Subgroups — C₃×#33719 #10652326

Order 768.

834 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#510	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#509	C₂	1	2	yes	maximal, complemented
3	C₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2251	C₂	1	2	yes	maximal, complemented
6	C₃×#2250	C₂	1	2	yes	maximal, complemented
7	C₃×#2249	C₂	1	4	yes	maximal, complemented
8	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#33719	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂×C₃×#135	—	2	1	no	—
11	C₂×C₃×#135	C₂²	1	2	yes	—
12	C₂²×C₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₃×SD₁₆	—	2	1	no	—
14	C₂²×C₃×D₈	C₂²	1	1	yes	characteristic
15	C₃×#532	—	2	1	no	—
16	C₂×C₃×#124	—	2	1	no	—
17	C₃×#510	—	2	8	no	—
18	C₃×#509	—	2	8	no	—
19	C₂×C₃×Q₃₂	C₂²	1	4	yes	complemented
20	C₂×C₃×SD₃₂	C₂²	1	4	yes	—
21	C₂×C₃×SD₃₂	C₂²	1	4	yes	complemented
22	C₂×C₃×D₁₆	C₂²	1	4	yes	—
23	C₂×C₃×M₃₂	C₂²	1	2	yes	—
24	C₃×#495	—	2	1	no	—
25	C₃×#495	—	2	1	no	—
26	C₃×#495	—	2	1	no	—
27	C₃×#495	C₂²	1	2	yes	—
28	C₃×#495	C₂²	1	2	yes	—
29	C₃×#495	C₂²	1	2	yes	complemented
30	C₃×#494	C₂²	1	1	yes	characteristic, complemented
31	C₃×#493	C₂²	1	1	yes	characteristic, complemented
32	C₃×#492	—	2	1	no	—
33	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
34	C₃×#346	C₂²	1	2	yes	complemented
35	C₃×C₄×C₁₆	C₂²	1	2	yes	complemented
36	C₂×#510	C₂×C₃	1	2	yes	complemented
37	C₂×#509	C₂×C₃	1	2	yes	complemented
38	#3147	C₂×C₃	1	1	yes	characteristic, complemented
39	C₂×#495	C₂×C₃	1	1	yes	characteristic, complemented
40	#2251	C₂×C₃	1	2	yes	complemented
41	#2250	C₂×C₃	1	2	yes	complemented
42	#2249	C₂×C₃	1	4	yes	complemented
43	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂×C₃×#41	—	2	2	no	—
45	C₂²×C₃×Q₈	—	2	1	no	—
46	C₂²×C₃×D₄	—	2	1	no	—
47	C₃×#135	—	2	4	no	—
48	C₃×#135	—	2	4	no	—
49	C₃×#135	—	4	4	no	—
50	C₂×C₃×Q₁₆	—	2	4	no	—
51	C₂×C₃×Q₁₆	—	2	1	no	—
52	C₂×C₃×Q₁₆	—	2	1	no	—
53	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
54	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
55	C₂×C₃×SD₁₆	—	2	2	no	—
56	C₂×C₃×SD₁₆	—	2	2	no	—
57	C₂×C₃×SD₁₆	—	2	2	no	—
58	C₂×C₃×SD₁₆	—	2	2	no	—
59	C₂×C₃×SD₁₆	—	4	4	no	—
60	C₂×C₃×D₈	—	2	4	no	—
61	C₂×C₃×D₈	—	2	1	no	—
62	C₂×C₃×D₈	—	2	1	no	—
63	C₂×C₃×D₈	C₂³	1	2	yes	—
64	C₂×C₃×D₈	C₂³	1	2	yes	—
65	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
66	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₄⋊Q₈	—	2	1	no	—
68	C₃×#127	—	2	1	no	—
69	C₃×#124	—	2	2	no	—
70	C₃×#124	—	2	2	no	—
71	C₃×#124	—	2	2	no	—
72	C₃×#124	—	4	2	no	—
73	C₃×#121	—	4	2	no	—
74	C₃×C₄×D₄	—	4	2	no	—
75	C₂×C₃×C₂²⋊C₄	—	4	2	no	—
76	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
77	C₃×Q₃₂	—	2	8	no	—
78	C₃×SD₃₂	—	2	8	no	—
79	C₃×SD₃₂	—	2	8	no	—
80	C₃×D₁₆	—	2	8	no	—
81	C₃×M₃₂	D₄	1	8	yes	complemented
82	C₂×C₃×C₁₆	C₂³	1	4	yes	—
83	C₃×C₄×C₈	D₄	1	1	yes	characteristic
84	C₃×C₄×C₈	D₄	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
87	C₂×#135	—	2	1	no	—
88	C₂×#135	C₂²×C₃	1	2	yes	—
89	C₂²×Q₁₆	C₂²×C₃	1	1	yes	characteristic
90	C₂²×SD₁₆	—	2	1	no	—
91	C₂²×D₈	C₂²×C₃	1	1	yes	characteristic
92	#532	—	2	1	no	—
93	C₂×#124	—	2	1	no	—
94	#510	—	2	8	no	—
95	#509	—	2	8	no	—
96	C₂×Q₃₂	C₂²×C₃	1	4	yes	complemented
97	C₂×SD₃₂	C₂²×C₃	1	4	yes	—
98	C₂×SD₃₂	C₂²×C₃	1	4	yes	complemented
99	C₂×D₁₆	C₂²×C₃	1	4	yes	—
100	C₂×M₃₂	C₂²×C₃	1	2	yes	—
101	#495	—	2	1	no	—
102	#495	—	2	1	no	—
103	#495	—	2	1	no	—
104	#495	C₂²×C₃	1	2	yes	—
105	#495	C₂²×C₃	1	2	yes	—
106	#495	C₂²×C₃	1	2	yes	complemented
107	#494	C₂²×C₃	1	1	yes	characteristic, complemented
108	#493	C₂²×C₃	1	1	yes	characteristic, complemented
109	#492	—	2	1	no	—
110	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
111	#346	C₂²×C₃	1	2	yes	complemented
112	C₄×C₁₆	C₂²×C₃	1	2	yes	complemented
113	C₂⁴×C₃	—	4	1	no	—
114	C₃×#41	—	4	4	no	—
115	C₃×#41	—	4	4	no	—
116	C₂×C₃×Q₈	—	2	2	no	—
117	C₂×C₃×Q₈	—	2	2	no	—
118	C₂×C₃×Q₈	—	4	1	no	—
119	C₂×C₃×Q₈	—	4	2	no	—
120	C₂×C₃×Q₈	—	4	2	no	—
121	C₂×C₃×D₄	—	2	2	no	—
122	C₂×C₃×D₄	—	2	2	no	—
123	C₂×C₃×D₄	—	4	2	no	—
124	C₂×C₃×D₄	—	4	2	no	—
125	C₂×C₃×D₄	—	4	2	no	—
126	C₂×C₃×D₄	—	4	2	no	—
127	C₂×C₃×D₄	—	4	1	no	—
128	C₂²×C₃×C₄	—	4	1	no	—
129	C₂²×C₃×C₄	—	4	2	no	—
130	C₂²×C₃×C₄	D₈	1	2	yes	—
131	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
132	C₃×Q₁₆	—	2	2	no	—
133	C₃×Q₁₆	—	2	4	no	—
134	C₃×Q₁₆	—	2	2	no	—
135	C₃×Q₁₆	—	4	2	no	—
136	C₃×Q₁₆	—	4	2	no	—
137	C₃×SD₁₆	—	4	4	no	—
138	C₃×SD₁₆	—	4	4	no	—
139	C₃×SD₁₆	—	4	4	no	—
140	C₃×SD₁₆	—	4	4	no	—
141	C₃×D₈	—	2	2	no	—
142	C₃×D₈	—	2	4	no	—
143	C₃×D₈	—	2	2	no	—
144	C₃×D₈	—	4	2	no	—
145	C₃×D₈	—	4	2	no	—
146	C₂×C₃×C₈	D₈	1	4	yes	—
147	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
148	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
149	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
151	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
152	C₃×C₄⋊C₄	—	4	2	no	—
153	C₃×C₂²⋊C₄	—	4	4	no	—
154	C₃×C₂²⋊C₄	—	4	4	no	—
155	C₃×C₂²⋊C₄	—	4	4	no	—
156	C₃×C₄²	D₈	1	2	yes	—
157	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
159	C₃×C₁₆	C₂×D₄	1	8	yes	—
160	C₂×#41	—	2	2	no	—
161	C₂²×Q₈	—	2	1	no	—
162	C₂²×D₄	—	2	1	no	—
163	#135	—	2	4	no	—
164	#135	—	2	4	no	—
165	#135	—	4	4	no	—
166	C₂×Q₁₆	—	2	4	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	—	2	1	no	—
169	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
170	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
171	C₂×SD₁₆	—	2	2	no	—
172	C₂×SD₁₆	—	2	2	no	—
173	C₂×SD₁₆	—	2	2	no	—
174	C₂×SD₁₆	—	2	2	no	—
175	C₂×SD₁₆	—	4	4	no	—
176	C₂×D₈	—	2	4	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	C₂³×C₃	1	2	yes	—
180	C₂×D₈	C₂³×C₃	1	2	yes	—
181	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	1	no	—
184	#127	—	2	1	no	—
185	#124	—	2	2	no	—
186	#124	—	2	2	no	—
187	#124	—	2	2	no	—
188	#124	—	4	2	no	—
189	#121	—	4	2	no	—
190	C₄×D₄	—	4	2	no	—
191	C₂×C₂²⋊C₄	—	4	2	no	—
192	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
193	Q₃₂	—	2	8	no	—
194	SD₃₂	—	2	8	no	—
195	SD₃₂	—	2	8	no	—
196	D₁₆	—	2	8	no	—
197	M₃₂	C₃×D₄	1	8	yes	complemented
198	C₂×C₁₆	C₂³×C₃	1	4	yes	—
199	C₄×C₈	C₃×D₄	1	1	yes	characteristic
200	C₄×C₈	C₃×D₄	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
203	C₂³×C₃	—	4	2	no	—
204	C₂³×C₃	—	4	2	no	—
205	C₂³×C₃	—	8	4	no	—
206	C₂³×C₃	—	8	2	no	—
207	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
208	C₃×Q₈	—	4	2	no	—
209	C₃×Q₈	—	4	2	no	—
210	C₃×Q₈	—	4	2	no	—
211	C₃×Q₈	—	4	2	no	—
212	C₃×Q₈	—	8	2	no	—
213	C₃×D₄	—	4	2	no	—
214	C₃×D₄	—	4	2	no	—
215	C₃×D₄	—	4	2	no	—
216	C₃×D₄	—	4	2	no	—
217	C₃×D₄	—	8	4	no	—
218	C₃×D₄	—	8	4	no	—
219	C₃×D₄	—	8	2	no	—
220	C₂×C₃×C₄	—	4	2	no	—
221	C₂×C₃×C₄	—	4	2	no	—
222	C₂×C₃×C₄	—	4	2	no	—
223	C₂×C₃×C₄	—	8	2	no	—
224	C₂×C₃×C₄	—	8	4	no	—
225	C₂×C₃×C₄	#127	1	2	yes	—
226	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
227	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
228	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
229	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
231	C₃×C₈	#127	1	2	yes	—
232	C₃×C₈	C₂×D₈	1	2	yes	—
233	C₃×C₈	C₂×D₈	1	2	yes	—
234	C₃×C₈	C₂²×D₄	1	2	yes	—
235	C₂⁴	—	4	1	no	—
236	#41	—	4	4	no	—
237	#41	—	4	4	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₃×D₈	1	2	yes	—
253	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	2	4	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	4	2	no	—
258	Q₁₆	—	4	2	no	—
259	SD₁₆	—	4	4	no	—
260	SD₁₆	—	4	4	no	—
261	SD₁₆	—	4	4	no	—
262	SD₁₆	—	4	4	no	—
263	D₈	—	2	2	no	—
264	D₈	—	2	4	no	—
265	D₈	—	2	2	no	—
266	D₈	—	4	2	no	—
267	D₈	—	4	2	no	—
268	C₂×C₈	C₃×D₈	1	4	yes	—
269	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
270	C₂×C₈	C₂×C₃×D₄	1	4	yes	—
271	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
272	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
273	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
274	C₄⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	4	no	—
276	C₂²⋊C₄	—	4	4	no	—
277	C₂²⋊C₄	—	4	4	no	—
278	C₄²	C₃×D₈	1	2	yes	—
279	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
280	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
281	C₁₆	C₂×C₃×D₄	1	8	yes	—
282	C₂²×C₃	—	2	2	no	—
283	C₂²×C₃	—	8	2	no	—
284	C₂²×C₃	—	8	2	no	—
285	C₂²×C₃	—	8	4	no	—
286	C₂²×C₃	—	8	4	no	—
287	C₂²×C₃	—	16	4	no	—
288	C₂²×C₃	#493	1	2	yes	—
289	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	2	no	—
292	C₃×C₄	—	8	2	no	—
293	C₃×C₄	—	8	2	no	—
294	C₃×C₄	#493	1	2	yes	—
295	C₃×C₄	C₂×#127	1	1	yes	characteristic
296	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	8	4	no	—
300	C₂³	—	8	2	no	—
301	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	8	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	8	4	no	—
319	C₂×C₄	C₃×#127	1	2	yes	—
320	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
321	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
322	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
323	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
325	C₈	C₃×#127	1	2	yes	—
326	C₈	C₂×C₃×D₈	1	2	yes	—
327	C₈	C₂×C₃×D₈	1	2	yes	—
328	C₈	C₂²×C₃×D₄	1	2	yes	—
329	C₂×C₃	—	2	2	no	—
330	C₂×C₃	—	16	2	no	—
331	C₂×C₃	—	16	2	no	—
332	C₂×C₃	C₂×#493	1	1	yes	characteristic
333	C₂×C₃	#3414	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	4	no	—
338	C₂²	—	8	4	no	—
339	C₂²	—	16	4	no	—
340	C₂²	C₃×#493	1	2	yes	—
341	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	C₃×#493	1	2	yes	—
347	C₄	C₂×C₃×#127	1	1	yes	characteristic
348	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
349	C₃	#33719	1	1	yes	characteristic, direct product factor, Sylow
350	C₂	—	2	2	no	—
351	C₂	—	16	2	no	—
352	C₂	—	16	2	no	—
353	C₂	C₂×C₃×#493	1	1	yes	characteristic
354	C₂	C₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#510	C₂	semidirect
2. C₂×C₃×#509	C₂	semidirect
3. C₃×#3147	C₂	semidirect
4. C₂×C₃×#495	C₂	semidirect
5. C₃×#2251	C₂	semidirect
6. C₃×#2250	C₂	semidirect
7. C₃×#2249	C₂	semidirect
8. C₃×C₄×M₃₂	C₂	semidirect
9. #33719	C₃	direct (N × H)
19. C₂×C₃×Q₃₂	C₂²	semidirect
21. C₂×C₃×SD₃₂	C₂²	semidirect
29. C₃×#495	C₂²	semidirect
30. C₃×#494	C₂²	semidirect
31. C₃×#493	C₂²	semidirect
34. C₃×#346	C₂²	semidirect
35. C₃×C₄×C₁₆	C₂²	semidirect
36. C₂×#510	C₂×C₃	semidirect
37. C₂×#509	C₂×C₃	semidirect
38. #3147	C₂×C₃	semidirect
39. C₂×#495	C₂×C₃	semidirect
40. #2251	C₂×C₃	semidirect
41. #2250	C₂×C₃	semidirect
42. #2249	C₂×C₃	semidirect
43. C₄×M₃₂	C₂×C₃	semidirect
81. C₃×M₃₂	D₄	semidirect
96. C₂×Q₃₂	C₂²×C₃	semidirect
98. C₂×SD₃₂	C₂²×C₃	semidirect
106. #495	C₂²×C₃	semidirect
107. #494	C₂²×C₃	semidirect
108. #493	C₂²×C₃	semidirect
111. #346	C₂²×C₃	semidirect
112. C₄×C₁₆	C₂²×C₃	semidirect
197. M₃₂	C₃×D₄	semidirect
349. C₃	#33719	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×#510	2
C₂×C₃×#509	2
C₃×#3147	2
C₂×C₃×#495	2
C₃×#2251	2
C₃×#2250	2
C₃×#2249	2
C₃×C₄×M₃₂	2
#33719	3

Subgroups — C3×#33719 #10652326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33719 #10652326