Subgroups — #94028

Order 512.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1397	C₂	1	1	yes	maximal, characteristic
2	C₂×#1373	C₂	1	2	yes	maximal
3	C₂²×C₃₂	C₄	1	1	yes	characteristic
4	C₂²×C₃₂	C₄	1	1	yes	characteristic
5	C₂×#364	—	2	2	no	—
6	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
7	#1397	—	2	2	no	—
8	#1373	—	2	2	no	—
9	#1373	C₄	1	4	yes	—
10	C₂×M₃₂	—	4	2	no	—
11	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₁₆	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₈	D₄	1	1	yes	characteristic
14	C₂×C₃₂	—	2	1	no	—
15	C₂×C₃₂	—	2	1	no	—
16	C₂×C₃₂	—	2	1	no	—
17	C₂×C₃₂	—	2	1	no	—
18	C₂×C₃₂	C₈	1	2	yes	—
19	C₂×C₃₂	C₈	1	2	yes	—
20	#364	—	2	4	no	—
21	#364	—	4	2	no	—
22	C₄×C₁₆	C₂×C₄	1	2	yes	—
23	C₄×C₁₆	D₄	1	2	yes	—
24	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×C₈	D₈	1	1	yes	characteristic
26	C₂×C₄²	Q₁₆	1	1	yes	characteristic
27	M₃₂	—	4	4	no	—
28	M₃₂	—	8	2	no	—
29	C₂×C₁₆	—	2	1	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	8	2	no	—
32	C₂×C₁₆	C₄²	1	1	yes	characteristic
33	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
35	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
36	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
37	C₂×C₁₆	M₁₆	1	1	yes	characteristic
38	C₂×C₁₆	M₁₆	1	1	yes	characteristic
39	C₄×C₈	C₂²⋊C₄	1	2	yes	—
40	C₄×C₈	SD₁₆	1	2	yes	—
41	C₃₂	—	2	2	no	—
42	C₃₂	—	2	2	no	—
43	C₂²×C₄	—	2	1	no	—
44	C₂²×C₄	#107	1	1	yes	characteristic
45	C₂×C₈	#95	1	1	yes	characteristic
46	C₂×C₈	#102	1	2	yes	—
47	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₈	#106	1	1	yes	characteristic
50	C₂×C₈	#108	1	1	yes	characteristic
51	C₂×C₈	#108	1	1	yes	characteristic
52	C₂×C₈	D₁₆	1	1	yes	characteristic
53	C₂×C₈	SD₃₂	1	2	yes	—
54	C₂×C₈	Q₃₂	1	1	yes	characteristic
55	C₄²	—	2	1	no	—
56	C₄²	#103	1	2	yes	—
57	C₁₆	—	2	2	no	—
58	C₁₆	—	8	4	no	—
59	C₁₆	C₄×C₈	1	1	yes	characteristic
60	C₁₆	#97	1	1	yes	characteristic
61	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
62	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
63	C₂³	#368	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	#335	1	1	yes	characteristic
67	C₂×C₄	#335	1	1	yes	characteristic
68	C₂×C₄	#340	1	1	yes	characteristic
69	C₂×C₄	#366	1	1	yes	characteristic
70	C₂×C₄	#367	1	1	yes	characteristic
71	C₂×C₄	#368	1	1	yes	characteristic
72	C₈	#334	1	1	yes	characteristic
73	C₈	#334	1	1	yes	characteristic
74	C₈	#336	1	1	yes	characteristic
75	C₈	#341	1	1	yes	characteristic
76	C₈	#357	1	2	yes	—
77	C₈	#358	1	2	yes	—
78	C₂²	—	2	2	no	—
79	C₂²	#1371	1	1	yes	characteristic
80	C₂²	#1372	1	1	yes	characteristic
81	C₂²	#1381	1	1	yes	characteristic
82	C₄	—	4	2	no	—
83	C₄	#1277	1	1	yes	characteristic
84	C₄	#1371	1	1	yes	characteristic
85	C₄	#1372	1	1	yes	characteristic
86	C₄	#1380	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	—	2	1	no	—
89	C₂	#7154	1	1	yes	characteristic
90	C₂	#7464	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#367
commutator	C₁₆	C₄×C₈
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1372
prime radical	C₂³	#368
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
C₂×#1397	2
C₂×#1373	2