Subgroups — C₂×#7151 #126642

Order 512.

614 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₂²×C₈²	C₂	1	1	yes	maximal, characteristic
3	#7151	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₁₆	—	2	2	no	—
5	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₁₆	—	2	2	no	—
9	C₂×C₄⋊C₁₆	—	2	4	no	—
10	C₂×C₄⋊C₁₆	C₄	1	8	yes	—
11	C₂×C₄⋊C₁₆	C₂²	1	4	yes	—
12	C₂×C₈²	—	2	4	no	—
13	C₂×C₈²	C₄	1	2	yes	—
14	C₂×C₈²	C₂²	1	2	yes	—
15	C₂³×C₈	—	2	1	no	—
16	C₂³×C₈	C₈	1	2	yes	—
17	C₂³×C₈	D₄	1	1	yes	characteristic
18	C₂³×C₈	Q₈	1	1	yes	characteristic
19	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	—	2	8	no	—
21	C₂²×C₁₆	—	2	4	no	—
22	C₂²×C₁₆	—	4	4	no	—
23	C₂²×C₁₆	—	4	4	no	—
24	C₂×C₄×C₈	—	2	2	no	—
25	C₂×C₄×C₈	—	2	2	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	—	2	2	no	—
28	C₂×C₄×C₈	C₈	1	2	yes	—
29	C₂×C₄×C₈	C₈	1	4	yes	—
30	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	4	yes	—
36	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊C₁₆	—	2	8	no	—
39	C₄⋊C₁₆	—	2	4	no	—
40	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
41	C₈²	—	2	4	no	—
42	C₈²	—	2	4	no	—
43	C₂³×C₄	C₄⋊C₄	1	1	yes	characteristic
44	C₂³×C₄	C₂×C₈	1	1	yes	characteristic
45	C₂³×C₄	M₁₆	1	1	yes	characteristic
46	C₂²×C₈	—	2	8	no	—
47	C₂²×C₈	—	2	4	no	—
48	C₂²×C₈	—	2	4	no	—
49	C₂²×C₈	—	2	2	no	—
50	C₂²×C₈	—	2	2	no	—
51	C₂²×C₈	—	2	4	no	—
52	C₂²×C₈	—	2	2	no	—
53	C₂²×C₈	C₁₆	1	8	yes	complemented
54	C₂²×C₈	C₂²⋊C₄	1	4	yes	—
55	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₄⋊C₄	1	4	yes	—
57	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×C₈	1	4	yes	—
59	C₂²×C₈	D₈	1	2	yes	—
60	C₂²×C₈	SD₁₆	1	4	yes	—
61	C₂²×C₈	Q₁₆	1	2	yes	—
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄²	C₄²	1	2	yes	—
65	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄²	C₂×C₈	1	2	yes	—
69	C₂×C₄²	M₁₆	1	1	yes	characteristic
70	C₂×C₄²	M₁₆	1	2	yes	—
71	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₁₆	—	2	8	no	—
73	C₂×C₁₆	—	4	16	no	—
74	C₂×C₁₆	—	4	4	no	—
75	C₂×C₁₆	—	4	4	no	—
76	C₄×C₈	—	2	4	no	—
77	C₄×C₈	—	2	4	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	—	2	2	no	—
81	C₄×C₈	C₄²	1	2	yes	—
82	C₄×C₈	C₂²⋊C₄	1	4	yes	—
83	C₄×C₈	C₂²⋊C₄	1	2	yes	—
84	C₄×C₈	C₄⋊C₄	1	4	yes	—
85	C₄×C₈	C₂×C₈	1	4	yes	—
86	C₄×C₈	C₂²×C₄	1	2	yes	—
87	C₄×C₈	C₂×D₄	1	2	yes	—
88	C₂⁴	C₄⋊C₈	1	1	yes	characteristic
89	C₂²×C₄	—	2	4	no	—
90	C₂²×C₄	#95	1	2	yes	—
91	C₂²×C₄	C₄×C₈	1	2	yes	—
92	C₂²×C₄	#97	1	2	yes	—
93	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
94	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
95	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
96	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
97	C₂²×C₄	C₄⋊C₈	1	2	yes	—
98	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
99	C₂²×C₄	#106	1	2	yes	—
100	C₂²×C₄	#107	1	2	yes	—
101	C₂²×C₄	#108	1	2	yes	—
102	C₂²×C₄	#108	1	2	yes	—
103	C₂²×C₄	C₂×C₁₆	1	4	yes	—
104	C₂²×C₄	M₃₂	1	4	yes	—
105	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
107	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
108	C₂×C₈	—	2	16	no	—
109	C₂×C₈	—	2	8	no	—
110	C₂×C₈	—	2	8	no	—
111	C₂×C₈	—	2	16	no	—
112	C₂×C₈	—	2	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	#95	1	2	yes	—
120	C₂×C₈	#95	1	2	yes	—
121	C₂×C₈	#102	1	8	yes	—
122	C₂×C₈	#103	1	8	yes	—
123	C₂×C₈	#106	1	2	yes	—
124	C₂×C₈	#107	1	2	yes	—
125	C₂×C₈	C₂×C₁₆	1	8	yes	complemented
126	C₂×C₈	C₂×C₂²⋊C₄	1	2	yes	—
127	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×D₈	1	1	yes	characteristic
129	C₂×C₈	C₂×SD₁₆	1	2	yes	—
130	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
131	C₄²	#95	1	2	yes	—
132	C₄²	C₄×C₈	1	2	yes	—
133	C₄²	#97	1	2	yes	—
134	C₄²	C₂²⋊C₈	1	2	yes	—
135	C₄²	C₂²⋊C₈	1	2	yes	—
136	C₄²	C₄⋊C₈	1	2	yes	—
137	C₄²	C₂×C₄²	1	1	yes	characteristic
138	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₄²	C₂²×C₈	1	1	yes	characteristic
140	C₄²	C₂×M₁₆	1	1	yes	characteristic
141	C₁₆	—	4	8	no	—
142	C₂³	#334	1	2	yes	—
143	C₂³	#335	1	2	yes	—
144	C₂³	#336	1	2	yes	—
145	C₂³	C₄⋊C₁₆	1	4	yes	—
146	C₂³	#364	1	4	yes	—
147	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	4	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	#334	1	4	yes	—
153	C₂×C₄	#334	1	2	yes	—
154	C₂×C₄	#335	1	4	yes	—
155	C₂×C₄	#335	1	2	yes	—
156	C₂×C₄	#336	1	2	yes	—
157	C₂×C₄	#336	1	2	yes	—
158	C₂×C₄	#336	1	2	yes	—
159	C₂×C₄	#340	1	4	yes	—
160	C₂×C₄	#341	1	4	yes	—
161	C₂×C₄	C₄×C₁₆	1	4	yes	—
162	C₂×C₄	#346	1	4	yes	—
163	C₂×C₄	C₂²⋊C₁₆	1	8	yes	—
164	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
165	C₂×C₄	C₂×#95	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
167	C₂×C₄	C₂×#97	1	1	yes	characteristic
168	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#106	1	1	yes	characteristic
172	C₂×C₄	C₂×#107	1	1	yes	characteristic
173	C₂×C₄	C₂×#108	1	1	yes	characteristic
174	C₂×C₄	C₂×#108	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₁₆	1	2	yes	—
176	C₂×C₄	C₂×M₃₂	1	2	yes	—
177	C₈	—	2	8	no	—
178	C₈	—	2	8	no	—
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₈	#340	1	4	yes	—
182	C₈	C₂×#102	1	2	yes	—
183	C₈	C₂×#103	1	2	yes	—
184	C₂²	#1277	1	4	yes	—
185	C₂²	#1367	1	8	yes	—
186	C₂²	#1370	1	8	yes	—
187	C₂²	#1374	1	4	yes	—
188	C₂²	#1376	1	4	yes	—
189	C₂²	C₂×#334	1	1	yes	characteristic
190	C₂²	C₂×#335	1	1	yes	characteristic
191	C₂²	C₂×#336	1	1	yes	characteristic
192	C₂²	C₂×C₄⋊C₁₆	1	2	yes	—
193	C₂²	C₂×#364	1	2	yes	—
194	C₄	—	2	1	no	—
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	#1277	1	4	yes	—
198	C₄	#1374	1	4	yes	—
199	C₄	C₂×#334	1	1	yes	characteristic
200	C₄	C₂×#335	1	1	yes	characteristic
201	C₄	C₂×#340	1	1	yes	characteristic
202	C₄	C₂×#341	1	1	yes	characteristic
203	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
204	C₄	C₂×#346	1	1	yes	characteristic
205	C₄	C₂×C₂²⋊C₁₆	1	2	yes	—
206	C₂	#7151	1	8	yes	direct product factor
207	C₂	C₂×#1277	1	1	yes	characteristic
208	C₂	C₂×#1367	1	2	yes	—
209	C₂	C₂×#1370	1	2	yes	—
210	C₂	C₂×#1374	1	1	yes	characteristic
211	C₂	C₂×#1376	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
3. #7151	C₂	direct (N × H)
53. C₂²×C₈	C₁₆	semidirect
125. C₂×C₈	C₂×C₁₆	semidirect
206. C₂	#7151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊C₁₆	2
C₂²×C₈²	2
#7151	2

Subgroups — C2×#7151 #126642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#7151 #126642