Subgroups — C₂×#7142 #126582

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₂×#1744	C₂	1	1	yes	maximal, characteristic
3	#7142	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₄³	C₄	1	1	yes	characteristic
7	#2162	—	2	2	no	—
8	#2162	—	2	2	no	—
9	#2162	C₄	1	4	yes	complemented
10	#2162	C₂²	1	4	yes	—
11	C₂×C₄⋊C₁₆	—	2	2	no	—
12	C₂×C₂²⋊C₁₆	—	2	2	no	—
13	C₂×C₄×C₁₆	C₄	1	4	yes	complemented
14	#1744	—	2	2	no	—
15	#1744	C₄	1	2	yes	—
16	#1744	C₂²	1	2	yes	—
17	C₂×#336	—	2	2	no	—
18	C₂³×C₈	—	2	1	no	—
19	C₂³×C₈	D₄	1	2	yes	—
20	C₂²×C₄²	—	2	1	no	—
21	C₂²×C₄²	—	2	1	no	—
22	C₂²×C₄²	C₈	1	2	yes	—
23	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₆	—	2	4	no	—
25	C₂×C₄×C₈	—	2	1	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	—	2	1	no	—
28	C₂×C₄×C₈	C₈	1	2	yes	—
29	C₂×C₄×C₈	C₈	1	2	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₈	D₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
38	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	C₄³	—	2	1	no	—
41	C₄³	—	2	1	no	—
42	C₄³	C₈	1	2	yes	—
43	C₄³	C₂×C₄	1	2	yes	—
44	C₄⋊C₁₆	—	2	4	no	—
45	C₄⋊C₁₆	—	2	4	no	—
46	C₂²⋊C₁₆	—	2	4	no	—
47	C₂²⋊C₁₆	—	2	4	no	—
48	C₄×C₁₆	—	2	4	no	—
49	C₄×C₁₆	C₂×C₄	1	8	yes	complemented
50	#336	—	2	4	no	—
51	#336	—	2	4	no	—
52	C₂³×C₄	—	2	2	no	—
53	C₂³×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂³×C₄	M₁₆	1	1	yes	characteristic
56	C₂²×C₈	—	2	2	no	—
57	C₂²×C₈	—	2	2	no	—
58	C₂²×C₈	—	2	2	no	—
59	C₂²×C₈	—	4	4	no	—
60	C₂²×C₈	—	4	4	no	—
61	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₈	C₄⋊C₄	1	2	yes	—
65	C₂²×C₈	C₄⋊C₄	1	2	yes	—
66	C₂²×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₄²	—	2	2	no	—
68	C₂×C₄²	—	2	4	no	—
69	C₂×C₄²	—	2	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	—	2	2	no	—
72	C₂×C₄²	—	2	2	no	—
73	C₂×C₄²	—	4	4	no	—
74	C₂×C₄²	—	4	4	no	—
75	C₂×C₄²	C₁₆	1	8	yes	complemented
76	C₂×C₄²	C₄²	1	1	yes	characteristic
77	C₂×C₄²	C₄²	1	1	yes	characteristic
78	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄²	C₂×C₈	1	4	yes	—
82	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
84	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
85	C₂×C₄²	M₁₆	1	1	yes	characteristic
86	C₂×C₄²	M₁₆	1	1	yes	characteristic
87	C₂×C₄²	M₁₆	1	1	yes	characteristic
88	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₁₆	—	2	8	no	—
90	C₂×C₁₆	—	4	8	no	—
91	C₄×C₈	—	2	2	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	C₄²	1	2	yes	—
95	C₄×C₈	C₂²⋊C₄	1	2	yes	—
96	C₄×C₈	C₂²⋊C₄	1	2	yes	—
97	C₄×C₈	C₄⋊C₄	1	2	yes	—
98	C₄×C₈	C₄⋊C₄	1	2	yes	—
99	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
101	C₄×C₈	C₂×C₈	1	4	yes	—
102	C₄×C₈	C₂²×C₄	1	2	yes	—
103	C₄×C₈	C₂×D₄	1	1	yes	characteristic
104	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
105	C₂⁴	C₂²⋊C₈	1	1	yes	characteristic
106	C₂²×C₄	—	2	4	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	2	8	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	4	4	no	—
111	C₂²×C₄	—	4	4	no	—
112	C₂²×C₄	#95	1	1	yes	characteristic
113	C₂²×C₄	#95	1	1	yes	characteristic
114	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
115	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
116	C₂²×C₄	#97	1	1	yes	characteristic
117	C₂²×C₄	#97	1	1	yes	characteristic
118	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
121	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
123	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
124	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
127	C₂²×C₄	C₂×C₁₆	1	4	yes	—
128	C₂²×C₄	M₃₂	1	4	yes	—
129	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
131	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	4	no	—
140	C₂×C₈	—	4	4	no	—
141	C₂×C₈	—	4	4	no	—
142	C₂×C₈	#95	1	2	yes	—
143	C₂×C₈	#95	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
145	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
146	C₄²	—	2	8	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	4	4	no	—
149	C₄²	—	4	4	no	—
150	C₄²	—	4	4	no	—
151	C₄²	—	4	4	no	—
152	C₄²	#95	1	1	yes	characteristic
153	C₄²	#95	1	1	yes	characteristic
154	C₄²	C₄×C₈	1	1	yes	characteristic
155	C₄²	C₄×C₈	1	1	yes	characteristic
156	C₄²	#97	1	1	yes	characteristic
157	C₄²	#97	1	1	yes	characteristic
158	C₄²	C₂²⋊C₈	1	1	yes	characteristic
159	C₄²	C₂²⋊C₈	1	1	yes	characteristic
160	C₄²	C₄⋊C₈	1	1	yes	characteristic
161	C₄²	C₄⋊C₈	1	1	yes	characteristic
162	C₄²	C₄⋊C₈	1	1	yes	characteristic
163	C₄²	C₄⋊C₈	1	1	yes	characteristic
164	C₄²	C₂×C₁₆	1	8	yes	complemented
165	C₄²	C₂×C₄²	1	1	yes	characteristic
166	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
167	C₄²	C₂²×C₈	1	1	yes	characteristic
168	C₄²	C₂×M₁₆	1	1	yes	characteristic
169	C₁₆	—	4	8	no	—
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	#336	1	1	yes	characteristic
173	C₂³	#336	1	1	yes	characteristic
174	C₂³	C₂²⋊C₁₆	1	4	yes	—
175	C₂³	C₂×C₂²⋊C₈	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	4	16	no	—
190	C₂×C₄	#336	1	1	yes	characteristic
191	C₂×C₄	#336	1	1	yes	characteristic
192	C₂×C₄	#336	1	1	yes	characteristic
193	C₂×C₄	#336	1	1	yes	characteristic
194	C₂×C₄	#336	1	1	yes	characteristic
195	C₂×C₄	#336	1	1	yes	characteristic
196	C₂×C₄	H(3,ℤ/4)	1	4	yes	—
197	C₂×C₄	C₄×C₁₆	1	2	yes	—
198	C₂×C₄	C₄×C₁₆	1	2	yes	—
199	C₂×C₄	#346	1	2	yes	—
200	C₂×C₄	#346	1	2	yes	—
201	C₂×C₄	#347	1	4	yes	—
202	C₂×C₄	C₂²⋊C₁₆	1	4	yes	—
203	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
204	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
205	C₂×C₄	C₂×#95	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×#97	1	1	yes	characteristic
208	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
209	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₁₆	1	2	yes	—
212	C₂×C₄	C₂×M₃₂	1	2	yes	—
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	—	4	4	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	8	no	—
223	C₂²	#1274	1	2	yes	—
224	C₂²	#1374	1	2	yes	—
225	C₂²	#1374	1	2	yes	—
226	C₂²	#1375	1	2	yes	—
227	C₂²	C₂×#336	1	1	yes	characteristic
228	C₂²	C₂×C₂²⋊C₁₆	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	8	no	—
234	C₄	#1274	1	2	yes	—
235	C₄	#1374	1	2	yes	—
236	C₄	#1374	1	2	yes	—
237	C₄	#1375	1	2	yes	—
238	C₄	C₂×H(3,ℤ/4)	1	2	yes	—
239	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
240	C₄	C₂×#346	1	1	yes	characteristic
241	C₄	C₂×#347	1	2	yes	—
242	C₄	C₂×C₄⋊C₁₆	1	2	yes	—
243	C₂	—	2	2	no	—
244	C₂	—	2	2	no	—
245	C₂	#7142	1	4	yes	direct product factor
246	C₂	C₂×#1274	1	1	yes	characteristic
247	C₂	C₂×#1374	1	1	yes	characteristic
248	C₂	C₂×#1375	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₄²
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ 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
3. #7142	C₂	direct (N × H)
9. #2162	C₄	semidirect
13. C₂×C₄×C₁₆	C₄	semidirect
49. C₄×C₁₆	C₂×C₄	semidirect
75. C₂×C₄²	C₁₆	semidirect
164. C₄²	C₂×C₁₆	semidirect
245. C₂	#7142	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
C₂×#1744	2
#7142	2

Subgroups — C2×#7142 #126582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7142 #126582