Subgroups — #94254

Order 512.

119 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13627	C₂	1	1	yes	maximal, characteristic
2	#12024	C₂	1	2	yes	maximal
3	C₂×M₆₄	C₄	1	2	yes	—
4	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
5	#1568	—	2	2	no	—
6	C₄⋊C₃₂	—	2	2	no	—
7	#1372	C₄	1	4	yes	—
8	#1371	—	2	2	no	—
9	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₁₆	Q₈	1	1	yes	characteristic
11	#432	—	4	2	no	—
12	C₂×C₄×C₈	D₄	1	1	yes	characteristic
13	M₆₄	—	4	2	no	—
14	C₂×C₃₂	—	2	2	no	—
15	C₄×C₁₆	C₂×C₄	1	2	yes	—
16	C₄×C₁₆	D₄	1	2	yes	—
17	#335	—	2	4	no	—
18	#334	—	4	2	no	—
19	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₈	Q₁₆	1	1	yes	characteristic
21	C₂×C₄²	D₈	1	1	yes	characteristic
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	—	2	1	no	—
24	C₂×C₁₆	—	4	1	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	C₄⋊C₈	—	4	4	no	—
29	C₂²⋊C₈	—	8	2	no	—
30	#97	—	8	2	no	—
31	C₄×C₈	C₂²⋊C₄	1	2	yes	—
32	C₄×C₈	SD₁₆	1	2	yes	—
33	C₃₂	—	4	2	no	—
34	C₂²×C₄	#107	1	1	yes	characteristic
35	C₂²×C₄	SD₃₂	1	2	yes	—
36	C₂×C₈	—	2	1	no	—
37	C₂×C₈	—	2	1	no	—
38	C₂×C₈	—	4	1	no	—
39	C₂×C₈	—	8	4	no	—
40	C₂×C₈	#95	1	1	yes	characteristic
41	C₂×C₈	#103	1	2	yes	—
42	C₂×C₈	#106	1	1	yes	characteristic
43	C₄²	#102	1	2	yes	—
44	C₄²	D₁₆	1	1	yes	characteristic
45	C₄²	Q₃₂	1	1	yes	characteristic
46	C₁₆	—	2	2	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	#100	1	1	yes	characteristic
49	C₁₆	#101	1	1	yes	characteristic
50	C₂³	#367	1	1	yes	characteristic
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	#340	1	1	yes	characteristic
55	C₂×C₄	#357	1	2	yes	—
56	C₂×C₄	#358	1	2	yes	—
57	C₂×C₄	#366	1	1	yes	characteristic
58	C₈	—	2	1	no	—
59	C₈	—	2	2	no	—
60	C₈	—	16	4	no	—
61	C₈	#338	1	1	yes	characteristic
62	C₈	#343	1	1	yes	characteristic
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	#1380	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	#1292	1	1	yes	characteristic
70	C₄	#1385	1	1	yes	characteristic
71	C₂	—	4	1	no	—
72	C₂	#7241	1	1	yes	characteristic
73	C₂	#7480	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
prime radical	C₂³	#367
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ 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
#13627	2
#12024	2