Subgroups — C₂×#11410 #479921

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2162	C₂	1	2	yes	maximal
2	C₂²×C₈²	C₂	1	1	yes	maximal, characteristic
3	#11410	C₂	1	8	yes	maximal, direct product factor
4	C₂×#1312	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	#2162	C₄	1	8	yes	—
10	#2162	C₂²	1	8	yes	—
11	C₂×C₄⋊C₁₆	C₄	1	4	yes	—
12	C₂×C₂²⋊C₁₆	C₄	1	4	yes	—
13	C₂×C₄×C₁₆	C₂²	1	4	yes	—
14	C₂×C₈²	C₄	1	1	yes	characteristic
15	C₂×C₈²	C₄	1	1	yes	characteristic
16	C₂×C₈²	C₄	1	4	yes	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	C₂×C₈²	C₂²	1	4	yes	—
19	C₂×C₈²	C₂²	1	1	yes	characteristic
20	#1312	C₂²	1	8	yes	complemented
21	C₈×C₁₆	C₂²	1	8	yes	complemented
22	C₂³×C₈	C₈	1	2	yes	—
23	C₂³×C₈	C₈	1	2	yes	—
24	C₂³×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	C₂×C₄	1	8	yes	—
27	C₂×C₄×C₈	C₈	1	4	yes	—
28	C₂×C₄×C₈	C₈	1	2	yes	—
29	C₂×C₄×C₈	C₈	1	2	yes	—
30	C₂×C₄×C₈	C₈	1	4	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
37	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
39	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	2	yes	—
43	C₄⋊C₁₆	C₂×C₄	1	16	yes	—
44	C₂²⋊C₁₆	C₂×C₄	1	16	yes	—
45	C₄×C₁₆	C₂×C₄	1	8	yes	—
46	C₄×C₁₆	C₂³	1	8	yes	—
47	C₈²	C₂×C₄	1	2	yes	—
48	C₈²	C₂×C₄	1	4	yes	—
49	C₈²	C₂×C₄	1	4	yes	—
50	C₈²	C₂×C₄	1	2	yes	—
51	C₈²	C₂³	1	2	yes	—
52	C₈²	C₂³	1	2	yes	—
53	C₂³×C₄	C₄²	1	1	yes	characteristic
54	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₈	—	2	8	no	—
57	C₂²×C₈	—	2	8	no	—
58	C₂²×C₈	C₁₆	1	16	yes	complemented
59	C₂²×C₈	C₄²	1	2	yes	—
60	C₂²×C₈	C₄²	1	4	yes	—
61	C₂²×C₈	C₂×C₈	1	2	yes	—
62	C₂²×C₈	C₂×C₈	1	8	yes	—
63	C₂²×C₈	C₂×C₈	1	2	yes	—
64	C₂²×C₈	C₂×C₈	1	2	yes	—
65	C₂²×C₈	C₂×C₈	1	8	yes	—
66	C₂²×C₈	C₂×C₈	1	2	yes	—
67	C₂²×C₈	C₂²×C₄	1	4	yes	—
68	C₂²×C₈	C₂²×C₄	1	2	yes	—
69	C₂×C₄²	C₄²	1	2	yes	—
70	C₂×C₄²	C₄²	1	1	yes	characteristic
71	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄²	C₂×C₈	1	2	yes	—
73	C₂×C₄²	C₂×C₈	1	2	yes	—
74	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄²	C₂²×C₄	1	2	yes	—
76	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₁₆	—	2	16	no	—
78	C₂×C₁₆	C₂²×C₄	1	16	yes	—
79	C₄×C₈	C₄²	1	2	yes	—
80	C₄×C₈	C₄²	1	4	yes	—
81	C₄×C₈	C₂×C₈	1	2	yes	—
82	C₄×C₈	C₂×C₈	1	8	yes	—
83	C₄×C₈	C₂×C₈	1	8	yes	—
84	C₄×C₈	C₂×C₈	1	2	yes	—
85	C₄×C₈	C₂×C₈	1	2	yes	—
86	C₄×C₈	C₂×C₈	1	2	yes	—
87	C₄×C₈	C₂²×C₄	1	2	yes	—
88	C₄×C₈	C₂²×C₄	1	4	yes	—
89	C₄×C₈	C₂²×C₄	1	2	yes	—
90	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₄×C₈	C₂²×C₄	1	2	yes	—
92	C₄×C₈	C₂²×C₄	1	2	yes	—
93	C₄×C₈	C₂²×C₄	1	2	yes	—
94	C₄×C₈	C₂²×C₄	1	2	yes	—
95	C₄×C₈	C₂⁴	1	1	yes	characteristic
96	C₂⁴	C₄×C₈	1	1	yes	characteristic
97	C₂²×C₄	—	2	4	no	—
98	C₂²×C₄	—	2	4	no	—
99	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
100	C₂²×C₄	C₄×C₈	1	2	yes	—
101	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
102	C₂²×C₄	C₄×C₈	1	2	yes	—
103	C₂²×C₄	C₄×C₈	1	2	yes	—
104	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₁₆	1	8	yes	—
106	C₂²×C₄	C₂×C₄²	1	2	yes	—
107	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
108	C₂²×C₄	C₂²×C₈	1	2	yes	—
109	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
110	C₂²×C₄	C₂²×C₈	1	2	yes	—
111	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	8	no	—
115	C₂×C₈	—	2	8	no	—
116	C₂×C₈	—	2	16	no	—
117	C₂×C₈	C₂×C₁₆	1	8	yes	—
118	C₂×C₈	C₂×C₁₆	1	8	yes	—
119	C₂×C₈	C₂×C₁₆	1	32	yes	complemented
120	C₂×C₈	M₃₂	1	8	yes	—
121	C₂×C₈	C₂×C₄²	1	4	yes	—
122	C₂×C₈	C₂×C₄²	1	2	yes	—
123	C₂×C₈	C₂²×C₈	1	4	yes	—
124	C₂×C₈	C₂²×C₈	1	4	yes	—
125	C₂×C₈	C₂²×C₈	1	4	yes	—
126	C₂×C₈	C₂²×C₈	1	4	yes	—
127	C₂×C₈	#131	1	8	yes	—
128	C₂×C₈	C₂³×C₄	1	2	yes	—
129	C₄²	C₄×C₈	1	2	yes	—
130	C₄²	C₄×C₈	1	2	yes	—
131	C₄²	C₄×C₈	1	2	yes	—
132	C₄²	C₂×C₄²	1	2	yes	—
133	C₄²	C₂×C₄²	1	1	yes	characteristic
134	C₄²	C₂²×C₈	1	2	yes	—
135	C₄²	C₂²×C₈	1	1	yes	characteristic
136	C₄²	C₂²×C₈	1	2	yes	—
137	C₄²	C₂²×C₈	1	1	yes	characteristic
138	C₄²	C₂³×C₄	1	1	yes	characteristic
139	C₁₆	—	2	16	no	—
140	C₂³	—	2	2	no	—
141	C₂³	—	2	2	no	—
142	C₂³	C₄×C₁₆	1	4	yes	—
143	C₂³	C₂×C₄×C₈	1	2	yes	—
144	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	4	no	—
151	C₂×C₄	C₄×C₁₆	1	8	yes	—
152	C₂×C₄	C₄×C₁₆	1	4	yes	—
153	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
154	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
155	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
156	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
157	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
158	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
159	C₂×C₄	#405	1	4	yes	—
160	C₂×C₄	C₂²×C₁₆	1	4	yes	—
161	C₂×C₄	C₂²×C₁₆	1	8	yes	—
162	C₂×C₄	C₂×M₃₂	1	4	yes	—
163	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
164	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
165	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
166	C₈	—	2	4	no	—
167	C₈	—	2	4	no	—
168	C₈	#405	1	4	yes	—
169	C₈	C₂²×C₁₆	1	8	yes	—
170	C₈	C₂²×C₁₆	1	8	yes	—
171	C₈	C₂×M₃₂	1	8	yes	—
172	C₈	C₂×#131	1	4	yes	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	#1450	1	2	yes	—
177	C₂²	C₂×C₄×C₁₆	1	4	yes	—
178	C₂²	C₂×C₄×C₁₆	1	2	yes	—
179	C₂²	C₄×M₃₂	1	2	yes	—
180	C₂²	C₂²×C₄×C₈	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	#1450	1	2	yes	—
184	C₄	C₂×C₄×C₁₆	1	2	yes	—
185	C₄	C₂×C₄×C₁₆	1	4	yes	—
186	C₄	C₄×M₃₂	1	2	yes	—
187	C₄	C₂×#405	1	2	yes	—
188	C₄	C₂³×C₁₆	1	2	yes	—
189	C₄	C₂²×M₃₂	1	2	yes	—
190	C₂	—	2	4	no	—
191	C₂	#11410	1	4	yes	direct product factor
192	C₂	C₂×#1450	1	1	yes	characteristic
193	C₂	C₂²×C₄×C₁₆	1	1	yes	characteristic
194	C₂	C₂×C₄×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈²	C₂²
commutator	C₂	C₂²×C₄×C₁₆
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×C₄×C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #11410	C₂	direct (N × H)
4. C₂×#1312	C₂	semidirect
5. C₂×C₈×C₁₆	C₂	semidirect
20. #1312	C₂²	semidirect
21. C₈×C₁₆	C₂²	semidirect
58. C₂²×C₈	C₁₆	semidirect
119. C₂×C₈	C₂×C₁₆	semidirect
191. C₂	#11410	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×#2162	2
C₂²×C₈²	2
#11410	2
C₂×#1312	2
C₂×C₈×C₁₆	2

Subgroups — C2×#11410 #479921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11410 #479921