Subgroups — C₂×#7141 #126558

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 257 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1450	C₂	1	1	yes	maximal, characteristic
4	#7141	C₂	1	4	yes	maximal, direct product factor
5	C₂²×M₃₂	C₄	1	2	yes	—
6	C₂³×C₁₆	C₄	1	2	yes	—
7	C₂×#431	C₄	1	2	yes	—
8	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
9	C₄×M₃₂	—	2	1	no	—
10	C₄×M₃₂	—	2	1	no	—
11	C₄×M₃₂	C₄	1	2	yes	—
12	C₄×M₃₂	C₂²	1	2	yes	—
13	C₂×#346	—	2	1	no	—
14	C₂×C₄×C₁₆	—	2	1	no	—
15	C₂×C₄×C₁₆	—	2	1	no	—
16	C₂×C₄×C₁₆	—	2	1	no	—
17	C₂×C₄×C₁₆	—	2	1	no	—
18	C₂×C₄×C₁₆	—	2	1	no	—
19	C₂×C₄×C₁₆	C₄	1	2	yes	—
20	C₂×C₄×C₁₆	C₂²	1	2	yes	—
21	#1450	—	2	1	no	—
22	#1450	—	2	1	no	—
23	#1450	C₄	1	2	yes	—
24	#1450	C₂²	1	2	yes	—
25	C₂×#322	—	2	1	no	—
26	C₂×C₈²	—	2	1	no	—
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	—	2	4	no	—
32	C₂×M₃₂	C₈	1	8	yes	complemented
33	C₂×M₃₂	C₂×C₄	1	4	yes	—
34	C₂²×C₁₆	—	2	2	no	—
35	C₂²×C₁₆	—	2	2	no	—
36	C₂²×C₁₆	—	2	2	no	—
37	C₂²×C₁₆	—	2	4	no	—
38	C₂²×C₁₆	C₈	1	8	yes	complemented
39	C₂²×C₁₆	C₂×C₄	1	4	yes	—
40	#431	—	2	2	no	—
41	#431	—	2	2	no	—
42	#431	C₈	1	4	yes	—
43	#431	C₂×C₄	1	4	yes	—
44	C₂×C₄⋊C₈	C₈	1	4	yes	—
45	C₂×C₂²⋊C₈	—	2	2	no	—
46	C₂×C₄×C₈	—	2	2	no	—
47	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×C₈	D₄	1	1	yes	characteristic
51	C₂×C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×C₄×C₈	D₄	1	1	yes	characteristic
53	C₂×C₄×C₈	D₄	1	1	yes	characteristic
54	C₂×C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×C₄×C₈	D₄	1	1	yes	characteristic
56	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
57	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
59	#346	—	2	2	no	—
60	#346	—	2	2	no	—
61	C₄×C₁₆	—	2	2	no	—
62	C₄×C₁₆	—	2	2	no	—
63	C₄×C₁₆	—	2	2	no	—
64	C₄×C₁₆	—	2	2	no	—
65	C₄×C₁₆	—	2	2	no	—
66	C₄×C₁₆	—	2	2	no	—
67	#322	—	2	2	no	—
68	#322	—	2	2	no	—
69	C₈²	—	2	2	no	—
70	C₈²	—	2	2	no	—
71	C₂³×C₄	C₄²	1	1	yes	characteristic
72	C₂³×C₄	M₁₆	1	2	yes	—
73	C₂²×C₈	—	2	4	no	—
74	C₂²×C₈	C₄²	1	1	yes	characteristic
75	C₂²×C₈	C₄²	1	1	yes	characteristic
76	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₈	C₂×C₈	1	4	yes	—
85	C₂²×C₈	C₂×C₈	1	4	yes	—
86	C₂²×C₈	M₁₆	1	4	yes	—
87	C₂²×C₈	M₁₆	1	4	yes	—
88	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄²	C₄²	1	1	yes	characteristic
91	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄²	C₂×C₈	1	2	yes	—
96	C₂×C₄²	C₂×C₈	1	2	yes	—
97	C₂×C₄²	M₁₆	1	2	yes	—
98	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
99	M₃₂	—	2	8	no	—
100	M₃₂	—	2	4	no	—
101	M₃₂	C₂×C₈	1	8	yes	complemented
102	C₂×C₁₆	—	2	8	no	—
103	C₂×C₁₆	—	2	8	no	—
104	C₂×C₁₆	—	2	8	no	—
105	C₂×C₁₆	—	2	4	no	—
106	C₂×C₁₆	—	2	4	no	—
107	C₂×C₁₆	—	2	4	no	—
108	C₂×C₁₆	C₂×C₈	1	8	yes	complemented
109	C₄⋊C₈	—	2	4	no	—
110	C₄⋊C₈	C₂×C₈	1	8	yes	—
111	C₂²⋊C₈	—	2	4	no	—
112	C₂²⋊C₈	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₄×C₈	—	2	4	no	—
115	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
116	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
119	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
121	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
123	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₈	C₂×D₄	1	1	yes	characteristic
128	C₄×C₈	C₂×D₄	1	1	yes	characteristic
129	C₄×C₈	C₂×D₄	1	1	yes	characteristic
130	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
131	C₂⁴	#97	1	1	yes	characteristic
132	C₂²×C₄	—	2	8	no	—
133	C₂²×C₄	#95	1	1	yes	characteristic
134	C₂²×C₄	#95	1	1	yes	characteristic
135	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
136	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
137	C₂²×C₄	C₄×C₈	1	2	yes	—
138	C₂²×C₄	C₄×C₈	1	2	yes	—
139	C₂²×C₄	#97	1	2	yes	—
140	C₂²×C₄	#97	1	1	yes	characteristic
141	C₂²×C₄	#97	1	2	yes	—
142	C₂²×C₄	#97	1	2	yes	—
143	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
144	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
145	C₂²×C₄	C₄⋊C₈	1	2	yes	—
146	C₂²×C₄	C₄⋊C₈	1	2	yes	—
147	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
148	C₂²×C₄	C₂×M₁₆	1	2	yes	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	4	8	no	—
151	C₂×C₈	#95	1	1	yes	characteristic
152	C₂×C₈	#95	1	1	yes	characteristic
153	C₂×C₈	#95	1	1	yes	characteristic
154	C₂×C₈	#95	1	1	yes	characteristic
155	C₂×C₈	C₄×C₈	1	2	yes	—
156	C₂×C₈	C₄×C₈	1	2	yes	—
157	C₂×C₈	#97	1	2	yes	—
158	C₂×C₈	#97	1	2	yes	—
159	C₂×C₈	C₂²⋊C₈	1	4	yes	—
160	C₂×C₈	C₂²⋊C₈	1	4	yes	—
161	C₂×C₈	C₂²⋊C₈	1	4	yes	—
162	C₂×C₈	C₂²⋊C₈	1	4	yes	—
163	C₂×C₈	C₄⋊C₈	1	4	yes	—
164	C₂×C₈	C₄⋊C₈	1	4	yes	—
165	C₂×C₈	C₄⋊C₈	1	4	yes	—
166	C₂×C₈	C₄⋊C₈	1	4	yes	—
167	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₈	C₂²×C₈	1	2	yes	—
172	C₂×C₈	C₂²×C₈	1	2	yes	—
173	C₂×C₈	C₂×M₁₆	1	2	yes	—
174	C₂×C₈	C₂×M₁₆	1	2	yes	—
175	C₄²	#95	1	1	yes	characteristic
176	C₄²	#95	1	1	yes	characteristic
177	C₄²	C₄×C₈	1	2	yes	—
178	C₄²	C₂²⋊C₈	1	2	yes	—
179	C₄²	C₂²⋊C₈	1	2	yes	—
180	C₄²	C₄⋊C₈	1	2	yes	—
181	C₄²	C₄⋊C₈	1	2	yes	—
182	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₄²	C₂²×C₈	1	2	yes	—
185	C₁₆	—	2	8	no	—
186	C₁₆	—	2	8	no	—
187	C₂³	—	2	4	no	—
188	C₂³	#322	1	2	yes	—
189	C₂³	#322	1	2	yes	—
190	C₂³	#336	1	1	yes	characteristic
191	C₂³	#336	1	1	yes	characteristic
192	C₂³	C₂×#97	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	C₈²	1	2	yes	—
199	C₂×C₄	C₈²	1	2	yes	—
200	C₂×C₄	#322	1	2	yes	—
201	C₂×C₄	#322	1	2	yes	—
202	C₂×C₄	#322	1	2	yes	—
203	C₂×C₄	#322	1	2	yes	—
204	C₂×C₄	#336	1	2	yes	—
205	C₂×C₄	#336	1	2	yes	—
206	C₂×C₄	#336	1	2	yes	—
207	C₂×C₄	#336	1	2	yes	—
208	C₂×C₄	#336	1	2	yes	—
209	C₂×C₄	#336	1	2	yes	—
210	C₂×C₄	#336	1	1	yes	characteristic
211	C₂×C₄	#336	1	1	yes	characteristic
212	C₂×C₄	#350	1	8	yes	—
213	C₂×C₄	C₂×#95	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
216	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
217	C₂×C₄	C₂×#97	1	1	yes	characteristic
218	C₂×C₄	C₂×#97	1	1	yes	characteristic
219	C₂×C₄	C₂×C₂²⋊C₈	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
221	C₈	—	4	8	no	—
222	C₈	#336	1	2	yes	—
223	C₈	#336	1	2	yes	—
224	C₈	#336	1	2	yes	—
225	C₈	#336	1	2	yes	—
226	C₈	C₂×C₂²⋊C₈	1	2	yes	—
227	C₈	C₂×C₂²⋊C₈	1	2	yes	—
228	C₈	C₂×C₄⋊C₈	1	2	yes	—
229	C₈	C₂×C₄⋊C₈	1	2	yes	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	2	4	no	—
234	C₂²	#1273	1	2	yes	—
235	C₂²	#1273	1	2	yes	—
236	C₂²	#1379	1	4	yes	—
237	C₂²	C₂×#322	1	1	yes	characteristic
238	C₂²	C₂×#322	1	1	yes	characteristic
239	C₂²	C₂×#336	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	#1273	1	2	yes	—
243	C₄	#1273	1	2	yes	—
244	C₄	#1379	1	4	yes	—
245	C₄	C₂×C₈²	1	1	yes	characteristic
246	C₄	C₂×#322	1	1	yes	characteristic
247	C₄	C₂×#336	1	1	yes	characteristic
248	C₄	C₂×#336	1	1	yes	characteristic
249	C₄	C₂×#350	1	4	yes	—
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	2	2	no	—
253	C₂	#7141	1	4	yes	direct product factor
254	C₂	C₂×#1273	1	1	yes	characteristic
255	C₂	C₂×#1379	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈	D₄
commutator	C₄	C₂×C₈²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	#336
prime radical	C₂⁴	#97
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×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
4. #7141	C₂	direct (N × H)
32. C₂×M₃₂	C₈	semidirect
38. C₂²×C₁₆	C₈	semidirect
101. M₃₂	C₂×C₈	semidirect
108. C₂×C₁₆	C₂×C₈	semidirect
253. C₂	#7141	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×C₄×M₃₂	2
C₂²×C₄×C₁₆	2
C₂×#1450	2
#7141	2

Subgroups — C2×#7141 #126558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7141 #126558