Subgroups — C₃×#5614 #10595711

Order 576.

523 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#938	C₂	1	2	yes	maximal, complemented
3	C₃×#938	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#935	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×#67	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#926	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#890	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#67⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₃×#879	C₂	1	2	yes	maximal, complemented
12	C₃×#553	—	3	1	no	maximal
13	#5614	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
15	C₂×C₃²×D₄	C₂²	1	2	yes	—
16	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
17	C₂×C₃×#67	C₂²	1	2	yes	—
18	C₂×C₃×#67	C₂²	1	1	yes	characteristic
19	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
22	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×#217	C₂²	1	2	yes	—
25	C₃×#217	C₂²	1	2	yes	—
26	C₃×#217	C₂²	1	2	yes	—
27	C₃×#217	C₂²	1	1	yes	characteristic
28	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
32	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₃×C₄×Dic₃	C₂²	1	2	yes	—
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×#126	—	3	2	no	—
38	C₃×#125	—	3	1	no	—
39	C₃×#123	—	3	2	no	—
40	C₃×#121	—	2	1	no	—
41	C₃×#121	—	3	2	no	—
42	C₃×#121	—	3	1	no	—
43	C₃×#121	D₃	1	1	yes	characteristic, complemented
44	C₃×#121	C₂×C₃	1	1	yes	characteristic, complemented
45	C₃×C₄×D₄	—	3	1	no	—
46	C₃×C₄×D₄	—	3	1	no	—
47	C₃×C₄×D₄	—	3	2	no	—
48	C₃×#117	—	3	1	no	—
49	C₂×C₃×C₄⋊C₄	—	3	1	no	—
50	#938	C₂×C₃	1	2	yes	complemented
51	#938	C₂×C₃	1	1	yes	characteristic, complemented
52	#935	C₂×C₃	1	2	yes	complemented
53	D₄×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
54	C₄×#67	C₂×C₃	1	1	yes	characteristic, complemented
55	#926	C₂×C₃	1	1	yes	characteristic, complemented
56	D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
57	#890	C₂×C₃	1	1	yes	characteristic, complemented
58	#67⋊C₄	C₂×C₃	1	2	yes	complemented
59	#879	C₂×C₃	1	2	yes	complemented
60	C₂³×C₃²	C₂³	1	1	yes	characteristic
61	C₂³×C₃²	C₂³	1	2	yes	—
62	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
63	C₃²×D₄	—	2	2	no	—
64	C₃²×D₄	—	4	2	no	—
65	C₂×C₃²×C₄	—	2	2	no	—
66	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂³	1	2	yes	—
68	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
69	C₃×#67	—	2	4	no	—
70	C₃×#67	—	2	2	no	—
71	C₂×C₃×Dic₃	—	4	2	no	—
72	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
73	C₂×C₃×Dic₃	C₂³	1	2	yes	—
74	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
75	C₂×C₃×Dic₃	C₂³	1	2	yes	—
76	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
77	C₃×C₄×D₃	—	2	2	no	—
78	C₃×C₄×D₃	—	4	2	no	—
79	#553	—	3	1	no	Sylow
80	C₂×C₃×D₄	—	2	1	no	—
81	C₂×C₃×D₄	—	2	2	no	—
82	C₂×C₃×D₄	—	3	1	no	—
83	C₂×C₃×D₄	—	3	2	no	—
84	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
85	C₂×C₃×D₄	C₂×D₃	1	2	yes	—
86	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₃×D₄	C₂²×C₃	1	2	yes	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	3	2	no	—
90	C₂²×C₃×C₄	—	3	1	no	—
91	C₂²×C₃×C₄	—	3	2	no	—
92	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
94	C₂×#67	C₂²×C₃	1	2	yes	—
95	C₂×#67	C₂²×C₃	1	1	yes	characteristic
96	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
97	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
99	C₃×C₄⋊C₄	—	2	1	no	—
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₄⋊C₄	—	3	2	no	—
102	C₃×C₄⋊C₄	—	3	1	no	—
103	C₃×C₄⋊C₄	—	3	1	no	—
104	C₃×C₄⋊C₄	—	3	1	no	—
105	C₃×C₄⋊C₄	—	3	2	no	—
106	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
107	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₃×C₂²⋊C₄	—	2	2	no	—
109	C₃×C₂²⋊C₄	—	3	2	no	—
110	C₃×C₂²⋊C₄	—	3	2	no	—
111	C₃×C₂²⋊C₄	—	3	1	no	—
112	C₃×C₂²⋊C₄	—	3	2	no	—
113	C₃×C₂²⋊C₄	—	3	1	no	—
114	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
115	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
116	C₃×C₄²	—	3	2	no	—
117	C₃×C₄²	—	3	1	no	—
118	#217	C₂²×C₃	1	2	yes	—
119	#217	C₂²×C₃	1	2	yes	—
120	#217	C₂²×C₃	1	2	yes	—
121	#217	C₂²×C₃	1	1	yes	characteristic
122	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
123	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
124	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
125	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
126	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
127	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
128	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
129	C₄×Dic₃	C₂²×C₃	1	2	yes	—
130	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
131	C₂²×C₃²	—	2	1	no	—
132	C₂²×C₃²	—	2	1	no	—
133	C₂²×C₃²	—	2	2	no	—
134	C₂²×C₃²	—	2	1	no	—
135	C₂²×C₃²	—	2	2	no	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
138	C₂×C₃×D₃	—	2	1	no	—
139	C₂×C₃×D₃	—	2	1	no	—
140	C₂×C₃×D₃	#41	1	2	yes	—
141	C₃²×C₄	—	2	1	no	—
142	C₃²×C₄	—	2	2	no	—
143	C₃²×C₄	#41	1	2	yes	—
144	C₃×Dic₃	—	2	1	no	—
145	C₃×Dic₃	—	2	2	no	—
146	C₃×Dic₃	—	2	1	no	—
147	C₃×Dic₃	—	2	2	no	—
148	C₃×Dic₃	—	2	1	no	—
149	#126	—	3	2	no	—
150	#125	—	3	1	no	—
151	#123	—	3	2	no	—
152	#121	—	3	2	no	—
153	#121	—	3	1	no	—
154	#121	C₃×D₃	1	1	yes	characteristic, complemented
155	C₄×D₄	—	3	1	no	—
156	C₄×D₄	—	3	1	no	—
157	C₄×D₄	—	3	2	no	—
158	#117	—	3	1	no	—
159	C₂×C₄⋊C₄	—	3	1	no	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	—	3	1	no	—
163	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
164	C₂³×C₃	C₂²×D₃	1	2	yes	—
165	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
166	C₂³×C₃	C₂³×C₃	1	2	yes	—
167	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
168	C₃×D₄	—	2	2	no	—
169	C₃×D₄	—	2	2	no	—
170	C₃×D₄	—	4	2	no	—
171	C₃×D₄	—	4	2	no	—
172	C₃×D₄	—	4	2	no	—
173	C₃×D₄	—	4	4	no	—
174	C₃×D₄	—	6	2	no	—
175	C₃×D₄	—	6	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	3	2	no	—
182	C₂×C₃×C₄	—	3	1	no	—
183	C₂×C₃×C₄	—	3	1	no	—
184	C₂×C₃×C₄	—	3	2	no	—
185	C₂×C₃×C₄	—	3	1	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	6	2	no	—
188	C₂×C₃×C₄	—	12	2	no	—
189	C₂×C₃×C₄	—	12	2	no	—
190	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
192	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
195	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
196	#67	—	2	4	no	—
197	#67	—	2	2	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
200	C₂×Dic₃	C₂³×C₃	1	2	yes	—
201	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
202	C₂×Dic₃	C₂³×C₃	1	2	yes	—
203	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
204	C₄×D₃	—	2	2	no	—
205	C₄×D₃	—	4	2	no	—
206	C₂×C₃²	—	4	1	no	—
207	C₂×C₃²	—	4	2	no	—
208	C₂×C₃²	C₂×#41	1	1	yes	characteristic
209	C₂×C₃²	C₂×#41	1	1	yes	characteristic
210	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
211	C₃×D₃	—	2	2	no	—
212	C₂×D₄	—	3	1	no	—
213	C₂×D₄	—	3	2	no	—
214	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
215	C₂×D₄	C₂×C₃×D₃	1	2	yes	—
216	C₂²×C₄	—	3	2	no	—
217	C₂²×C₄	—	3	1	no	—
218	C₂²×C₄	—	3	2	no	—
219	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
220	C₄⋊C₄	—	3	1	no	—
221	C₄⋊C₄	—	3	2	no	—
222	C₄⋊C₄	—	3	1	no	—
223	C₄⋊C₄	—	3	2	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	3	2	no	—
226	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
227	C₂²⋊C₄	—	3	2	no	—
228	C₂²⋊C₄	—	3	2	no	—
229	C₂²⋊C₄	—	3	1	no	—
230	C₂²⋊C₄	—	3	2	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
233	C₄²	—	3	2	no	—
234	C₄²	—	3	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	3	2	no	—
250	C₂²×C₃	—	4	1	no	—
251	C₂²×C₃	—	4	1	no	—
252	C₂²×C₃	—	4	2	no	—
253	C₂²×C₃	—	4	2	no	—
254	C₂²×C₃	—	4	2	no	—
255	C₂²×C₃	—	6	1	no	—
256	C₂²×C₃	—	6	1	no	—
257	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
258	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
259	C₂×D₃	—	2	1	no	—
260	C₂×D₃	—	2	1	no	—
261	C₂×D₃	C₃×#41	1	2	yes	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	4	1	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	—	6	1	no	—
270	C₃×C₄	—	6	2	no	—
271	C₃×C₄	—	6	1	no	—
272	C₃×C₄	—	6	2	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	#237	1	2	yes	—
275	C₃×C₄	C₃×#41	1	2	yes	—
276	Dic₃	—	2	1	no	—
277	Dic₃	—	2	2	no	—
278	Dic₃	—	2	1	no	—
279	Dic₃	—	2	2	no	—
280	Dic₃	—	2	1	no	—
281	C₃²	#553	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	3	1	no	—
283	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
284	C₂³	C₂²×C₃×D₃	1	2	yes	—
285	D₄	—	2	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	6	2	no	—
288	D₄	—	6	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	3	2	no	—
291	C₂×C₄	—	3	1	no	—
292	C₂×C₄	—	3	2	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	1	no	—
295	C₂×C₄	—	6	2	no	—
296	C₂×C₄	—	12	2	no	—
297	C₂×C₄	—	12	2	no	—
298	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
300	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	4	2	no	—
305	C₂×C₃	—	4	1	no	—
306	C₂×C₃	—	4	2	no	—
307	C₂×C₃	—	4	1	no	—
308	C₂×C₃	—	4	2	no	—
309	C₂×C₃	—	6	2	no	—
310	C₂×C₃	—	8	2	no	—
311	C₂×C₃	C₂×#237	1	1	yes	characteristic
312	C₂×C₃	#1004	1	1	yes	characteristic
313	C₂×C₃	D₃×#41	1	1	yes	characteristic
314	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
315	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
316	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
317	D₃	—	2	2	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	3	2	no	—
325	C₂²	—	6	1	no	—
326	C₂²	—	6	1	no	—
327	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	6	1	no	—
331	C₄	—	6	2	no	—
332	C₄	—	6	1	no	—
333	C₄	—	6	2	no	—
334	C₄	—	6	1	no	—
335	C₄	C₃×#237	1	2	yes	—
336	C₃	—	2	1	no	—
337	C₃	#5614	1	1	yes	characteristic, direct product factor
338	C₃	C₃×#553	1	1	yes	characteristic, complemented
339	C₂	—	4	1	no	—
340	C₂	—	4	2	no	—
341	C₂	—	6	2	no	—
342	C₂	C₂×C₃×#237	1	1	yes	characteristic
343	C₂	C₃×#1004	1	1	yes	characteristic
344	C₂	C₃×D₃×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×#121	C₂
socle	C₂²×C₃²	C₂⁴
hypercenter	C₃×#121	D₃
hypocenter	C₃	C₃×#553
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#121

Complements

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

N	H ≅ G/N	type
1. C₃²×#121	C₂	semidirect
2. C₃×#938	C₂	semidirect
3. C₃×#938	C₂	semidirect
4. C₃×#935	C₂	semidirect
5. C₃×D₄×Dic₃	C₂	semidirect
6. C₃×C₄×#67	C₂	semidirect
7. C₃×#926	C₂	semidirect
8. C₃×D₃×C₄⋊C₄	C₂	semidirect
9. C₃×#890	C₂	semidirect
10. C₃×#67⋊C₄	C₂	semidirect
11. C₃×#879	C₂	semidirect
13. #5614	C₃	direct (N × H)
43. C₃×#121	D₃	semidirect
44. C₃×#121	C₂×C₃	semidirect
50. #938	C₂×C₃	semidirect
51. #938	C₂×C₃	semidirect
52. #935	C₂×C₃	semidirect
53. D₄×Dic₃	C₂×C₃	semidirect
54. C₄×#67	C₂×C₃	semidirect
55. #926	C₂×C₃	semidirect
56. D₃×C₄⋊C₄	C₂×C₃	semidirect
57. #890	C₂×C₃	semidirect
58. #67⋊C₄	C₂×C₃	semidirect
59. #879	C₂×C₃	semidirect
154. #121	C₃×D₃	semidirect
281. C₃²	#553	semidirect
337. C₃	#5614	direct (N × H)
338. C₃	C₃×#553	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#121	2
C₃×#938	2
C₃×#935	2
C₃×D₄×Dic₃	2
C₃×C₄×#67	2
C₃×#926	2
C₃×D₃×C₄⋊C₄	2
C₃×#890	2
C₃×#67⋊C₄	2
C₃×#879	2
C₃×#553	3
#5614	3

Subgroups — C3×#5614 #10595711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5614 #10595711