Subgroups — #126521

Order 512.

410 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#346	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	4	yes	maximal
4	#7841	C₂	1	1	yes	maximal, characteristic
5	C₂×#346	C₄	1	4	yes	—
6	C₂×C₄×C₁₆	C₄	1	4	yes	—
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₂²	1	2	yes	—
9	#1744	C₄	1	2	yes	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#322	C₄	1	2	yes	—
12	C₂×C₈²	C₂²	1	2	yes	—
13	#1312	C₄	1	16	yes	complemented
14	C₂²×C₁₆	C₂×C₄	1	4	yes	—
15	C₂²×C₁₆	C₂×C₄	1	4	yes	—
16	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	C₄³	C₂×C₄	1	1	yes	characteristic
22	#346	—	2	8	no	—
23	C₄×C₁₆	—	2	8	no	—
24	C₄×C₁₆	C₂×C₄	1	8	yes	—
25	C₄×C₁₆	C₂×C₄	1	8	yes	—
26	#336	C₈	1	8	yes	—
27	#322	C₈	1	8	yes	—
28	C₈²	C₂×C₄	1	8	yes	—
29	C₂²×C₈	C₄²	1	2	yes	—
30	C₂²×C₈	C₂×C₈	1	8	yes	—
31	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄²	C₄²	1	2	yes	—
34	C₂×C₄²	C₂×C₈	1	4	yes	—
35	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₁₆	—	2	8	no	—
37	C₂×C₁₆	—	2	8	no	—
38	C₂×C₁₆	C₄²	1	8	yes	—
39	C₂×C₁₆	C₄²	1	8	yes	—
40	C₄×C₈	C₄²	1	8	yes	—
41	C₄×C₈	C₂×C₈	1	8	yes	—
42	C₄×C₈	C₂×C₈	1	4	yes	—
43	C₄×C₈	M₁₆	1	8	yes	—
44	C₄×C₈	M₁₆	1	8	yes	—
45	C₄×C₈	C₂²×C₄	1	4	yes	—
46	C₂²×C₄	C₄×C₈	1	4	yes	—
47	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
48	C₂²×C₄	C₂²×C₈	1	2	yes	—
49	C₂×C₈	—	2	16	no	—
50	C₂×C₈	C₄×C₈	1	16	yes	—
51	C₂×C₈	C₄×C₈	1	4	yes	—
52	C₂×C₈	#97	1	4	yes	—
53	C₂×C₈	#97	1	4	yes	—
54	C₂×C₈	C₂×C₄²	1	2	yes	—
55	C₂×C₈	C₂×C₄²	1	2	yes	—
56	C₂×C₈	C₂×M₁₆	1	4	yes	—
57	C₂×C₈	C₂×M₁₆	1	4	yes	—
58	C₄²	—	2	8	no	—
59	C₄²	#97	1	4	yes	—
60	C₄²	#97	1	4	yes	—
61	C₄²	C₂×C₄²	1	2	yes	—
62	C₄²	C₂²×C₈	1	2	yes	—
63	C₁₆	—	2	8	no	—
64	C₁₆	—	2	8	no	—
65	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×C₄	—	2	8	no	—
67	C₂×C₄	#322	1	8	yes	—
68	C₂×C₄	C₄³	1	1	yes	characteristic
69	C₂×C₄	C₂×C₄×C₈	1	4	yes	—
70	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
71	C₂×C₄	C₂×#97	1	2	yes	—
72	C₂×C₄	C₂×#97	1	2	yes	—
73	C₂×C₄	#504	1	8	yes	—
74	C₈	—	2	16	no	—
75	C₈	#322	1	8	yes	—
76	C₈	C₄×M₁₆	1	4	yes	—
77	C₈	C₄×M₁₆	1	4	yes	—
78	C₂²	C₂×#322	1	2	yes	—
79	C₂²	C₄²×C₈	1	1	yes	characteristic
80	C₂²	#2108	1	4	yes	—
81	C₄	—	2	8	no	—
82	C₄	C₂×#322	1	2	yes	—
83	C₄	C₄×#97	1	1	yes	characteristic
84	C₄	C₄×#97	1	1	yes	characteristic
85	C₄	#2108	1	8	yes	—
86	C₂	C₄×#322	1	1	yes	characteristic
87	C₂	#11401	1	4	yes	—
88	C₂	#12539	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄×C₈	C₂³
commutator	C₂²	C₄²×C₈
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
13. #1312	C₄	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
C₄²×C₁₆	2
C₂×#1312	2
#7841	2