Subgroups — #94021

Order 512.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#1371	C₂	1	2	yes	maximal
3	C₂²×C₃₂	C₄	1	2	yes	—
4	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×#335	—	2	2	no	—
6	C₄×C₃₂	—	2	2	no	—
7	#1371	—	2	2	no	—
8	#1371	C₄	1	4	yes	—
9	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₁₆	Q₈	1	1	yes	characteristic
11	C₂×C₄⋊C₈	—	4	2	no	—
12	C₂×C₄×C₈	D₄	1	1	yes	characteristic
13	C₂×C₃₂	—	2	2	no	—
14	C₂×C₃₂	—	2	2	no	—
15	C₂×C₃₂	C₈	1	4	yes	complemented
16	C₄×C₁₆	C₂×C₄	1	2	yes	—
17	C₄×C₁₆	D₄	1	2	yes	—
18	#335	—	2	4	no	—
19	#335	—	4	2	no	—
20	C₂²×C₈	—	8	2	no	—
21	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
22	C₂²×C₈	Q₁₆	1	1	yes	characteristic
23	C₂×C₄²	D₈	1	1	yes	characteristic
24	C₂×C₁₆	—	2	2	no	—
25	C₂×C₁₆	C₄²	1	1	yes	characteristic
26	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
28	C₂×C₁₆	C₂×C₈	1	2	yes	—
29	C₂×C₁₆	M₁₆	1	2	yes	—
30	C₄⋊C₈	—	4	4	no	—
31	C₄⋊C₈	—	8	2	no	—
32	C₄×C₈	C₂²⋊C₄	1	2	yes	—
33	C₄×C₈	SD₁₆	1	2	yes	—
34	C₃₂	—	2	4	no	—
35	C₂²×C₄	#107	1	1	yes	characteristic
36	C₂²×C₄	D₁₆	1	1	yes	characteristic
37	C₂²×C₄	Q₃₂	1	1	yes	characteristic
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	—	8	4	no	—
40	C₂×C₈	—	16	2	no	—
41	C₂×C₈	—	16	2	no	—
42	C₂×C₈	#95	1	1	yes	characteristic
43	C₂×C₈	#103	1	2	yes	—
44	C₂×C₈	C₄⋊C₈	1	2	yes	—
45	C₂×C₈	#106	1	1	yes	characteristic
46	C₂×C₈	#108	1	2	yes	—
47	C₄²	#102	1	2	yes	—
48	C₄²	SD₃₂	1	2	yes	—
49	C₁₆	—	2	2	no	—
50	C₁₆	C₄×C₈	1	1	yes	characteristic
51	C₁₆	#97	1	1	yes	characteristic
52	C₁₆	C₂²⋊C₈	1	2	yes	—
53	C₂³	#366	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	#335	1	2	yes	—
56	C₂×C₄	#340	1	1	yes	characteristic
57	C₂×C₄	#357	1	2	yes	—
58	C₂×C₄	#358	1	2	yes	—
59	C₂×C₄	#367	1	1	yes	characteristic
60	C₂×C₄	#368	1	2	yes	—
61	C₂×C₄	D₃₂	1	1	yes	characteristic
62	C₂×C₄	SD₆₄	1	2	yes	—
63	C₂×C₄	Q₆₄	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	—	16	4	no	—
66	C₈	#334	1	2	yes	—
67	C₈	#336	1	1	yes	characteristic
68	C₈	#341	1	1	yes	characteristic
69	C₂²	#1371	1	2	yes	—
70	C₂²	#1380	1	1	yes	characteristic
71	C₂²	#1423	1	1	yes	characteristic
72	C₂²	#1424	1	1	yes	characteristic
73	C₂²	#1425	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	#1277	1	1	yes	characteristic
76	C₄	#1372	1	2	yes	—
77	C₄	#1381	1	1	yes	characteristic
78	C₄	#1415	1	2	yes	—
79	C₄	#1416	1	2	yes	—
80	C₂	#7154	1	1	yes	characteristic
81	C₂	#7460	1	2	yes	—
82	C₂	#7461	1	2	yes	—
83	C₂	#7471	1	1	yes	characteristic
84	C₂	#7472	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₁₆
commutator	C₁₆	C₄×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle, prime radical	C₂³	#366
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
15. C₂×C₃₂	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃₂	2
C₂×#1371	2