Subgroups — #135803

Order 512.

209 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12020	C₂	1	4	yes	maximal
2	#11402	C₂	1	1	yes	maximal, characteristic
3	#8824	C₂	1	2	yes	maximal
4	C₄×M₃₂	C₂²	1	2	yes	—
5	#1837	—	2	2	no	—
6	#1567	—	2	2	no	—
7	#1566	C₂²	1	4	yes	—
8	C₂×C₈²	C₂²	1	1	yes	characteristic
9	#1368	—	2	8	no	—
10	#1312	C₄	1	4	yes	—
11	C₂×M₃₂	Q₈	1	4	yes	—
12	#423	—	2	4	no	—
13	C₂×C₄×C₈	D₄	1	2	yes	—
14	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
15	#346	D₄	1	4	yes	—
16	C₄×C₁₆	C₂×C₄	1	4	yes	—
17	#341	—	4	4	no	—
18	#335	—	4	4	no	—
19	#334	—	2	8	no	—
20	C₈²	C₂×C₄	1	2	yes	—
21	C₈²	Q₈	1	2	yes	—
22	C₂×M₁₆	—	4	4	no	—
23	C₂²×C₈	SD₁₆	1	4	yes	—
24	C₂²×C₈	C₂×Q₈	1	2	yes	—
25	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
26	M₃₂	—	4	4	no	—
27	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
28	C₄⋊C₈	—	4	4	no	—
29	C₄⋊C₈	—	4	4	no	—
30	C₄×C₈	C₄⋊C₄	1	4	yes	—
31	C₄×C₈	SD₁₆	1	4	yes	—
32	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
33	C₄×C₈	C₂×D₄	1	2	yes	—
34	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
36	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
37	M₁₆	—	8	8	no	—
38	C₂×C₈	—	2	4	no	—
39	C₂×C₈	—	4	4	no	—
40	C₂×C₈	—	8	4	no	—
41	C₂×C₈	#106	1	8	yes	—
42	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₈	C₄⋊Q₈	1	2	yes	—
44	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄²	#127	1	1	yes	characteristic
46	C₄²	C₂×SD₁₆	1	2	yes	—
47	C₁₆	—	2	8	no	—
48	C₂³	#498	1	1	yes	characteristic
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	#384	1	1	yes	characteristic
52	C₂×C₄	C₂×#106	1	2	yes	—
53	C₂×C₄	#492	1	2	yes	—
54	C₂×C₄	#498	1	1	yes	characteristic
55	C₈	—	2	2	no	—
56	C₈	—	2	8	no	—
57	C₈	—	16	4	no	—
58	C₈	#430	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	—	2	1	no	—
61	C₂²	#1844	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	2	1	no	—
64	C₄	#1849	1	2	yes	—
65	C₄	#2155	1	4	yes	—
66	C₂	—	4	1	no	—
67	C₂	#8825	1	1	yes	characteristic
68	C₂	#12904	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#1844
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#498
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Maximal subgroups

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

subgroup	index
#12020	2
#11402	2
#8824	2