Subgroups — C₂×#11408 #479912

Order 512.

678 conjugacy classes of subgroups.

The subgroup lattice has 223 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	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#11408	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1312	C₂	1	2	yes	maximal, complemented
6	C₂×#1311	C₂	1	2	yes	maximal, complemented
7	C₂³×C₁₆	C₄	1	2	yes	—
8	C₂×#431	C₄	1	1	yes	characteristic
9	C₂×C₄×M₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
11	#2162	C₄	1	4	yes	—
12	#2162	C₂²	1	4	yes	—
13	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
14	C₂×C₂²⋊C₁₆	C₄	1	2	yes	—
15	C₂×C₄×C₁₆	C₄	1	4	yes	—
16	C₂×C₄×C₁₆	C₄	1	2	yes	—
17	C₂×C₄×C₁₆	C₂²	1	4	yes	—
18	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₁₆	C₂²	1	2	yes	—
21	C₈×M₁₆	C₄	1	4	yes	—
22	C₈×M₁₆	C₂²	1	4	yes	—
23	C₂×#322	C₄	1	2	yes	—
24	C₂×C₈²	C₂²	1	2	yes	—
25	#1312	C₂²	1	8	yes	complemented
26	#1311	C₂²	1	8	yes	complemented
27	C₂²×M₁₆	C₈	1	2	yes	—
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₁₆	—	2	8	no	—
32	C₂²×C₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₁₆	C₂×C₄	1	8	yes	—
34	C₂²×C₁₆	C₂×C₄	1	4	yes	—
35	C₂²×C₁₆	C₂×C₄	1	2	yes	—
36	#431	C₈	1	4	yes	—
37	#431	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₈	C₈	1	2	yes	—
39	C₂×C₂²⋊C₈	C₈	1	2	yes	—
40	C₄×M₁₆	C₈	1	4	yes	—
41	C₄×M₁₆	C₂×C₄	1	4	yes	—
42	C₂×#97	C₈	1	2	yes	—
43	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
49	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
50	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
51	C₂×C₄×C₈	C₂³	1	2	yes	—
52	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
53	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
54	C₂²⋊C₁₆	C₂×C₄	1	8	yes	—
55	C₄×C₁₆	C₂×C₄	1	8	yes	—
56	C₄×C₁₆	C₂×C₄	1	2	yes	—
57	C₄×C₁₆	C₂×C₄	1	2	yes	—
58	C₄×C₁₆	C₂×C₄	1	4	yes	—
59	C₄×C₁₆	C₂³	1	2	yes	—
60	C₄×C₁₆	C₂³	1	4	yes	—
61	C₄×C₁₆	C₂³	1	2	yes	—
62	#322	C₂×C₄	1	8	yes	—
63	C₈²	C₂×C₄	1	4	yes	—
64	C₈²	C₂³	1	4	yes	—
65	C₂³×C₄	C₄²	1	1	yes	characteristic
66	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
67	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂×M₁₆	—	2	8	no	—
69	C₂×M₁₆	C₂×C₈	1	8	yes	—
70	C₂²×C₈	—	2	4	no	—
71	C₂²×C₈	—	2	4	no	—
72	C₂²×C₈	C₄²	1	1	yes	characteristic
73	C₂²×C₈	C₄²	1	1	yes	characteristic
74	C₂²×C₈	C₄²	1	2	yes	—
75	C₂²×C₈	C₄²	1	2	yes	—
76	C₂²×C₈	C₂×C₈	1	4	yes	—
77	C₂²×C₈	C₂×C₈	1	4	yes	—
78	C₂²×C₈	C₂²×C₄	1	2	yes	—
79	C₂²×C₈	C₂²×C₄	1	2	yes	—
80	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄²	C₄²	1	2	yes	—
83	C₂×C₄²	C₄²	1	1	yes	characteristic
84	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄²	C₂×C₈	1	2	yes	—
86	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄²	C₂×C₈	1	2	yes	—
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄²	C₂²×C₄	1	2	yes	—
90	C₂×C₁₆	—	2	4	no	—
91	C₂×C₁₆	—	2	8	no	—
92	C₂×C₁₆	—	2	16	no	—
93	C₂×C₁₆	M₁₆	1	8	yes	complemented
94	C₂×C₁₆	C₂²×C₄	1	4	yes	—
95	C₂×C₁₆	C₂²×C₄	1	8	yes	—
96	C₂×C₁₆	C₂²×C₄	1	4	yes	—
97	C₄⋊C₈	C₂×C₈	1	8	yes	—
98	C₂²⋊C₈	C₂×C₈	1	8	yes	—
99	#97	C₂×C₈	1	8	yes	—
100	C₄×C₈	C₄²	1	2	yes	—
101	C₄×C₈	C₄²	1	2	yes	—
102	C₄×C₈	C₄²	1	2	yes	—
103	C₄×C₈	C₂×C₈	1	4	yes	—
104	C₄×C₈	C₂×C₈	1	4	yes	—
105	C₄×C₈	C₂²×C₄	1	2	yes	—
106	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₄×C₈	C₂²×C₄	1	2	yes	—
109	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
110	C₄×C₈	C₂²×C₄	1	2	yes	—
111	C₄×C₈	C₂²×C₄	1	4	yes	—
112	C₄×C₈	C₂²×C₄	1	4	yes	—
113	C₄×C₈	C₂⁴	1	1	yes	characteristic
114	C₂⁴	C₄×C₈	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
119	C₂²×C₄	C₄×C₈	1	2	yes	—
120	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
121	C₂²×C₄	C₄×C₈	1	2	yes	—
122	C₂²×C₄	C₄×C₈	1	2	yes	—
123	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
124	C₂²×C₄	M₃₂	1	8	yes	—
125	C₂²×C₄	C₂×C₄²	1	2	yes	—
126	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
127	C₂²×C₄	C₂²×C₈	1	2	yes	—
128	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²×C₈	1	2	yes	—
130	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
131	M₁₆	—	2	16	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	8	no	—
135	C₂×C₈	—	2	8	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₂×C₄²	1	2	yes	—
141	C₂×C₈	C₂×C₄²	1	2	yes	—
142	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₈	C₂²×C₈	1	8	yes	—
145	C₂×C₈	C₂²×C₈	1	8	yes	—
146	C₂×C₈	C₂×M₁₆	1	4	yes	—
147	C₂×C₈	#131	1	4	yes	—
148	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
150	C₄²	C₄×C₈	1	2	yes	—
151	C₄²	C₄×C₈	1	2	yes	—
152	C₄²	C₄×C₈	1	2	yes	—
153	C₄²	C₂×C₄²	1	2	yes	—
154	C₄²	C₂×C₄²	1	1	yes	characteristic
155	C₄²	C₂²×C₈	1	1	yes	characteristic
156	C₄²	C₂²×C₈	1	1	yes	characteristic
157	C₄²	C₂²×C₈	1	2	yes	—
158	C₄²	C₂²×C₈	1	2	yes	—
159	C₄²	C₂³×C₄	1	1	yes	characteristic
160	C₁₆	—	2	4	no	—
161	C₁₆	—	2	8	no	—
162	C₁₆	C₂×M₁₆	1	8	yes	complemented
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	#346	1	4	yes	—
166	C₂³	C₂×C₄×C₈	1	2	yes	—
167	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	#346	1	8	yes	—
176	C₂×C₄	#346	1	4	yes	—
177	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
178	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
179	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×C₈	1	2	yes	—
181	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
183	C₂×C₄	C₄×M₁₆	1	2	yes	—
184	C₂×C₄	#405	1	2	yes	—
185	C₂×C₄	C₂×M₃₂	1	4	yes	—
186	C₂×C₄	C₂×M₃₂	1	8	yes	—
187	C₂×C₄	#504	1	4	yes	—
188	C₂×C₄	C₂²×C₄²	1	1	yes	characteristic
189	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
190	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	8	no	—
194	C₈	—	2	8	no	—
195	C₈	C₄×M₁₆	1	2	yes	—
196	C₈	#405	1	2	yes	—
197	C₈	C₂²×M₁₆	1	2	yes	—
198	C₈	C₂×#131	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	2	2	no	—
202	C₂²	C₈×M₁₆	1	2	yes	—
203	C₂²	C₂×#346	1	4	yes	—
204	C₂²	C₂×#346	1	2	yes	—
205	C₂²	#2108	1	2	yes	—
206	C₂²	C₂²×C₄×C₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	C₈×M₁₆	1	2	yes	—
210	C₄	C₂×#346	1	2	yes	—
211	C₄	C₂×#346	1	4	yes	—
212	C₄	#2108	1	2	yes	—
213	C₄	C₂×C₄×M₁₆	1	1	yes	characteristic
214	C₄	C₂×#405	1	1	yes	characteristic
215	C₄	C₂²×M₃₂	1	2	yes	—
216	C₄	C₂×#504	1	2	yes	—
217	C₂	—	2	4	no	—
218	C₂	#11408	1	4	yes	direct product factor
219	C₂	C₂×C₈×M₁₆	1	1	yes	characteristic
220	C₂	C₂²×#346	1	1	yes	characteristic
221	C₂	C₂×#2108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂³
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂³	C₂×C₄×C₈
prime radical	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ 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
4. #11408	C₂	direct (N × H)
5. C₂×#1312	C₂	semidirect
6. C₂×#1311	C₂	semidirect
25. #1312	C₂²	semidirect
26. #1311	C₂²	semidirect
93. C₂×C₁₆	M₁₆	semidirect
162. C₁₆	C₂×M₁₆	semidirect
218. C₂	#11408	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×C₄×C₁₆	2
C₂×C₈×M₁₆	2
#11408	2
C₂×#1312	2
C₂×#1311	2

Subgroups — C2×#11408 #479912

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11408 #479912