Subgroups — C₂×D₄×C₃⋊C₁₆ #10948337

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	C₂×C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×#4549	C₂	1	2	yes	maximal, complemented
4	D₄×C₃⋊C₁₆	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×C₁₆	—	3	1	no	maximal, Sylow
8	C₂³×C₃×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄×D₄	C₄	1	1	yes	characteristic
10	C₃×C₈×D₄	C₄	1	4	yes	—
11	C₃×C₈×D₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₃×C₂²⋊C₈	C₄	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₁₆	—	2	8	no	—
16	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
17	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₃⋊C₁₆	C₂²	1	8	yes	complemented
20	#4549	C₂²	1	8	yes	complemented
21	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	complemented
22	C₄×C₃⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂³×C₁₆	—	3	2	no	—
24	C₂×C₈×D₄	D₃	1	1	yes	characteristic
25	D₄×C₁₆	—	3	8	no	—
26	C₂×C₄⋊C₁₆	—	3	1	no	—
27	C₂×C₂²⋊C₁₆	—	3	2	no	—
28	C₂×C₄×C₁₆	—	3	1	no	—
29	C₂²×C₃×D₄	C₈	1	1	yes	characteristic
30	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₈	—	2	4	no	—
32	C₂²×C₃×C₈	—	2	4	no	—
33	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₈	C₂³	1	4	yes	—
37	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
38	C₃×C₄×D₄	C₈	1	4	yes	—
39	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
41	C₂×C₃×C₂²⋊C₄	C₈	1	2	yes	—
42	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
45	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
46	C₃×C₄×C₈	C₂³	1	2	yes	—
47	C₂×C₃⋊C₁₆	—	2	4	no	—
48	C₂×C₃⋊C₁₆	—	2	2	no	—
49	C₂×C₃⋊C₁₆	—	2	16	no	—
50	C₂×C₃⋊C₁₆	D₄	1	4	yes	direct product factor
51	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
53	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
54	C₂³×C₈	C₂×D₃	1	2	yes	—
55	C₂×C₄×D₄	Dic₃	1	1	yes	characteristic
56	C₂²×C₁₆	—	3	1	no	—
57	C₂²×C₁₆	—	3	8	no	—
58	C₂²×C₁₆	—	3	1	no	—
59	C₂²×C₁₆	—	3	2	no	—
60	C₂²×C₁₆	—	6	8	no	—
61	C₈×D₄	Dic₃	1	4	yes	—
62	C₈×D₄	C₂×D₃	1	4	yes	—
63	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
64	C₂×C₂²⋊C₈	Dic₃	1	2	yes	—
65	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
66	C₄⋊C₁₆	—	3	4	no	—
67	C₂²⋊C₁₆	—	3	8	no	—
68	C₄×C₁₆	—	3	4	no	—
69	C₂⁴×C₃	C₂×C₈	1	2	yes	—
70	C₂×C₃×D₄	C₁₆	1	8	yes	complemented
71	C₂×C₃×D₄	C₂×C₈	1	4	yes	—
72	C₂²×C₃×C₄	—	2	4	no	—
73	C₂²×C₃×C₄	—	2	4	no	—
74	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
76	C₂²×C₃×C₄	C₂×C₈	1	4	yes	—
77	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
79	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₈	—	2	4	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	8	no	—
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	—	2	4	no	—
85	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
87	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₈	#41	1	2	yes	—
90	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
91	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
92	C₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
93	C₃×C₄²	C₂×C₈	1	2	yes	—
94	C₃×C₄²	C₂²×C₄	1	2	yes	—
95	C₃⋊C₁₆	—	2	4	no	—
96	C₃⋊C₁₆	—	2	2	no	—
97	C₃⋊C₁₆	C₂×D₄	1	4	yes	direct product factor
98	C₂²×D₄	C₃⋊C₈	1	1	yes	characteristic
99	C₂³×C₄	C₂×Dic₃	1	2	yes	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	—	2	4	no	—
102	C₂²×C₈	C₂×Dic₃	1	2	yes	—
103	C₂²×C₈	C₂×Dic₃	1	4	yes	—
104	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₃	1	4	yes	—
106	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₄×D₄	C₃⋊C₈	1	4	yes	—
108	C₄×D₄	C₂×Dic₃	1	4	yes	—
109	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
110	C₂×C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
111	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
112	C₂×C₁₆	—	3	4	no	—
113	C₂×C₁₆	—	3	2	no	—
114	C₂×C₁₆	—	3	2	no	—
115	C₂×C₁₆	—	3	4	no	—
116	C₂×C₁₆	—	6	16	no	—
117	C₂×C₁₆	—	6	4	no	—
118	C₂×C₁₆	—	6	2	no	—
119	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
120	C₂²⋊C₈	C₂×Dic₃	1	8	yes	—
121	C₄×C₈	C₂×Dic₃	1	2	yes	—
122	C₄×C₈	C₂²×D₃	1	2	yes	—
123	C₂³×C₃	—	2	4	no	—
124	C₂³×C₃	—	2	4	no	—
125	C₂³×C₃	C₂×C₁₆	1	8	yes	complemented
126	C₂³×C₃	C₂²×C₈	1	4	yes	—
127	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
128	C₃×D₄	C₂×C₁₆	1	16	yes	complemented
129	C₂×C₃×C₄	—	2	4	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	8	no	—
134	C₂×C₃×C₄	C₂×C₁₆	1	4	yes	complemented
135	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
136	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
137	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
138	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
140	C₂×C₃×C₄	#131	1	2	yes	—
141	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
142	C₃×C₈	—	2	4	no	—
143	C₃×C₈	—	2	2	no	—
144	C₃×C₈	C₄×D₄	1	2	yes	—
145	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
146	C₃×C₈	C₂×#41	1	1	yes	characteristic
147	C₂⁴	C₂×C₃⋊C₈	1	2	yes	—
148	C₂×D₄	C₃⋊C₁₆	1	8	yes	direct product factor
149	C₂×D₄	C₂×C₃⋊C₈	1	4	yes	—
150	C₂²×C₄	—	2	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
154	C₂²×C₄	C₂×C₃⋊C₈	1	4	yes	—
155	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₃	1	4	yes	—
157	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	D₃×D₄	1	2	yes	—
164	C₂×C₈	#237	1	2	yes	—
165	C₂×C₈	C₂²×Dic₃	1	2	yes	—
166	C₂×C₈	C₂²×Dic₃	1	4	yes	—
167	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
168	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
169	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
170	C₂²⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
171	C₄²	C₂×C₃⋊C₈	1	2	yes	—
172	C₄²	C₂²×Dic₃	1	2	yes	—
173	C₁₆	—	3	4	no	—
174	C₁₆	—	6	4	no	—
175	C₁₆	—	6	2	no	—
176	C₂²×C₃	—	2	4	no	—
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	2	16	no	—
179	C₂²×C₃	C₈×D₄	1	2	yes	—
180	C₂²×C₃	C₂²×C₁₆	1	2	yes	—
181	C₂²×C₃	C₂²×C₁₆	1	8	yes	complemented
182	C₂²×C₃	#504	1	2	yes	—
183	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	4	no	—
186	C₃×C₄	C₈×D₄	1	2	yes	—
187	C₃×C₄	C₂²×C₁₆	1	4	yes	complemented
188	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
189	C₃×C₄	C₂×#131	1	1	yes	characteristic
190	C₂³	—	2	4	no	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×C₃⋊C₁₆	1	8	yes	complemented
193	C₂³	C₂²×C₃⋊C₈	1	4	yes	—
194	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
195	D₄	C₂×C₃⋊C₁₆	1	16	yes	direct product factor
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	C₂×C₃⋊C₁₆	1	4	yes	complemented
202	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
203	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
204	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
206	C₂×C₄	D₄×Dic₃	1	2	yes	—
207	C₂×C₄	#948	1	2	yes	—
208	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
209	C₈	—	2	4	no	—
210	C₈	—	2	2	no	—
211	C₈	D₄×Dic₃	1	2	yes	—
212	C₈	C₂×D₃×D₄	1	1	yes	characteristic
213	C₈	C₂×#237	1	1	yes	characteristic
214	C₂×C₃	—	2	8	no	—
215	C₂×C₃	D₄×C₁₆	1	4	yes	complemented
216	C₂×C₃	C₂×C₈×D₄	1	1	yes	characteristic
217	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
218	C₂×C₃	C₂×#504	1	1	yes	characteristic
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	16	no	—
222	C₂²	D₄×C₃⋊C₈	1	2	yes	—
223	C₂²	C₂²×C₃⋊C₁₆	1	2	yes	—
224	C₂²	C₂²×C₃⋊C₁₆	1	8	yes	complemented
225	C₂²	#5140	1	2	yes	—
226	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	4	no	—
229	C₄	D₄×C₃⋊C₈	1	2	yes	—
230	C₄	C₂²×C₃⋊C₁₆	1	4	yes	complemented
231	C₄	C₂×D₄×Dic₃	1	1	yes	characteristic
232	C₄	C₂×#948	1	1	yes	characteristic
233	C₃	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
234	C₂	—	2	8	no	—
235	C₂	D₄×C₃⋊C₁₆	1	4	yes	direct product factor
236	C₂	C₂×D₄×C₃⋊C₈	1	1	yes	characteristic
237	C₂	C₂³×C₃⋊C₁₆	1	1	yes	characteristic
238	C₂	C₂×#5140	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×C₈×D₄	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂²×C₃×D₄	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₈×D₄	D₃
hypocenter	C₃	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊C₁₆	C₂	semidirect
3. C₂×#4549	C₂	semidirect
4. D₄×C₃⋊C₁₆	C₂	direct (N × H)
5. C₂×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
6. C₂×C₄×C₃⋊C₁₆	C₂	semidirect
19. C₂²×C₃⋊C₁₆	C₂²	semidirect
20. #4549	C₂²	semidirect
21. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
22. C₄×C₃⋊C₁₆	C₂²	semidirect
50. C₂×C₃⋊C₁₆	D₄	direct (N × H)
70. C₂×C₃×D₄	C₁₆	semidirect
97. C₃⋊C₁₆	C₂×D₄	direct (N × H)
125. C₂³×C₃	C₂×C₁₆	semidirect
128. C₃×D₄	C₂×C₁₆	semidirect
134. C₂×C₃×C₄	C₂×C₁₆	semidirect
148. C₂×D₄	C₃⋊C₁₆	direct (N × H)
181. C₂²×C₃	C₂²×C₁₆	semidirect
187. C₃×C₄	C₂²×C₁₆	semidirect
192. C₂³	C₂×C₃⋊C₁₆	semidirect
195. D₄	C₂×C₃⋊C₁₆	direct (N × H)
201. C₂×C₄	C₂×C₃⋊C₁₆	semidirect
215. C₂×C₃	D₄×C₁₆	semidirect
224. C₂²	C₂²×C₃⋊C₁₆	semidirect
230. C₄	C₂²×C₃⋊C₁₆	semidirect
233. C₃	C₂×D₄×C₁₆	semidirect
235. C₂	D₄×C₃⋊C₁₆	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×C₃×C₈×D₄	2
C₂³×C₃⋊C₁₆	2
C₂×#4549	2
D₄×C₃⋊C₁₆	2
C₂×C₄⋊(C₃⋊C₁₆)	2
C₂×C₄×C₃⋊C₁₆	2
C₂×D₄×C₁₆	3

Subgroups — C2×D4×C3⋊C16 #10948337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₃⋊C₁₆ #10948337