Subgroups — C₃×#69075 #11797477

Order 1152.

355 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#798	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
4	C₃²×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2168	—	3	1	no	maximal
9	#69075	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×D₁₂	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
17	C₃×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	C₃×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
21	C₃×#798	—	2	2	no	—
22	C₃×D₃×C₁₆	—	2	2	no	—
23	C₂×C₃×M₃₂	—	3	1	no	—
24	C₂²×C₃×C₁₆	—	3	1	no	—
25	C₃×C₈×D₄	—	3	1	no	—
26	C₂×#798	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂×D₃×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
28	C₈×D₁₂	C₂×C₃	1	1	yes	characteristic
29	C₃×C₄⋊C₁₆	—	3	1	no	—
30	C₃×C₂²⋊C₁₆	—	3	1	no	—
31	C₃×C₂²⋊C₁₆	—	3	1	no	—
32	C₃×#346	—	2	1	no	—
33	C₃×#346	D₃	1	1	yes	characteristic, complemented
34	C₃×#346	C₂×C₃	1	1	yes	characteristic, complemented
35	(C₂×D₃)⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
36	(C₂×D₃)⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄⋊(C₃⋊C₁₆)	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₃×D₁₂	C₈	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
43	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
45	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
46	C₃×C₄⋊Dic₃	C₈	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×D₃×C₈	—	2	1	no	—
50	C₃×D₃×C₈	—	2	1	no	—
51	C₃×D₃×C₈	—	2	1	no	—
52	C₃×D₃×C₈	—	2	1	no	—
53	C₃²×C₁₆	—	2	1	no	—
54	C₃²×C₁₆	D₄	1	2	yes	complemented
55	C₃×C₃⋊C₁₆	—	2	1	no	—
56	C₃×C₃⋊C₁₆	—	2	1	no	—
57	#2168	—	3	1	no	Sylow
58	C₂²×C₃×C₈	—	3	1	no	—
59	C₂²×C₃×C₈	—	3	1	no	—
60	C₃×C₄×D₄	—	3	1	no	—
61	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₄×D₁₂	C₃×C₄	1	1	yes	characteristic
64	C₃×M₃₂	—	6	2	no	—
65	C₂×C₃×C₁₆	—	2	1	no	—
66	C₂×C₃×C₁₆	—	2	1	no	—
67	C₂×C₃×C₁₆	—	3	1	no	—
68	C₂×C₃×C₁₆	—	3	1	no	—
69	C₂×C₃×C₁₆	—	6	2	no	—
70	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
71	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₃×C₄⋊C₈	—	3	1	no	—
75	C₃×C₂²⋊C₈	—	3	1	no	—
76	C₃×C₂²⋊C₈	—	3	1	no	—
77	C₃×C₄×C₈	—	2	1	no	—
78	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
79	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
80	(C₂×D₃)⋊C₈	C₃×C₄	1	1	yes	characteristic
81	(C₂×D₃)⋊C₈	C₃×C₄	1	1	yes	characteristic
82	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
84	C₄⋊(C₃⋊C₈)	C₃×C₄	1	1	yes	characteristic
85	#798	—	2	2	no	—
86	D₃×C₁₆	—	2	2	no	—
87	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
93	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
94	C₃×D₁₂	—	2	2	no	—
95	C₃×C₄×D₃	—	2	1	no	—
96	C₃×C₄×D₃	—	2	1	no	—
97	C₃×C₄×D₃	—	2	1	no	—
98	C₃×C₄×D₃	—	2	1	no	—
99	C₃²×C₈	—	2	1	no	—
100	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₃²×C₈	#41	1	1	yes	characteristic
102	C₃×C₃⋊C₈	—	2	1	no	—
103	C₃×C₃⋊C₈	—	2	1	no	—
104	C₂×M₃₂	—	3	1	no	—
105	C₂²×C₁₆	—	3	1	no	—
106	C₈×D₄	—	3	1	no	—
107	C₄⋊C₁₆	—	3	1	no	—
108	C₂²⋊C₁₆	—	3	1	no	—
109	C₂²⋊C₁₆	—	3	1	no	—
110	#346	C₃×D₃	1	1	yes	characteristic, complemented
111	C₂×C₃×D₄	—	3	1	no	—
112	C₂²×C₃×C₄	—	3	1	no	—
113	C₂²×C₃×C₄	—	3	1	no	—
114	C₂×D₁₂	C₃×C₈	1	1	yes	characteristic
115	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	3	1	no	—
120	C₂×C₃×C₈	—	3	1	no	—
121	C₂×C₃×C₈	—	6	1	no	—
122	C₂×C₃×C₈	—	6	1	no	—
123	C₂×C₃×C₈	—	6	1	no	—
124	C₂×C₃×C₈	—	6	1	no	—
125	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
129	C₃×C₄⋊C₄	—	3	1	no	—
130	C₃×C₂²⋊C₄	—	3	1	no	—
131	C₃×C₂²⋊C₄	—	3	1	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
134	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
135	(C₂×D₃)⋊C₄	C₃×C₈	1	1	yes	characteristic
136	(C₂×D₃)⋊C₄	C₃×C₈	1	1	yes	characteristic
137	C₄⋊Dic₃	C₃×C₈	1	1	yes	characteristic
138	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
139	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
140	D₃×C₈	—	2	1	no	—
141	D₃×C₈	—	2	1	no	—
142	D₃×C₈	—	2	1	no	—
143	D₃×C₈	—	2	1	no	—
144	C₃×C₁₆	—	2	2	no	—
145	C₃×C₁₆	—	2	1	no	—
146	C₃×C₁₆	—	2	1	no	—
147	C₃×C₁₆	—	2	2	no	—
148	C₃×C₁₆	—	6	1	no	—
149	C₃×C₁₆	—	6	1	no	—
150	C₃×C₁₆	D₁₂	1	2	yes	complemented
151	C₃×C₁₆	C₃×D₄	1	2	yes	complemented
152	C₃⋊C₁₆	—	2	1	no	—
153	C₃⋊C₁₆	—	2	1	no	—
154	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₃×D₃	—	2	1	no	—
156	C₂×C₃×D₃	—	2	1	no	—
157	C₂×C₃×D₃	—	2	1	no	—
158	C₂×C₃×D₃	—	2	1	no	—
159	C₂×C₃×D₃	—	2	1	no	—
160	C₂×C₃×D₃	M₃₂	1	2	yes	complemented
161	C₃²×C₄	—	2	1	no	—
162	C₃²×C₄	—	2	1	no	—
163	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
164	C₃²×C₄	#131	1	1	yes	characteristic
165	C₃×Dic₃	—	2	1	no	—
166	C₃×Dic₃	—	2	1	no	—
167	C₂²×C₈	—	3	1	no	—
168	C₂²×C₈	—	3	1	no	—
169	C₄×D₄	—	3	1	no	—
170	M₃₂	—	6	2	no	—
171	C₂×C₁₆	—	3	1	no	—
172	C₂×C₁₆	—	3	1	no	—
173	C₂×C₁₆	—	6	2	no	—
174	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
176	C₄⋊C₈	—	3	1	no	—
177	C₂²⋊C₈	—	3	1	no	—
178	C₂²⋊C₈	—	3	1	no	—
179	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
180	C₂³×C₃	—	3	1	no	—
181	C₂³×C₃	—	3	1	no	—
182	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
183	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
184	C₃×D₄	—	6	2	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	3	1	no	—
189	C₂×C₃×C₄	—	3	1	no	—
190	C₂×C₃×C₄	—	6	1	no	—
191	C₂×C₃×C₄	—	6	1	no	—
192	C₂×C₃×C₄	—	6	1	no	—
193	C₂×C₃×C₄	—	6	1	no	—
194	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
195	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
200	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
201	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
202	D₁₂	—	2	2	no	—
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	2	1	no	—
206	C₄×D₃	—	2	1	no	—
207	C₃×C₈	—	2	2	no	—
208	C₃×C₈	—	2	1	no	—
209	C₃×C₈	—	2	1	no	—
210	C₃×C₈	—	2	1	no	—
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	6	1	no	—
213	C₃×C₈	—	6	1	no	—
214	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
215	C₃×C₈	#235	1	1	yes	characteristic
216	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
217	C₃×C₈	C₃×#41	1	1	yes	characteristic
218	C₃⋊C₈	—	2	1	no	—
219	C₃⋊C₈	—	2	1	no	—
220	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
221	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
222	C₂×C₃²	#504	1	1	yes	characteristic
223	C₃×D₃	—	2	2	no	—
224	C₃×D₃	—	4	1	no	—
225	C₂×D₄	—	3	1	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	3	1	no	—
228	C₂×C₈	—	3	1	no	—
229	C₂×C₈	—	3	1	no	—
230	C₂×C₈	—	6	1	no	—
231	C₂×C₈	—	6	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	—	6	1	no	—
234	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
235	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
236	C₄⋊C₄	—	3	1	no	—
237	C₂²⋊C₄	—	3	1	no	—
238	C₂²⋊C₄	—	3	1	no	—
239	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
240	C₁₆	—	2	1	no	—
241	C₁₆	—	6	1	no	—
242	C₁₆	—	6	1	no	—
243	C₁₆	C₃×D₁₂	1	2	yes	complemented
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	3	2	no	—
246	C₂²×C₃	—	6	1	no	—
247	C₂²×C₃	—	6	1	no	—
248	C₂²×C₃	—	6	1	no	—
249	C₂²×C₃	—	6	1	no	—
250	C₂²×C₃	—	6	1	no	—
251	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
252	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
253	C₂×D₃	—	2	1	no	—
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	—	2	1	no	—
258	C₂×D₃	C₃×M₃₂	1	2	yes	complemented
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	4	1	no	—
266	C₃×C₄	—	4	1	no	—
267	C₃×C₄	—	6	1	no	—
268	C₃×C₄	—	6	1	no	—
269	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
270	C₃×C₄	#901	1	1	yes	characteristic
271	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
272	C₃×C₄	C₃×#131	1	1	yes	characteristic
273	Dic₃	—	2	1	no	—
274	Dic₃	—	2	1	no	—
275	C₃²	#2168	1	1	yes	characteristic, complemented, Sylow
276	C₂³	—	3	1	no	—
277	C₂³	—	3	1	no	—
278	D₄	—	6	2	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	6	1	no	—
284	C₂×C₄	—	6	1	no	—
285	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
286	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
287	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	6	1	no	—
290	C₈	—	6	1	no	—
291	C₈	C₂×C₃×D₁₂	1	1	yes	characteristic
292	C₈	C₃×#235	1	1	yes	characteristic
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	6	2	no	—
297	C₂×C₃	—	12	1	no	—
298	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
299	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
300	C₂×C₃	#4907	1	1	yes	characteristic
301	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
303	C₂×C₃	C₃×#504	1	1	yes	characteristic
304	D₃	—	2	2	no	—
305	D₃	—	4	1	no	—
306	C₂²	—	3	2	no	—
307	C₂²	—	6	1	no	—
308	C₂²	—	6	1	no	—
309	C₂²	—	6	1	no	—
310	C₂²	—	6	1	no	—
311	C₂²	—	6	1	no	—
312	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	6	1	no	—
316	C₄	—	6	1	no	—
317	C₄	C₃×C₄×D₁₂	1	1	yes	characteristic
318	C₄	C₃×#901	1	1	yes	characteristic
319	C₃	—	2	1	no	—
320	C₃	#69075	1	1	yes	characteristic, direct product factor
321	C₃	C₃×#2168	1	1	yes	characteristic, complemented
322	C₂	—	6	2	no	—
323	C₂	—	12	1	no	—
324	C₂	C₃×C₈×D₁₂	1	1	yes	characteristic
325	C₂	C₃×D₃×M₃₂	1	1	yes	characteristic
326	C₂	C₃×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×#346	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃×D₁₂	C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#346	D₃
hypocenter	C₃	C₃×#2168
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×#346

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#798	C₂	semidirect
2. C₂×C₃×D₃×C₁₆	C₂	semidirect
4. C₃²×#346	C₂	semidirect
5. C₃×(C₂×D₃)⋊C₁₆	C₂	semidirect
6. C₃×(C₂×D₃)⋊C₁₆	C₂	semidirect
7. C₃×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
9. #69075	C₃	direct (N × H)
26. C₂×#798	C₂×C₃	semidirect
27. C₂×D₃×C₁₆	C₂×C₃	semidirect
33. C₃×#346	D₃	semidirect
34. C₃×#346	C₂×C₃	semidirect
35. (C₂×D₃)⋊C₁₆	C₂×C₃	semidirect
36. (C₂×D₃)⋊C₁₆	C₂×C₃	semidirect
37. C₄⋊(C₃⋊C₁₆)	C₂×C₃	semidirect
54. C₃²×C₁₆	D₄	semidirect
110. #346	C₃×D₃	semidirect
150. C₃×C₁₆	D₁₂	semidirect
151. C₃×C₁₆	C₃×D₄	semidirect
160. C₂×C₃×D₃	M₃₂	semidirect
243. C₁₆	C₃×D₁₂	semidirect
258. C₂×D₃	C₃×M₃₂	semidirect
275. C₃²	#2168	semidirect
320. C₃	#69075	direct (N × H)
321. C₃	C₃×#2168	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#798	2
C₂×C₃×D₃×C₁₆	2
C₃×C₈×D₁₂	2
C₃²×#346	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×C₄⋊(C₃⋊C₁₆)	2
C₃×#2168	3
#69075	3

Subgroups — C3×#69075 #11797477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69075 #11797477