Subgroups — C₄×#504 #33325

Order 256.

417 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#504	C₂	1	2	yes	maximal
2	C₄×#131	C₂	1	1	yes	maximal, characteristic
3	#2108	C₂	1	6	yes	maximal, complemented
4	C₄×M₃₂	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₁₆	C₂	1	3	yes	maximal, complemented
6	C₂×#131	C₄	1	1	yes	characteristic
7	C₂×#131	C₂²	1	1	yes	characteristic
8	C₄×#41	C₄	1	1	yes	characteristic
9	#504	C₄	1	16	yes	direct product factor
10	C₂×M₃₂	C₂²	1	6	yes	—
11	C₂²×C₁₆	C₂²	1	6	yes	—
12	#405	C₄	1	3	yes	—
13	#405	C₂²	1	3	yes	—
14	C₄×M₁₆	C₄	1	3	yes	—
15	C₂×C₄×C₈	C₂²	1	3	yes	—
16	#346	C₂²	1	6	yes	—
17	C₄×C₁₆	C₂²	1	3	yes	—
18	C₄×C₁₆	C₂²	1	6	yes	—
19	C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×#41	C₂×C₄	1	1	yes	characteristic
21	#131	C₈	1	8	yes	—
22	#131	C₂×C₄	1	8	yes	—
23	C₂×M₁₆	C₂×C₄	1	3	yes	—
24	C₂×M₁₆	C₂×C₄	1	3	yes	—
25	C₂²×C₈	C₂×C₄	1	3	yes	—
26	C₂²×C₈	C₂³	1	3	yes	—
27	C₄×Q₈	C₈	1	2	yes	—
28	C₄×D₄	C₈	1	6	yes	—
29	#117	C₂×C₄	1	3	yes	—
30	C₂×C₄²	C₂×C₄	1	3	yes	—
31	M₃₂	C₂×C₄	1	24	yes	complemented
32	C₂×C₁₆	C₂×C₄	1	24	yes	complemented
33	C₂×C₁₆	C₂³	1	2	yes	—
34	C₂×C₁₆	C₂³	1	6	yes	—
35	#97	C₂×C₄	1	6	yes	—
36	C₄×C₈	C₂×C₄	1	3	yes	—
37	C₄×C₈	C₂×C₄	1	3	yes	—
38	C₄×C₈	C₂³	1	1	yes	characteristic
39	C₄×C₈	C₂³	1	3	yes	—
40	#41	C₄²	1	4	yes	—
41	#41	C₂×C₈	1	4	yes	—
42	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
43	C₂×D₄	C₂×C₈	1	3	yes	—
44	C₂²×C₄	C₂×C₈	1	6	yes	—
45	C₂²×C₄	C₂²×C₄	1	3	yes	—
46	M₁₆	C₄²	1	12	yes	—
47	M₁₆	C₂×C₈	1	12	yes	—
48	C₂×C₈	C₂×C₈	1	12	yes	—
49	C₂×C₈	C₂²×C₄	1	3	yes	—
50	C₂×C₈	C₂²×C₄	1	12	yes	—
51	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈	C₂²×C₄	1	3	yes	—
53	C₂×C₈	C₂⁴	1	1	yes	characteristic
54	C₄⋊C₄	C₂×C₈	1	6	yes	—
55	C₂²⋊C₄	C₂×C₈	1	6	yes	—
56	C₄²	C₂×C₈	1	6	yes	—
57	C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₄²	C₂²×C₄	1	3	yes	—
59	C₁₆	C₂²×C₄	1	4	yes	—
60	C₁₆	C₂²×C₄	1	12	yes	—
61	C₂³	C₂²×C₈	1	3	yes	—
62	Q₈	C₄×C₈	1	4	yes	—
63	D₄	C₄×C₈	1	12	yes	—
64	C₂×C₄	—	2	12	no	—
65	C₂×C₄	C₂×C₄²	1	6	yes	—
66	C₂×C₄	C₂²×C₈	1	6	yes	—
67	C₂×C₄	C₂²×C₈	1	6	yes	—
68	C₂×C₄	C₂²×C₈	1	3	yes	—
69	C₂×C₄	C₂²×C₈	1	2	yes	—
70	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
71	C₈	C₂×C₄²	1	6	yes	—
72	C₈	C₂²×C₈	1	2	yes	—
73	C₈	C₂²×C₈	1	6	yes	—
74	C₈	C₂³×C₄	1	2	yes	—
75	C₂²	—	2	6	no	—
76	C₂²	C₂×C₄×C₈	1	6	yes	—
77	C₂²	C₂³×C₈	1	1	yes	characteristic
78	C₄	—	2	6	no	—
79	C₄	C₂×C₄×C₈	1	6	yes	—
80	C₄	#504	1	4	yes	direct product factor
81	C₄	C₂²×C₄²	1	1	yes	characteristic
82	C₄	C₂³×C₈	1	1	yes	characteristic
83	C₂	—	2	6	no	—
84	C₂	C₂²×C₄×C₈	1	1	yes	characteristic
85	C₂	C₂×#504	1	2	yes	—

	subgroup	G/H
center	C₄×C₁₆	C₂²
commutator	C₂	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂²	C₂³×C₈
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₆ ◁ 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. #2108	C₂	semidirect
4. C₄×M₃₂	C₂	semidirect
5. C₂×C₄×C₁₆	C₂	semidirect
9. #504	C₄	direct (N × H)
31. M₃₂	C₂×C₄	semidirect
32. C₂×C₁₆	C₂×C₄	semidirect
80. C₄	#504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#504	2
C₄×#131	2
#2108	2
C₄×M₃₂	2
C₂×C₄×C₁₆	2

Subgroups — C4×#504 #33325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#504 #33325