Subgroups — #126531

Order 512.

402 conjugacy classes of subgroups.

The subgroup lattice has 119 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#12590	C₂	1	2	yes	maximal
2	C₂×#1312	C₂	1	2	yes	maximal
3	C₂×#1311	C₂	1	2	yes	maximal
4	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₆	C₂²	1	4	yes	—
6	C₄²×C₈	C₄	1	1	yes	characteristic
7	C₄²×C₈	C₄	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₈²	C₄	1	1	yes	characteristic
10	C₂×C₈²	C₄	1	1	yes	characteristic
11	C₂×C₈²	C₂²	1	1	yes	characteristic
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1374	C₄	1	8	yes	—
14	#1312	C₄	1	8	yes	complemented
15	#1311	C₄	1	8	yes	complemented
16	C₂²×C₁₆	C₂×C₄	1	8	yes	—
17	C₂×C₄×C₈	C₈	1	4	yes	—
18	C₂×C₄×C₈	C₈	1	4	yes	—
19	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	C₄³	C₂×C₄	1	1	yes	characteristic
27	C₄×C₁₆	C₂×C₄	1	8	yes	—
28	C₄×C₁₆	C₂×C₄	1	8	yes	—
29	C₈²	C₈	1	4	yes	—
30	C₈²	C₈	1	4	yes	—
31	C₈²	C₂×C₄	1	4	yes	—
32	C₈²	C₂×C₄	1	4	yes	—
33	C₂²×C₈	C₄²	1	2	yes	—
34	C₂²×C₈	C₂×C₈	1	2	yes	—
35	C₂²×C₈	C₂×C₈	1	2	yes	—
36	C₂²×C₈	C₂×C₈	1	2	yes	—
37	C₂²×C₈	C₂×C₈	1	2	yes	—
38	C₂²×C₈	C₂²×C₄	1	2	yes	—
39	C₂×C₄²	C₄²	1	2	yes	—
40	C₂×C₄²	C₂×C₈	1	2	yes	—
41	C₂×C₄²	C₂×C₈	1	2	yes	—
42	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₁₆	—	2	16	no	—
44	C₂×C₁₆	C₄²	1	16	yes	—
45	C₄×C₈	—	2	8	no	—
46	C₄×C₈	—	2	8	no	—
47	C₄×C₈	—	2	8	no	—
48	C₄×C₈	C₄²	1	2	yes	—
49	C₄×C₈	C₄²	1	2	yes	—
50	C₄×C₈	C₄²	1	2	yes	—
51	C₄×C₈	C₄²	1	2	yes	—
52	C₄×C₈	C₂×C₈	1	2	yes	—
53	C₄×C₈	C₂×C₈	1	2	yes	—
54	C₄×C₈	C₂×C₈	1	2	yes	—
55	C₄×C₈	C₂×C₈	1	2	yes	—
56	C₄×C₈	C₂×C₈	1	2	yes	—
57	C₄×C₈	C₂×C₈	1	2	yes	—
58	C₄×C₈	C₂²×C₄	1	2	yes	—
59	C₄×C₈	C₂²×C₄	1	2	yes	—
60	C₂²×C₄	C₄×C₈	1	4	yes	—
61	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₈	—	2	4	no	—
65	C₂×C₈	—	2	4	no	—
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	C₄×C₈	1	4	yes	—
69	C₂×C₈	C₄×C₈	1	4	yes	—
70	C₂×C₈	C₄×C₈	1	4	yes	—
71	C₂×C₈	C₄×C₈	1	4	yes	—
72	C₂×C₈	C₄×C₈	1	4	yes	—
73	C₂×C₈	M₃₂	1	8	yes	—
74	C₂×C₈	C₂×C₄²	1	4	yes	—
75	C₂×C₈	#131	1	8	yes	—
76	C₄²	—	2	4	no	—
77	C₄²	—	2	4	no	—
78	C₄²	M₃₂	1	8	yes	—
79	C₄²	C₂×C₄²	1	1	yes	characteristic
80	C₄²	C₂×C₄²	1	1	yes	characteristic
81	C₄²	C₂²×C₈	1	1	yes	characteristic
82	C₄²	C₂²×C₈	1	1	yes	characteristic
83	C₁₆	—	2	16	no	—
84	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	#346	1	8	yes	—
88	C₂×C₄	C₄³	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂×C₄	#405	1	4	yes	—
95	C₂×C₄	C₂×M₃₂	1	4	yes	—
96	C₂×C₄	C₂×M₃₂	1	4	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	2	4	no	—
99	C₈	—	2	4	no	—
100	C₈	—	2	4	no	—
101	C₈	#346	1	8	yes	—
102	C₈	#405	1	8	yes	—
103	C₂²	#1450	1	2	yes	—
104	C₂²	C₄²×C₈	1	1	yes	characteristic
105	C₂²	C₂×#346	1	2	yes	—
106	C₂²	C₄×M₃₂	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	#1450	1	2	yes	—
109	C₄	#1726	1	2	yes	—
110	C₄	C₂×#346	1	2	yes	—
111	C₄	C₄×M₃₂	1	2	yes	—
112	C₄	C₄×M₃₂	1	2	yes	—
113	C₄	C₄×M₃₂	1	2	yes	—
114	C₂	#7841	1	1	yes	characteristic
115	C₂	#11411	1	4	yes	—
116	C₂	C₄×#346	1	1	yes	characteristic
117	C₂	C₄×#346	1	1	yes	characteristic

	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
14. #1312	C₄	semidirect
15. #1311	C₄	semidirect

Maximal subgroups

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

subgroup	index
#12590	2
C₂×#1312	2
C₂×#1311	2
C₄×C₈²	2