Subgroups — C₃×#72016 #11785615

Order 1152.

661 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₃×#5232	C₂	1	2	yes	maximal
3	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#818	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4554	C₂	1	2	yes	maximal, complemented
6	C₃×#1816	—	3	1	no	maximal
7	#72016	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
9	C₃²×#131	—	2	2	no	—
10	C₃²×#131	C₄	1	4	yes	complemented
11	C₂×C₃²×M₁₆	—	2	1	no	—
12	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×C₈	—	2	1	no	—
14	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₃×#208	C₂²	1	2	yes	—
17	C₃×#835	—	2	2	no	—
18	C₃×#832	—	2	2	no	—
19	C₃×#819	—	2	2	no	—
20	C₃×#818	—	2	1	no	—
21	C₃×#818	—	2	1	no	—
22	C₂×C₃×#131	—	2	1	no	—
23	C₂×C₃×#131	D₃	1	1	yes	characteristic
24	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×#108	—	3	1	no	—
26	C₂×C₃×#107	—	3	1	no	—
27	C₃×#417	—	3	2	no	—
28	#5232	C₂×C₃	1	2	yes	—
29	C₂×#819	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×#818	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#343	—	3	2	no	—
32	#4554	C₂×C₃	1	2	yes	complemented
33	C₃²×#41	—	2	1	no	—
34	C₃²×#41	—	2	1	no	—
35	C₃²×#41	C₂×C₄	1	2	yes	—
36	C₃²×#41	Q₈	1	2	yes	—
37	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
38	C₂×C₃²×D₄	—	2	1	no	—
39	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
40	C₂²×C₃²×C₄	—	2	1	no	—
41	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₂²×C₃×Dic₃	—	2	2	no	—
43	C₃²×M₁₆	—	2	1	no	—
44	C₃²×M₁₆	—	2	4	no	—
45	C₃²×M₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₃²×C₈	—	2	1	no	—
47	C₂×C₃²×C₈	—	2	4	no	—
48	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
49	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
50	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
51	C₂×C₃²×C₈	D₄	1	2	yes	—
52	C₃×C₄⋊Dic₃	—	2	2	no	—
53	C₃×C₄⋊Dic₃	—	2	2	no	—
54	C₃×#208	—	2	2	no	—
55	C₃×#208	—	2	2	no	—
56	C₂×C₃×C₃⋊C₈	—	2	2	no	—
57	#1816	—	3	1	no	Sylow
58	C₂×C₃×#41	—	2	1	no	—
59	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
61	C₃×#131	—	2	4	no	—
62	C₃×#131	—	2	2	no	—
63	C₃×#131	—	2	2	no	—
64	C₃×#131	—	2	4	no	—
65	C₃×#131	Dic₃	1	4	yes	complemented
66	C₃×#131	C₃×C₄	1	4	yes	complemented
67	C₂×C₃×M₁₆	—	2	2	no	—
68	C₂×C₃×M₁₆	—	2	1	no	—
69	C₂×C₃×M₁₆	—	2	1	no	—
70	C₂×C₃×M₁₆	—	2	1	no	—
71	C₂×C₃×M₁₆	—	3	2	no	—
72	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₃×C₈	—	2	2	no	—
75	C₂²×C₃×C₈	—	2	1	no	—
76	C₂²×C₃×C₈	—	2	1	no	—
77	C₂²×C₃×C₈	—	2	1	no	—
78	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
80	C₂×C₃×C₄⋊C₄	—	3	2	no	—
81	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
82	C₂×#208	C₂²×C₃	1	2	yes	—
83	C₃×#108	—	6	2	no	—
84	C₃×#107	—	6	1	no	—
85	C₃×#107	—	6	1	no	—
86	C₃×#103	—	6	2	no	—
87	C₃×#102	—	6	2	no	—
88	#835	—	2	2	no	—
89	#832	—	2	2	no	—
90	#819	—	2	2	no	—
91	#818	—	2	1	no	—
92	#818	—	2	1	no	—
93	C₂³×C₃²	—	2	1	no	—
94	C₂³×C₃²	#41	1	1	yes	characteristic
95	C₃²×Q₈	—	2	1	no	—
96	C₃²×Q₈	C₄⋊C₄	1	2	yes	—
97	C₃²×D₄	—	2	2	no	—
98	C₃²×D₄	—	2	2	no	—
99	C₃²×D₄	—	2	1	no	—
100	C₃²×D₄	C₄⋊C₄	1	2	yes	—
101	C₂×C₃²×C₄	—	2	1	no	—
102	C₂×C₃²×C₄	—	2	2	no	—
103	C₂×C₃²×C₄	—	2	2	no	—
104	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₃²×C₄	#41	1	1	yes	characteristic
110	C₂×C₃×Dic₃	—	4	2	no	—
111	C₂×C₃×Dic₃	—	8	2	no	—
112	C₃²×C₈	—	2	2	no	—
113	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
114	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
115	C₃×C₃⋊C₈	—	4	2	no	—
116	C₂×#131	C₃×D₃	1	1	yes	characteristic
117	C₂×#108	—	3	1	no	—
118	C₂×#107	—	3	1	no	—
119	#417	—	3	2	no	—
120	#343	—	3	2	no	—
121	C₃×#41	—	2	2	no	—
122	C₃×#41	—	2	2	no	—
123	C₃×#41	—	2	1	no	—
124	C₃×#41	—	2	1	no	—
125	C₃×#41	—	2	1	no	—
126	C₃×#41	—	2	1	no	—
127	C₃×#41	—	2	2	no	—
128	C₃×#41	—	2	2	no	—
129	C₃×#41	Dic₆	1	2	yes	—
130	C₃×#41	C₂×Dic₃	1	2	yes	—
131	C₃×#41	C₂×C₃×C₄	1	2	yes	—
132	C₃×#41	C₃×Q₈	1	2	yes	—
133	C₂×C₃×Q₈	—	2	1	no	—
134	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
135	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
136	C₂×C₃×D₄	—	2	2	no	—
137	C₂×C₃×D₄	—	2	1	no	—
138	C₂×C₃×D₄	—	2	1	no	—
139	C₂×C₃×D₄	—	2	1	no	—
140	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
141	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
142	C₂²×C₃×C₄	—	2	2	no	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂²×C₃×C₄	—	2	1	no	—
145	C₂²×C₃×C₄	—	2	1	no	—
146	C₂²×C₃×C₄	—	6	2	no	—
147	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×Dic₃	—	2	2	no	—
150	C₃×M₁₆	—	2	2	no	—
151	C₃×M₁₆	—	2	8	no	—
152	C₃×M₁₆	—	2	1	no	—
153	C₃×M₁₆	—	2	1	no	—
154	C₃×M₁₆	—	2	4	no	—
155	C₃×M₁₆	—	2	4	no	—
156	C₃×M₁₆	—	2	2	no	—
157	C₃×M₁₆	—	6	2	no	—
158	C₃×M₁₆	—	6	2	no	—
159	C₃×M₁₆	C₂×Dic₃	1	2	yes	—
160	C₃×M₁₆	C₂×C₃×C₄	1	2	yes	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₂×C₃×C₈	—	2	8	no	—
163	C₂×C₃×C₈	—	2	1	no	—
164	C₂×C₃×C₈	—	2	1	no	—
165	C₂×C₃×C₈	—	2	4	no	—
166	C₂×C₃×C₈	—	2	4	no	—
167	C₂×C₃×C₈	—	2	2	no	—
168	C₂×C₃×C₈	—	2	1	no	—
169	C₂×C₃×C₈	—	2	1	no	—
170	C₂×C₃×C₈	—	2	2	no	—
171	C₂×C₃×C₈	—	6	2	no	—
172	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
173	C₂×C₃×C₈	#67	1	1	yes	characteristic
174	C₂×C₃×C₈	#67	1	1	yes	characteristic
175	C₂×C₃×C₈	#67	1	2	yes	—
176	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
177	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
178	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
180	C₃×C₄⋊C₄	—	6	2	no	—
181	C₃×C₄⋊C₄	—	6	2	no	—
182	C₄⋊Dic₃	—	2	2	no	—
183	C₄⋊Dic₃	—	2	2	no	—
184	#208	—	2	2	no	—
185	#208	—	2	2	no	—
186	C₂×C₃⋊C₈	—	2	2	no	—
187	C₂²×C₃²	—	2	2	no	—
188	C₂²×C₃²	—	2	1	no	—
189	C₂²×C₃²	—	2	1	no	—
190	C₂²×C₃²	—	4	1	no	—
191	C₂²×C₃²	—	4	1	no	—
192	C₂²×C₃²	#117	1	1	yes	characteristic
193	C₂²×C₃²	#121	1	1	yes	characteristic
194	C₂²×C₃²	C₂²⋊Q₈	1	1	yes	characteristic
195	C₃²×C₄	—	2	2	no	—
196	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
198	C₃²×C₄	#121	1	1	yes	characteristic
199	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
200	C₃×Dic₃	—	8	2	no	—
201	C₂×#41	C₂×C₃×D₃	1	1	yes	characteristic
202	#131	—	2	2	no	—
203	#131	C₃×Dic₃	1	4	yes	complemented
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	3	2	no	—
206	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	3	2	no	—
210	#108	—	6	2	no	—
211	#107	—	6	1	no	—
212	#107	—	6	1	no	—
213	#103	—	6	2	no	—
214	#102	—	6	2	no	—
215	C₂³×C₃	—	2	2	no	—
216	C₂³×C₃	—	2	1	no	—
217	C₂³×C₃	—	2	1	no	—
218	C₂³×C₃	—	2	1	no	—
219	C₂³×C₃	#235	1	1	yes	characteristic
220	C₂³×C₃	C₃×#41	1	1	yes	characteristic
221	C₃×Q₈	—	2	2	no	—
222	C₃×Q₈	—	2	1	no	—
223	C₃×Q₈	—	2	1	no	—
224	C₃×Q₈	—	2	2	no	—
225	C₃×Q₈	C₄⋊Dic₃	1	2	yes	—
226	C₃×Q₈	C₃×C₄⋊C₄	1	2	yes	—
227	C₃×D₄	—	2	4	no	—
228	C₃×D₄	—	2	4	no	—
229	C₃×D₄	—	2	2	no	—
230	C₃×D₄	—	2	2	no	—
231	C₃×D₄	—	2	2	no	—
232	C₃×D₄	—	2	2	no	—
233	C₃×D₄	—	2	2	no	—
234	C₃×D₄	—	2	1	no	—
235	C₃×D₄	—	2	1	no	—
236	C₃×D₄	—	2	2	no	—
237	C₃×D₄	C₄⋊Dic₃	1	2	yes	—
238	C₃×D₄	C₃×C₄⋊C₄	1	2	yes	—
239	C₂×C₃×C₄	—	2	4	no	—
240	C₂×C₃×C₄	—	2	2	no	—
241	C₂×C₃×C₄	—	2	4	no	—
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	2	2	no	—
245	C₂×C₃×C₄	—	2	2	no	—
246	C₂×C₃×C₄	—	2	2	no	—
247	C₂×C₃×C₄	—	2	2	no	—
248	C₂×C₃×C₄	—	2	1	no	—
249	C₂×C₃×C₄	—	2	1	no	—
250	C₂×C₃×C₄	—	2	1	no	—
251	C₂×C₃×C₄	—	2	1	no	—
252	C₂×C₃×C₄	—	2	1	no	—
253	C₂×C₃×C₄	—	2	1	no	—
254	C₂×C₃×C₄	—	12	2	no	—
255	C₂×C₃×C₄	—	24	2	no	—
256	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
257	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
258	C₂×C₃×C₄	#235	1	1	yes	characteristic
259	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
260	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
261	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
262	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
263	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
264	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
265	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
266	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
267	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
268	C₂×Dic₃	—	4	2	no	—
269	C₂×Dic₃	—	8	2	no	—
270	C₃×C₈	—	2	4	no	—
271	C₃×C₈	—	2	2	no	—
272	C₃×C₈	—	2	2	no	—
273	C₃×C₈	—	2	2	no	—
274	C₃×C₈	—	2	2	no	—
275	C₃×C₈	—	12	2	no	—
276	C₃×C₈	#217	1	2	yes	—
277	C₃×C₈	#217	1	2	yes	—
278	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
279	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
280	C₃⋊C₈	—	4	2	no	—
281	C₂×C₃²	—	2	1	no	—
282	C₂×C₃²	—	2	1	no	—
283	C₂×C₃²	—	4	2	no	—
284	C₂×C₃²	#380	1	1	yes	characteristic
285	C₂×C₃²	#472	1	1	yes	characteristic
286	C₂×C₃²	#473	1	1	yes	characteristic
287	#41	—	2	1	no	—
288	#41	—	2	1	no	—
289	#41	C₃×Dic₆	1	2	yes	—
290	#41	C₂×C₃×Dic₃	1	2	yes	—
291	C₂×Q₈	C₃×D₁₂	1	1	yes	characteristic
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	C₃×D₁₂	1	1	yes	characteristic
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	6	2	no	—
296	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
297	M₁₆	—	2	1	no	—
298	M₁₆	—	2	4	no	—
299	M₁₆	—	6	2	no	—
300	M₁₆	—	6	2	no	—
301	M₁₆	C₂×C₃×Dic₃	1	2	yes	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	4	no	—
304	C₂×C₈	—	6	2	no	—
305	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
306	C₂×C₈	C₃×#67	1	1	yes	characteristic
307	C₂×C₈	C₃×#67	1	1	yes	characteristic
308	C₂×C₈	C₃×#67	1	2	yes	—
309	C₄⋊C₄	—	6	2	no	—
310	C₄⋊C₄	—	6	2	no	—
311	C₂²×C₃	—	2	1	no	—
312	C₂²×C₃	—	2	1	no	—
313	C₂²×C₃	—	2	1	no	—
314	C₂²×C₃	—	2	1	no	—
315	C₂²×C₃	—	2	4	no	—
316	C₂²×C₃	—	2	2	no	—
317	C₂²×C₃	—	2	2	no	—
318	C₂²×C₃	—	2	1	no	—
319	C₂²×C₃	—	2	1	no	—
320	C₂²×C₃	—	2	1	no	—
321	C₂²×C₃	—	4	1	no	—
322	C₂²×C₃	—	4	1	no	—
323	C₂²×C₃	—	4	2	no	—
324	C₂²×C₃	—	4	2	no	—
325	C₂²×C₃	—	4	1	no	—
326	C₂²×C₃	—	4	1	no	—
327	C₂²×C₃	—	4	1	no	—
328	C₂²×C₃	—	4	1	no	—
329	C₂²×C₃	#924	1	1	yes	characteristic
330	C₂²×C₃	#926	1	1	yes	characteristic
331	C₂²×C₃	#930	1	1	yes	characteristic
332	C₂²×C₃	C₃×#117	1	1	yes	characteristic
333	C₂²×C₃	C₃×#121	1	1	yes	characteristic
334	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
335	C₃×C₄	—	2	4	no	—
336	C₃×C₄	—	2	2	no	—
337	C₃×C₄	—	2	2	no	—
338	C₃×C₄	—	2	1	no	—
339	C₃×C₄	—	2	1	no	—
340	C₃×C₄	—	2	1	no	—
341	C₃×C₄	—	2	1	no	—
342	C₃×C₄	—	24	2	no	—
343	C₃×C₄	#924	1	1	yes	characteristic
344	C₃×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
345	C₃×C₄	#930	1	1	yes	characteristic
346	C₃×C₄	C₂×#217	1	1	yes	characteristic
347	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
348	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
349	C₃×C₄	C₃×#121	1	1	yes	characteristic
350	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
351	Dic₃	—	8	2	no	—
352	C₃²	#1816	1	1	yes	characteristic, complemented, Sylow
353	C₂³	—	2	1	no	—
354	C₂³	C₃×#235	1	1	yes	characteristic
355	Q₈	—	2	1	no	—
356	Q₈	C₃×C₄⋊Dic₃	1	2	yes	—
357	D₄	—	2	2	no	—
358	D₄	—	2	2	no	—
359	D₄	—	2	1	no	—
360	D₄	C₃×C₄⋊Dic₃	1	2	yes	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	12	2	no	—
365	C₂×C₄	—	24	2	no	—
366	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
367	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
368	C₂×C₄	C₃×#235	1	1	yes	characteristic
369	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
370	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
371	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
372	C₈	—	2	2	no	—
373	C₈	—	12	2	no	—
374	C₈	C₃×#217	1	2	yes	—
375	C₈	C₃×#217	1	2	yes	—
376	C₂×C₃	—	2	1	no	—
377	C₂×C₃	—	2	1	no	—
378	C₂×C₃	—	2	1	no	—
379	C₂×C₃	—	2	1	no	—
380	C₂×C₃	—	2	1	no	—
381	C₂×C₃	—	2	1	no	—
382	C₂×C₃	—	2	1	no	—
383	C₂×C₃	—	4	1	no	—
384	C₂×C₃	—	4	1	no	—
385	C₂×C₃	—	4	4	no	—
386	C₂×C₃	—	4	2	no	—
387	C₂×C₃	—	4	2	no	—
388	C₂×C₃	#5142	1	1	yes	characteristic
389	C₂×C₃	#5143	1	1	yes	characteristic
390	C₂×C₃	#5212	1	1	yes	characteristic
391	C₂×C₃	C₃×#380	1	1	yes	characteristic
392	C₂×C₃	C₃×#472	1	1	yes	characteristic
393	C₂×C₃	C₃×#473	1	1	yes	characteristic
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	2	no	—
397	C₂²	—	4	1	no	—
398	C₂²	—	4	1	no	—
399	C₂²	C₃×#924	1	1	yes	characteristic
400	C₂²	C₃×#926	1	1	yes	characteristic
401	C₂²	C₃×#930	1	1	yes	characteristic
402	C₄	—	2	2	no	—
403	C₄	—	24	2	no	—
404	C₄	C₃×#924	1	1	yes	characteristic
405	C₄	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
406	C₄	C₃×#930	1	1	yes	characteristic
407	C₄	C₂×C₃×#217	1	1	yes	characteristic
408	C₃	—	2	1	no	—
409	C₃	#72016	1	1	yes	characteristic, direct product factor
410	C₃	C₃×#1816	1	1	yes	characteristic, complemented
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	4	2	no	—
414	C₂	C₃×#5142	1	1	yes	characteristic
415	C₂	C₃×#5143	1	1	yes	characteristic
416	C₂	C₃×#5212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#930
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂²×C₃²	#121
prime radical	C₂×C₃²×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#131	D₃
hypocenter	C₃	C₃×#1816
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#131

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#819	C₂	semidirect
4. C₂×C₃×#818	C₂	semidirect
5. C₃×#4554	C₂	semidirect
7. #72016	C₃	direct (N × H)
10. C₃²×#131	C₄	semidirect
29. C₂×#819	C₂×C₃	semidirect
30. C₂×#818	C₂×C₃	semidirect
32. #4554	C₂×C₃	semidirect
65. C₃×#131	Dic₃	semidirect
66. C₃×#131	C₃×C₄	semidirect
203. #131	C₃×Dic₃	semidirect
352. C₃²	#1816	semidirect
409. C₃	#72016	direct (N × H)
410. C₃	C₃×#1816	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₃×#5232	2
C₂×C₃×#819	2
C₂×C₃×#818	2
C₃×#4554	2
C₃×#1816	3
#72016	3

Subgroups — C3×#72016 #11785615

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72016 #11785615