Subgroups — #94855

Order 512.

530 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#8131	C₂	1	4	yes	maximal
2	C₄×#322	C₂	1	2	yes	maximal
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
4	C₄²×C₈	C₂²	1	1	yes	characteristic
5	C₄²×C₈	C₂²	1	2	yes	—
6	C₄²×C₈	C₂²	1	4	yes	—
7	C₂×#322	C₄	1	8	yes	—
8	C₂×C₈²	C₄	1	4	yes	—
9	#1273	C₄	1	16	yes	—
10	C₂×C₄×C₈	C₂×C₄	1	8	yes	—
11	C₂×C₄×C₈	C₂×C₄	1	16	yes	—
12	C₂×C₄×C₈	C₂×C₄	1	8	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
14	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
16	C₄³	C₂³	1	1	yes	characteristic
17	#322	—	2	16	no	—
18	C₈²	C₈	1	16	yes	complemented
19	C₂²×C₈	C₄²	1	16	yes	—
20	C₂²×C₈	C₄²	1	8	yes	—
21	C₂²×C₈	C₄²	1	4	yes	—
22	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄²	C₂²×C₄	1	2	yes	—
24	C₂×C₄²	C₂²×C₄	1	4	yes	—
25	C₄×C₈	—	2	16	no	—
26	C₄×C₈	—	2	16	no	—
27	C₄×C₈	—	2	32	no	—
28	C₄×C₈	C₂×C₈	1	16	yes	—
29	C₄×C₈	C₂×C₈	1	8	yes	—
30	C₄×C₈	M₁₆	1	16	yes	—
31	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
32	C₂²×C₄	C₂×C₄²	1	2	yes	—
33	C₂²×C₄	C₂×C₄²	1	4	yes	—
34	C₂×C₈	—	2	16	no	—
35	C₂×C₈	—	2	16	no	—
36	C₂×C₈	—	2	32	no	—
37	C₂×C₈	C₄×C₈	1	16	yes	—
38	C₂×C₈	C₄×C₈	1	8	yes	—
39	C₂×C₈	#97	1	16	yes	—
40	C₄²	C₂²×C₈	1	4	yes	—
41	C₄²	C₂×M₁₆	1	8	yes	—
42	C₄²	#131	1	16	yes	—
43	C₂³	C₄³	1	1	yes	characteristic
44	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
45	C₂×C₄	C₂×C₄×C₈	1	4	yes	—
46	C₂×C₄	C₂×#97	1	4	yes	—
47	C₂×C₄	C₄×M₁₆	1	8	yes	—
48	C₂×C₄	#405	1	8	yes	—
49	C₂×C₄	#405	1	16	yes	—
50	C₈	—	2	16	no	—
51	C₈	—	4	16	no	—
52	C₈	#322	1	16	yes	—
53	C₂²	C₄²×C₈	1	1	yes	characteristic
54	C₂²	C₄×#97	1	2	yes	—
55	C₂²	#1726	1	4	yes	—
56	C₄	C₂×#322	1	4	yes	—
57	C₄	#1450	1	8	yes	—
58	C₄	#1453	1	16	yes	—
59	C₂	C₄×#322	1	1	yes	characteristic
60	C₂	#7841	1	2	yes	—
61	C₂	#7844	1	4	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₄³	C₂³
commutator	C₂²	C₄²×C₈
socle, prime radical	C₂³	C₄³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
18. C₈²	C₈	semidirect

Maximal subgroups

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

subgroup	index
#8131	2
C₄×#322	2
C₄×C₈²	2