Subgroups — C₃×#72213956 #72226108

Order 3072.

2414 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#123411	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#123403	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#123419	C₂	1	3	yes	maximal, complemented
4	C₃×#123412	C₂	1	3	yes	maximal, complemented
5	C₃×#122211	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#491500	C₂	1	3	yes	maximal, complemented
7	C₃×#491469	C₂	1	3	yes	maximal, complemented
8	#72213956	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₃×#33748	—	2	6	no	—
10	C₃×#33717	—	2	6	no	—
11	C₃×#19603	C₂²	1	3	yes	—
12	C₂×C₃×#1712	C₂²	1	1	yes	characteristic
13	C₃×#12523	C₂²	1	3	yes	complemented
14	C₃×#12519	—	2	6	no	—
15	C₃×#12517	C₂²	1	3	yes	—
16	C₃×#12517	C₂²	1	3	yes	—
17	C₃×#12515	C₂²	1	3	yes	—
18	C₃×#12515	C₂²	1	3	yes	—
19	C₃×#12515	C₂²	1	3	yes	complemented
20	C₃×#12509	C₂²	1	6	yes	complemented
21	C₃×#12506	—	2	6	no	—
22	C₃×C₈×M₃₂	C₂²	1	3	yes	—
23	C₃×#7052	C₂²	1	3	yes	complemented
24	C₃×C₁₆²	C₂²	1	1	yes	characteristic, complemented
25	#491500	C₂×C₃	1	3	yes	complemented
26	#491469	C₂×C₃	1	3	yes	complemented
27	#123419	C₂×C₃	1	3	yes	complemented
28	#123412	C₂×C₃	1	3	yes	complemented
29	#123411	C₂×C₃	1	1	yes	characteristic, complemented
30	#123403	C₂×C₃	1	1	yes	characteristic, complemented
31	#122211	C₂×C₃	1	1	yes	characteristic, complemented
32	C₂×C₃×#509	—	4	12	no	—
33	C₃×#3169	—	2	6	no	—
34	C₃×#3147	—	2	3	no	—
35	C₂×C₃×#493	—	2	3	no	—
36	C₃×#2254	—	2	6	no	—
37	C₃×#2250	—	4	12	no	—
38	C₃×#2248	—	2	6	no	—
39	C₃×#2248	—	2	6	no	—
40	C₃×#2246	—	2	6	no	—
41	C₃×#2246	—	2	6	no	—
42	C₃×#2246	—	2	6	no	—
43	C₃×#2240	—	2	12	no	—
44	C₃×#2218	—	4	12	no	—
45	C₃×#2177	—	4	12	no	—
46	C₃×C₄×M₃₂	D₄	1	6	yes	—
47	C₃×#1713	C₂³	1	3	yes	—
48	C₃×#1712	—	2	3	no	—
49	C₃×#1712	C₂³	1	1	yes	characteristic
50	C₃×#1712	C₂³	1	1	yes	characteristic
51	C₃×#1712	C₂³	1	3	yes	—
52	C₃×#1709	—	2	3	no	—
53	C₃×#1706	—	2	6	no	—
54	C₂×C₃×C₈²	C₂³	1	1	yes	characteristic
55	C₃×#1311	D₄	1	6	yes	—
56	C₃×C₈×C₁₆	C₂³	1	3	yes	—
57	C₃×C₈×C₁₆	C₂³	1	3	yes	—
58	#33748	—	2	6	no	—
59	#33717	—	2	6	no	—
60	#19603	C₂²×C₃	1	3	yes	—
61	C₂×#1712	C₂²×C₃	1	1	yes	characteristic
62	#12523	C₂²×C₃	1	3	yes	complemented
63	#12519	—	2	6	no	—
64	#12517	C₂²×C₃	1	3	yes	—
65	#12517	C₂²×C₃	1	3	yes	—
66	#12515	C₂²×C₃	1	3	yes	—
67	#12515	C₂²×C₃	1	3	yes	—
68	#12515	C₂²×C₃	1	3	yes	complemented
69	#12509	C₂²×C₃	1	6	yes	complemented
70	#12506	—	2	6	no	—
71	C₈×M₃₂	C₂²×C₃	1	3	yes	—
72	#7052	C₂²×C₃	1	3	yes	complemented
73	C₁₆²	C₂²×C₃	1	1	yes	characteristic, complemented
74	C₂×C₃×#135	—	4	6	no	—
75	C₂²×C₃×D₈	—	4	6	no	—
76	C₃×#532	—	4	3	no	—
77	C₂×C₃×#127	—	4	1	no	—
78	C₃×#509	—	8	48	no	—
79	C₂×C₃×SD₃₂	—	4	12	no	—
80	C₂×C₃×SD₃₂	—	4	12	no	—
81	C₂×C₃×D₁₆	—	4	12	no	—
82	C₂×C₃×D₁₆	—	4	12	no	—
83	C₂×C₃×D₁₆	—	4	12	no	—
84	C₂×C₃×M₃₂	D₈	1	12	yes	—
85	C₃×#500	—	2	6	no	—
86	C₃×#495	—	4	6	no	—
87	C₃×#494	—	2	3	no	—
88	C₃×#493	—	2	3	no	—
89	C₃×#493	—	2	3	no	—
90	C₃×#493	—	2	3	no	—
91	C₃×#493	—	2	6	no	—
92	C₃×#493	—	4	3	no	—
93	C₃×#493	—	4	6	no	—
94	C₃×#492	—	4	3	no	—
95	C₃×#486	—	4	6	no	—
96	C₃×#466	—	4	12	no	—
97	C₃×C₄×D₈	—	4	12	no	—
98	C₂×C₃×C₄×C₈	C₂×D₄	1	3	yes	—
99	C₃×#367	—	4	12	no	—
100	C₃×#357	—	4	24	no	—
101	C₃×#346	D₈	1	12	yes	—
102	C₃×C₄×C₁₆	C₂×D₄	1	6	yes	—
103	C₃×C₄×C₁₆	C₂×D₄	1	6	yes	—
104	C₃×C₈²	C₂×D₄	1	3	yes	—
105	C₃×C₈²	C₂⁴	1	1	yes	characteristic
106	C₂×#509	—	4	12	no	—
107	#3169	—	2	6	no	—
108	#3147	—	2	3	no	—
109	C₂×#493	—	2	3	no	—
110	#2254	—	2	6	no	—
111	#2250	—	4	12	no	—
112	#2248	—	2	6	no	—
113	#2248	—	2	6	no	—
114	#2246	—	2	6	no	—
115	#2246	—	2	6	no	—
116	#2246	—	2	6	no	—
117	#2240	—	2	12	no	—
118	#2218	—	4	12	no	—
119	#2177	—	4	12	no	—
120	C₄×M₃₂	C₃×D₄	1	6	yes	—
121	#1713	C₂³×C₃	1	3	yes	—
122	#1712	—	2	3	no	—
123	#1712	C₂³×C₃	1	1	yes	characteristic
124	#1712	C₂³×C₃	1	1	yes	characteristic
125	#1712	C₂³×C₃	1	3	yes	—
126	#1709	—	2	3	no	—
127	#1706	—	2	6	no	—
128	C₂×C₈²	C₂³×C₃	1	1	yes	characteristic
129	#1311	C₃×D₄	1	6	yes	—
130	C₈×C₁₆	C₂³×C₃	1	3	yes	—
131	C₈×C₁₆	C₂³×C₃	1	3	yes	—
132	C₂×C₃×#41	—	8	3	no	—
133	C₂²×C₃×D₄	—	8	3	no	—
134	C₃×#135	—	8	24	no	—
135	C₂×C₃×Q₁₆	—	4	6	no	—
136	C₂×C₃×SD₁₆	—	8	6	no	—
137	C₂×C₃×D₈	—	4	6	no	—
138	C₂×C₃×D₈	—	4	6	no	—
139	C₂×C₃×D₈	—	4	6	no	—
140	C₂×C₃×D₈	—	4	12	no	—
141	C₂×C₃×D₈	—	8	24	no	—
142	C₂×C₃×D₈	—	8	6	no	—
143	C₂²×C₃×C₈	C₂×D₈	1	6	yes	—
144	C₃×C₄⋊Q₈	—	4	3	no	—
145	C₃×#127	—	4	1	no	—
146	C₃×#127	—	4	1	no	—
147	C₃×#127	—	4	3	no	—
148	C₃×#127	—	8	3	no	—
149	C₃×#124	—	8	3	no	—
150	C₃×#121	—	8	6	no	—
151	C₃×C₄×D₄	—	8	6	no	—
152	C₂×C₃×C₄²	#127	1	1	yes	characteristic
153	C₃×SD₃₂	—	8	24	no	—
154	C₃×SD₃₂	—	8	24	no	—
155	C₃×D₁₆	—	8	24	no	—
156	C₃×D₁₆	—	8	24	no	—
157	C₃×D₁₆	—	8	24	no	—
158	C₃×M₃₂	—	2	24	no	—
159	C₂×C₃×C₁₆	C₂×D₈	1	12	yes	—
160	C₂×C₃×C₁₆	C₂×D₈	1	12	yes	—
161	C₃×#107	—	4	12	no	—
162	C₃×#102	—	8	12	no	—
163	C₃×C₄×C₈	#127	1	3	yes	—
164	C₃×C₄×C₈	C₂×D₈	1	6	yes	—
165	C₃×C₄×C₈	C₂²×D₄	1	3	yes	—
166	C₂×#135	—	4	6	no	—
167	C₂²×D₈	—	4	6	no	—
168	#532	—	4	3	no	—
169	C₂×#127	—	4	1	no	—
170	#509	—	8	48	no	—
171	C₂×SD₃₂	—	4	12	no	—
172	C₂×SD₃₂	—	4	12	no	—
173	C₂×D₁₆	—	4	12	no	—
174	C₂×D₁₆	—	4	12	no	—
175	C₂×D₁₆	—	4	12	no	—
176	C₂×M₃₂	C₃×D₈	1	12	yes	—
177	#500	—	2	6	no	—
178	#495	—	4	6	no	—
179	#494	—	2	3	no	—
180	#493	—	2	3	no	—
181	#493	—	2	3	no	—
182	#493	—	2	3	no	—
183	#493	—	2	6	no	—
184	#493	—	4	3	no	—
185	#493	—	4	6	no	—
186	#492	—	4	3	no	—
187	#486	—	4	6	no	—
188	#466	—	4	12	no	—
189	C₄×D₈	—	4	12	no	—
190	C₂×C₄×C₈	C₂×C₃×D₄	1	3	yes	—
191	#367	—	4	12	no	—
192	#357	—	4	24	no	—
193	#346	C₃×D₈	1	12	yes	—
194	C₄×C₁₆	C₂×C₃×D₄	1	6	yes	—
195	C₄×C₁₆	C₂×C₃×D₄	1	6	yes	—
196	C₈²	C₂×C₃×D₄	1	3	yes	—
197	C₈²	C₂⁴×C₃	1	1	yes	characteristic
198	C₂⁴×C₃	—	16	1	no	—
199	C₃×#41	—	16	12	no	—
200	C₂×C₃×Q₈	—	8	3	no	—
201	C₂×C₃×D₄	—	8	6	no	—
202	C₂×C₃×D₄	—	8	3	no	—
203	C₂×C₃×D₄	—	8	3	no	—
204	C₂×C₃×D₄	—	8	3	no	—
205	C₂×C₃×D₄	—	16	12	no	—
206	C₂×C₃×D₄	—	16	3	no	—
207	C₂×C₃×D₄	—	16	3	no	—
208	C₂²×C₃×C₄	—	16	3	no	—
209	C₂²×C₃×C₄	#493	1	3	yes	—
210	C₃×Q₁₆	—	8	12	no	—
211	C₃×SD₁₆	—	16	12	no	—
212	C₃×D₈	—	8	12	no	—
213	C₃×D₈	—	8	12	no	—
214	C₃×D₈	—	8	12	no	—
215	C₃×D₈	—	8	24	no	—
216	C₃×D₈	—	16	12	no	—
217	C₂×C₃×C₈	—	2	12	no	—
218	C₂×C₃×C₈	#493	1	6	yes	—
219	C₂×C₃×C₈	C₂²×D₈	1	6	yes	—
220	C₃×C₄⋊C₄	—	8	6	no	—
221	C₃×C₂²⋊C₄	—	16	6	no	—
222	C₃×C₄²	#493	1	3	yes	—
223	C₃×C₄²	C₂×#127	1	1	yes	characteristic
224	C₃×C₁₆	—	2	12	no	—
225	C₃×C₁₆	#509	1	24	yes	complemented
226	C₂×#41	—	8	3	no	—
227	C₂²×D₄	—	8	3	no	—
228	#135	—	8	24	no	—
229	C₂×Q₁₆	—	4	6	no	—
230	C₂×SD₁₆	—	8	6	no	—
231	C₂×D₈	—	4	6	no	—
232	C₂×D₈	—	4	6	no	—
233	C₂×D₈	—	4	6	no	—
234	C₂×D₈	—	4	12	no	—
235	C₂×D₈	—	8	24	no	—
236	C₂×D₈	—	8	6	no	—
237	C₂²×C₈	C₂×C₃×D₈	1	6	yes	—
238	C₄⋊Q₈	—	4	3	no	—
239	#127	—	4	1	no	—
240	#127	—	4	1	no	—
241	#127	—	4	3	no	—
242	#127	—	8	3	no	—
243	#124	—	8	3	no	—
244	#121	—	8	6	no	—
245	C₄×D₄	—	8	6	no	—
246	C₂×C₄²	C₃×#127	1	1	yes	characteristic
247	SD₃₂	—	8	24	no	—
248	SD₃₂	—	8	24	no	—
249	D₁₆	—	8	24	no	—
250	D₁₆	—	8	24	no	—
251	D₁₆	—	8	24	no	—
252	M₃₂	—	2	24	no	—
253	C₂×C₁₆	C₂×C₃×D₈	1	12	yes	—
254	C₂×C₁₆	C₂×C₃×D₈	1	12	yes	—
255	#107	—	4	12	no	—
256	#102	—	8	12	no	—
257	C₄×C₈	C₃×#127	1	3	yes	—
258	C₄×C₈	C₂×C₃×D₈	1	6	yes	—
259	C₄×C₈	C₂²×C₃×D₄	1	3	yes	—
260	C₂³×C₃	—	16	3	no	—
261	C₂³×C₃	—	16	1	no	—
262	C₂³×C₃	—	16	1	no	—
263	C₂³×C₃	—	32	6	no	—
264	C₂³×C₃	#1712	1	1	yes	characteristic
265	C₃×Q₈	—	16	6	no	—
266	C₃×D₄	—	16	6	no	—
267	C₃×D₄	—	16	6	no	—
268	C₃×D₄	—	16	6	no	—
269	C₃×D₄	—	16	12	no	—
270	C₃×D₄	—	32	6	no	—
271	C₃×D₄	—	32	6	no	—
272	C₂×C₃×C₄	—	2	6	no	—
273	C₂×C₃×C₄	—	16	3	no	—
274	C₂×C₃×C₄	—	32	6	no	—
275	C₂×C₃×C₄	#1712	1	3	yes	—
276	C₂×C₃×C₄	C₂×#493	1	3	yes	—
277	C₃×C₈	—	2	6	no	—
278	C₃×C₈	C₂×#509	1	12	yes	—
279	C₂⁴	—	16	1	no	—
280	#41	—	16	12	no	—
281	C₂×Q₈	—	8	3	no	—
282	C₂×D₄	—	8	6	no	—
283	C₂×D₄	—	8	3	no	—
284	C₂×D₄	—	8	3	no	—
285	C₂×D₄	—	8	3	no	—
286	C₂×D₄	—	16	12	no	—
287	C₂×D₄	—	16	3	no	—
288	C₂×D₄	—	16	3	no	—
289	C₂²×C₄	—	16	3	no	—
290	C₂²×C₄	C₃×#493	1	3	yes	—
291	Q₁₆	—	8	12	no	—
292	SD₁₆	—	16	12	no	—
293	D₈	—	8	12	no	—
294	D₈	—	8	12	no	—
295	D₈	—	8	12	no	—
296	D₈	—	8	24	no	—
297	D₈	—	16	12	no	—
298	C₂×C₈	—	2	12	no	—
299	C₂×C₈	C₃×#493	1	6	yes	—
300	C₂×C₈	C₂²×C₃×D₈	1	6	yes	—
301	C₄⋊C₄	—	8	6	no	—
302	C₂²⋊C₄	—	16	6	no	—
303	C₄²	C₃×#493	1	3	yes	—
304	C₄²	C₂×C₃×#127	1	1	yes	characteristic
305	C₁₆	—	2	12	no	—
306	C₁₆	C₃×#509	1	24	yes	complemented
307	C₂²×C₃	—	2	3	no	—
308	C₂²×C₃	—	32	3	no	—
309	C₂²×C₃	—	32	3	no	—
310	C₂²×C₃	—	32	3	no	—
311	C₂²×C₃	—	32	6	no	—
312	C₂²×C₃	—	64	4	no	—
313	C₂²×C₃	C₂×#1712	1	1	yes	characteristic
314	C₃×C₄	—	2	3	no	—
315	C₃×C₄	—	32	3	no	—
316	C₃×C₄	#33717	1	6	yes	—
317	C₂³	—	16	3	no	—
318	C₂³	—	16	1	no	—
319	C₂³	—	16	1	no	—
320	C₂³	—	32	6	no	—
321	C₂³	C₃×#1712	1	1	yes	characteristic
322	Q₈	—	16	6	no	—
323	D₄	—	16	6	no	—
324	D₄	—	16	6	no	—
325	D₄	—	16	6	no	—
326	D₄	—	16	12	no	—
327	D₄	—	32	6	no	—
328	D₄	—	32	6	no	—
329	C₂×C₄	—	2	6	no	—
330	C₂×C₄	—	16	3	no	—
331	C₂×C₄	—	32	6	no	—
332	C₂×C₄	C₃×#1712	1	3	yes	—
333	C₂×C₄	C₂×C₃×#493	1	3	yes	—
334	C₈	—	2	6	no	—
335	C₈	C₂×C₃×#509	1	12	yes	—
336	C₂×C₃	—	4	1	no	—
337	C₂×C₃	—	64	3	no	—
338	C₂×C₃	—	64	1	no	—
339	C₂×C₃	—	64	1	no	—
340	C₂×C₃	#491469	1	3	yes	—
341	C₂²	—	2	3	no	—
342	C₂²	—	32	3	no	—
343	C₂²	—	32	3	no	—
344	C₂²	—	32	3	no	—
345	C₂²	—	32	6	no	—
346	C₂²	—	64	4	no	—
347	C₂²	C₂×C₃×#1712	1	1	yes	characteristic
348	C₄	—	2	3	no	—
349	C₄	—	32	3	no	—
350	C₄	C₃×#33717	1	6	yes	—
351	C₃	#72213956	1	1	yes	characteristic, direct product factor, Sylow
352	C₂	—	4	1	no	—
353	C₂	—	64	1	no	—
354	C₂	—	64	1	no	—
355	C₂	—	64	3	no	—
356	C₂	C₃×#491469	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#1712
commutator, Frattini, cosocle	C₈²	C₂⁴×C₃
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₈² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#123411	C₂	semidirect
2. C₃×#123403	C₂	semidirect
3. C₃×#123419	C₂	semidirect
4. C₃×#123412	C₂	semidirect
5. C₃×#122211	C₂	semidirect
6. C₃×#491500	C₂	semidirect
7. C₃×#491469	C₂	semidirect
8. #72213956	C₃	direct (N × H)
13. C₃×#12523	C₂²	semidirect
19. C₃×#12515	C₂²	semidirect
20. C₃×#12509	C₂²	semidirect
23. C₃×#7052	C₂²	semidirect
24. C₃×C₁₆²	C₂²	semidirect
25. #491500	C₂×C₃	semidirect
26. #491469	C₂×C₃	semidirect
27. #123419	C₂×C₃	semidirect
28. #123412	C₂×C₃	semidirect
29. #123411	C₂×C₃	semidirect
30. #123403	C₂×C₃	semidirect
31. #122211	C₂×C₃	semidirect
62. #12523	C₂²×C₃	semidirect
68. #12515	C₂²×C₃	semidirect
69. #12509	C₂²×C₃	semidirect
72. #7052	C₂²×C₃	semidirect
73. C₁₆²	C₂²×C₃	semidirect
225. C₃×C₁₆	#509	semidirect
306. C₁₆	C₃×#509	semidirect
351. C₃	#72213956	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#123411	2
C₃×#123403	2
C₃×#123419	2
C₃×#123412	2
C₃×#122211	2
C₃×#491500	2
C₃×#491469	2
#72213956	3

Subgroups — C3×#72213956 #72226108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72213956 #72226108