Subgroups — #92902

Order 512.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11410	C₂	1	2	yes	maximal
2	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#2162	C₄	1	4	yes	—
4	C₂×C₄×C₁₆	C₄	1	2	yes	—
5	C₂×C₈²	C₂²	1	1	yes	characteristic
6	#1312	—	2	2	no	—
7	C₈×C₁₆	—	2	2	no	—
8	C₈×C₁₆	—	2	2	no	—
9	C₂²×C₁₆	C₈	1	4	yes	—
10	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
11	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
12	C₄⋊C₁₆	C₈	1	8	yes	—
13	C₂²⋊C₁₆	—	2	4	no	—
14	C₄×C₁₆	—	2	2	no	—
15	C₄×C₁₆	—	2	4	no	—
16	C₄×C₁₆	—	2	2	no	—
17	C₈²	D₄	1	2	yes	—
18	C₈²	D₄	1	1	yes	characteristic
19	C₈²	Q₈	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₈	1	2	yes	—
21	C₂²×C₈	M₁₆	1	4	yes	—
22	C₂×C₄²	C₄²	1	1	yes	characteristic
23	C₂×C₁₆	—	2	4	no	—
24	C₂×C₁₆	—	2	4	no	—
25	C₂×C₁₆	—	2	8	no	—
26	C₂×C₁₆	C₁₆	1	8	yes	complemented
27	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
28	C₄×C₈	C₂²⋊C₄	1	2	yes	—
29	C₄×C₈	C₄⋊C₄	1	2	yes	—
30	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₄×C₈	C₂×C₈	1	4	yes	—
32	C₄×C₈	M₁₆	1	2	yes	—
33	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
34	C₂²×C₄	#97	1	2	yes	—
35	C₂×C₈	—	2	8	no	—
36	C₂×C₈	C₂²⋊C₈	1	2	yes	—
37	C₂×C₈	C₂²⋊C₈	1	4	yes	—
38	C₂×C₈	C₄⋊C₈	1	4	yes	—
39	C₂×C₈	C₄⋊C₈	1	2	yes	—
40	C₂×C₈	C₂×C₁₆	1	4	yes	—
41	C₂×C₈	M₃₂	1	4	yes	—
42	C₄²	#95	1	1	yes	characteristic
43	C₄²	C₄×C₈	1	2	yes	—
44	C₄²	#97	1	1	yes	characteristic
45	C₁₆	—	2	8	no	—
46	C₁₆	—	4	8	no	—
47	C₂³	#322	1	1	yes	characteristic
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	C₈²	1	1	yes	characteristic
50	C₂×C₄	#322	1	2	yes	—
51	C₂×C₄	#336	1	1	yes	characteristic
52	C₂×C₄	#336	1	2	yes	—
53	C₂×C₄	C₄×C₁₆	1	2	yes	—
54	C₂×C₄	#346	1	2	yes	—
55	C₈	—	2	4	no	—
56	C₈	C₂²⋊C₁₆	1	4	yes	—
57	C₈	#350	1	8	yes	—
58	C₈	C₄⋊C₁₆	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	#1273	1	1	yes	characteristic
61	C₂²	#1312	1	2	yes	—
62	C₄	—	2	2	no	—
63	C₄	C₈×C₁₆	1	1	yes	characteristic
64	C₄	#1311	1	1	yes	characteristic
65	C₄	#1374	1	2	yes	—
66	C₄	#1379	1	4	yes	—
67	C₂	—	2	2	no	—
68	C₂	#7136	1	1	yes	characteristic
69	C₂	#7141	1	2	yes	—

	subgroup	G/H
center	C₈²	D₄
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
prime radical	C₂³	#322
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
26. C₂×C₁₆	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#11410	2
C₂×C₈×C₁₆	2