Subgroups — C₃×#71790 #11791871

Order 1152.

665 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×#4924	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#208	C₂	1	1	yes	maximal, characteristic
4	C₃×#4530	C₂	1	4	yes	maximal
5	C₃×#1997	—	3	1	no	maximal
6	#71790	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×Q₈	C₄	1	2	yes	—
8	C₃²×C₄⋊Q₈	—	2	1	no	—
9	C₃²×C₄⋊Q₈	—	2	1	no	—
10	C₃²×C₄⋊Q₈	—	2	1	no	—
11	C₃²×C₄⋊Q₈	—	2	1	no	—
12	C₂×C₃²×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
14	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
15	C₂×C₃×#208	C₂²	1	2	yes	—
16	C₂×C₃×#208	C₂²	1	2	yes	—
17	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₃×#803	—	2	1	no	—
20	C₃×C₄×C₃⋊C₈	—	2	1	no	—
21	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
22	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
23	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#423	—	3	1	no	—
25	C₃×C₄×M₁₆	—	3	1	no	—
26	#4924	C₂×C₃	1	1	yes	characteristic
27	C₄×#208	C₂×C₃	1	1	yes	characteristic
28	C₃×#343	—	3	4	no	—
29	#4530	C₂×C₃	1	4	yes	—
30	C₂×C₃²×Q₈	—	2	2	no	—
31	C₂×C₃²×Q₈	—	2	2	no	—
32	C₂×C₃²×Q₈	—	4	4	no	—
33	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃²×C₄	C₂×C₄	1	4	yes	—
36	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
37	C₃²×C₄⋊C₄	—	2	2	no	—
38	C₃²×C₄⋊C₄	—	2	2	no	—
39	C₃²×C₄⋊C₄	—	2	2	no	—
40	C₃²×C₄⋊C₄	—	2	2	no	—
41	C₃²×C₄²	D₄	1	1	yes	characteristic
42	C₃²×C₄²	D₄	1	1	yes	characteristic
43	C₃²×C₄²	D₄	1	1	yes	characteristic
44	C₃²×C₄²	D₄	1	1	yes	characteristic
45	C₃×#208	—	2	2	no	—
46	C₃×#208	—	2	4	no	—
47	C₃×#208	Q₈	1	4	yes	complemented
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	C₂×C₃×C₃⋊C₈	—	2	2	no	—
50	#1997	—	3	1	no	Sylow
51	C₂²×C₃×Q₈	—	2	2	no	—
52	C₂²×C₃×Q₈	Dic₃	1	2	yes	—
53	C₂²×C₃×Q₈	C₃×C₄	1	2	yes	—
54	C₂×C₃×M₁₆	—	3	2	no	—
55	C₂×C₃×M₁₆	—	3	2	no	—
56	C₃×C₄⋊Q₈	—	2	2	no	—
57	C₃×C₄⋊Q₈	—	2	2	no	—
58	C₃×C₄⋊Q₈	—	2	2	no	—
59	C₃×C₄⋊Q₈	—	2	2	no	—
60	C₃×C₄⋊Q₈	—	2	1	no	—
61	C₃×C₄⋊Q₈	—	2	1	no	—
62	C₃×C₄⋊Q₈	—	2	1	no	—
63	C₃×C₄⋊Q₈	—	2	1	no	—
64	C₃×C₄⋊Q₈	—	2	1	no	—
65	C₃×C₄⋊Q₈	—	2	1	no	—
66	C₃×C₄⋊Q₈	—	2	1	no	—
67	C₃×C₄⋊Q₈	—	2	1	no	—
68	C₂×C₃×C₄⋊C₄	—	2	2	no	—
69	C₂×C₃×C₄⋊C₄	—	2	2	no	—
70	C₂×C₃×C₄⋊C₄	Dic₃	1	2	yes	—
71	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
72	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
73	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
74	C₂×C₃×C₄²	—	2	1	no	—
75	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
77	C₂×#208	C₂²×C₃	1	2	yes	—
78	C₂×#208	C₂²×C₃	1	2	yes	—
79	C₃×C₄⋊C₈	—	6	1	no	—
80	C₃×C₄⋊C₈	—	6	1	no	—
81	C₃×#97	—	6	1	no	—
82	C₃×C₄×C₈	—	6	1	no	—
83	C₄⋊(C₃⋊C₈)	—	2	1	no	—
84	C₄⋊(C₃⋊C₈)	—	2	1	no	—
85	#803	—	2	1	no	—
86	C₄×C₃⋊C₈	—	2	1	no	—
87	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
88	C₃²×Q₈	—	4	4	no	—
89	C₃²×Q₈	—	4	4	no	—
90	C₂×C₃²×C₄	—	2	2	no	—
91	C₂×C₃²×C₄	—	2	2	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	—	4	4	no	—
94	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
101	C₂×C₃²×C₄	#41	1	2	yes	—
102	C₃×C₃⋊C₈	—	2	4	no	—
103	C₃×C₃⋊C₈	—	4	2	no	—
104	C₂×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
105	#423	—	3	1	no	—
106	C₄×M₁₆	—	3	1	no	—
107	#343	—	3	4	no	—
108	C₂×C₃×Q₈	—	2	4	no	—
109	C₂×C₃×Q₈	—	2	4	no	—
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×Q₈	—	2	2	no	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₂×C₃×Q₈	—	4	4	no	—
115	C₂×C₃×Q₈	—	4	4	no	—
116	C₂×C₃×Q₈	—	4	8	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	—	2	4	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
124	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×C₃×C₄	1	4	yes	—
127	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₃×M₁₆	—	3	4	no	—
130	C₃×M₁₆	—	6	2	no	—
131	C₃×M₁₆	—	6	4	no	—
132	C₂×C₃×C₈	—	6	2	no	—
133	C₂×C₃×C₈	—	6	2	no	—
134	C₃×C₄⋊C₄	—	2	4	no	—
135	C₃×C₄⋊C₄	—	2	4	no	—
136	C₃×C₄⋊C₄	—	2	4	no	—
137	C₃×C₄⋊C₄	—	2	4	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	—	2	1	no	—
149	C₃×C₄²	—	2	1	no	—
150	C₃×C₄²	#67	1	1	yes	characteristic
151	C₃×C₄²	#67	1	1	yes	characteristic
152	C₃×C₄²	#67	1	1	yes	characteristic
153	C₃×C₄²	#67	1	1	yes	characteristic
154	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
155	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
156	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
157	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
158	#208	—	2	2	no	—
159	#208	—	2	4	no	—
160	#208	C₃×Q₈	1	4	yes	complemented
161	C₂×C₃⋊C₈	—	2	2	no	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂²×C₃²	—	2	1	no	—
164	C₂²×C₃²	—	2	1	no	—
165	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₃²	C₄×Q₈	1	2	yes	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	—	2	1	no	—
169	C₃²×C₄	—	4	4	no	—
170	C₃²×C₄	#100	1	2	yes	—
171	C₃²×C₄	#101	1	2	yes	—
172	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
173	C₃²×C₄	#124	1	1	yes	characteristic
174	C₃²×C₄	C₄⋊Q₈	1	1	yes	characteristic
175	C₂²×Q₈	C₃×Dic₃	1	2	yes	—
176	C₂×M₁₆	—	3	2	no	—
177	C₂×M₁₆	—	3	2	no	—
178	C₄⋊Q₈	—	2	1	no	—
179	C₄⋊Q₈	—	2	1	no	—
180	C₄⋊Q₈	—	2	1	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₂×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
184	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
185	C₄⋊C₈	—	6	1	no	—
186	C₄⋊C₈	—	6	1	no	—
187	#97	—	6	1	no	—
188	C₄×C₈	—	6	1	no	—
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
191	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
192	C₃×Q₈	—	4	8	no	—
193	C₃×Q₈	—	4	4	no	—
194	C₃×Q₈	—	4	4	no	—
195	C₃×Q₈	—	4	8	no	—
196	C₃×Q₈	—	4	4	no	—
197	C₃×Q₈	—	4	4	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	8	no	—
205	C₂×C₃×C₄	—	2	4	no	—
206	C₂×C₃×C₄	—	2	4	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	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	4	8	no	—
216	C₂×C₃×C₄	—	4	4	no	—
217	C₂×C₃×C₄	—	4	4	no	—
218	C₂×C₃×C₄	#217	1	1	yes	characteristic
219	C₂×C₃×C₄	#217	1	1	yes	characteristic
220	C₂×C₃×C₄	#217	1	1	yes	characteristic
221	C₂×C₃×C₄	#217	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	C₃×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
227	C₂×C₃×C₄	#239	1	2	yes	—
228	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
233	C₂×C₃×C₄	C₃×#41	1	2	yes	—
234	C₃×C₈	—	6	4	no	—
235	C₃×C₈	—	12	2	no	—
236	C₃⋊C₈	—	2	4	no	—
237	C₃⋊C₈	—	4	2	no	—
238	C₂×C₃²	—	2	2	no	—
239	C₂×C₃²	#391	1	1	yes	characteristic
240	C₂×C₃²	C₂×#100	1	1	yes	characteristic
241	C₂×C₃²	C₂×#101	1	1	yes	characteristic
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	4	no	—
245	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₃×Dic₃	1	4	yes	—
248	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
249	M₁₆	—	3	4	no	—
250	M₁₆	—	6	2	no	—
251	M₁₆	—	6	4	no	—
252	C₂×C₈	—	6	2	no	—
253	C₂×C₈	—	6	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄²	C₃×#67	1	1	yes	characteristic
259	C₄²	C₃×#67	1	1	yes	characteristic
260	C₄²	C₃×#67	1	1	yes	characteristic
261	C₄²	C₃×#67	1	1	yes	characteristic
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	2	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	Q₈×Dic₃	1	2	yes	—
271	C₂²×C₃	C₂×#217	1	1	yes	characteristic
272	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
273	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	4	8	no	—
286	C₃×C₄	—	4	4	no	—
287	C₃×C₄	—	4	4	no	—
288	C₃×C₄	#833	1	2	yes	—
289	C₃×C₄	#836	1	2	yes	—
290	C₃×C₄	C₃×#100	1	2	yes	—
291	C₃×C₄	C₃×#101	1	2	yes	—
292	C₃×C₄	#944	1	1	yes	characteristic
293	C₃×C₄	#946	1	2	yes	—
294	C₃×C₄	#947	1	1	yes	characteristic
295	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
296	C₃×C₄	C₃×#124	1	1	yes	characteristic
297	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
298	C₃²	#1997	1	1	yes	characteristic, complemented, Sylow
299	C₂³	C₂²×C₃×Dic₃	1	1	yes	characteristic
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	C₃×#217	1	1	yes	characteristic
307	C₂×C₄	C₃×#217	1	1	yes	characteristic
308	C₂×C₄	C₃×#217	1	1	yes	characteristic
309	C₂×C₄	C₃×#217	1	1	yes	characteristic
310	C₂×C₄	C₃×D₃×Q₈	1	2	yes	—
311	C₂×C₄	C₃×#239	1	2	yes	—
312	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
313	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
314	C₈	—	6	4	no	—
315	C₈	—	12	2	no	—
316	C₂×C₃	—	2	4	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	C₂×#833	1	1	yes	characteristic
323	C₂×C₃	C₂×#836	1	1	yes	characteristic
324	C₂×C₃	#5229	1	1	yes	characteristic
325	C₂×C₃	C₃×#391	1	1	yes	characteristic
326	C₂×C₃	C₂×C₃×#100	1	1	yes	characteristic
327	C₂×C₃	C₂×C₃×#101	1	1	yes	characteristic
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	C₃×Q₈×Dic₃	1	2	yes	—
331	C₂²	C₂×C₃×#217	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	4	no	—
335	C₄	C₃×#833	1	2	yes	—
336	C₄	C₃×#836	1	2	yes	—
337	C₄	C₃×#944	1	1	yes	characteristic
338	C₄	C₃×#946	1	2	yes	—
339	C₄	C₃×#947	1	1	yes	characteristic
340	C₃	—	2	1	no	—
341	C₃	#71790	1	1	yes	characteristic, direct product factor
342	C₃	C₃×#1997	1	1	yes	characteristic, complemented
343	C₂	—	2	2	no	—
344	C₂	C₂×C₃×#833	1	1	yes	characteristic
345	C₂	C₂×C₃×#836	1	1	yes	characteristic
346	C₂	C₃×#5229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#217
commutator	C₂³×C₃	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄⋊Q₈	C₂
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#1997
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. #71790	C₃	direct (N × H)
47. C₃×#208	Q₈	semidirect
160. #208	C₃×Q₈	semidirect
298. C₃²	#1997	semidirect
341. C₃	#71790	direct (N × H)
342. C₃	C₃×#1997	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄⋊Q₈	2
C₃×#4924	2
C₃×C₄×#208	2
C₃×#4530	2
C₃×#1997	3
#71790	3

Subgroups — C3×#71790 #11791871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71790 #11791871