Subgroups — #135774

Order 512.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12891	C₂	1	4	yes	maximal
2	C₂×#1311	C₂	1	1	yes	maximal, characteristic
3	#8823	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆	C₂²	1	2	yes	—
5	#1848	C₂²	1	4	yes	—
6	#1567	—	2	4	no	—
7	C₂×C₈²	C₂²	1	1	yes	characteristic
8	#1381	—	2	8	no	—
9	#1311	C₄	1	4	yes	—
10	C₂²×C₁₆	D₄	1	4	yes	—
11	C₂×#108	—	2	8	no	—
12	#423	—	2	4	no	—
13	C₂×C₄×C₈	D₄	1	2	yes	—
14	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
15	C₄×C₁₆	C₂×C₄	1	4	yes	—
16	C₄×C₁₆	Q₈	1	4	yes	—
17	#335	—	4	8	no	—
18	C₈²	C₂×C₄	1	2	yes	—
19	C₈²	Q₈	1	2	yes	—
20	C₂×M₁₆	—	4	4	no	—
21	C₂²×C₈	Q₁₆	1	4	yes	—
22	C₂²×C₈	C₂×D₄	1	2	yes	—
23	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
24	C₂×C₁₆	—	2	8	no	—
25	C₂×C₁₆	C₄⋊C₄	1	8	yes	—
26	#108	—	4	16	no	—
27	C₄⋊C₈	—	4	8	no	—
28	C₄×C₈	C₄⋊C₄	1	4	yes	—
29	C₄×C₈	D₈	1	4	yes	—
30	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
32	C₄×C₈	C₂×Q₈	1	2	yes	—
33	C₂²×C₄	#127	1	1	yes	characteristic
34	C₂²×C₄	C₂×Q₁₆	1	2	yes	—
35	M₁₆	—	8	8	no	—
36	C₂×C₈	—	4	4	no	—
37	C₂×C₈	—	8	4	no	—
38	C₂×C₈	#107	1	8	yes	—
39	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₈	C₄⋊Q₈	1	2	yes	—
41	C₂×C₈	#135	1	8	yes	—
42	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₄²	C₄⋊Q₈	1	1	yes	characteristic
44	C₄²	C₂×D₈	1	2	yes	—
45	C₁₆	—	2	8	no	—
46	C₂³	#494	1	1	yes	characteristic
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	#384	1	1	yes	characteristic
49	C₂×C₄	C₂×#107	1	2	yes	—
50	C₂×C₄	#493	1	1	yes	characteristic
51	C₂×C₄	#495	1	4	yes	—
52	C₂×C₄	#500	1	2	yes	—
53	C₈	—	2	8	no	—
54	C₈	—	16	4	no	—
55	C₈	#427	1	4	yes	—
56	C₈	#499	1	4	yes	—
57	C₂²	—	2	2	no	—
58	C₂²	#1710	1	2	yes	—
59	C₂²	#1845	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#1709	1	2	yes	—
62	C₄	#1846	1	2	yes	—
63	C₄	#2161	1	4	yes	—
64	C₂	—	2	2	no	—
65	C₂	#8816	1	1	yes	characteristic
66	C₂	#12910	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1845
prime radical	C₂³	#494
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ 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
#12891	2
C₂×#1311	2
#8823	2