Subgroups — C₃×#83334 #11873482

Order 1152.

911 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#5477	C₂	1	1	yes	maximal, characteristic
3	C₃×#5093	C₂	1	2	yes	maximal, complemented
4	C₃×#5088	C₂	1	2	yes	maximal, complemented
5	C₃×#5068	C₂	1	4	yes	maximal, complemented
6	C₃×#5002	C₂	1	4	yes	maximal, complemented
7	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3131	—	3	1	no	maximal
9	#83334	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#235	C₂²	1	2	yes	—
11	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃²×#125	C₂²	1	2	yes	—
13	C₃²×C₂²⋊Q₈	—	2	2	no	—
14	C₃²×C₄×Q₈	—	2	2	no	—
15	C₃²×#117	C₂²	1	2	yes	—
16	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
17	C₃×#930	—	2	2	no	—
18	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
19	C₃×#875	C₂²	1	2	yes	—
20	C₃×#874	C₂²	1	1	yes	characteristic
21	C₃×C₄×D₁₂	—	2	2	no	—
22	C₃×#869	C₂²	1	1	yes	characteristic
23	C₃×#810	C₂²	1	8	yes	—
24	C₃×#809	C₂²	1	8	yes	—
25	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
26	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
27	C₃×#533	—	2	1	no	—
28	C₃×#533	D₃	1	1	yes	characteristic, complemented
29	C₃×#533	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×#532	—	3	1	no	—
31	#5477	C₂×C₃	1	1	yes	characteristic
32	C₃×#488	—	3	4	no	—
33	C₃×#487	—	3	2	no	—
34	C₃×#486	—	3	2	no	—
35	C₃×#416	—	3	4	no	—
36	C₂×C₃×C₄×C₈	—	3	1	no	—
37	#5093	C₂×C₃	1	2	yes	complemented
38	#5088	C₂×C₃	1	2	yes	complemented
39	#5068	C₂×C₃	1	4	yes	complemented
40	#5002	C₂×C₃	1	4	yes	complemented
41	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₃²×Q₈	—	2	2	no	—
43	C₂²×C₃²×C₄	D₄	1	2	yes	—
44	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
45	C₂×C₃×#67	—	2	2	no	—
46	C₃×#235	—	2	8	no	—
47	C₂×C₃×D₁₂	—	2	2	no	—
48	C₂×C₃×D₁₂	C₂³	1	2	yes	—
49	C₂×C₃×C₄×D₃	—	2	2	no	—
50	C₂×C₃×Dic₆	C₂³	1	2	yes	—
51	C₃²×C₄⋊C₄	—	2	4	no	—
52	C₃²×C₄⋊C₄	—	2	2	no	—
53	C₃²×C₄⋊C₄	C₂³	1	4	yes	—
54	C₃²×C₂²⋊C₄	—	2	2	no	—
55	C₃²×C₄²	—	2	2	no	—
56	C₃²×C₄²	D₄	1	2	yes	—
57	C₃²×C₄²	C₂³	1	1	yes	characteristic
58	C₃²×C₄²	C₂³	1	1	yes	characteristic
59	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
60	C₃×C₄⋊Dic₃	—	2	2	no	—
61	C₂×C₃×C₃⋊C₈	—	2	4	no	—
62	C₂×C₃×C₃⋊C₈	C₂³	1	4	yes	—
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	—	3	2	no	—
67	C₃×C₄⋊Q₈	—	2	2	no	—
68	C₃×C₄⋊Q₈	—	3	1	no	—
69	C₃×C₄⋊Q₈	C₂×D₃	1	2	yes	—
70	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
71	C₃×#127	—	3	1	no	—
72	C₃×#125	—	2	2	no	—
73	C₃×#125	C₂×D₃	1	2	yes	—
74	C₃×#125	C₂²×C₃	1	2	yes	—
75	C₃×#124	—	3	2	no	—
76	C₃×C₂²⋊Q₈	—	2	4	no	—
77	C₃×C₂²⋊Q₈	—	2	2	no	—
78	C₃×C₂²⋊Q₈	—	2	2	no	—
79	C₃×#121	—	6	2	no	—
80	C₃×C₄×Q₈	—	2	4	no	—
81	C₃×C₄×Q₈	—	2	2	no	—
82	C₃×C₄×Q₈	—	2	2	no	—
83	C₃×C₄×D₄	—	6	2	no	—
84	C₃×#117	—	2	2	no	—
85	C₃×#117	C₂×D₃	1	2	yes	—
86	C₃×#117	C₂²×C₃	1	2	yes	—
87	C₂×C₃×C₄²	—	2	1	no	—
88	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
89	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
90	#930	—	2	2	no	—
91	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
92	#875	C₂²×C₃	1	2	yes	—
93	#874	C₂²×C₃	1	1	yes	characteristic
94	C₄×D₁₂	—	2	2	no	—
95	#869	C₂²×C₃	1	1	yes	characteristic
96	C₃×#103	—	3	8	no	—
97	C₃×#102	—	3	8	no	—
98	C₃×C₄×C₈	—	3	2	no	—
99	C₃×C₄×C₈	—	3	2	no	—
100	#810	C₂²×C₃	1	8	yes	—
101	#809	C₂²×C₃	1	8	yes	—
102	C₄×C₃⋊C₈	C₂²×C₃	1	2	yes	—
103	C₄×C₃⋊C₈	C₂²×C₃	1	2	yes	—
104	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₃×D₃	—	2	2	no	—
106	C₃²×Q₈	—	4	4	no	—
107	C₂×C₃²×C₄	—	2	4	no	—
108	C₂×C₃²×C₄	—	2	4	no	—
109	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
110	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
111	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
112	C₂×C₃²×C₄	#41	1	4	yes	—
113	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
114	C₃×#67	—	4	4	no	—
115	C₂×C₃×Dic₃	—	2	2	no	—
116	C₃×D₁₂	—	2	4	no	—
117	C₃×D₁₂	—	4	4	no	—
118	C₃×C₄×D₃	—	4	4	no	—
119	C₃×Dic₆	—	2	4	no	—
120	C₃×C₃⋊C₈	—	2	4	no	—
121	#533	C₃×D₃	1	1	yes	characteristic, complemented
122	#532	—	3	1	no	—
123	#488	—	3	4	no	—
124	#487	—	3	2	no	—
125	#486	—	3	2	no	—
126	#416	—	3	4	no	—
127	C₂×C₄×C₈	—	3	1	no	—
128	C₃×#41	—	6	8	no	—
129	C₂×C₃×Q₈	—	2	4	no	—
130	C₂×C₃×Q₈	—	2	2	no	—
131	C₂×C₃×Q₈	—	2	2	no	—
132	C₂×C₃×Q₈	—	3	2	no	—
133	C₂×C₃×D₄	—	3	2	no	—
134	C₂×C₃×D₄	—	6	2	no	—
135	C₂×C₃×D₄	—	6	2	no	—
136	C₂²×C₃×C₄	—	2	2	no	—
137	C₂²×C₃×C₄	—	2	1	no	—
138	C₂²×C₃×C₄	—	6	2	no	—
139	C₂²×C₃×C₄	#67	1	2	yes	—
140	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
141	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
143	C₂×#67	—	2	2	no	—
144	#235	—	2	8	no	—
145	C₂×D₁₂	—	2	2	no	—
146	C₂×D₁₂	C₂³×C₃	1	2	yes	—
147	C₂×C₄×D₃	—	2	2	no	—
148	C₂×Dic₆	C₂³×C₃	1	2	yes	—
149	C₂×C₃×C₈	—	3	4	no	—
150	C₂×C₃×C₈	—	6	4	no	—
151	C₃×C₄⋊C₄	—	2	8	no	—
152	C₃×C₄⋊C₄	—	2	4	no	—
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₄⋊C₄	—	2	2	no	—
155	C₃×C₄⋊C₄	—	2	4	no	—
156	C₃×C₄⋊C₄	—	2	2	no	—
157	C₃×C₄⋊C₄	—	2	4	no	—
158	C₃×C₄⋊C₄	—	6	2	no	—
159	C₃×C₄⋊C₄	C₂²×D₃	1	4	yes	—
160	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
161	C₃×C₂²⋊C₄	—	2	4	no	—
162	C₃×C₂²⋊C₄	—	2	2	no	—
163	C₃×C₂²⋊C₄	—	2	2	no	—
164	C₃×C₂²⋊C₄	—	6	4	no	—
165	C₃×C₄²	—	2	4	no	—
166	C₃×C₄²	—	2	2	no	—
167	C₃×C₄²	—	2	2	no	—
168	C₃×C₄²	—	2	2	no	—
169	C₃×C₄²	—	2	1	no	—
170	C₃×C₄²	—	2	1	no	—
171	C₃×C₄²	#67	1	2	yes	—
172	C₃×C₄²	C₃×D₄	1	2	yes	—
173	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
174	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
175	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
176	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
177	(C₂×D₃)⋊C₄	—	2	4	no	—
178	C₄⋊Dic₃	—	2	2	no	—
179	C₂×C₃⋊C₈	—	2	4	no	—
180	C₂×C₃⋊C₈	C₂³×C₃	1	4	yes	—
181	C₂²×C₃²	—	2	2	no	—
182	C₂²×C₃²	#124	1	2	yes	—
183	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₃×D₃	—	4	2	no	—
185	C₂×C₃×D₃	—	4	4	no	—
186	C₃²×C₄	—	2	2	no	—
187	C₃²×C₄	—	4	4	no	—
188	C₃²×C₄	#124	1	2	yes	—
189	C₃²×C₄	#135	1	4	yes	—
190	C₃²×C₄	C₂×#41	1	2	yes	—
191	C₃×Dic₃	—	4	2	no	—
192	C₂×#41	—	3	2	no	—
193	C₂²×C₈	—	3	2	no	—
194	C₄⋊Q₈	—	3	1	no	—
195	C₄⋊Q₈	C₂×C₃×D₃	1	2	yes	—
196	#127	—	3	1	no	—
197	#125	C₂×C₃×D₃	1	2	yes	—
198	#124	—	3	2	no	—
199	C₂²⋊Q₈	—	2	2	no	—
200	#121	—	6	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×D₄	—	6	2	no	—
203	#117	C₂×C₃×D₃	1	2	yes	—
204	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
205	#103	—	3	8	no	—
206	#102	—	3	8	no	—
207	C₄×C₈	—	3	2	no	—
208	C₄×C₈	—	3	2	no	—
209	C₂³×C₃	—	2	1	no	—
210	C₂³×C₃	—	6	2	no	—
211	C₂³×C₃	C₂×#67	1	1	yes	characteristic
212	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂²×D₃	—	2	2	no	—
214	C₃×Q₈	—	4	8	no	—
215	C₃×Q₈	—	4	4	no	—
216	C₃×Q₈	—	4	4	no	—
217	C₃×Q₈	—	6	4	no	—
218	C₃×D₄	—	6	4	no	—
219	C₃×D₄	—	12	4	no	—
220	C₃×D₄	—	12	4	no	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	8	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	—	2	1	no	—
227	C₂×C₃×C₄	—	2	4	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	2	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	4	4	no	—
232	C₂×C₃×C₄	—	6	2	no	—
233	C₂×C₃×C₄	—	12	4	no	—
234	C₂×C₃×C₄	#239	1	4	yes	—
235	C₂×C₃×C₄	C₂×#67	1	2	yes	—
236	C₂×C₃×C₄	C₂×#67	1	2	yes	—
237	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
239	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
241	C₂×C₃×C₄	C₃×#41	1	4	yes	—
242	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
244	#67	—	4	4	no	—
245	C₂×Dic₃	—	2	2	no	—
246	D₁₂	—	2	4	no	—
247	D₁₂	—	4	4	no	—
248	C₄×D₃	—	4	4	no	—
249	Dic₆	—	2	4	no	—
250	C₃×C₈	—	6	4	no	—
251	C₃⋊C₈	—	2	4	no	—
252	C₂×C₃²	—	2	2	no	—
253	C₂×C₃²	C₂×#124	1	1	yes	characteristic
254	C₂×C₃²	C₂×#135	1	2	yes	—
255	C₃×D₃	—	8	2	no	—
256	#41	—	6	8	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	3	2	no	—
259	C₂×D₄	—	3	2	no	—
260	C₂×D₄	—	6	2	no	—
261	C₂×D₄	—	6	2	no	—
262	C₂²×C₄	—	6	2	no	—
263	C₂²×C₄	C₃×#67	1	2	yes	—
264	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
265	C₂×C₈	—	3	4	no	—
266	C₂×C₈	—	6	4	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	6	2	no	—
270	C₄⋊C₄	C₂²×C₃×D₃	1	4	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	6	4	no	—
273	C₄²	—	2	2	no	—
274	C₄²	C₃×#67	1	2	yes	—
275	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
276	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
277	C₂²×C₃	—	2	2	no	—
278	C₂²×C₃	—	2	2	no	—
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	—	2	1	no	—
281	C₂²×C₃	—	4	2	no	—
282	C₂²×C₃	—	12	2	no	—
283	C₂²×C₃	—	12	4	no	—
284	C₂²×C₃	#947	1	2	yes	—
285	C₂²×C₃	C₃×#124	1	2	yes	—
286	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
287	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
288	C₂×D₃	—	4	4	no	—
289	C₂×D₃	—	4	2	no	—
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	4	no	—
294	C₃×C₄	—	2	2	no	—
295	C₃×C₄	—	4	2	no	—
296	C₃×C₄	—	4	8	no	—
297	C₃×C₄	—	4	4	no	—
298	C₃×C₄	—	4	4	no	—
299	C₃×C₄	—	12	2	no	—
300	C₃×C₄	#947	1	2	yes	—
301	C₃×C₄	#950	1	4	yes	—
302	C₃×C₄	C₃×#124	1	2	yes	—
303	C₃×C₄	C₃×#135	1	4	yes	—
304	C₃×C₄	C₂×#239	1	2	yes	—
305	C₃×C₄	C₂×C₃×#41	1	2	yes	—
306	Dic₃	—	4	2	no	—
307	C₃²	#3131	1	1	yes	characteristic, complemented, Sylow
308	C₂³	—	6	2	no	—
309	C₂³	C₂×C₃×#67	1	1	yes	characteristic
310	Q₈	—	4	4	no	—
311	Q₈	—	6	4	no	—
312	D₄	—	6	4	no	—
313	D₄	—	12	4	no	—
314	D₄	—	12	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	6	2	no	—
318	C₂×C₄	—	12	4	no	—
319	C₂×C₄	C₃×#239	1	4	yes	—
320	C₂×C₄	C₂×C₃×#67	1	2	yes	—
321	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
322	C₂×C₄	C₂×C₃×#67	1	2	yes	—
323	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
324	C₈	—	6	4	no	—
325	C₂×C₃	—	2	2	no	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	2	no	—
329	C₂×C₃	—	4	2	no	—
330	C₂×C₃	—	24	2	no	—
331	C₂×C₃	C₂×#947	1	1	yes	characteristic
332	C₂×C₃	C₂×#950	1	2	yes	—
333	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
334	C₂×C₃	C₂×C₃×#135	1	2	yes	—
335	D₃	—	8	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	12	2	no	—
338	C₂²	—	12	4	no	—
339	C₂²	C₃×#947	1	2	yes	—
340	C₂²	C₂²×C₃×#67	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	4	4	no	—
343	C₄	—	12	2	no	—
344	C₄	C₃×#947	1	2	yes	—
345	C₄	C₃×#950	1	4	yes	—
346	C₄	C₂×C₃×#239	1	2	yes	—
347	C₃	—	2	1	no	—
348	C₃	#83334	1	1	yes	characteristic, direct product factor
349	C₃	C₃×#3131	1	1	yes	characteristic, complemented
350	C₂	—	2	2	no	—
351	C₂	—	24	2	no	—
352	C₂	C₂×C₃×#947	1	1	yes	characteristic
353	C₂	C₂×C₃×#950	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×#533	C₂
socle	C₂²×C₃²	C₂²×D₄
prime radical	C₃×#5477	C₂
hypercenter	C₃×#533	D₃
hypocenter	C₃	C₃×#3131
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#533

Complements

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

N	H ≅ G/N	type
1. C₃²×#533	C₂	semidirect
3. C₃×#5093	C₂	semidirect
4. C₃×#5088	C₂	semidirect
5. C₃×#5068	C₂	semidirect
6. C₃×#5002	C₂	semidirect
7. C₂×C₃×C₄×C₃⋊C₈	C₂	semidirect
9. #83334	C₃	direct (N × H)
28. C₃×#533	D₃	semidirect
29. C₃×#533	C₂×C₃	semidirect
37. #5093	C₂×C₃	semidirect
38. #5088	C₂×C₃	semidirect
39. #5068	C₂×C₃	semidirect
40. #5002	C₂×C₃	semidirect
41. C₂×C₄×C₃⋊C₈	C₂×C₃	semidirect
121. #533	C₃×D₃	semidirect
307. C₃²	#3131	semidirect
348. C₃	#83334	direct (N × H)
349. 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
C₃²×#533	2
C₃×#5477	2
C₃×#5093	2
C₃×#5088	2
C₃×#5068	2
C₃×#5002	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×#3131	3
#83334	3

Subgroups — C3×#83334 #11873482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#83334 #11873482