Subgroups — C₃×#33065 #10651672

Order 768.

746 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#3080	C₂	1	1	yes	maximal, characteristic
3	C₃×#3039	C₂	1	1	yes	maximal, characteristic
4	C₃×#2875	C₂	1	1	yes	maximal, characteristic
5	C₃×#1998	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1966	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1930	C₂	1	4	yes	maximal, complemented
8	C₃×#1908	C₂	1	4	yes	maximal
9	C₃×#1838	C₂	1	1	yes	maximal, characteristic, complemented
10	#33065	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#137	C₂²	1	2	yes	—
12	C₂×C₃×#131	C₂²	1	2	yes	—
13	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃×#125	C₂²	1	1	yes	characteristic
15	C₃×C₄×#41	C₂²	1	1	yes	characteristic
16	C₃×#491	—	2	1	no	—
17	C₃×#491	—	2	1	no	—
18	C₃×#491	—	2	2	no	—
19	C₃×#479	—	2	2	no	—
20	C₃×#478	—	2	2	no	—
21	C₃×#462	—	2	2	no	—
22	C₃×#461	—	2	2	no	—
23	C₃×#428	C₂²	1	2	yes	—
24	C₃×#423	C₂²	1	1	yes	characteristic
25	C₃×#423	C₂²	1	1	yes	characteristic
26	C₃×#419	C₂²	1	2	yes	—
27	C₃×#417	C₂²	1	2	yes	—
28	C₂×C₃×#103	C₂²	1	4	yes	—
29	C₂×C₃×#101	C₂²	1	4	yes	—
30	C₃×#405	—	2	2	no	—
31	C₃×#405	C₂²	1	2	yes	—
32	C₃×C₄×M₁₆	—	2	1	no	—
33	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
34	C₂×C₃×#97	—	2	1	no	—
35	C₂×C₃×#97	C₂²	1	1	yes	characteristic
36	C₃×#343	C₂²	1	4	yes	—
37	C₃×#341	C₂²	1	4	yes	—
38	C₂×#491	C₂×C₃	1	1	yes	characteristic, complemented
39	#3080	C₂×C₃	1	1	yes	characteristic
40	#3039	C₂×C₃	1	1	yes	characteristic
41	#2875	C₂×C₃	1	1	yes	characteristic
42	#1998	C₂×C₃	1	1	yes	characteristic, complemented
43	#1966	C₂×C₃	1	1	yes	characteristic, complemented
44	#1930	C₂×C₃	1	4	yes	complemented
45	#1908	C₂×C₃	1	4	yes	—
46	#1838	C₂×C₃	1	1	yes	characteristic, complemented
47	C₂×C₃×#41	C₂³	1	1	yes	characteristic
48	C₂²×C₃×Q₈	C₂³	1	2	yes	—
49	C₃×#137	—	2	8	no	—
50	C₂×C₃×Q₁₆	—	2	2	no	—
51	C₂×C₃×SD₁₆	—	2	2	no	—
52	C₃×#131	—	2	4	no	—
53	C₃×#131	—	2	4	no	—
54	C₂×C₃×M₁₆	—	2	2	no	—
55	C₂×C₃×M₁₆	C₂³	1	2	yes	—
56	C₂×C₃×M₁₆	C₂³	1	2	yes	—
57	C₂×C₃×M₁₆	C₂³	1	2	yes	—
58	C₂²×C₃×C₈	—	2	2	no	—
59	C₂²×C₃×C₈	C₂³	1	2	yes	—
60	C₃×C₄⋊Q₈	—	2	2	no	—
61	C₃×C₄⋊Q₈	—	2	1	no	—
62	C₃×C₄⋊Q₈	—	2	1	no	—
63	C₃×#125	—	2	2	no	—
64	C₃×#125	—	2	1	no	—
65	C₃×#125	—	2	1	no	—
66	C₃×C₄×Q₈	D₄	1	2	yes	—
67	C₃×C₄×D₄	—	2	2	no	—
68	C₃×C₄×D₄	D₄	1	2	yes	—
69	C₃×#117	—	2	1	no	—
70	C₃×#117	C₂³	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	—	2	2	no	—
72	C₂×C₃×C₄⋊C₄	—	2	2	no	—
73	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₂×C₃×C₄²	—	2	1	no	—
75	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
76	C₃×#107	—	2	2	no	—
77	C₃×#106	—	2	2	no	—
78	C₃×C₄⋊C₈	—	2	2	no	—
79	C₃×C₄⋊C₈	—	2	2	no	—
80	C₃×#103	—	2	2	no	—
81	C₃×#103	—	2	4	no	—
82	C₃×#103	—	2	4	no	—
83	C₃×#103	—	2	4	no	—
84	C₃×#102	—	2	2	no	—
85	C₃×#101	—	2	8	no	—
86	C₃×#97	—	2	2	no	—
87	C₃×#97	—	2	1	no	—
88	C₃×#97	D₄	1	1	yes	characteristic
89	C₃×#97	D₄	1	1	yes	characteristic
90	C₃×#97	D₄	1	2	yes	—
91	C₃×C₄×C₈	—	2	2	no	—
92	C₃×C₄×C₈	—	2	1	no	—
93	C₂×#137	C₂²×C₃	1	2	yes	—
94	C₂×#131	C₂²×C₃	1	2	yes	—
95	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
96	C₂×#125	C₂²×C₃	1	1	yes	characteristic
97	C₄×#41	C₂²×C₃	1	1	yes	characteristic
98	#491	—	2	1	no	—
99	#491	—	2	1	no	—
100	#491	—	2	2	no	—
101	#479	—	2	2	no	—
102	#478	—	2	2	no	—
103	#462	—	2	2	no	—
104	#461	—	2	2	no	—
105	#428	C₂²×C₃	1	2	yes	—
106	#423	C₂²×C₃	1	1	yes	characteristic
107	#423	C₂²×C₃	1	1	yes	characteristic
108	#419	C₂²×C₃	1	2	yes	—
109	#417	C₂²×C₃	1	2	yes	—
110	C₂×#103	C₂²×C₃	1	4	yes	—
111	C₂×#101	C₂²×C₃	1	4	yes	—
112	#405	—	2	2	no	—
113	#405	C₂²×C₃	1	2	yes	—
114	C₄×M₁₆	—	2	1	no	—
115	C₄×M₁₆	C₂²×C₃	1	1	yes	characteristic
116	C₂×#97	—	2	1	no	—
117	C₂×#97	C₂²×C₃	1	1	yes	characteristic
118	#343	C₂²×C₃	1	4	yes	—
119	#341	C₂²×C₃	1	4	yes	—
120	C₃×#41	—	2	2	no	—
121	C₃×#41	#41	1	4	yes	—
122	C₂×C₃×Q₈	—	2	2	no	—
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	2	4	no	—
125	C₂×C₃×Q₈	—	2	4	no	—
126	C₂×C₃×Q₈	C₂×D₄	1	1	yes	characteristic
127	C₂×C₃×D₄	—	2	1	no	—
128	C₂×C₃×D₄	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	2	no	—
132	C₂²×C₃×C₄	—	2	2	no	—
133	C₂²×C₃×C₄	#41	1	2	yes	—
134	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
135	C₃×Q₁₆	—	4	4	no	—
136	C₃×SD₁₆	—	4	4	no	—
137	C₃×M₁₆	—	2	8	no	—
138	C₃×M₁₆	—	2	2	no	—
139	C₃×M₁₆	—	2	4	no	—
140	C₃×M₁₆	—	2	4	no	—
141	C₃×M₁₆	#41	1	4	yes	—
142	C₂×C₃×C₈	—	2	8	no	—
143	C₂×C₃×C₈	—	2	2	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	2	2	no	—
146	C₂×C₃×C₈	—	2	2	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
149	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	—	4	2	no	—
154	C₃×C₄⋊C₄	—	4	2	no	—
155	C₃×C₄⋊C₄	—	4	2	no	—
156	C₃×C₄⋊C₄	—	4	2	no	—
157	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
158	C₃×C₂²⋊C₄	—	2	1	no	—
159	C₃×C₂²⋊C₄	—	2	2	no	—
160	C₃×C₄²	—	2	2	no	—
161	C₃×C₄²	—	2	1	no	—
162	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
163	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
164	C₃×C₄²	C₂×D₄	1	2	yes	—
165	C₂×#41	C₂³×C₃	1	1	yes	characteristic
166	C₂²×Q₈	C₂³×C₃	1	2	yes	—
167	#137	—	2	8	no	—
168	C₂×Q₁₆	—	2	2	no	—
169	C₂×SD₁₆	—	2	2	no	—
170	#131	—	2	4	no	—
171	#131	—	2	4	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	C₂³×C₃	1	2	yes	—
174	C₂×M₁₆	C₂³×C₃	1	2	yes	—
175	C₂×M₁₆	C₂³×C₃	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	C₂³×C₃	1	2	yes	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	1	no	—
180	C₄⋊Q₈	—	2	1	no	—
181	#125	—	2	2	no	—
182	#125	—	2	1	no	—
183	#125	—	2	1	no	—
184	C₄×Q₈	C₃×D₄	1	2	yes	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	C₃×D₄	1	2	yes	—
187	#117	—	2	1	no	—
188	#117	C₂³×C₃	1	1	yes	characteristic
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
194	#107	—	2	2	no	—
195	#106	—	2	2	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	#103	—	2	2	no	—
199	#103	—	2	4	no	—
200	#103	—	2	4	no	—
201	#103	—	2	4	no	—
202	#102	—	2	2	no	—
203	#101	—	2	8	no	—
204	#97	—	2	2	no	—
205	#97	—	2	1	no	—
206	#97	C₃×D₄	1	1	yes	characteristic
207	#97	C₃×D₄	1	1	yes	characteristic
208	#97	C₃×D₄	1	2	yes	—
209	C₄×C₈	—	2	2	no	—
210	C₄×C₈	—	2	1	no	—
211	C₂³×C₃	—	2	1	no	—
212	C₂³×C₃	C₂×#41	1	1	yes	characteristic
213	C₃×Q₈	—	2	1	no	—
214	C₃×Q₈	—	4	4	no	—
215	C₃×Q₈	—	4	2	no	—
216	C₃×Q₈	—	4	2	no	—
217	C₃×Q₈	#124	1	2	yes	—
218	C₃×D₄	—	2	4	no	—
219	C₃×D₄	—	2	1	no	—
220	C₃×D₄	#124	1	2	yes	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	4	2	no	—
225	C₂×C₃×C₄	—	4	2	no	—
226	C₂×C₃×C₄	—	4	2	no	—
227	C₂×C₃×C₄	—	4	4	no	—
228	C₂×C₃×C₄	—	8	2	no	—
229	C₂×C₃×C₄	—	8	2	no	—
230	C₂×C₃×C₄	#121	1	2	yes	—
231	C₂×C₃×C₄	#121	1	2	yes	—
232	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×#41	1	2	yes	—
235	C₂×C₃×C₄	C₂×#41	1	2	yes	—
236	C₃×C₈	—	2	2	no	—
237	C₃×C₈	—	2	4	no	—
238	C₃×C₈	—	2	2	no	—
239	C₃×C₈	—	4	2	no	—
240	C₃×C₈	—	4	2	no	—
241	#41	—	2	2	no	—
242	#41	C₃×#41	1	4	yes	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	C₂×C₃×D₄	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	C₃×#41	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
256	Q₁₆	—	4	4	no	—
257	SD₁₆	—	4	4	no	—
258	M₁₆	—	2	8	no	—
259	M₁₆	—	2	2	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	C₃×#41	1	4	yes	—
263	C₂×C₈	—	2	8	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
270	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	1	no	—
283	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
284	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
285	C₄²	C₂×C₃×D₄	1	2	yes	—
286	C₂²×C₃	—	2	2	no	—
287	C₂²×C₃	—	2	2	no	—
288	C₂²×C₃	—	4	2	no	—
289	C₂²×C₃	C₂×#121	1	1	yes	characteristic
290	C₂²×C₃	#529	1	2	yes	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	—	4	2	no	—
295	C₃×C₄	—	8	2	no	—
296	C₃×C₄	—	8	2	no	—
297	C₃×C₄	#525	1	2	yes	—
298	C₃×C₄	C₂×#124	1	1	yes	characteristic
299	C₃×C₄	#532	1	1	yes	characteristic
300	C₂³	—	2	1	no	—
301	C₂³	C₂×C₃×#41	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	4	4	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	C₃×#124	1	2	yes	—
307	D₄	—	2	4	no	—
308	D₄	—	2	1	no	—
309	D₄	C₃×#124	1	2	yes	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	C₃×#121	1	2	yes	—
320	C₂×C₄	C₃×#121	1	2	yes	—
321	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
323	C₂×C₄	C₂×C₃×#41	1	2	yes	—
324	C₂×C₄	C₂×C₃×#41	1	2	yes	—
325	C₈	—	2	2	no	—
326	C₈	—	2	4	no	—
327	C₈	—	2	2	no	—
328	C₈	—	4	2	no	—
329	C₈	—	4	2	no	—
330	C₂×C₃	—	2	2	no	—
331	C₂×C₃	—	4	2	no	—
332	C₂×C₃	#2850	1	1	yes	characteristic
333	C₂×C₃	#3069	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	C₂×C₃×#121	1	1	yes	characteristic
338	C₂²	C₃×#529	1	2	yes	—
339	C₄	—	2	2	no	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	8	2	no	—
344	C₄	—	8	2	no	—
345	C₄	C₃×#525	1	2	yes	—
346	C₄	C₂×C₃×#124	1	1	yes	characteristic
347	C₄	C₃×#532	1	1	yes	characteristic
348	C₃	#33065	1	1	yes	characteristic, direct product factor, Sylow
349	C₂	—	2	2	no	—
350	C₂	—	4	2	no	—
351	C₂	C₃×#2850	1	1	yes	characteristic
352	C₂	C₃×#3069	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
prime radical	C₂×C₃×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#491	C₂	semidirect
5. C₃×#1998	C₂	semidirect
6. C₃×#1966	C₂	semidirect
7. C₃×#1930	C₂	semidirect
9. C₃×#1838	C₂	semidirect
10. #33065	C₃	direct (N × H)
38. C₂×#491	C₂×C₃	semidirect
42. #1998	C₂×C₃	semidirect
43. #1966	C₂×C₃	semidirect
44. #1930	C₂×C₃	semidirect
46. #1838	C₂×C₃	semidirect
348. C₃	#33065	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#491	2
C₃×#3080	2
C₃×#3039	2
C₃×#2875	2
C₃×#1998	2
C₃×#1966	2
C₃×#1930	2
C₃×#1908	2
C₃×#1838	2
#33065	3

Subgroups — C3×#33065 #10651672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33065 #10651672