Subgroups — C₃×#78463 #11873476

Order 1152.

881 conjugacy classes of subgroups.

The subgroup lattice has 347 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	C₃×#5470	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5113	C₂	1	4	yes	maximal, complemented
5	C₃×#4697	C₂	1	4	yes	maximal, complemented
6	C₃×#4694	C₂	1	2	yes	maximal, complemented
7	C₃×#4681	C₂	1	2	yes	maximal, complemented
8	C₃×#3131	—	3	1	no	maximal
9	#78463	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#235	C₂²	1	2	yes	—
11	C₂²×C₃²×C₈	C₂²	1	2	yes	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#930	—	2	2	no	—
14	C₃×#926	C₂²	1	2	yes	—
15	C₃×#924	—	2	2	no	—
16	C₃×#875	C₂²	1	2	yes	—
17	C₃×#874	C₂²	1	1	yes	characteristic
18	C₃×C₄×D₁₂	—	2	2	no	—
19	C₃×#870	C₂²	1	2	yes	—
20	C₃×#869	C₂²	1	1	yes	characteristic
21	C₃×#869	C₂²	1	2	yes	—
22	C₃×C₄×Dic₆	—	2	2	no	—
23	C₃²×C₄×C₈	C₂²	1	2	yes	—
24	C₃²×C₄×C₈	C₂²	1	2	yes	—
25	C₃×#821	C₂²	1	8	yes	—
26	C₃×#816	C₂²	1	8	yes	—
27	C₃×#533	—	3	1	no	—
28	C₃×#532	—	3	1	no	—
29	#5477	C₂×C₃	1	1	yes	characteristic
30	#5470	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#488	—	3	4	no	—
32	C₃×#487	—	3	2	no	—
33	C₃×#486	—	3	2	no	—
34	C₃×#416	—	3	4	no	—
35	C₂×C₃×C₄×C₈	—	2	1	no	—
36	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic, complemented
37	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
38	#5113	C₂×C₃	1	4	yes	complemented
39	#4697	C₂×C₃	1	4	yes	complemented
40	#4694	C₂×C₃	1	2	yes	complemented
41	#4681	C₂×C₃	1	2	yes	complemented
42	C₂²×C₃²×C₄	D₄	1	2	yes	—
43	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
44	C₂×C₃×#67	—	2	2	no	—
45	C₃×#235	—	2	8	no	—
46	C₂×C₃×D₁₂	—	2	2	no	—
47	C₂×C₃×D₁₂	C₂³	1	2	yes	—
48	C₂×C₃×C₄×D₃	—	2	2	no	—
49	C₂×C₃×Dic₆	—	2	2	no	—
50	C₂×C₃×Dic₆	C₂³	1	2	yes	—
51	C₂×C₃²×C₈	—	2	4	no	—
52	C₂×C₃²×C₈	C₂³	1	4	yes	—
53	C₃²×C₄²	D₄	1	2	yes	—
54	C₃²×C₄²	C₂³	1	1	yes	characteristic
55	C₃²×C₄²	C₂³	1	1	yes	characteristic
56	C₃×#217	—	2	2	no	—
57	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
58	C₃×C₄⋊Dic₃	—	2	2	no	—
59	C₃×C₄⋊Dic₃	—	2	2	no	—
60	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
61	C₃×Dic₃⋊C₄	—	2	4	no	—
62	C₃×C₄×Dic₃	—	2	2	no	—
63	#3131	—	3	1	no	Sylow
64	C₂×C₃×#41	—	3	2	no	—
65	C₂×#235	C₂²×C₃	1	2	yes	—
66	C₂²×C₃×C₈	—	2	2	no	—
67	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
68	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
69	C₃×C₄⋊Q₈	—	3	2	no	—
70	C₃×C₄⋊Q₈	—	3	1	no	—
71	C₃×#127	—	3	1	no	—
72	C₃×#125	—	3	2	no	—
73	C₃×#124	—	3	2	no	—
74	C₃×C₂²⋊Q₈	—	6	2	no	—
75	C₃×#121	—	6	2	no	—
76	C₃×C₄×Q₈	—	6	2	no	—
77	C₃×C₄×D₄	—	6	2	no	—
78	C₃×#117	—	3	2	no	—
79	C₂×C₃×C₄²	—	2	1	no	—
80	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
82	#930	—	2	2	no	—
83	#926	C₂²×C₃	1	2	yes	—
84	#924	—	2	2	no	—
85	#875	C₂²×C₃	1	2	yes	—
86	#874	C₂²×C₃	1	1	yes	characteristic
87	C₄×D₁₂	—	2	2	no	—
88	#870	C₂²×C₃	1	2	yes	—
89	#869	C₂²×C₃	1	1	yes	characteristic
90	#869	C₂²×C₃	1	2	yes	—
91	C₄×Dic₆	—	2	2	no	—
92	C₃×#103	—	3	8	no	—
93	C₃×#102	—	3	8	no	—
94	C₃×C₄×C₈	—	2	2	no	—
95	C₃×C₄×C₈	—	2	2	no	—
96	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
97	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
98	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
99	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
100	#821	C₂²×C₃	1	8	yes	—
101	#816	C₂²×C₃	1	8	yes	—
102	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₃×D₃	—	2	2	no	—
104	C₂×C₃²×C₄	—	2	4	no	—
105	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
106	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
107	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃²×C₄	#41	1	4	yes	—
109	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
110	C₃×#67	—	4	4	no	—
111	C₂×C₃×Dic₃	—	2	4	no	—
112	C₂×C₃×Dic₃	—	2	2	no	—
113	C₃×D₁₂	—	2	4	no	—
114	C₃×D₁₂	—	4	4	no	—
115	C₃×C₄×D₃	—	4	4	no	—
116	C₃×Dic₆	—	2	4	no	—
117	C₃×Dic₆	—	4	4	no	—
118	C₃²×C₈	—	2	4	no	—
119	#533	—	3	1	no	—
120	#532	—	3	1	no	—
121	#488	—	3	4	no	—
122	#487	—	3	2	no	—
123	#486	—	3	2	no	—
124	#416	—	3	4	no	—
125	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic, complemented
126	C₃×#41	—	6	8	no	—
127	C₂×C₃×Q₈	—	3	2	no	—
128	C₂×C₃×Q₈	—	6	2	no	—
129	C₂×C₃×D₄	—	3	2	no	—
130	C₂×C₃×D₄	—	6	2	no	—
131	C₂×C₃×D₄	—	6	2	no	—
132	C₂²×C₃×C₄	—	2	2	no	—
133	C₂²×C₃×C₄	—	2	1	no	—
134	C₂²×C₃×C₄	—	6	2	no	—
135	C₂²×C₃×C₄	D₁₂	1	2	yes	—
136	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
137	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₂×#67	—	2	2	no	—
140	#235	—	2	8	no	—
141	C₂×D₁₂	—	2	2	no	—
142	C₂×D₁₂	C₂³×C₃	1	2	yes	—
143	C₂×C₄×D₃	—	2	2	no	—
144	C₂×Dic₆	—	2	2	no	—
145	C₂×Dic₆	C₂³×C₃	1	2	yes	—
146	C₂×C₃×C₈	—	2	8	no	—
147	C₂×C₃×C₈	—	2	4	no	—
148	C₂×C₃×C₈	—	2	4	no	—
149	C₂×C₃×C₈	—	2	4	no	—
150	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
151	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
152	C₃×C₄⋊C₄	—	3	4	no	—
153	C₃×C₄⋊C₄	—	6	4	no	—
154	C₃×C₄⋊C₄	—	6	2	no	—
155	C₃×C₄⋊C₄	—	6	2	no	—
156	C₃×C₂²⋊C₄	—	6	2	no	—
157	C₃×C₂²⋊C₄	—	6	4	no	—
158	C₃×C₄²	—	2	2	no	—
159	C₃×C₄²	—	2	1	no	—
160	C₃×C₄²	—	2	1	no	—
161	C₃×C₄²	—	6	2	no	—
162	C₃×C₄²	D₁₂	1	2	yes	—
163	C₃×C₄²	C₃×D₄	1	2	yes	—
164	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
165	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
166	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
167	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
168	#217	—	2	2	no	—
169	(C₂×D₃)⋊C₄	—	2	4	no	—
170	C₄⋊Dic₃	—	2	2	no	—
171	C₄⋊Dic₃	—	2	2	no	—
172	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
173	Dic₃⋊C₄	—	2	4	no	—
174	C₄×Dic₃	—	2	2	no	—
175	C₂²×C₃²	—	2	2	no	—
176	C₂²×C₃²	#124	1	2	yes	—
177	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₃×D₃	—	4	2	no	—
179	C₂×C₃×D₃	—	4	4	no	—
180	C₃²×C₄	—	2	2	no	—
181	C₃²×C₄	#124	1	2	yes	—
182	C₃²×C₄	#135	1	4	yes	—
183	C₃²×C₄	C₂×#41	1	2	yes	—
184	C₃×Dic₃	—	4	2	no	—
185	C₃×Dic₃	—	4	4	no	—
186	C₂×#41	—	3	2	no	—
187	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
188	C₄⋊Q₈	—	3	2	no	—
189	C₄⋊Q₈	—	3	1	no	—
190	#127	—	3	1	no	—
191	#125	—	3	2	no	—
192	#124	—	3	2	no	—
193	C₂²⋊Q₈	—	6	2	no	—
194	#121	—	6	2	no	—
195	C₄×Q₈	—	6	2	no	—
196	C₄×D₄	—	6	2	no	—
197	#117	—	3	2	no	—
198	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
199	#103	—	3	8	no	—
200	#102	—	3	8	no	—
201	C₄×C₈	C₂×C₃×D₃	1	2	yes	—
202	C₄×C₈	C₂×C₃×D₃	1	2	yes	—
203	C₂³×C₃	—	2	1	no	—
204	C₂³×C₃	—	6	2	no	—
205	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
206	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂²×D₃	—	2	2	no	—
208	C₃×Q₈	—	6	4	no	—
209	C₃×Q₈	—	12	4	no	—
210	C₃×D₄	—	6	4	no	—
211	C₃×D₄	—	12	4	no	—
212	C₃×D₄	—	12	4	no	—
213	C₂×C₃×C₄	—	2	8	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	4	no	—
221	C₂×C₃×C₄	—	6	4	no	—
222	C₂×C₃×C₄	—	6	2	no	—
223	C₂×C₃×C₄	—	12	4	no	—
224	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
225	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
226	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
227	C₂×C₃×C₄	#235	1	4	yes	—
228	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
229	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
231	C₂×C₃×C₄	C₃×#41	1	4	yes	—
232	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
234	#67	—	4	4	no	—
235	C₂×Dic₃	—	2	4	no	—
236	C₂×Dic₃	—	2	2	no	—
237	D₁₂	—	2	4	no	—
238	D₁₂	—	4	4	no	—
239	C₄×D₃	—	4	4	no	—
240	Dic₆	—	2	4	no	—
241	Dic₆	—	4	4	no	—
242	C₃×C₈	—	2	8	no	—
243	C₃×C₈	—	2	4	no	—
244	C₃×C₈	—	2	4	no	—
245	C₂×C₃²	—	2	2	no	—
246	C₂×C₃²	C₂×#124	1	1	yes	characteristic
247	C₂×C₃²	C₂×#135	1	2	yes	—
248	C₃×D₃	—	8	2	no	—
249	#41	—	6	8	no	—
250	C₂×Q₈	—	3	2	no	—
251	C₂×Q₈	—	6	2	no	—
252	C₂×D₄	—	3	2	no	—
253	C₂×D₄	—	6	2	no	—
254	C₂×D₄	—	6	2	no	—
255	C₂²×C₄	—	6	2	no	—
256	C₂²×C₄	C₃×D₁₂	1	2	yes	—
257	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	C₂²×C₃×D₃	1	4	yes	—
260	C₄⋊C₄	—	3	4	no	—
261	C₄⋊C₄	—	6	2	no	—
262	C₄⋊C₄	—	6	4	no	—
263	C₄⋊C₄	—	6	2	no	—
264	C₂²⋊C₄	—	6	2	no	—
265	C₂²⋊C₄	—	6	4	no	—
266	C₄²	—	6	2	no	—
267	C₄²	C₃×D₁₂	1	2	yes	—
268	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
269	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
270	C₂²×C₃	—	2	4	no	—
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	—	12	2	no	—
276	C₂²×C₃	—	12	4	no	—
277	C₂²×C₃	#875	1	2	yes	—
278	C₂²×C₃	C₃×#124	1	2	yes	—
279	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
280	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
281	C₂×D₃	—	4	4	no	—
282	C₂×D₃	—	4	2	no	—
283	C₃×C₄	—	2	4	no	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	2	4	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	12	2	no	—
290	C₃×C₄	—	12	4	no	—
291	C₃×C₄	#875	1	2	yes	—
292	C₃×C₄	#904	1	4	yes	—
293	C₃×C₄	C₃×#124	1	2	yes	—
294	C₃×C₄	C₃×#135	1	4	yes	—
295	C₃×C₄	C₂×#235	1	2	yes	—
296	C₃×C₄	C₂×C₃×#41	1	2	yes	—
297	Dic₃	—	4	4	no	—
298	Dic₃	—	4	2	no	—
299	C₃²	#3131	1	1	yes	characteristic, complemented, Sylow
300	C₂³	—	6	2	no	—
301	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
302	Q₈	—	6	4	no	—
303	Q₈	—	12	4	no	—
304	D₄	—	6	4	no	—
305	D₄	—	12	4	no	—
306	D₄	—	12	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	6	4	no	—
309	C₂×C₄	—	6	2	no	—
310	C₂×C₄	—	12	4	no	—
311	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
312	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
313	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
314	C₂×C₄	C₃×#235	1	4	yes	—
315	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
316	C₈	—	2	4	no	—
317	C₂×C₃	—	2	4	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	2	no	—
322	C₂×C₃	—	24	2	no	—
323	C₂×C₃	C₂×#875	1	1	yes	characteristic
324	C₂×C₃	C₂×#904	1	2	yes	—
325	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
326	C₂×C₃	C₂×C₃×#135	1	2	yes	—
327	D₃	—	8	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	12	2	no	—
330	C₂²	—	12	4	no	—
331	C₂²	C₃×#875	1	2	yes	—
332	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	12	2	no	—
335	C₄	—	12	4	no	—
336	C₄	C₃×#875	1	2	yes	—
337	C₄	C₃×#904	1	4	yes	—
338	C₄	C₂×C₃×#235	1	2	yes	—
339	C₃	—	2	1	no	—
340	C₃	#78463	1	1	yes	characteristic, direct product factor
341	C₃	C₃×#3131	1	1	yes	characteristic, complemented
342	C₂	—	2	2	no	—
343	C₂	—	24	2	no	—
344	C₂	C₂×C₃×#875	1	1	yes	characteristic
345	C₂	C₂×C₃×#904	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₃×#5477	C₂
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#3131
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×#5470	C₂	semidirect
3. C₂×C₃²×C₄×C₈	C₂	semidirect
4. C₃×#5113	C₂	semidirect
5. C₃×#4697	C₂	semidirect
6. C₃×#4694	C₂	semidirect
7. C₃×#4681	C₂	semidirect
9. #78463	C₃	direct (N × H)
30. #5470	C₂×C₃	semidirect
36. C₂×C₃×C₄×C₈	D₃	semidirect
37. C₂×C₃×C₄×C₈	C₂×C₃	semidirect
38. #5113	C₂×C₃	semidirect
39. #4697	C₂×C₃	semidirect
40. #4694	C₂×C₃	semidirect
41. #4681	C₂×C₃	semidirect
125. C₂×C₄×C₈	C₃×D₃	semidirect
299. C₃²	#3131	semidirect
340. C₃	#78463	direct (N × H)
341. C₃	C₃×#3131	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
C₃×#5470	2
C₂×C₃²×C₄×C₈	2
C₃×#5113	2
C₃×#4697	2
C₃×#4694	2
C₃×#4681	2
C₃×#3131	3
#78463	3

Subgroups — C3×#78463 #11873476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#78463 #11873476