆	D₄	1	3	yes	—
20	C₂²×C₈	C₂³	1	3	yes	—
21	M₃₂	C₂×C₄	1	12	yes	—
22	C₂×C₁₆	C₂×C₄	1	12	yes	—
23	C₂×C₁₆	C₂³	1	1	yes	characteristic
24	C₂×C₁₆	C₂³	1	3	yes	—
25	#108	—	2	12	no	—
26	#108	C₂³	1	8	yes	—
27	#41	C₄⋊C₄	1	4	yes	—
28	#41	D₈	1	4	yes	—
29	C₂×Q₈	Q₁₆	1	1	yes	characteristic
30	C₂×D₄	Q₁₆	1	3	yes	—
31	C₂²×C₄	C₂×D₄	1	3	yes	—
32	M₁₆	—	2	8	no	—
33	M₁₆	—	4	12	no	—
34	M₁₆	C₄⋊C₄	1	12	yes	—
35	C₂×C₈	—	4	12	no	—
36	C₂×C₈	C₂²×C₄	1	6	yes	—
37	C₂×C₈	C₂×D₄	1	3	yes	—
38	C₂×C₈	C₂×Q₈	1	6	yes	—
39	C₂×C₈	C₂⁴	1	1	yes	characteristic
40	C₁₆	C₂²×C₄	1	2	yes	—
41	C₁₆	C₂²×C₄	1	6	yes	—
42	C₂³	C₂×Q₁₆	1	3	yes	—
43	Q₈	#107	1	4	yes	—
44	D₄	#107	1	12	yes	—
45	C₂×C₄	C₂×C₄⋊C₄	1	6	yes	—
46	C₂×C₄	C₂×D₈	1	6	yes	—
47	C₂×C₄	C₂×Q₁₆	1	3	yes	—
48	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
49	C₈	—	4	8	no	—
50	C₈	C₂×C₄⋊C₄	1	6	yes	—
51	C₈	C₂³×C₄	1	1	yes	characteristic
52	C₈	C₂²×Q₈	1	1	yes	characteristic
53	C₂²	—	4	3	no	—
54	C₂²	C₂×#107	1	6	yes	—
55	C₂²	C₂²×Q₁₆	1	1	yes	characteristic
56	C₄	C₂×#107	1	6	yes	—
57	C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
58	C₄	C₂²×D₈	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	—	2	6	no	—
61	C₂	C₂²×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₈
commutator	C₈	C₂³×C₄
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂	C₂²×#107
prime radical	C₂×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #2161	C₂	semidirect
4. C₂×#368	C₂	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
#2916	2
#2161	2
C₂×#368	2