Subgroups — C₃×#69798 #11775867

Order 1152.

585 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5477	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#4926	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4464	C₂	1	4	yes	maximal, complemented
5	C₃×#1530	—	3	1	no	maximal
6	#69798	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃×#235	C₄	1	2	yes	—
8	C₂²×C₃²×C₈	—	2	1	no	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₃×#930	—	2	1	no	—
11	C₃×#930	—	2	1	no	—
12	C₃×#875	C₂²	1	2	yes	—
13	C₃×#874	C₄	1	1	yes	characteristic
14	C₃×C₄×D₁₂	—	2	1	no	—
15	C₃×C₄×D₁₂	—	2	1	no	—
16	C₃×#869	C₄	1	1	yes	characteristic
17	C₃²×#97	—	2	1	no	—
18	C₃²×#97	C₂²	1	2	yes	—
19	C₃×#830	—	2	1	no	—
20	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
21	C₃×#803	C₂²	1	2	yes	—
22	C₃×#532	—	3	1	no	—
23	#5477	C₂×C₃	1	1	yes	characteristic
24	C₃×#432	—	3	1	no	—
25	C₂×C₃×#97	—	2	1	no	—
26	C₂×C₃×#97	D₃	1	1	yes	characteristic, complemented
27	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic, complemented
28	#4926	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#329	—	3	4	no	—
30	#4464	C₂×C₃	1	4	yes	complemented
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
34	C₂×C₃×#67	—	2	2	no	—
35	C₃×#235	—	4	4	no	—
36	C₂×C₃×D₁₂	—	2	1	no	—
37	C₂×C₃×D₁₂	—	2	1	no	—
38	C₂×C₃×D₁₂	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₄×D₃	—	2	2	no	—
40	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
41	C₂×C₃²×C₈	—	2	4	no	—
42	C₂×C₃²×C₈	—	2	2	no	—
43	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
44	C₃²×C₄²	D₄	1	1	yes	characteristic
45	C₃²×C₄²	D₄	1	1	yes	characteristic
46	C₃²×C₄²	C₂³	1	1	yes	characteristic
47	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
48	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
49	C₃×C₄⋊Dic₃	—	2	1	no	—
50	C₃×C₄⋊Dic₃	—	2	1	no	—
51	C₂×C₃×C₃⋊C₈	—	2	2	no	—
52	#1530	—	3	1	no	Sylow
53	C₂×C₃×#41	—	3	2	no	—
54	C₂×#235	C₃×C₄	1	2	yes	—
55	C₂²×C₃×C₈	—	2	2	no	—
56	C₂²×C₃×C₈	—	2	1	no	—
57	C₂²×C₃×C₈	—	2	1	no	—
58	C₃×C₄⋊Q₈	—	3	1	no	—
59	C₃×#127	—	3	1	no	—
60	C₃×#124	—	3	2	no	—
61	C₃×#121	—	6	1	no	—
62	C₃×#121	—	6	1	no	—
63	C₃×C₄×D₄	—	6	1	no	—
64	C₃×C₄×D₄	—	6	1	no	—
65	C₂×C₃×C₄²	—	2	1	no	—
66	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
68	#930	—	2	1	no	—
69	#930	—	2	1	no	—
70	#875	C₂²×C₃	1	2	yes	—
71	#874	C₃×C₄	1	1	yes	characteristic
72	C₄×D₁₂	—	2	1	no	—
73	C₄×D₁₂	—	2	1	no	—
74	#869	C₃×C₄	1	1	yes	characteristic
75	C₃×C₄⋊C₈	—	6	1	no	—
76	C₃×C₂²⋊C₈	—	6	1	no	—
77	C₃×#97	—	2	2	no	—
78	C₃×#97	—	2	1	no	—
79	C₃×#97	—	2	1	no	—
80	C₃×#97	—	2	2	no	—
81	C₃×#97	—	3	2	no	—
82	C₃×#97	C₂×D₃	1	2	yes	—
83	C₃×#97	C₂²×C₃	1	2	yes	—
84	#830	—	2	1	no	—
85	C₄⋊(C₃⋊C₈)	—	2	1	no	—
86	#803	C₂²×C₃	1	2	yes	—
87	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₃×D₃	—	2	2	no	—
89	C₂×C₃²×C₄	—	2	2	no	—
90	C₂×C₃²×C₄	—	2	2	no	—
91	C₂×C₃²×C₄	—	4	1	no	—
92	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₃×#67	—	4	4	no	—
99	C₂×C₃×Dic₃	—	2	2	no	—
100	C₃×D₁₂	—	4	2	no	—
101	C₃×D₁₂	—	4	2	no	—
102	C₃×D₁₂	—	4	2	no	—
103	C₃×C₄×D₃	—	4	4	no	—
104	C₃×Dic₆	—	4	2	no	—
105	C₃²×C₈	—	2	4	no	—
106	C₃×C₃⋊C₈	—	4	2	no	—
107	#532	—	3	1	no	—
108	#432	—	3	1	no	—
109	C₂×#97	C₃×D₃	1	1	yes	characteristic, complemented
110	#329	—	3	4	no	—
111	C₃×#41	—	12	4	no	—
112	C₂×C₃×Q₈	—	3	2	no	—
113	C₂×C₃×D₄	—	3	2	no	—
114	C₂×C₃×D₄	—	6	2	no	—
115	C₂×C₃×D₄	—	6	1	no	—
116	C₂×C₃×D₄	—	6	1	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	—	6	2	no	—
121	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
122	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
123	C₂²×C₃×C₄	#67	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
127	C₂×#67	—	2	2	no	—
128	#235	—	4	4	no	—
129	C₂×D₁₂	—	2	1	no	—
130	C₂×D₁₂	—	2	1	no	—
131	C₂×D₁₂	C₂×C₃×C₄	1	2	yes	—
132	C₂×C₄×D₃	—	2	2	no	—
133	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
134	C₂×C₃×C₈	—	2	4	no	—
135	C₂×C₃×C₈	—	2	8	no	—
136	C₂×C₃×C₈	—	2	4	no	—
137	C₂×C₃×C₈	—	2	4	no	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	6	2	no	—
141	C₃×C₄⋊C₄	—	6	1	no	—
142	C₃×C₄⋊C₄	—	6	1	no	—
143	C₃×C₂²⋊C₄	—	6	2	no	—
144	C₃×C₂²⋊C₄	—	6	2	no	—
145	C₃×C₄²	—	2	1	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	—	2	1	no	—
149	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
150	C₃×C₄²	D₁₂	1	1	yes	characteristic
151	C₃×C₄²	#67	1	1	yes	characteristic
152	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
153	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
154	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
155	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
156	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
157	(C₂×D₃)⋊C₄	—	2	2	no	—
158	(C₂×D₃)⋊C₄	—	2	2	no	—
159	C₄⋊Dic₃	—	2	1	no	—
160	C₄⋊Dic₃	—	2	1	no	—
161	C₂×C₃⋊C₈	—	2	2	no	—
162	C₂²×C₃²	—	2	1	no	—
163	C₂²×C₃²	—	2	1	no	—
164	C₂²×C₃²	#100	1	2	yes	—
165	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂×C₃×D₃	—	4	2	no	—
167	C₂×C₃×D₃	—	4	2	no	—
168	C₂×C₃×D₃	—	4	2	no	—
169	C₃²×C₄	—	2	1	no	—
170	C₃²×C₄	—	2	1	no	—
171	C₃²×C₄	—	2	1	no	—
172	C₃²×C₄	—	2	1	no	—
173	C₃²×C₄	—	2	1	no	—
174	C₃²×C₄	#104	1	2	yes	—
175	C₃×Dic₃	—	4	2	no	—
176	C₂×#41	—	3	2	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₄⋊Q₈	—	3	1	no	—
179	#127	—	3	1	no	—
180	#124	—	3	2	no	—
181	#121	—	6	1	no	—
182	#121	—	6	1	no	—
183	C₄×D₄	—	6	1	no	—
184	C₄×D₄	—	6	1	no	—
185	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
186	C₄⋊C₈	—	6	1	no	—
187	C₂²⋊C₈	—	6	1	no	—
188	#97	—	2	1	no	—
189	#97	—	3	2	no	—
190	#97	C₂×C₃×D₃	1	2	yes	—
191	C₂³×C₃	—	2	1	no	—
192	C₂³×C₃	—	6	2	no	—
193	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₂³×C₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²×D₃	—	2	2	no	—
196	C₃×Q₈	—	12	2	no	—
197	C₃×D₄	—	12	2	no	—
198	C₃×D₄	—	12	2	no	—
199	C₃×D₄	—	12	4	no	—
200	C₃×D₄	—	12	2	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	6	2	no	—
217	C₂×C₃×C₄	—	12	4	no	—
218	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
219	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
220	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
230	#67	—	4	4	no	—
231	C₂×Dic₃	—	2	2	no	—
232	D₁₂	—	4	2	no	—
233	D₁₂	—	4	2	no	—
234	D₁₂	—	4	2	no	—
235	C₄×D₃	—	4	4	no	—
236	Dic₆	—	4	2	no	—
237	C₃×C₈	—	2	8	no	—
238	C₃×C₈	—	2	4	no	—
239	C₃×C₈	—	2	4	no	—
240	C₃×C₈	—	12	2	no	—
241	C₃⋊C₈	—	4	2	no	—
242	C₂×C₃²	—	2	2	no	—
243	C₂×C₃²	C₂×#100	1	1	yes	characteristic
244	C₂×C₃²	C₂×#104	1	1	yes	characteristic
245	C₂×C₃²	#421	1	1	yes	characteristic
246	C₃×D₃	—	8	2	no	—
247	#41	—	12	4	no	—
248	C₂×Q₈	—	3	2	no	—
249	C₂×D₄	—	3	2	no	—
250	C₂×D₄	—	6	2	no	—
251	C₂×D₄	—	6	1	no	—
252	C₂×D₄	—	6	1	no	—
253	C₂²×C₄	—	6	2	no	—
254	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
255	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
256	C₂²×C₄	C₃×#67	1	1	yes	characteristic
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	6	2	no	—
260	C₄⋊C₄	—	6	1	no	—
261	C₄⋊C₄	—	6	1	no	—
262	C₂²⋊C₄	—	6	2	no	—
263	C₂²⋊C₄	—	6	2	no	—
264	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
265	C₄²	C₃×D₁₂	1	1	yes	characteristic
266	C₄²	C₃×#67	1	1	yes	characteristic
267	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	2	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	2	no	—
276	C₂²×C₃	—	12	2	no	—
277	C₂²×C₃	—	12	2	no	—
278	C₂²×C₃	—	12	2	no	—
279	C₂²×C₃	#823	1	2	yes	—
280	C₂²×C₃	C₃×#100	1	2	yes	—
281	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
282	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×D₃	—	4	2	no	—
284	C₂×D₃	—	4	2	no	—
285	C₂×D₃	—	4	2	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	1	no	—
288	C₃×C₄	—	2	1	no	—
289	C₃×C₄	—	2	1	no	—
290	C₃×C₄	—	2	1	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	2	1	no	—
297	C₃×C₄	—	2	2	no	—
298	C₃×C₄	—	4	1	no	—
299	C₃×C₄	—	4	1	no	—
300	C₃×C₄	—	4	1	no	—
301	C₃×C₄	—	4	1	no	—
302	C₃×C₄	—	12	2	no	—
303	C₃×C₄	#805	1	2	yes	—
304	C₃×C₄	C₃×#104	1	2	yes	—
305	Dic₃	—	4	2	no	—
306	C₃²	#1530	1	1	yes	characteristic, complemented, Sylow
307	C₂³	—	6	2	no	—
308	C₂³	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
309	Q₈	—	12	2	no	—
310	D₄	—	12	2	no	—
311	D₄	—	12	2	no	—
312	D₄	—	12	4	no	—
313	D₄	—	12	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	6	2	no	—
318	C₂×C₄	—	12	4	no	—
319	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
324	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	12	2	no	—
327	C₂×C₃	—	2	4	no	—
328	C₂×C₃	—	2	2	no	—
329	C₂×C₃	—	2	2	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	24	2	no	—
334	C₂×C₃	C₂×#805	1	1	yes	characteristic
335	C₂×C₃	C₂×#823	1	1	yes	characteristic
336	C₂×C₃	#5139	1	1	yes	characteristic
337	C₂×C₃	C₂×C₃×#100	1	1	yes	characteristic
338	C₂×C₃	C₂×C₃×#104	1	1	yes	characteristic
339	C₂×C₃	C₃×#421	1	1	yes	characteristic
340	D₃	—	8	2	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	12	2	no	—
344	C₂²	—	12	2	no	—
345	C₂²	—	12	2	no	—
346	C₂²	C₃×#823	1	2	yes	—
347	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	12	2	no	—
354	C₄	C₃×#805	1	2	yes	—
355	C₃	—	2	1	no	—
356	C₃	#69798	1	1	yes	characteristic, direct product factor
357	C₃	C₃×#1530	1	1	yes	characteristic, complemented
358	C₂	—	2	2	no	—
359	C₂	—	24	2	no	—
360	C₂	C₂×C₃×#805	1	1	yes	characteristic
361	C₂	C₂×C₃×#823	1	1	yes	characteristic
362	C₂	C₃×#5139	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₃×#5477	C₂
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#1530
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×#97	C₂	semidirect
3. C₃×#4926	C₂	semidirect
4. C₃×#4464	C₂	semidirect
6. #69798	C₃	direct (N × H)
26. C₂×C₃×#97	D₃	semidirect
27. C₂×C₃×#97	C₂×C₃	semidirect
28. #4926	C₂×C₃	semidirect
30. #4464	C₂×C₃	semidirect
109. C₂×#97	C₃×D₃	semidirect
306. C₃²	#1530	semidirect
356. C₃	#69798	direct (N × H)
357. C₃	C₃×#1530	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5477	2
C₂×C₃²×#97	2
C₃×#4926	2
C₃×#4464	2
C₃×#1530	3
#69798	3

Subgroups — C3×#69798 #11775867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69798 #11775867