Subgroups — C₃×#71970 #11785347

Order 1152.

841 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#5212	C₂	1	2	yes	maximal
3	C₂×C₃×#817	C₂	1	2	yes	maximal, complemented
4	C₃×#4551	C₂	1	2	yes	maximal, complemented
5	C₃×#1807	—	3	1	no	maximal
6	#71970	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂²×C₃²×C₈	—	2	4	no	—
9	C₂²×C₃²×C₈	C₄	1	4	yes	complemented
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×#217	—	2	2	no	—
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
14	C₃×C₃⋊#95	—	2	2	no	—
15	C₃×#817	—	2	4	no	—
16	C₂³×C₃×C₈	—	2	1	no	—
17	C₂³×C₃×C₈	D₃	1	1	yes	characteristic
18	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#106	—	3	2	no	—
20	C₃×#380	—	3	2	no	—
21	#5212	C₂×C₃	1	2	yes	—
22	C₂×#817	C₂×C₃	1	2	yes	complemented
23	C₃×#340	—	3	2	no	—
24	#4551	C₂×C₃	1	2	yes	complemented
25	C₂⁴×C₃²	D₄	1	1	yes	characteristic
26	C₂²×C₃²×C₄	—	2	2	no	—
27	C₂²×C₃²×C₄	—	2	2	no	—
28	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	Q₈	1	2	yes	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	—	2	4	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	—	2	8	no	—
35	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	D₄	1	4	yes	—
38	C₃×#217	—	4	4	no	—
39	C₃×C₄⋊Dic₃	—	2	4	no	—
40	C₃×C₄⋊Dic₃	—	2	4	no	—
41	#1807	—	3	1	no	Sylow
42	C₂³×C₃×C₄	—	2	1	no	—
43	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
44	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₃×C₈	—	2	8	no	—
46	C₂²×C₃×C₈	—	2	4	no	—
47	C₂²×C₃×C₈	—	2	4	no	—
48	C₂²×C₃×C₈	—	2	4	no	—
49	C₂²×C₃×C₈	—	2	1	no	—
50	C₂²×C₃×C₈	—	2	1	no	—
51	C₂²×C₃×C₈	Dic₃	1	4	yes	complemented
52	C₂²×C₃×C₈	C₃×C₄	1	4	yes	complemented
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₂×C₃×C₄⋊C₄	—	3	4	no	—
58	C₂×C₃×C₂²⋊C₄	—	6	2	no	—
59	C₂×#217	—	2	2	no	—
60	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
61	C₃×#106	—	6	4	no	—
62	C₃×#95	—	6	2	no	—
63	C₃⋊#95	—	2	2	no	—
64	#817	—	2	4	no	—
65	C₂³×C₃²	—	2	2	no	—
66	C₂³×C₃²	—	2	2	no	—
67	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
68	C₂³×C₃²	SD₁₆	1	4	yes	—
69	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	—	2	2	no	—
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	—	2	4	no	—
74	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
77	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃²×C₄	#41	1	2	yes	—
79	C₂×C₃×Dic₃	—	4	4	no	—
80	C₂×C₃×Dic₃	—	4	4	no	—
81	C₂×C₃×Dic₃	—	4	4	no	—
82	C₃²×C₈	—	2	2	no	—
83	C₃²×C₈	C₂²⋊C₄	1	4	yes	complemented
84	C₂³×C₈	C₃×D₃	1	1	yes	characteristic
85	C₂×#106	—	3	2	no	—
86	#380	—	3	2	no	—
87	#340	—	3	2	no	—
88	C₂⁴×C₃	—	2	1	no	—
89	C₂⁴×C₃	D₁₂	1	1	yes	characteristic
90	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	—	2	4	no	—
92	C₂²×C₃×C₄	—	2	4	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	—	2	2	no	—
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	1	no	—
99	C₂²×C₃×C₄	—	2	2	no	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	—	6	4	no	—
102	C₂²×C₃×C₄	Dic₆	1	2	yes	—
103	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
105	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
106	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
108	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
110	C₂²×Dic₃	—	2	4	no	—
111	C₂×C₃×C₈	—	2	4	no	—
112	C₂×C₃×C₈	—	2	16	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	8	no	—
116	C₂×C₃×C₈	—	2	8	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	4	no	—
120	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
121	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
122	C₂×C₃×C₈	#67	1	4	yes	—
123	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
125	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
126	C₃×C₄⋊C₄	—	6	4	no	—
127	C₃×C₄⋊C₄	—	6	4	no	—
128	C₃×C₂²⋊C₄	—	12	4	no	—
129	#217	—	4	4	no	—
130	C₄⋊Dic₃	—	2	4	no	—
131	C₄⋊Dic₃	—	2	4	no	—
132	C₂²×C₃²	—	2	2	no	—
133	C₂²×C₃²	—	2	2	no	—
134	C₂²×C₃²	—	2	8	no	—
135	C₂²×C₃²	#106	1	4	yes	complemented
136	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂²×C₃²	#121	1	2	yes	—
138	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
139	C₂²×C₃²	#135	1	2	yes	—
140	C₃²×C₄	—	2	2	no	—
141	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₃²×C₄	#117	1	1	yes	characteristic
143	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
144	C₃×Dic₃	—	8	4	no	—
145	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	C₃×Dic₃	1	4	yes	complemented
148	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
149	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	3	4	no	—
151	C₂×C₂²⋊C₄	—	6	2	no	—
152	#106	—	6	4	no	—
153	#95	—	6	2	no	—
154	C₂³×C₃	—	2	4	no	—
155	C₂³×C₃	—	2	4	no	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	—	2	2	no	—
158	C₂³×C₃	—	2	2	no	—
159	C₂³×C₃	—	2	2	no	—
160	C₂³×C₃	—	2	4	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	—	2	1	no	—
163	C₂³×C₃	#204	1	4	yes	—
164	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
165	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
166	C₂³×C₃	C₃×SD₁₆	1	4	yes	—
167	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
168	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	8	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	12	4	no	—
187	C₂×C₃×C₄	—	12	4	no	—
188	C₂×C₃×C₄	—	12	4	no	—
189	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
190	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
191	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
192	C₂×C₃×C₄	#235	1	2	yes	—
193	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×#67	1	2	yes	—
195	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
198	C₂×C₃×C₄	C₃×#41	1	2	yes	—
199	C₂×Dic₃	—	4	4	no	—
200	C₂×Dic₃	—	4	4	no	—
201	C₂×Dic₃	—	4	4	no	—
202	C₃×C₈	—	2	4	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	2	4	no	—
206	C₃×C₈	#217	1	4	yes	complemented
207	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	complemented
208	C₂×C₃²	—	2	4	no	—
209	C₂×C₃²	#380	1	1	yes	characteristic
210	C₂×C₃²	C₂×#106	1	1	yes	characteristic
211	C₂×C₃²	#427	1	1	yes	characteristic
212	C₂×C₃²	#465	1	4	yes	—
213	C₂⁴	C₃×D₁₂	1	1	yes	characteristic
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	6	4	no	—
217	C₂²×C₄	C₃×Dic₆	1	2	yes	—
218	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
220	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	8	no	—
223	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
224	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
225	C₂×C₈	C₃×#67	1	4	yes	—
226	C₄⋊C₄	—	6	4	no	—
227	C₄⋊C₄	—	6	4	no	—
228	C₂²⋊C₄	—	12	4	no	—
229	C₂²×C₃	—	2	4	no	—
230	C₂²×C₃	—	2	4	no	—
231	C₂²×C₃	—	2	16	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	8	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	8	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	4	no	—
243	C₂²×C₃	#817	1	4	yes	complemented
244	C₂²×C₃	C₃×#106	1	4	yes	complemented
245	C₂²×C₃	C₂×#204	1	2	yes	—
246	C₂²×C₃	#904	1	2	yes	—
247	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
248	C₂²×C₃	#930	1	2	yes	—
249	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
250	C₂²×C₃	C₃×#121	1	2	yes	—
251	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
252	C₂²×C₃	C₃×#135	1	2	yes	—
253	C₃×C₄	—	2	4	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	2	2	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	24	4	no	—
260	C₃×C₄	#924	1	2	yes	—
261	C₃×C₄	#926	1	1	yes	characteristic
262	C₃×C₄	C₂×#217	1	1	yes	characteristic
263	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
264	C₃×C₄	C₃×#117	1	1	yes	characteristic
265	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
266	Dic₃	—	8	4	no	—
267	C₃²	#1807	1	1	yes	characteristic, complemented, Sylow
268	C₂³	—	2	2	no	—
269	C₂³	—	2	2	no	—
270	C₂³	C₃×#204	1	4	yes	—
271	C₂³	C₃×C₄⋊Dic₃	1	2	yes	—
272	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	12	4	no	—
278	C₂×C₄	—	12	4	no	—
279	C₂×C₄	—	12	4	no	—
280	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
281	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
282	C₂×C₄	C₃×#235	1	2	yes	—
283	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
284	C₂×C₄	C₂×C₃×#67	1	2	yes	—
285	C₈	—	2	2	no	—
286	C₈	C₃×#217	1	4	yes	complemented
287	C₂×C₃	—	2	8	no	—
288	C₂×C₃	—	2	4	no	—
289	C₂×C₃	—	2	4	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	4	no	—
294	C₂×C₃	C₂×#817	1	1	yes	characteristic
295	C₂×C₃	#5106	1	1	yes	characteristic
296	C₂×C₃	#5114	1	4	yes	—
297	C₂×C₃	#5212	1	1	yes	characteristic
298	C₂×C₃	C₃×#380	1	1	yes	characteristic
299	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
300	C₂×C₃	C₃×#427	1	1	yes	characteristic
301	C₂×C₃	C₃×#465	1	4	yes	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	8	no	—
305	C₂²	C₃×#817	1	4	yes	complemented
306	C₂²	C₂×C₃×#204	1	2	yes	—
307	C₂²	C₃×#904	1	2	yes	—
308	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
309	C₂²	C₃×#930	1	2	yes	—
310	C₄	—	2	2	no	—
311	C₄	—	24	4	no	—
312	C₄	C₃×#924	1	2	yes	—
313	C₄	C₃×#926	1	1	yes	characteristic
314	C₄	C₂×C₃×#217	1	1	yes	characteristic
315	C₃	—	2	1	no	—
316	C₃	#71970	1	1	yes	characteristic, direct product factor
317	C₃	C₃×#1807	1	1	yes	characteristic, complemented
318	C₂	—	2	4	no	—
319	C₂	C₂×C₃×#817	1	1	yes	characteristic
320	C₂	C₃×#5106	1	1	yes	characteristic
321	C₂	C₃×#5114	1	4	yes	—
322	C₂	C₃×#5212	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂³×C₃²×C₈	C₂
socle	C₂³×C₃²	C₂×D₄
prime radical	C₂⁴×C₃²	D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂³×C₃×C₈	D₃
hypocenter	C₃	C₃×#1807
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#817	C₂	semidirect
4. C₃×#4551	C₂	semidirect
6. #71970	C₃	direct (N × H)
9. C₂²×C₃²×C₈	C₄	semidirect
22. C₂×#817	C₂×C₃	semidirect
24. #4551	C₂×C₃	semidirect
51. C₂²×C₃×C₈	Dic₃	semidirect
52. C₂²×C₃×C₈	C₃×C₄	semidirect
83. C₃²×C₈	C₂²⋊C₄	semidirect
135. C₂²×C₃²	#106	semidirect
147. C₂²×C₈	C₃×Dic₃	semidirect
206. C₃×C₈	#217	semidirect
207. C₃×C₈	C₃×C₂²⋊C₄	semidirect
243. C₂²×C₃	#817	semidirect
244. C₂²×C₃	C₃×#106	semidirect
267. C₃²	#1807	semidirect
286. C₈	C₃×#217	semidirect
305. C₂²	C₃×#817	semidirect
316. C₃	#71970	direct (N × H)
317. C₃	C₃×#1807	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃²×C₈	2
C₃×#5212	2
C₂×C₃×#817	2
C₃×#4551	2
C₃×#1807	3
#71970	3

Subgroups — C3×#71970 #11785347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71970 #11785347