Subgroups — #128204

Order 512.

394 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#346	C₂	1	2	yes	maximal
2	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
3	#7136	C₂	1	4	yes	maximal
4	C₂×#346	C₄	1	4	yes	—
5	C₂×#346	C₄	1	4	yes	—
6	C₂×C₄×C₁₆	C₂²	1	4	yes	—
7	C₄²×C₈	C₄	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈²	C₄	1	2	yes	—
10	C₂×C₈²	C₂²	1	2	yes	—
11	C₂²×C₁₆	C₂×C₄	1	8	yes	—
12	C₂×C₄×C₈	C₈	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	2	yes	—
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
18	C₄³	C₂×C₄	1	1	yes	characteristic
19	#346	—	2	8	no	—
20	#346	C₈	1	16	yes	complemented
21	C₄×C₁₆	—	2	8	no	—
22	C₈²	—	2	4	no	—
23	C₈²	—	2	4	no	—
24	C₂²×C₈	C₄²	1	2	yes	—
25	C₂²×C₈	C₂×C₈	1	8	yes	—
26	C₂²×C₈	C₂²×C₄	1	2	yes	—
27	C₂×C₄²	C₄²	1	1	yes	characteristic
28	C₂×C₄²	C₄²	1	1	yes	characteristic
29	C₂×C₄²	C₂×C₈	1	4	yes	—
30	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₁₆	—	2	8	no	—
32	C₂×C₁₆	—	2	8	no	—
33	C₂×C₁₆	C₂×C₈	1	16	yes	—
34	C₄×C₈	—	2	16	no	—
35	C₄×C₈	—	2	4	no	—
36	C₄×C₈	—	2	4	no	—
37	C₄×C₈	C₂×C₈	1	8	yes	—
38	C₄×C₈	M₁₆	1	8	yes	—
39	C₄×C₈	M₁₆	1	4	yes	—
40	C₄×C₈	#41	1	4	yes	—
41	C₂²×C₄	C₄×C₈	1	2	yes	—
42	C₂²×C₄	C₄×C₈	1	2	yes	—
43	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₄	C₂²×C₈	1	2	yes	—
45	C₂×C₈	—	2	8	no	—
46	C₂×C₈	—	2	8	no	—
47	C₂×C₈	C₄×C₈	1	4	yes	—
48	C₂×C₈	#97	1	4	yes	—
49	C₂×C₈	#97	1	4	yes	—
50	C₂×C₈	M₃₂	1	16	yes	—
51	C₂×C₈	#117	1	4	yes	—
52	C₂×C₈	C₂²×C₈	1	4	yes	—
53	C₂×C₈	C₂×M₁₆	1	4	yes	—
54	C₄²	—	2	8	no	—
55	C₄²	C₄×C₈	1	4	yes	—
56	C₄²	#97	1	4	yes	—
57	C₄²	#117	1	2	yes	—
58	C₄²	C₂×M₁₆	1	2	yes	—
59	C₁₆	—	2	16	no	—
60	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
61	C₂×C₄	—	2	4	no	—
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	C₈²	1	4	yes	—
64	C₂×C₄	#322	1	4	yes	—
65	C₂×C₄	#376	1	1	yes	characteristic
66	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
67	C₂×C₄	C₂×#97	1	1	yes	characteristic
68	C₂×C₄	C₂×#97	1	2	yes	—
69	C₂×C₄	#431	1	4	yes	—
70	C₂×C₄	C₂×M₃₂	1	8	yes	—
71	C₈	—	2	16	no	—
72	C₈	#322	1	8	yes	—
73	C₈	#431	1	8	yes	—
74	C₂²	C₂×C₈²	1	1	yes	characteristic
75	C₂²	C₂×#322	1	1	yes	characteristic
76	C₂²	#1744	1	1	yes	characteristic
77	C₂²	C₄×M₃₂	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	2	4	no	—
80	C₄	C₂×#322	1	2	yes	—
81	C₄	#1744	1	2	yes	—
82	C₄	#2163	1	8	yes	—
83	C₂	#8131	1	1	yes	characteristic
84	C₂	C₈×M₃₂	1	2	yes	—
85	C₂	#11402	1	2	yes	—
86	C₂	#12591	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
20. #346	C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#346	2
C₄×C₈²	2
#7136	2