Subgroups — C₂²×#1424 #512713

Order 512.

482 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#366	C₂	1	2	yes	maximal
3	C₂×#1424	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₁₆	C₂²	1	1	yes	characteristic
5	C₂²×#107	—	2	2	no	—
6	C₂²×C₃₂	C₄	1	8	yes	complemented
7	C₂²×C₃₂	C₂²	1	6	yes	—
8	C₂×#366	—	2	6	no	—
9	C₂×#366	C₂²	1	12	yes	—
10	#1424	C₂²	1	16	yes	direct product factor
11	C₂³×C₈	D₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄	—	4	2	no	—
13	C₂²×C₁₆	C₂×C₄	1	4	yes	—
14	C₂²×C₁₆	D₄	1	3	yes	—
15	C₂²×C₁₆	Q₈	1	4	yes	—
16	C₂²×C₁₆	C₂³	1	3	yes	—
17	C₂×#107	—	2	12	no	—
18	C₂×#107	—	4	6	no	—
19	C₂×C₃₂	C₂×C₄	1	24	yes	complemented
20	C₂×C₃₂	C₂³	1	4	yes	—
21	#366	—	2	12	no	—
22	#366	C₂³	1	8	yes	—
23	C₂³×C₄	—	8	2	no	—
24	C₂³×C₄	D₈	1	1	yes	characteristic
25	C₂²×C₈	C₄⋊C₄	1	4	yes	—
26	C₂²×C₈	D₈	1	3	yes	—
27	C₂²×C₈	Q₁₆	1	4	yes	—
28	C₂²×C₈	C₂×D₄	1	3	yes	—
29	C₂×C₄⋊C₄	—	4	12	no	—
30	C₂×C₄⋊C₄	—	8	6	no	—
31	C₂×C₁₆	C₄⋊C₄	1	12	yes	—
32	C₂×C₁₆	C₂²×C₄	1	6	yes	—
33	C₂×C₁₆	C₂×D₄	1	3	yes	—
34	C₂×C₁₆	C₂×Q₈	1	6	yes	—
35	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
36	#107	—	2	8	no	—
37	#107	—	4	12	no	—
38	C₃₂	C₂²×C₄	1	8	yes	complemented
39	C₂⁴	D₁₆	1	1	yes	characteristic
40	C₂²×C₄	—	8	12	no	—
41	C₂²×C₄	—	16	8	no	—
42	C₂²×C₄	#107	1	4	yes	—
43	C₂²×C₄	D₁₆	1	3	yes	—
44	C₂²×C₄	Q₃₂	1	4	yes	—
45	C₂²×C₄	C₂×D₈	1	3	yes	—
46	C₂×C₈	#107	1	12	yes	—
47	C₂×C₈	C₂×C₄⋊C₄	1	6	yes	—
48	C₂×C₈	C₂×D₈	1	3	yes	—
49	C₂×C₈	C₂×Q₁₆	1	6	yes	—
50	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
51	C₄⋊C₄	—	4	8	no	—
52	C₄⋊C₄	—	8	12	no	—
53	C₁₆	C₂×C₄⋊C₄	1	6	yes	—
54	C₁₆	C₂³×C₄	1	1	yes	characteristic
55	C₁₆	C₂²×Q₈	1	1	yes	characteristic
56	C₂³	#366	1	4	yes	—
57	C₂³	SD₆₄	1	8	yes	—
58	C₂³	C₂×D₁₆	1	3	yes	—
59	C₂×C₄	—	8	8	no	—
60	C₂×C₄	—	16	24	no	—
61	C₂×C₄	#366	1	12	yes	—
62	C₂×C₄	C₂×#107	1	6	yes	—
63	C₂×C₄	C₂×D₁₆	1	3	yes	—
64	C₂×C₄	C₂×Q₃₂	1	6	yes	—
65	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
66	C₈	C₂×#107	1	6	yes	—
67	C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
68	C₈	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂²	#1424	1	16	yes	direct product factor
70	C₂²	C₂×#366	1	6	yes	—
71	C₂²	C₂×SD₆₄	1	12	yes	—
72	C₂²	C₂²×D₁₆	1	1	yes	characteristic
73	C₄	—	16	8	no	—
74	C₄	C₂×#366	1	6	yes	—
75	C₄	C₂²×#107	1	1	yes	characteristic
76	C₄	C₂²×Q₃₂	1	1	yes	characteristic
77	C₂	C₂×#1424	1	12	yes	direct product factor
78	C₂	C₂²×#366	1	1	yes	characteristic
79	C₂	C₂²×SD₆₄	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴	D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×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. C₂×#1424	C₂	direct (N × H)
6. C₂²×C₃₂	C₄	semidirect
10. #1424	C₂²	direct (N × H)
19. C₂×C₃₂	C₂×C₄	semidirect
38. C₃₂	C₂²×C₄	semidirect
69. C₂²	#1424	direct (N × H)
77. C₂	C₂×#1424	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃₂	2
C₂²×#366	2
C₂×#1424	2

Subgroups — C22×#1424 #512713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#1424 #512713