Subgroups — #151123

Order 512.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₆₄	C₂	1	1	yes	maximal, characteristic
2	#12083	C₂	1	2	yes	maximal
3	#7460	C₂	1	1	yes	maximal, characteristic
4	#7384	C₂	1	2	yes	maximal
5	C₈×C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₆₄	C₄	1	1	yes	characteristic
7	C₄×Q₃₂	C₂²	1	2	yes	—
8	#1591	—	2	2	no	—
9	#1423	C₄	1	1	yes	characteristic
10	#1416	C₄	1	2	yes	—
11	C₄×C₃₂	C₂²	1	1	yes	characteristic
12	C₄×C₃₂	C₂²	1	1	yes	characteristic
13	#1371	C₂²	1	2	yes	—
14	#1332	—	2	2	no	—
15	C₈×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂	C₂×C₄	1	2	yes	—
17	#446	—	4	2	no	—
18	C₄×Q₁₆	—	2	2	no	—
19	Q₆₄	—	2	2	no	—
20	C₂×C₃₂	—	2	1	no	—
21	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
23	#366	C₂×C₄	1	2	yes	—
24	#358	—	2	2	no	—
25	C₄×C₁₆	D₄	1	1	yes	characteristic
26	C₄×C₁₆	C₂³	1	1	yes	characteristic
27	#335	—	2	2	no	—
28	Q₈⋊C₈	—	4	2	no	—
29	C₈²	D₄	1	1	yes	characteristic
30	C₂×Q₁₆	—	2	2	no	—
31	C₄×Q₈	—	4	2	no	—
32	Q₃₂	—	2	2	no	—
33	Q₃₂	—	2	2	no	—
34	C₂×C₁₆	—	2	1	no	—
35	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₁₆	#41	1	1	yes	characteristic
37	#107	—	2	2	no	—
38	C₄⋊C₈	—	4	2	no	—
39	C₄⋊C₈	—	8	2	no	—
40	#103	—	4	2	no	—
41	#97	—	8	2	no	—
42	C₄×C₈	D₈	1	1	yes	characteristic
43	C₄×C₈	D₈	1	1	yes	characteristic
44	C₄×C₈	C₂×D₄	1	1	yes	characteristic
45	C₃₂	—	2	2	no	—
46	C₃₂	—	2	1	no	—
47	C₃₂	M₁₆	1	2	yes	—
48	C₂×Q₈	—	4	2	no	—
49	Q₁₆	—	4	2	no	—
50	Q₁₆	—	4	2	no	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	—	8	2	no	—
53	C₂×C₈	D₁₆	1	1	yes	characteristic
54	C₂×C₈	D₁₆	1	1	yes	characteristic
55	C₂×C₈	C₄×D₄	1	1	yes	characteristic
56	C₂×C₈	#135	1	1	yes	characteristic
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	8	2	no	—
59	C₄²	—	8	2	no	—
60	C₄²	C₂×D₈	1	1	yes	characteristic
61	C₁₆	—	2	2	no	—
62	C₁₆	—	2	1	no	—
63	C₁₆	C₂×M₁₆	1	1	yes	characteristic
64	C₁₆	#131	1	1	yes	characteristic
65	Q₈	—	8	2	no	—
66	Q₈	—	8	2	no	—
67	C₂×C₄	—	8	2	no	—
68	C₂×C₄	—	8	2	no	—
69	C₂×C₄	C₄×D₈	1	1	yes	characteristic
70	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
71	C₂×C₄	#508	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₈	—	2	1	no	—
74	C₈	—	2	1	no	—
75	C₈	—	16	2	no	—
76	C₈	Q₆₄	1	2	yes	complemented
77	C₈	#436	1	1	yes	characteristic
78	C₈	#443	1	1	yes	characteristic
79	C₂²	C₄×D₁₆	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	16	2	no	—
82	C₄	—	16	2	no	—
83	C₄	#1589	1	1	yes	characteristic
84	C₄	#2178	1	1	yes	characteristic
85	C₄	C₂×Q₆₄	1	1	yes	characteristic
86	C₄	#2262	1	1	yes	characteristic
87	C₂	#12081	1	1	yes	characteristic
88	C₂	C₄×Q₆₄	1	1	yes	characteristic
89	C₂	#13655	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
socle, prime radical	C₂²	C₄×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
76. C₈	Q₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₆₄	2
#12083	2
#7460	2
#7384	2
C₈×C₃₂	2