Subgroups — C₂×#12542 #492833

Order 512.

852 conjugacy classes of subgroups.

The subgroup lattice has 143 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2110	C₂	1	3	yes	maximal
2	C₂×#2108	C₂	1	3	yes	maximal
3	C₂×#1747	C₂	1	1	yes	maximal, characteristic
4	#12542	C₂	1	8	yes	maximal, direct product factor
5	C₂²×M₃₂	C₂²	1	3	yes	—
6	C₂³×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#405	C₂²	1	3	yes	—
8	#2110	C₄	1	12	yes	complemented
9	#2110	C₂²	1	12	yes	—
10	C₂×#347	C₄	1	12	yes	complemented
11	#2108	C₄	1	12	yes	complemented
12	#2108	C₂²	1	12	yes	—
13	C₂×#346	C₄	1	6	yes	complemented
14	C₂×C₄×C₁₆	C₄	1	6	yes	complemented
15	#1747	C₄	1	4	yes	—
16	#1747	C₂²	1	4	yes	—
17	C₂×#338	C₄	1	1	yes	characteristic
18	C₂×H(3,ℤ/4)	C₄	1	3	yes	—
19	C₂²×M₁₆	C₂×C₄	1	3	yes	—
20	C₂³×C₈	C₂³	1	1	yes	characteristic
21	C₂×#117	C₂×C₄	1	3	yes	—
22	C₂×M₃₂	C₂×C₄	1	12	yes	—
23	C₂×M₃₂	C₂×C₄	1	12	yes	—
24	C₂×M₃₂	C₂×C₄	1	6	yes	—
25	C₂×M₃₂	C₂³	1	6	yes	—
26	C₂²×C₁₆	C₂×C₄	1	6	yes	—
27	C₂²×C₁₆	C₂×C₄	1	3	yes	—
28	C₂²×C₁₆	C₂×C₄	1	6	yes	—
29	C₂²×C₁₆	C₂×C₄	1	3	yes	—
30	C₂²×C₁₆	C₂³	1	2	yes	—
31	#405	C₂×C₄	1	6	yes	—
32	#405	C₂×C₄	1	6	yes	—
33	#405	C₂×C₄	1	6	yes	—
34	#405	C₂³	1	6	yes	—
35	C₂×#97	C₂×C₄	1	6	yes	—
36	C₂×C₄×C₈	C₂×C₄	1	6	yes	—
37	#347	C₂×C₄	1	48	yes	complemented
38	#346	C₂×C₄	1	24	yes	complemented
39	C₄×C₁₆	C₂×C₄	1	24	yes	complemented
40	#338	C₂×C₄	1	4	yes	—
41	H(3,ℤ/4)	C₂×C₄	1	12	yes	—
42	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂×M₁₆	—	2	6	no	—
44	C₂×M₁₆	—	2	6	no	—
45	C₂×M₁₆	C₄²	1	6	yes	—
46	C₂×M₁₆	C₂²×C₄	1	6	yes	—
47	C₂²×C₈	C₄²	1	6	yes	—
48	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²×C₄	1	3	yes	—
50	C₂²×C₈	C₂²×C₄	1	3	yes	—
51	C₂²×C₈	C₂²×C₄	1	3	yes	—
52	C₂²×C₈	C₂²×C₄	1	3	yes	—
53	C₂²×C₈	C₂⁴	1	1	yes	characteristic
54	#117	—	2	3	no	—
55	#117	—	2	3	no	—
56	#117	C₄²	1	6	yes	—
57	#117	C₂²×C₄	1	6	yes	—
58	C₂×C₄⋊C₄	—	2	3	no	—
59	C₂×C₂²⋊C₄	—	2	3	no	—
60	C₂×C₄²	C₄²	1	6	yes	—
61	M₃₂	C₄²	1	24	yes	—
62	M₃₂	C₂²×C₄	1	24	yes	—
63	C₂×C₁₆	C₄²	1	6	yes	—
64	C₂×C₁₆	C₄²	1	3	yes	—
65	C₂×C₁₆	C₄²	1	6	yes	—
66	C₂×C₁₆	C₄²	1	12	yes	—
67	C₂×C₁₆	C₄²	1	12	yes	—
68	C₂×C₁₆	C₄²	1	1	yes	characteristic
69	C₂×C₁₆	C₂²×C₄	1	6	yes	—
70	C₂×C₁₆	C₂²×C₄	1	12	yes	—
71	C₂×C₁₆	C₂²×C₄	1	6	yes	—
72	#97	C₄²	1	12	yes	—
73	#97	C₂²×C₄	1	12	yes	—
74	C₄×C₈	C₄²	1	12	yes	—
75	C₄×C₈	C₂²×C₄	1	12	yes	—
76	C₂⁴	#131	1	1	yes	characteristic
77	C₂²×C₄	—	2	6	no	—
78	C₂²×C₄	C₂×C₄²	1	3	yes	—
79	C₂²×C₄	C₂×C₄²	1	3	yes	—
80	C₂²×C₄	#131	1	3	yes	—
81	C₂²×C₄	#131	1	3	yes	—
82	C₂²×C₄	#131	1	1	yes	characteristic
83	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
84	M₁₆	—	2	12	no	—
85	M₁₆	—	2	12	no	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	C₂×C₄²	1	12	yes	—
89	C₂×C₈	C₂×C₄²	1	3	yes	—
90	C₂×C₈	C₂×C₄²	1	3	yes	—
91	C₂×C₈	C₂×C₄²	1	3	yes	—
92	C₂×C₈	C₂×C₄²	1	3	yes	—
93	C₂×C₈	C₂×C₄²	1	3	yes	—
94	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₈	C₂×C₄²	1	6	yes	—
96	C₂×C₈	C₂³×C₄	1	3	yes	—
97	C₂×C₈	C₂³×C₄	1	3	yes	—
98	C₄⋊C₄	—	2	6	no	—
99	C₄⋊C₄	—	2	6	no	—
100	C₂²⋊C₄	—	2	6	no	—
101	C₂²⋊C₄	—	2	6	no	—
102	C₄²	—	2	6	no	—
103	C₄²	C₂×C₄²	1	12	yes	—
104	C₁₆	C₂×C₄²	1	6	yes	—
105	C₁₆	C₂×C₄²	1	2	yes	—
106	C₁₆	C₂×C₄²	1	24	yes	—
107	C₂³	—	4	2	no	—
108	C₂³	#405	1	3	yes	—
109	C₂³	#405	1	3	yes	—
110	C₂³	C₂×#131	1	1	yes	characteristic
111	C₂×C₄	—	2	12	no	—
112	C₂×C₄	—	4	12	no	—
113	C₂×C₄	C₄³	1	3	yes	—
114	C₂×C₄	C₄³	1	1	yes	characteristic
115	C₂×C₄	#405	1	3	yes	—
116	C₂×C₄	#405	1	3	yes	—
117	C₂×C₄	#405	1	6	yes	—
118	C₂×C₄	#405	1	3	yes	—
119	C₂×C₄	#405	1	3	yes	—
120	C₂×C₄	C₂²×C₄²	1	3	yes	—
121	C₂×C₄	C₂×#131	1	3	yes	—
122	C₈	—	2	12	no	—
123	C₈	C₄³	1	1	yes	characteristic
124	C₈	C₄³	1	3	yes	—
125	C₈	C₂²×C₄²	1	1	yes	characteristic
126	C₈	C₂²×C₄²	1	3	yes	—
127	C₂²	—	2	6	no	—
128	C₂²	—	4	1	no	—
129	C₂²	—	4	3	no	—
130	C₂²	#1726	1	3	yes	—
131	C₂²	#1726	1	1	yes	characteristic
132	C₂²	C₂×#405	1	3	yes	—
133	C₄	—	4	12	no	—
134	C₄	#1726	1	1	yes	characteristic
135	C₄	#1726	1	3	yes	—
136	C₄	C₂×C₄³	1	1	yes	characteristic
137	C₄	C₂×#405	1	3	yes	—
138	C₂	—	2	3	no	—
139	C₂	—	2	3	no	—
140	C₂	#12542	1	2	yes	direct product factor
141	C₂	C₂×#1726	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#1726
prime radical	C₂⁴	#131
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #12542	C₂	direct (N × H)
8. #2110	C₄	semidirect
10. C₂×#347	C₄	semidirect
11. #2108	C₄	semidirect
13. C₂×#346	C₄	semidirect
14. C₂×C₄×C₁₆	C₄	semidirect
37. #347	C₂×C₄	semidirect
38. #346	C₂×C₄	semidirect
39. C₄×C₁₆	C₂×C₄	semidirect
140. C₂	#12542	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#2110	2
C₂×#2108	2
C₂×#1747	2
#12542	2

Subgroups — C2×#12542 #492833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#12542 #492833