Subgroups — C₄×D₃×C₃⋊C₁₆ #11795814

Order 1152.

461 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×C₃⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₃⋊C₁₆	—	3	1	no	maximal
7	C₄×D₃×C₁₆	—	3	1	no	maximal
8	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
9	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄²×D₃	C₄	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
13	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
14	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
17	D₃×C₃⋊C₁₆	C₄	1	8	yes	direct product factor
18	C₂×C₃×C₄×C₈	—	3	1	no	—
19	C₂²×C₃⋊C₁₆	—	3	2	no	—
20	C₂×D₃×C₁₆	—	3	2	no	—
21	C₄×D₃×C₈	D₃	1	1	yes	characteristic
22	C₃×C₄×C₁₆	—	3	1	no	—
23	Dic₃×C₁₆	—	3	2	no	—
24	C₄×C₃⋊C₁₆	—	3	1	no	—
25	C₄×C₃⋊C₁₆	—	3	1	no	—
26	C₄×C₃⋊C₁₆	—	3	2	no	—
27	C₄×C₃⋊C₁₆	—	6	1	no	—
28	C₄×C₃⋊C₁₆	D₃	1	1	yes	characteristic, direct product factor
29	C₂×C₃×C₄×D₃	C₈	1	2	yes	—
30	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
33	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₈	1	2	yes	—
35	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×D₃×C₈	C₈	1	4	yes	—
39	C₃×D₃×C₈	C₂×C₄	1	4	yes	—
40	C₃⋊(C₃⋊C₁₆)	C₂×C₄	1	4	yes	complemented
41	C₃×C₃⋊C₁₆	C₂×C₄	1	4	yes	complemented
42	C₂×C₄×C₁₆	—	9	1	no	Sylow
43	C₂²×C₃×C₈	—	3	1	no	—
44	C₂²×C₃×C₈	—	3	1	no	—
45	C₂×C₃×C₄²	—	3	1	no	—
46	C₂×D₃×C₈	Dic₃	1	1	yes	characteristic
47	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₄²×D₃	Dic₃	1	1	yes	characteristic
49	C₂×C₃×C₁₆	—	3	2	no	—
50	C₃×C₄×C₈	—	2	1	no	—
51	C₃×C₄×C₈	—	3	1	no	—
52	C₃×C₄×C₈	—	3	1	no	—
53	C₃×C₄×C₈	—	3	1	no	—
54	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	C₈×Dic₃	Dic₃	1	1	yes	characteristic
57	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃⋊C₁₆	—	3	2	no	—
59	C₂×C₃⋊C₁₆	—	3	2	no	—
60	C₂×C₃⋊C₁₆	—	3	8	no	—
61	C₂×C₃⋊C₁₆	—	6	2	no	—
62	C₂×C₃⋊C₁₆	C₂×D₃	1	2	yes	—
63	C₄×C₃⋊C₈	Dic₃	1	1	yes	characteristic
64	D₃×C₁₆	—	3	8	no	—
65	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
70	C₃×C₄×D₃	C₁₆	1	8	yes	complemented
71	C₃×C₄×D₃	C₄²	1	2	yes	—
72	C₃×C₄×D₃	C₂×C₈	1	2	yes	—
73	C₃²×C₈	C₂×C₈	1	2	yes	—
74	C₃²×C₈	C₂²×C₄	1	2	yes	—
75	C₃×C₃⋊C₈	C₄²	1	2	yes	—
76	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	—
77	C₂²×C₁₆	—	9	2	no	—
78	C₂×C₄×C₈	—	3	1	no	—
79	C₄×C₁₆	—	3	1	no	—
80	C₄×C₁₆	—	9	2	no	—
81	C₄×C₁₆	—	9	1	no	—
82	C₂²×C₃×C₄	—	3	2	no	—
83	C₂²×C₃×C₄	—	3	1	no	—
84	C₂×C₄×D₃	C₃⋊C₈	1	2	yes	—
85	C₂×C₄×D₃	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	3	1	no	—
89	C₂×C₃×C₈	—	3	1	no	—
90	C₂×C₃×C₈	—	3	4	no	—
91	C₂×C₃×C₈	—	3	4	no	—
92	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	3	2	no	—
98	C₃×C₄²	—	3	1	no	—
99	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
100	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
101	C₄×Dic₃	C₃⋊C₈	1	2	yes	—
102	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
103	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
105	D₃×C₈	C₃⋊C₈	1	4	yes	—
106	D₃×C₈	C₄×D₃	1	4	yes	—
107	C₃×C₁₆	—	3	4	no	—
108	C₃⋊C₁₆	—	3	4	no	—
109	C₃⋊C₁₆	—	3	4	no	—
110	C₃⋊C₁₆	—	6	4	no	—
111	C₃⋊C₁₆	C₄×D₃	1	4	yes	direct product factor
112	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₃×D₃	C₄×C₈	1	2	yes	—
114	C₂×C₃×D₃	C₂×C₁₆	1	4	yes	—
115	C₃²×C₄	C₂×C₁₆	1	4	yes	complemented
116	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
117	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
118	C₃×Dic₃	C₄×C₈	1	2	yes	—
119	C₃×Dic₃	C₂×C₁₆	1	4	yes	complemented
120	C₂²×C₈	—	3	1	no	—
121	C₂²×C₈	—	3	1	no	—
122	C₂×C₄²	—	3	1	no	—
123	C₂×C₁₆	—	3	2	no	—
124	C₂×C₁₆	—	9	8	no	—
125	C₂×C₁₆	—	9	2	no	—
126	C₄×C₈	—	3	1	no	—
127	C₄×C₈	—	3	1	no	—
128	C₄×C₈	—	3	1	no	—
129	C₄×C₈	D₃²	1	1	yes	characteristic
130	C₂³×C₃	—	3	1	no	—
131	C₂²×D₃	C₂×C₃⋊C₈	1	1	yes	characteristic
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	3	2	no	—
135	C₂×C₃×C₄	—	3	1	no	—
136	C₂×C₃×C₄	—	3	8	no	—
137	C₂×C₃×C₄	—	3	2	no	—
138	C₂×C₃×C₄	—	3	2	no	—
139	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
143	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
144	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
145	C₄×D₃	C₃⋊C₁₆	1	8	yes	direct product factor
146	C₄×D₃	C₂×C₃⋊C₈	1	2	yes	—
147	C₄×D₃	C₄×Dic₃	1	2	yes	—
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	—	2	2	no	—
150	C₃×C₈	—	3	2	no	—
151	C₃×C₈	—	3	2	no	—
152	C₃×C₈	D₃×C₈	1	2	yes	—
153	C₃×C₈	C₂×C₃⋊C₈	1	2	yes	—
154	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
155	C₃×C₈	C₂×C₄×D₃	1	2	yes	—
156	C₃⋊C₈	C₂×C₃⋊C₈	1	2	yes	—
157	C₃⋊C₈	C₄×Dic₃	1	2	yes	—
158	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
159	C₂×C₃²	C₂²×C₁₆	1	2	yes	—
160	C₃×D₃	C₄×C₁₆	1	4	yes	complemented
161	C₂²×C₄	—	3	2	no	—
162	C₂²×C₄	—	3	1	no	—
163	C₂×C₈	—	3	1	no	—
164	C₂×C₈	—	3	1	no	—
165	C₂×C₈	—	3	4	no	—
166	C₂×C₈	—	3	4	no	—
167	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
168	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
169	C₄²	—	3	2	no	—
170	C₄²	—	3	1	no	—
171	C₄²	D₃×Dic₃	1	1	yes	characteristic
172	C₁₆	—	3	4	no	—
173	C₁₆	—	9	4	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	3	4	no	—
176	C₂²×C₃	—	3	2	no	—
177	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
178	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
179	C₂×D₃	C₄×C₃⋊C₈	1	2	yes	—
180	C₂×D₃	C₂×C₃⋊C₁₆	1	4	yes	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	3	4	no	—
185	C₃×C₄	—	3	2	no	—
186	C₃×C₄	D₃×C₁₆	1	4	yes	complemented
187	C₃×C₄	C₂×C₃⋊C₁₆	1	4	yes	complemented
188	C₃×C₄	C₄²×D₃	1	1	yes	characteristic
189	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
190	C₃×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
191	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
192	Dic₃	C₄×C₃⋊C₈	1	2	yes	—
193	Dic₃	C₂×C₃⋊C₁₆	1	4	yes	complemented
194	C₃²	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	3	1	no	—
196	C₂×C₄	—	3	1	no	—
197	C₂×C₄	—	3	2	no	—
198	C₂×C₄	—	3	8	no	—
199	C₂×C₄	—	3	2	no	—
200	C₂×C₄	—	3	2	no	—
201	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
202	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
203	C₈	—	3	2	no	—
204	C₈	—	3	2	no	—
205	C₈	D₃×C₃⋊C₈	1	2	yes	—
206	C₈	C₄×D₃²	1	2	yes	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	3	4	no	—
210	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
211	C₂×C₃	C₂×D₃×C₁₆	1	2	yes	—
212	C₂×C₃	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
213	C₂×C₃	C₂²×C₃⋊C₁₆	1	2	yes	—
214	D₃	C₄×C₃⋊C₁₆	1	4	yes	direct product factor
215	C₂²	—	3	2	no	—
216	C₂²	—	3	4	no	—
217	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
218	C₄	—	3	2	no	—
219	C₄	—	3	4	no	—
220	C₄	D₃×C₃⋊C₁₆	1	4	yes	direct product factor
221	C₄	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
222	C₄	C₄×D₃×Dic₃	1	1	yes	characteristic
223	C₃	—	2	1	no	—
224	C₃	C₄×D₃×C₁₆	1	1	yes	characteristic, complemented
225	C₃	C₂×C₄×C₃⋊C₁₆	1	1	yes	characteristic, complemented
226	C₂	—	3	4	no	—
227	C₂	C₄×D₃×C₃⋊C₈	1	1	yes	characteristic
228	C₂	C₂×D₃×C₃⋊C₁₆	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₄×C₈	D₃²
commutator, hypocenter	C₃²	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₃²×C₄×C₈	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃²×C₈	C₂³
LCS	G ▷ C₃²
UCS	C₁ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
4. C₃×C₄×C₃⋊C₁₆	C₂	semidirect
5. Dic₃×C₃⋊C₁₆	C₂	semidirect
17. D₃×C₃⋊C₁₆	C₄	direct (N × H)
28. C₄×C₃⋊C₁₆	D₃	direct (N × H)
40. C₃⋊(C₃⋊C₁₆)	C₂×C₄	semidirect
41. C₃×C₃⋊C₁₆	C₂×C₄	semidirect
70. C₃×C₄×D₃	C₁₆	semidirect
111. C₃⋊C₁₆	C₄×D₃	direct (N × H)
115. C₃²×C₄	C₂×C₁₆	semidirect
119. C₃×Dic₃	C₂×C₁₆	semidirect
145. C₄×D₃	C₃⋊C₁₆	direct (N × H)
160. C₃×D₃	C₄×C₁₆	semidirect
186. C₃×C₄	D₃×C₁₆	semidirect
187. C₃×C₄	C₂×C₃⋊C₁₆	semidirect
193. Dic₃	C₂×C₃⋊C₁₆	semidirect
194. C₃²	C₂×C₄×C₁₆	semidirect
214. D₃	C₄×C₃⋊C₁₆	direct (N × H)
220. C₄	D₃×C₃⋊C₁₆	direct (N × H)
224. C₃	C₄×D₃×C₁₆	semidirect
225. C₃	C₂×C₄×C₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₃×C₈	2
C₂×D₃×C₃⋊C₁₆	2
C₄×C₃⋊(C₃⋊C₁₆)	2
C₃×C₄×C₃⋊C₁₆	2
Dic₃×C₃⋊C₁₆	2
C₂×C₄×C₃⋊C₁₆	3
C₄×D₃×C₁₆	3

Subgroups — C4×D3×C3⋊C16 #11795814

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₃×C₃⋊C₁₆ #11795814