Subgroups — C₃×D₄×C₃⋊C₁₆ #11797436

Order 1152.

415 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×#4549	C₂	1	2	yes	maximal, complemented
4	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₄×C₁₆	—	3	1	no	maximal
7	D₄×C₃⋊C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₃²×C₄×D₄	C₄	1	1	yes	characteristic
10	C₃²×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₃²×C₂²⋊C₈	C₄	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₃⋊C₁₆	—	2	4	no	—
14	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₁₆	—	3	2	no	—
18	C₃×C₈×D₄	—	2	1	no	—
19	C₃×C₈×D₄	D₃	1	1	yes	characteristic
20	C₃×C₈×D₄	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₃⋊C₁₆	C₂×C₃	1	2	yes	complemented
22	C₃×C₄⋊C₁₆	—	3	1	no	—
23	C₃×C₂²⋊C₁₆	—	3	2	no	—
24	C₃×C₄×C₁₆	—	3	1	no	—
25	#4549	C₂×C₃	1	2	yes	complemented
26	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	1	yes	characteristic, complemented
27	C₄×C₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
35	C₃²×C₄⋊C₄	C₈	1	1	yes	characteristic
36	C₃²×C₂²⋊C₄	C₈	1	2	yes	—
37	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₃×C₃⋊C₁₆	—	2	2	no	—
39	C₃×C₃⋊C₁₆	—	2	1	no	—
40	C₃×C₃⋊C₁₆	D₄	1	2	yes	direct product factor
41	D₄×C₁₆	—	3	1	no	Sylow
42	C₂²×C₃×C₈	—	2	2	no	—
43	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
44	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
45	C₃×C₄×D₄	—	2	1	no	—
46	C₃×C₄×D₄	Dic₃	1	1	yes	characteristic
47	C₃×C₄×D₄	C₃×C₄	1	1	yes	characteristic
48	C₂×C₃×C₁₆	—	3	1	no	—
49	C₂×C₃×C₁₆	—	3	2	no	—
50	C₂×C₃×C₁₆	—	3	1	no	—
51	C₂×C₃×C₁₆	—	6	4	no	—
52	C₃×C₄⋊C₈	—	2	1	no	—
53	C₃×C₄⋊C₈	Dic₃	1	1	yes	characteristic
54	C₃×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
55	C₃×C₂²⋊C₈	—	2	2	no	—
56	C₃×C₂²⋊C₈	Dic₃	1	2	yes	—
57	C₃×C₂²⋊C₈	C₃×C₄	1	2	yes	—
58	C₃×C₄×C₈	—	2	1	no	—
59	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃⋊C₁₆	—	2	4	no	—
62	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
64	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
65	C₂³×C₃²	C₂×C₈	1	2	yes	—
66	C₃²×D₄	C₁₆	1	4	yes	complemented
67	C₂×C₃²×C₄	—	2	2	no	—
68	C₂×C₃²×C₄	—	2	2	no	—
69	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
71	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
73	C₃²×C₈	—	2	2	no	—
74	C₃²×C₈	—	2	1	no	—
75	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
76	C₃²×C₈	#41	1	1	yes	characteristic
77	C₂²×C₁₆	—	3	2	no	—
78	C₈×D₄	C₃×D₃	1	1	yes	characteristic
79	C₄⋊C₁₆	—	3	1	no	—
80	C₂²⋊C₁₆	—	3	2	no	—
81	C₄×C₁₆	—	3	1	no	—
82	C₂×C₃×D₄	—	2	1	no	—
83	C₂×C₃×D₄	C₃⋊C₈	1	1	yes	characteristic
84	C₂×C₃×D₄	C₃×C₈	1	1	yes	characteristic
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
87	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₃×C₈	—	2	2	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	4	2	no	—
96	C₂×C₃×C₈	—	4	2	no	—
97	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
99	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
100	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
101	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
103	C₃×C₄⋊C₄	—	2	1	no	—
104	C₃×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
105	C₃×C₄⋊C₄	C₃×C₈	1	1	yes	characteristic
106	C₃×C₂²⋊C₄	—	2	2	no	—
107	C₃×C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
108	C₃×C₂²⋊C₄	C₃×C₈	1	2	yes	—
109	C₃×C₄²	—	2	1	no	—
110	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
112	C₃×C₁₆	—	3	2	no	—
113	C₃×C₁₆	—	6	2	no	—
114	C₃×C₁₆	—	6	1	no	—
115	C₃⋊C₁₆	—	2	2	no	—
116	C₃⋊C₁₆	—	2	1	no	—
117	C₃⋊C₁₆	C₃×D₄	1	2	yes	direct product factor
118	C₂²×C₃²	—	2	2	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	C₂×C₁₆	1	4	yes	complemented
121	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
122	C₃²×C₄	—	2	2	no	—
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	C₂×C₁₆	1	2	yes	complemented
125	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
126	C₃²×C₄	#131	1	1	yes	characteristic
127	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
128	C₄×D₄	C₃×Dic₃	1	1	yes	characteristic
129	C₂×C₁₆	—	3	1	no	—
130	C₂×C₁₆	—	3	1	no	—
131	C₂×C₁₆	—	3	2	no	—
132	C₂×C₁₆	—	6	4	no	—
133	C₄⋊C₈	C₃×Dic₃	1	1	yes	characteristic
134	C₂²⋊C₈	C₃×Dic₃	1	2	yes	—
135	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
136	C₂³×C₃	—	2	2	no	—
137	C₂³×C₃	C₂×C₃⋊C₈	1	2	yes	—
138	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
139	C₃×D₄	—	2	4	no	—
140	C₃×D₄	C₃⋊C₁₆	1	4	yes	direct product factor
141	C₃×D₄	C₃×C₁₆	1	4	yes	complemented
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	4	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
160	C₃×C₈	—	2	2	no	—
161	C₃×C₈	—	2	1	no	—
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	—	2	1	no	—
164	C₃×C₈	—	2	1	no	—
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	4	1	no	—
167	C₃×C₈	—	4	2	no	—
168	C₃×C₈	D₃×D₄	1	1	yes	characteristic
169	C₃×C₈	#237	1	1	yes	characteristic
170	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
171	C₃×C₈	C₃×#41	1	1	yes	characteristic
172	C₂×C₃²	—	2	4	no	—
173	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
174	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
175	C₂×C₃²	#504	1	1	yes	characteristic
176	C₂×D₄	C₃×C₃⋊C₈	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
181	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
182	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
183	C₄⋊C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
184	C₂²⋊C₄	C₃×C₃⋊C₈	1	2	yes	—
185	C₄²	C₂×C₃×Dic₃	1	1	yes	characteristic
186	C₁₆	—	3	2	no	—
187	C₁₆	—	6	2	no	—
188	C₁₆	—	6	1	no	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	2	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	—	2	2	no	—
193	C₂²×C₃	—	2	1	no	—
194	C₂²×C₃	—	2	4	no	—
195	C₂²×C₃	—	4	2	no	—
196	C₂²×C₃	—	4	2	no	—
197	C₂²×C₃	C₂×C₃⋊C₁₆	1	4	yes	complemented
198	C₂²×C₃	C₂×C₃×C₁₆	1	4	yes	complemented
199	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
200	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	4	2	no	—
209	C₃×C₄	—	4	1	no	—
210	C₃×C₄	C₂×C₃⋊C₁₆	1	2	yes	complemented
211	C₃×C₄	C₂×C₃×C₁₆	1	2	yes	complemented
212	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
213	C₃×C₄	#948	1	1	yes	characteristic
214	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
215	C₃×C₄	C₃×#131	1	1	yes	characteristic
216	C₃²	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
217	C₂³	C₂×C₃×C₃⋊C₈	1	2	yes	—
218	D₄	C₃×C₃⋊C₁₆	1	4	yes	direct product factor
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
222	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
223	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	2	1	no	—
227	C₈	C₃×D₃×D₄	1	1	yes	characteristic
228	C₈	C₃×#237	1	1	yes	characteristic
229	C₂×C₃	—	2	4	no	—
230	C₂×C₃	—	2	4	no	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	1	no	—
234	C₂×C₃	—	4	4	no	—
235	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
236	C₂×C₃	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
237	C₂×C₃	#5140	1	1	yes	characteristic
238	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
239	C₂×C₃	C₂²×C₃×C₁₆	1	1	yes	characteristic
240	C₂×C₃	C₃×#504	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₂×C₃×C₃⋊C₁₆	1	4	yes	complemented
244	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	C₂×C₃×C₃⋊C₁₆	1	2	yes	complemented
248	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
249	C₄	C₃×#948	1	1	yes	characteristic
250	C₃	—	2	1	no	—
251	C₃	D₄×C₃⋊C₁₆	1	1	yes	characteristic, direct product factor
252	C₃	C₃×D₄×C₁₆	1	1	yes	characteristic, complemented
253	C₂	—	2	4	no	—
254	C₂	C₃×D₄×C₃⋊C₈	1	1	yes	characteristic
255	C₂	C₂²×C₃×C₃⋊C₁₆	1	1	yes	characteristic
256	C₂	C₃×#5140	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃×C₁₆
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₈×D₄	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃²×D₄	C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₈×D₄	D₃
hypocenter	C₃	C₃×D₄×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₃⋊C₁₆	C₂	semidirect
3. C₃×#4549	C₂	semidirect
4. C₃×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
5. C₃×C₄×C₃⋊C₁₆	C₂	semidirect
7. D₄×C₃⋊C₁₆	C₃	direct (N × H)
21. C₂²×C₃⋊C₁₆	C₂×C₃	semidirect
25. #4549	C₂×C₃	semidirect
26. C₄⋊(C₃⋊C₁₆)	C₂×C₃	semidirect
27. C₄×C₃⋊C₁₆	C₂×C₃	semidirect
40. C₃×C₃⋊C₁₆	D₄	direct (N × H)
66. C₃²×D₄	C₁₆	semidirect
117. C₃⋊C₁₆	C₃×D₄	direct (N × H)
120. C₂²×C₃²	C₂×C₁₆	semidirect
124. C₃²×C₄	C₂×C₁₆	semidirect
140. C₃×D₄	C₃⋊C₁₆	direct (N × H)
141. C₃×D₄	C₃×C₁₆	semidirect
197. C₂²×C₃	C₂×C₃⋊C₁₆	semidirect
198. C₂²×C₃	C₂×C₃×C₁₆	semidirect
210. C₃×C₄	C₂×C₃⋊C₁₆	semidirect
211. C₃×C₄	C₂×C₃×C₁₆	semidirect
216. C₃²	D₄×C₁₆	semidirect
218. D₄	C₃×C₃⋊C₁₆	direct (N × H)
243. C₂²	C₂×C₃×C₃⋊C₁₆	semidirect
247. C₄	C₂×C₃×C₃⋊C₁₆	semidirect
251. C₃	D₄×C₃⋊C₁₆	direct (N × H)
252. C₃	C₃×D₄×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₈×D₄	2
C₂²×C₃×C₃⋊C₁₆	2
C₃×#4549	2
C₃×C₄⋊(C₃⋊C₁₆)	2
C₃×C₄×C₃⋊C₁₆	2
C₃×D₄×C₁₆	3
D₄×C₃⋊C₁₆	3

Subgroups — C3×D4×C3⋊C16 #11797436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₄×C₃⋊C₁₆ #11797436