Subgroups — #94256

Order 512.

101 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13628	C₂	1	1	yes	maximal, characteristic
2	#12027	C₂	1	2	yes	maximal
3	C₂×M₆₄	C₄	1	1	yes	characteristic
4	C₂²×C₃₂	C₄	1	1	yes	characteristic
5	#2166	—	2	2	no	—
6	#2108	C₂²	1	1	yes	characteristic
7	C₄⋊C₃₂	—	2	2	no	—
8	#1373	—	2	2	no	—
9	#1369	C₄	1	4	yes	—
10	C₂×M₃₂	Q₈	1	1	yes	characteristic
11	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
12	#405	D₄	1	1	yes	characteristic
13	M₆₄	—	4	1	no	—
14	C₂×C₃₂	—	2	1	no	—
15	C₂×C₃₂	—	2	1	no	—
16	C₂×C₃₂	—	4	1	no	—
17	#364	—	2	4	no	—
18	#349	—	4	2	no	—
19	#347	—	4	2	no	—
20	#346	C₂×C₄	1	2	yes	—
21	C₄×C₁₆	D₄	1	2	yes	—
22	C₂×M₁₆	D₈	1	1	yes	characteristic
23	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
24	#117	Q₁₆	1	1	yes	characteristic
25	M₃₂	—	4	1	no	—
26	M₃₂	—	4	4	no	—
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	—	2	1	no	—
29	C₂×C₁₆	C₄²	1	1	yes	characteristic
30	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
32	#97	SD₁₆	1	2	yes	—
33	C₄×C₈	C₂²⋊C₄	1	2	yes	—
34	C₃₂	—	4	1	no	—
35	C₃₂	—	4	1	no	—
36	C₂²×C₄	#107	1	1	yes	characteristic
37	M₁₆	—	4	1	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	#95	1	1	yes	characteristic
41	C₂×C₈	#102	1	2	yes	—
42	C₂×C₈	#106	1	1	yes	characteristic
43	C₄⋊C₄	—	2	1	no	—
44	C₂²⋊C₄	—	2	1	no	—
45	C₄²	#103	1	2	yes	—
46	C₁₆	—	2	2	no	—
47	C₁₆	—	2	1	no	—
48	C₁₆	—	8	4	no	—
49	C₁₆	#100	1	1	yes	characteristic
50	C₁₆	#101	1	1	yes	characteristic
51	C₂³	#368	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	#340	1	1	yes	characteristic
56	C₂×C₄	#368	1	1	yes	characteristic
57	C₈	—	2	1	no	—
58	C₈	—	2	2	no	—
59	C₈	#338	1	1	yes	characteristic
60	C₈	#343	1	1	yes	characteristic
61	C₂²	—	2	1	no	—
62	C₂²	—	4	1	no	—
63	C₂²	#1381	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	4	2	no	—
66	C₄	#1292	1	1	yes	characteristic
67	C₄	#1384	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	#7242	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#1384
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	#2108	C₂²
socle	C₂	#7242
prime radical	C₂³	#368
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ #2108 ▷ 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
#13628	2
#12027	2