Subgroups — C₂×#69277 #10948336

Order 768.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₃×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂×D₃)⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#69277	C₂	1	8	yes	maximal, direct product factor
7	C₂×D₄×C₁₆	—	3	1	no	maximal, Sylow
8	C₂²×D₃×C₈	C₂²	1	2	yes	—
9	C₂×C₄×D₁₂	C₄	1	1	yes	characteristic
10	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
13	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
16	C₂×D₃×C₁₆	—	2	8	no	—
17	C₂×D₃×C₁₆	C₂²	1	8	yes	complemented
18	C₈×D₁₂	C₄	1	4	yes	—
19	C₈×D₁₂	C₂²	1	4	yes	—
20	C₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
21	(C₂×D₃)⋊C₁₆	C₂²	1	8	yes	complemented
22	C₄×C₃⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂³×C₁₆	—	3	2	no	—
24	C₂×C₈×D₄	—	3	1	no	—
25	D₄×C₁₆	—	3	8	no	—
26	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
27	C₂×C₂²⋊C₁₆	—	3	2	no	—
28	C₂×C₄×C₁₆	—	3	1	no	—
29	C₂²×D₁₂	C₈	1	1	yes	characteristic
30	C₂²×C₄×D₃	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂×(C₂×D₃)⋊C₄	C₈	1	2	yes	—
35	C₂×C₄⋊Dic₃	C₈	1	1	yes	characteristic
36	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×D₃×C₈	—	2	4	no	—
38	C₂×D₃×C₈	—	2	4	no	—
39	C₂×D₃×C₈	C₂×C₄	1	4	yes	—
40	C₂×D₃×C₈	C₂³	1	4	yes	—
41	C₄×D₁₂	C₈	1	4	yes	—
42	C₄×D₁₂	C₂×C₄	1	4	yes	—
43	C₂×C₃×C₁₆	—	2	4	no	—
44	C₂×C₃×C₁₆	C₂³	1	4	yes	—
45	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
46	C₃×C₄×C₈	C₂³	1	2	yes	—
47	(C₂×D₃)⋊C₈	C₂×C₄	1	8	yes	—
48	C₂×C₃⋊C₁₆	—	2	2	no	—
49	C₂×C₃⋊C₁₆	D₄	1	4	yes	complemented
50	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
51	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
52	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
53	D₃×C₁₆	—	2	16	no	—
54	C₂³×C₈	—	3	2	no	—
55	C₂×C₄×D₄	—	3	1	no	—
56	C₂²×C₁₆	—	3	1	no	—
57	C₂²×C₁₆	—	3	1	no	—
58	C₂²×C₁₆	—	3	8	no	—
59	C₂²×C₁₆	—	6	8	no	—
60	C₂²×C₁₆	C₂×D₃	1	2	yes	—
61	C₈×D₄	—	3	4	no	—
62	C₈×D₄	—	3	4	no	—
63	C₂×C₄⋊C₈	—	3	1	no	—
64	C₂×C₂²⋊C₈	—	3	2	no	—
65	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
66	C₄⋊C₁₆	C₂×D₃	1	4	yes	complemented
67	C₂²⋊C₁₆	—	3	8	no	—
68	C₄×C₁₆	—	3	4	no	—
69	C₂³×D₃	C₂×C₈	1	2	yes	—
70	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₂²×Dic₃	C₂×C₈	1	2	yes	—
74	C₂×D₁₂	C₁₆	1	8	yes	complemented
75	C₂×D₁₂	C₂×C₈	1	4	yes	—
76	C₂×C₄×D₃	—	2	4	no	—
77	C₂×C₄×D₃	—	2	4	no	—
78	C₂×C₄×D₃	C₂×C₈	1	4	yes	—
79	C₂×C₄×D₃	C₂²×C₄	1	4	yes	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
82	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₈	#41	1	2	yes	—
85	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
86	C₃×C₄²	C₂×C₈	1	2	yes	—
87	C₃×C₄²	C₂²×C₄	1	2	yes	—
88	(C₂×D₃)⋊C₄	C₂×C₈	1	8	yes	—
89	C₄⋊Dic₃	C₂×C₈	1	4	yes	—
90	C₂×C₃⋊C₈	—	2	4	no	—
91	C₂×C₃⋊C₈	C₂²×C₄	1	4	yes	—
92	D₃×C₈	—	2	8	no	—
93	D₃×C₈	—	2	4	no	—
94	D₃×C₈	—	2	4	no	—
95	C₃×C₁₆	—	2	4	no	—
96	C₃⋊C₁₆	—	2	2	no	—
97	C₃⋊C₁₆	C₂×D₄	1	4	yes	complemented
98	C₂²×D₄	—	3	1	no	—
99	C₂³×C₄	—	3	2	no	—
100	C₂²×C₈	—	3	2	no	—
101	C₂²×C₈	—	3	4	no	—
102	C₂²×C₈	—	3	4	no	—
103	C₂²×C₈	—	6	4	no	—
104	C₂²×C₈	—	6	4	no	—
105	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₄×D₄	—	3	4	no	—
108	C₄×D₄	—	3	4	no	—
109	C₂×C₄⋊C₄	—	3	1	no	—
110	C₂×C₂²⋊C₄	—	3	2	no	—
111	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	—	3	4	no	—
114	C₂×C₁₆	—	3	2	no	—
115	C₂×C₁₆	—	3	2	no	—
116	C₂×C₁₆	—	6	16	no	—
117	C₂×C₁₆	—	6	2	no	—
118	C₂×C₁₆	C₂²×D₃	1	4	yes	—
119	C₄⋊C₈	—	3	4	no	—
120	C₂²⋊C₈	—	3	8	no	—
121	C₄×C₈	C₄×D₃	1	2	yes	—
122	C₄×C₈	C₂²×D₃	1	2	yes	—
123	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
124	C₂²×D₃	—	2	4	no	—
125	C₂²×D₃	—	2	4	no	—
126	C₂²×D₃	C₂×C₁₆	1	8	yes	complemented
127	C₂²×D₃	C₂²×C₈	1	4	yes	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	C₂×C₁₆	1	4	yes	complemented
130	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
131	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
132	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
134	C₂×C₃×C₄	#131	1	2	yes	—
135	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
136	C₂×Dic₃	—	2	4	no	—
137	C₂×Dic₃	C₂²×C₈	1	4	yes	—
138	D₁₂	C₂×C₁₆	1	16	yes	complemented
139	C₄×D₃	—	2	4	no	—
140	C₄×D₃	—	2	4	no	—
141	C₄×D₃	—	2	8	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	C₄×D₄	1	2	yes	—
144	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
145	C₃×C₈	C₂×#41	1	1	yes	characteristic
146	C₃⋊C₈	—	2	4	no	—
147	C₂⁴	—	3	2	no	—
148	C₂×D₄	—	3	8	no	—
149	C₂×D₄	—	3	4	no	—
150	C₂²×C₄	—	3	2	no	—
151	C₂²×C₄	—	3	4	no	—
152	C₂²×C₄	—	3	4	no	—
153	C₂²×C₄	—	6	4	no	—
154	C₂²×C₄	—	6	4	no	—
155	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
156	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	3	4	no	—
160	C₂×C₈	—	6	4	no	—
161	C₂×C₈	—	6	8	no	—
162	C₂×C₈	—	6	4	no	—
163	C₂×C₈	—	6	4	no	—
164	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
165	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂×C₈	D₃×D₄	1	2	yes	—
167	C₂×C₈	#239	1	2	yes	—
168	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
169	C₄⋊C₄	—	3	4	no	—
170	C₂²⋊C₄	—	3	8	no	—
171	C₄²	D₃×C₈	1	2	yes	—
172	C₄²	C₂×C₄×D₃	1	2	yes	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	3	4	no	—
175	C₁₆	—	6	2	no	—
176	C₂²×C₃	C₈×D₄	1	2	yes	—
177	C₂²×C₃	C₂²×C₁₆	1	2	yes	—
178	C₂²×C₃	#504	1	2	yes	—
179	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
180	C₂×D₃	—	2	4	no	—
181	C₂×D₃	—	2	4	no	—
182	C₂×D₃	—	2	16	no	—
183	C₂×D₃	C₂²×C₁₆	1	8	yes	complemented
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	C₈×D₄	1	2	yes	—
186	C₃×C₄	C₂²×C₁₆	1	4	yes	complemented
187	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
188	C₃×C₄	C₂×#131	1	1	yes	characteristic
189	Dic₃	—	2	4	no	—
190	C₂³	—	3	8	no	—
191	C₂³	—	3	4	no	—
192	C₂³	—	6	4	no	—
193	C₂³	—	6	4	no	—
194	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
195	D₄	—	3	16	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	3	4	no	—
198	C₂×C₄	—	6	4	no	—
199	C₂×C₄	—	6	4	no	—
200	C₂×C₄	—	6	4	no	—
201	C₂×C₄	—	6	8	no	—
202	C₂×C₄	D₃×C₁₆	1	4	yes	complemented
203	C₂×C₄	#893	1	2	yes	—
204	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
205	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
206	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
207	C₂×C₄	#907	1	2	yes	—
208	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	6	4	no	—
211	C₈	#893	1	2	yes	—
212	C₈	C₂×D₃×D₄	1	1	yes	characteristic
213	C₈	C₂×#239	1	1	yes	characteristic
214	C₂×C₃	D₄×C₁₆	1	4	yes	complemented
215	C₂×C₃	C₂×C₈×D₄	1	1	yes	characteristic
216	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
217	C₂×C₃	C₂×#504	1	1	yes	characteristic
218	D₃	—	2	8	no	—
219	C₂²	—	3	8	no	—
220	C₂²	—	6	4	no	—
221	C₂²	—	6	4	no	—
222	C₂²	—	6	16	no	—
223	C₂²	#4834	1	2	yes	—
224	C₂²	C₂×D₃×C₁₆	1	2	yes	—
225	C₂²	#4907	1	2	yes	—
226	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	6	4	no	—
229	C₄	#4834	1	2	yes	—
230	C₄	C₂×D₃×C₁₆	1	4	yes	complemented
231	C₄	C₂×#893	1	1	yes	characteristic
232	C₄	C₂×#907	1	1	yes	characteristic
233	C₃	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	6	8	no	—
235	C₂	#69277	1	4	yes	direct product factor
236	C₂	C₂×#4834	1	1	yes	characteristic
237	C₂	C₂²×D₃×C₁₆	1	1	yes	characteristic
238	C₂	C₂×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₃
commutator	C₂×C₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂²×D₁₂	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₃×C₁₆	C₂	semidirect
3. C₂×C₃×C₄⋊C₁₆	C₂	semidirect
4. C₂×(C₂×D₃)⋊C₁₆	C₂	semidirect
5. C₂×C₄×C₃⋊C₁₆	C₂	semidirect
6. #69277	C₂	direct (N × H)
17. C₂×D₃×C₁₆	C₂²	semidirect
20. C₃×C₄⋊C₁₆	C₂²	semidirect
21. (C₂×D₃)⋊C₁₆	C₂²	semidirect
22. C₄×C₃⋊C₁₆	C₂²	semidirect
26. C₂×C₄⋊C₁₆	D₃	semidirect
49. C₂×C₃⋊C₁₆	D₄	semidirect
66. C₄⋊C₁₆	C₂×D₃	semidirect
74. C₂×D₁₂	C₁₆	semidirect
97. C₃⋊C₁₆	C₂×D₄	semidirect
126. C₂²×D₃	C₂×C₁₆	semidirect
129. C₂×C₃×C₄	C₂×C₁₆	semidirect
138. D₁₂	C₂×C₁₆	semidirect
183. C₂×D₃	C₂²×C₁₆	semidirect
186. C₃×C₄	C₂²×C₁₆	semidirect
202. C₂×C₄	D₃×C₁₆	semidirect
214. C₂×C₃	D₄×C₁₆	semidirect
230. C₄	C₂×D₃×C₁₆	semidirect
233. C₃	C₂×D₄×C₁₆	semidirect
235. C₂	#69277	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×D₃×C₁₆	2
C₂×C₈×D₁₂	2
C₂×C₃×C₄⋊C₁₆	2
C₂×(C₂×D₃)⋊C₁₆	2
C₂×C₄×C₃⋊C₁₆	2
#69277	2
C₂×D₄×C₁₆	3

Subgroups — C2×#69277 #10948336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69277 #10948336