Subgroups — C₃×#5562 #10595659

Order 576.

527 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×C₄×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#938	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#930	C₂	1	2	yes	maximal, complemented
5	C₃×#929	C₂	1	2	yes	maximal, complemented
6	C₃×#895	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#890	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#884	C₂	1	2	yes	maximal, complemented
9	C₃×#883	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#872	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#540	—	3	1	no	maximal
13	#5562	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	2	yes	—
16	C₂×C₃×#67	C₂²	1	2	yes	—
17	C₂×C₃×#67	C₂²	1	2	yes	—
18	C₂×C₃×#67	C₂²	1	2	yes	—
19	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
20	C₂×C₃×D₁₂	C₂²	1	2	yes	—
21	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
23	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₃²×C₄²	C₂²	1	1	yes	characteristic
26	C₃×#217	C₂²	1	2	yes	—
27	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
29	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
32	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
33	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×#127	—	3	1	no	—
38	C₃×#125	—	3	1	no	—
39	C₃×#123	—	3	2	no	—
40	C₃×#123	—	3	2	no	—
41	C₃×#121	—	3	1	no	—
42	C₃×#121	—	3	2	no	—
43	C₃×#121	—	3	1	no	—
44	C₃×#121	—	3	2	no	—
45	C₃×C₄×D₄	—	2	1	no	—
46	C₃×C₄×D₄	—	3	1	no	—
47	C₃×C₄×D₄	D₃	1	1	yes	characteristic, complemented
48	C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic, complemented
49	C₃×#117	—	3	1	no	—
50	#940	C₂×C₃	1	1	yes	characteristic, complemented
51	#938	C₂×C₃	1	1	yes	characteristic, complemented
52	#930	C₂×C₃	1	2	yes	complemented
53	#929	C₂×C₃	1	2	yes	complemented
54	#895	C₂×C₃	1	1	yes	characteristic, complemented
55	#890	C₂×C₃	1	1	yes	characteristic, complemented
56	#884	C₂×C₃	1	2	yes	complemented
57	#883	C₂×C₃	1	2	yes	complemented
58	C₄×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
59	#872	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂³×C₃²	C₂³	1	2	yes	—
61	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
62	C₂²×C₃×D₃	C₂³	1	2	yes	—
63	C₃²×D₄	—	2	2	no	—
64	C₂×C₃²×C₄	—	4	2	no	—
65	C₂×C₃²×C₄	C₂³	1	2	yes	—
66	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
69	C₃×#67	—	4	2	no	—
70	C₃×#67	—	4	2	no	—
71	C₃×#67	—	4	2	no	—
72	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
73	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
75	C₂×C₃×Dic₃	C₂³	1	2	yes	—
76	C₃×D₁₂	—	2	2	no	—
77	C₃×D₁₂	—	4	2	no	—
78	C₃×C₄×D₃	—	2	2	no	—
79	C₃×C₄×D₃	—	4	2	no	—
80	#540	—	3	1	no	Sylow
81	C₂×C₃×D₄	—	2	1	no	—
82	C₂×C₃×D₄	—	3	2	no	—
83	C₂×C₃×D₄	—	3	2	no	—
84	C₂×C₃×D₄	—	3	2	no	—
85	C₂×C₃×D₄	—	3	1	no	—
86	C₂×C₃×D₄	—	3	2	no	—
87	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
88	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	3	1	no	—
91	C₂²×C₃×C₄	—	3	2	no	—
92	C₂²×C₃×C₄	C₂×D₃	1	2	yes	—
93	C₂²×C₃×C₄	C₂²×C₃	1	2	yes	—
94	C₂×#67	C₂²×C₃	1	2	yes	—
95	C₂×#67	C₂²×C₃	1	2	yes	—
96	C₂×#67	C₂²×C₃	1	2	yes	—
97	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
98	C₂×D₁₂	C₂²×C₃	1	2	yes	—
99	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
100	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
101	C₃×C₄⋊C₄	—	2	1	no	—
102	C₃×C₄⋊C₄	—	3	1	no	—
103	C₃×C₄⋊C₄	—	3	1	no	—
104	C₃×C₄⋊C₄	—	3	2	no	—
105	C₃×C₄⋊C₄	—	3	1	no	—
106	C₃×C₄⋊C₄	—	3	2	no	—
107	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
108	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
109	C₃×C₂²⋊C₄	—	2	2	no	—
110	C₃×C₂²⋊C₄	—	3	2	no	—
111	C₃×C₂²⋊C₄	—	3	1	no	—
112	C₃×C₂²⋊C₄	—	3	1	no	—
113	C₃×C₂²⋊C₄	—	3	2	no	—
114	C₃×C₂²⋊C₄	—	3	2	no	—
115	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
116	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	3	1	no	—
119	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
120	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
121	#217	C₂²×C₃	1	2	yes	—
122	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
123	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
124	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
125	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
126	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
127	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
128	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
129	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
130	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
131	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
132	C₂²×C₃²	—	2	2	no	—
133	C₂²×C₃²	—	2	2	no	—
134	C₂²×C₃²	—	2	2	no	—
135	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
136	C₂×C₃×D₃	—	2	1	no	—
137	C₂×C₃×D₃	—	2	2	no	—
138	C₂×C₃×D₃	—	2	2	no	—
139	C₂×C₃×D₃	—	2	1	no	—
140	C₂×C₃×D₃	—	2	2	no	—
141	C₂×C₃×D₃	—	2	1	no	—
142	C₃²×C₄	—	2	1	no	—
143	C₃²×C₄	—	2	1	no	—
144	C₃²×C₄	—	2	2	no	—
145	C₃²×C₄	#41	1	2	yes	—
146	C₃×Dic₃	—	2	1	no	—
147	C₃×Dic₃	—	2	2	no	—
148	C₃×Dic₃	—	2	1	no	—
149	C₃×Dic₃	—	2	1	no	—
150	#127	—	3	1	no	—
151	#125	—	3	1	no	—
152	#123	—	3	2	no	—
153	#123	—	3	2	no	—
154	#121	—	3	1	no	—
155	#121	—	3	2	no	—
156	#121	—	3	1	no	—
157	#121	—	3	2	no	—
158	C₄×D₄	—	3	1	no	—
159	C₄×D₄	C₃×D₃	1	1	yes	characteristic, complemented
160	#117	—	3	1	no	—
161	C₂³×C₃	—	2	2	no	—
162	C₂³×C₃	—	3	1	no	—
163	C₂³×C₃	—	3	2	no	—
164	C₂³×C₃	C₂²×D₃	1	2	yes	—
165	C₂³×C₃	C₂³×C₃	1	2	yes	—
166	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
167	C₂²×D₃	C₂³×C₃	1	2	yes	—
168	C₃×D₄	—	2	4	no	—
169	C₃×D₄	—	2	2	no	—
170	C₃×D₄	—	2	2	no	—
171	C₃×D₄	—	6	2	no	—
172	C₃×D₄	—	12	2	no	—
173	C₃×D₄	—	12	2	no	—
174	C₃×D₄	—	12	2	no	—
175	C₃×D₄	—	12	2	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	3	1	no	—
181	C₂×C₃×C₄	—	3	1	no	—
182	C₂×C₃×C₄	—	3	1	no	—
183	C₂×C₃×C₄	—	3	2	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	4	no	—
187	C₂×C₃×C₄	—	6	2	no	—
188	C₂×C₃×C₄	—	12	2	no	—
189	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
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	2	yes	—
194	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
197	#67	—	4	2	no	—
198	#67	—	4	2	no	—
199	#67	—	4	2	no	—
200	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
201	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
202	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
203	C₂×Dic₃	C₂³×C₃	1	2	yes	—
204	D₁₂	—	2	2	no	—
205	D₁₂	—	4	2	no	—
206	C₄×D₃	—	2	2	no	—
207	C₄×D₃	—	4	2	no	—
208	C₂×C₃²	—	4	2	no	—
209	C₂×C₃²	C₂×#41	1	1	yes	characteristic
210	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
212	C₃×D₃	—	4	1	no	—
213	C₃×D₃	—	4	2	no	—
214	C₂×D₄	—	3	2	no	—
215	C₂×D₄	—	3	2	no	—
216	C₂×D₄	—	3	2	no	—
217	C₂×D₄	—	3	1	no	—
218	C₂×D₄	—	3	2	no	—
219	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
220	C₂²×C₄	—	3	1	no	—
221	C₂²×C₄	—	3	2	no	—
222	C₂²×C₄	C₂×C₃×D₃	1	2	yes	—
223	C₄⋊C₄	—	3	1	no	—
224	C₄⋊C₄	—	3	1	no	—
225	C₄⋊C₄	—	3	1	no	—
226	C₄⋊C₄	—	3	2	no	—
227	C₄⋊C₄	—	3	2	no	—
228	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
229	C₂²⋊C₄	—	3	2	no	—
230	C₂²⋊C₄	—	3	1	no	—
231	C₂²⋊C₄	—	3	1	no	—
232	C₂²⋊C₄	—	3	2	no	—
233	C₂²⋊C₄	—	3	2	no	—
234	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
235	C₄²	—	3	1	no	—
236	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
237	C₂²×C₃	—	2	4	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	—	2	2	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	4	2	no	—
246	C₂²×C₃	—	4	2	no	—
247	C₂²×C₃	—	6	1	no	—
248	C₂²×C₃	—	6	2	no	—
249	C₂²×C₃	—	6	1	no	—
250	C₂²×C₃	—	6	2	no	—
251	C₂²×C₃	—	6	2	no	—
252	C₂²×C₃	—	6	1	no	—
253	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
254	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	2	no	—
257	C₂×D₃	—	2	2	no	—
258	C₂×D₃	—	2	1	no	—
259	C₂×D₃	—	2	2	no	—
260	C₂×D₃	—	2	1	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	4	2	no	—
270	C₃×C₄	—	4	1	no	—
271	C₃×C₄	—	6	1	no	—
272	C₃×C₄	—	6	1	no	—
273	C₃×C₄	—	6	2	no	—
274	C₃×C₄	—	6	1	no	—
275	C₃×C₄	#235	1	2	yes	—
276	C₃×C₄	C₃×#41	1	2	yes	—
277	Dic₃	—	2	1	no	—
278	Dic₃	—	2	2	no	—
279	Dic₃	—	2	1	no	—
280	Dic₃	—	2	1	no	—
281	C₃²	#540	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	3	1	no	—
283	C₂³	—	3	2	no	—
284	C₂³	C₂²×C₃×D₃	1	2	yes	—
285	D₄	—	2	2	no	—
286	D₄	—	6	2	no	—
287	D₄	—	12	2	no	—
288	D₄	—	12	2	no	—
289	D₄	—	12	2	no	—
290	D₄	—	12	2	no	—
291	C₂×C₄	—	3	1	no	—
292	C₂×C₄	—	3	1	no	—
293	C₂×C₄	—	3	1	no	—
294	C₂×C₄	—	3	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	6	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	1	yes	characteristic
300	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
301	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	4	4	no	—
306	C₂×C₃	—	4	2	no	—
307	C₂×C₃	—	4	2	no	—
308	C₂×C₃	—	12	1	no	—
309	C₂×C₃	—	12	2	no	—
310	C₂×C₃	C₂×#235	1	1	yes	characteristic
311	C₂×C₃	#1004	1	1	yes	characteristic
312	C₂×C₃	#1009	1	1	yes	characteristic
313	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
314	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
315	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
316	D₃	—	4	1	no	—
317	D₃	—	4	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	6	1	no	—
322	C₂²	—	6	2	no	—
323	C₂²	—	6	1	no	—
324	C₂²	—	6	2	no	—
325	C₂²	—	6	2	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	—	2	1	no	—
331	C₄	—	6	1	no	—
332	C₄	—	6	1	no	—
333	C₄	—	6	2	no	—
334	C₄	—	6	1	no	—
335	C₄	C₃×#235	1	2	yes	—
336	C₃	—	2	1	no	—
337	C₃	#5562	1	1	yes	characteristic, direct product factor
338	C₃	C₃×#540	1	1	yes	characteristic, complemented
339	C₂	—	4	2	no	—
340	C₂	—	12	1	no	—
341	C₂	—	12	2	no	—
342	C₂	C₂×C₃×#235	1	1	yes	characteristic
343	C₂	C₃×#1004	1	1	yes	characteristic
344	C₂	C₃×#1009	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×C₄×D₄	C₂
socle	C₂²×C₃²	C₂⁴
hypercenter	C₃×C₄×D₄	D₃
hypocenter	C₃	C₃×#540
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₃²×C₄×D₄	C₂	semidirect
2. C₃×#940	C₂	semidirect
3. C₃×#938	C₂	semidirect
4. C₃×#930	C₂	semidirect
5. C₃×#929	C₂	semidirect
6. C₃×#895	C₂	semidirect
7. C₃×#890	C₂	semidirect
8. C₃×#884	C₂	semidirect
9. C₃×#883	C₂	semidirect
10. C₃×C₄×D₁₂	C₂	semidirect
11. C₃×#872	C₂	semidirect
13. #5562	C₃	direct (N × H)
47. C₃×C₄×D₄	D₃	semidirect
48. C₃×C₄×D₄	C₂×C₃	semidirect
50. #940	C₂×C₃	semidirect
51. #938	C₂×C₃	semidirect
52. #930	C₂×C₃	semidirect
53. #929	C₂×C₃	semidirect
54. #895	C₂×C₃	semidirect
55. #890	C₂×C₃	semidirect
56. #884	C₂×C₃	semidirect
57. #883	C₂×C₃	semidirect
58. C₄×D₁₂	C₂×C₃	semidirect
59. #872	C₂×C₃	semidirect
159. C₄×D₄	C₃×D₃	semidirect
281. C₃²	#540	semidirect
337. C₃	#5562	direct (N × H)
338. C₃	C₃×#540	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₄	2
C₃×#940	2
C₃×#938	2
C₃×#930	2
C₃×#929	2
C₃×#895	2
C₃×#890	2
C₃×#884	2
C₃×#883	2
C₃×C₄×D₁₂	2
C₃×#872	2
C₃×#540	3
#5562	3

Subgroups — C3×#5562 #10595659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5562 #10595659