Subgroups — #513296

Order 512.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#33962	C₂	1	2	yes	maximal
2	#13726	C₂	1	3	yes	maximal
3	#13688	C₂	1	6	yes	maximal
4	Q₈×C₃₂	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₆₄	C₂	1	3	yes	maximal
6	#3395	—	2	2	no	—
7	C₂×Q₆₄	C₂²	1	3	yes	—
8	C₂×Q₆₄	C₂²	1	6	yes	—
9	#2247	C₂²	1	6	yes	—
10	#2209	—	2	6	no	—
11	Q₈×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×Q₃₂	C₂²	1	6	yes	—
13	#1423	C₂²	1	1	yes	characteristic
14	C₄⋊C₃₂	C₂²	1	3	yes	—
15	#1416	C₂²	1	6	yes	—
16	C₄×C₃₂	C₂²	1	3	yes	—
17	#557	—	4	2	no	—
18	C₂×Q₃₂	—	2	6	no	—
19	C₂×Q₃₂	C₂³	1	6	yes	—
20	#494	—	2	6	no	—
21	#462	—	4	6	no	—
22	C₈×Q₈	D₄	1	1	yes	characteristic
23	C₄×Q₁₆	—	2	6	no	—
24	Q₆₄	—	2	6	no	—
25	Q₆₄	—	4	6	no	—
26	C₂×C₃₂	C₂³	1	3	yes	—
27	C₂×C₃₂	C₂³	1	1	yes	characteristic
28	#366	C₂³	1	2	yes	—
29	C₄⋊C₁₆	D₄	1	3	yes	—
30	#358	—	2	6	no	—
31	C₄×C₁₆	C₂³	1	3	yes	—
32	C₂×Q₁₆	—	2	6	no	—
33	C₂×Q₁₆	—	4	6	no	—
34	C₄⋊Q₈	—	4	6	no	—
35	#125	—	8	6	no	—
36	C₄×Q₈	—	4	6	no	—
37	C₄×Q₈	—	8	2	no	—
38	C₄×Q₈	D₈	1	1	yes	characteristic
39	Q₃₂	—	2	6	no	—
40	Q₃₂	—	2	6	no	—
41	Q₃₂	—	8	6	no	—
42	C₂×C₁₆	C₂×D₄	1	3	yes	—
43	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
44	#107	—	2	2	no	—
45	C₄⋊C₈	D₈	1	3	yes	—
46	#103	—	4	6	no	—
47	C₄×C₈	C₂×D₄	1	3	yes	—
48	C₃₂	—	2	3	no	—
49	C₃₂	#41	1	2	yes	—
50	C₂×Q₈	—	4	6	no	—
51	C₂×Q₈	D₁₆	1	1	yes	characteristic
52	Q₁₆	—	4	6	no	—
53	Q₁₆	—	4	6	no	—
54	Q₁₆	—	16	6	no	—
55	C₂×C₈	C₂×D₈	1	3	yes	—
56	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	8	6	no	—
59	C₄⋊C₄	—	8	6	no	—
60	C₄⋊C₄	—	8	6	no	—
61	C₄⋊C₄	D₁₆	1	3	yes	—
62	C₄²	—	8	6	no	—
63	C₄²	C₂×D₈	1	3	yes	—
64	C₁₆	—	2	3	no	—
65	C₁₆	C₂×#41	1	1	yes	characteristic
66	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
67	Q₈	—	8	6	no	—
68	Q₈	—	8	6	no	—
69	Q₈	Q₆₄	1	4	yes	complemented
70	C₂×C₄	—	8	2	no	—
71	C₂×C₄	—	8	6	no	—
72	C₂×C₄	C₂×D₁₆	1	3	yes	—
73	C₂×C₄	C₂×D₁₆	1	3	yes	—
74	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
75	C₈	—	2	3	no	—
76	C₈	#550	1	1	yes	characteristic
77	C₈	#576	1	1	yes	characteristic
78	C₂²	C₂²×D₁₆	1	1	yes	characteristic
79	C₄	—	2	3	no	—
80	C₄	—	16	6	no	—
81	C₄	—	16	2	no	—
82	C₄	C₂×Q₆₄	1	6	yes	—
83	C₄	#3391	1	1	yes	characteristic
84	C₄	#3415	1	1	yes	characteristic
85	C₂	#33958	1	1	yes	characteristic
86	C₂	C₂²×Q₆₄	1	1	yes	characteristic
87	C₂	#33984	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×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
69. Q₈	Q₆₄	semidirect

Maximal subgroups

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

subgroup	index
#33962	2
#13726	2
#13688	2
Q₈×C₃₂	2
C₄×Q₆₄	2