Subgroups — C₃×#5600 #10595697

Order 576.

515 conjugacy classes of subgroups.

The subgroup lattice has 330 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
2	C₃×#936	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#935	C₂	1	2	yes	maximal, complemented
4	C₃×D₄×Dic₃	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#890	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#887	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#879	C₂	1	2	yes	maximal, complemented
11	C₃×#877	C₂	1	2	yes	maximal, complemented
12	C₃×#529	—	3	1	no	maximal
13	#5600	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×D₄	C₂²	1	2	yes	—
15	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
16	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
18	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
19	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
21	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₃×#217	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×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
33	C₃×C₄×Dic₃	C₂²	1	2	yes	—
34	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×#126	—	3	2	no	—
38	C₃×#125	—	3	1	no	—
39	C₃×#124	—	3	1	no	—
40	C₃×#123	—	3	2	no	—
41	C₃×C₂²⋊Q₈	—	3	1	no	—
42	C₃×#121	—	2	1	no	—
43	C₃×#121	D₃	1	1	yes	characteristic
44	C₃×#121	C₂×C₃	1	1	yes	characteristic
45	C₃×C₄×Q₈	—	3	1	no	—
46	C₃×C₄×D₄	—	3	1	no	—
47	C₃×C₄×D₄	—	3	2	no	—
48	C₃×#117	—	3	2	no	—
49	C₂×C₃×C₄²	—	3	1	no	—
50	#936	C₂×C₃	1	1	yes	characteristic, complemented
51	#935	C₂×C₃	1	2	yes	complemented
52	D₄×Dic₃	C₂×C₃	1	2	yes	complemented
53	D₄×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
54	#924	C₂×C₃	1	1	yes	characteristic, complemented
55	C₂×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
56	#890	C₂×C₃	1	1	yes	characteristic, complemented
57	#887	C₂×C₃	1	1	yes	characteristic, complemented
58	#879	C₂×C₃	1	2	yes	complemented
59	#877	C₂×C₃	1	2	yes	complemented
60	C₂³×C₃²	C₂³	1	2	yes	—
61	C₂³×C₃²	C₂³	1	1	yes	characteristic
62	C₃²×D₄	—	2	4	no	—
63	C₃²×D₄	—	2	2	no	—
64	C₂×C₃²×C₄	—	2	2	no	—
65	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂³	1	2	yes	—
67	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
68	C₂×C₃×Dic₃	—	2	2	no	—
69	C₂×C₃×Dic₃	—	2	2	no	—
70	C₂×C₃×Dic₃	—	2	4	no	—
71	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
72	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
73	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₂³	1	2	yes	—
75	C₂×C₃×Dic₃	C₂³	1	2	yes	—
76	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
77	C₃×Dic₆	—	2	2	no	—
78	#529	—	3	1	no	Sylow
79	C₂×C₃×Q₈	—	3	1	no	—
80	C₂×C₃×D₄	—	2	2	no	—
81	C₂×C₃×D₄	—	2	1	no	—
82	C₂×C₃×D₄	C₂×D₃	1	2	yes	—
83	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
84	C₂×C₃×D₄	C₂²×C₃	1	2	yes	—
85	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂²×C₃×C₄	—	3	1	no	—
89	C₂²×C₃×C₄	—	3	2	no	—
90	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
92	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
93	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
94	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
95	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
96	C₃×C₄⋊C₄	—	2	1	no	—
97	C₃×C₄⋊C₄	—	3	2	no	—
98	C₃×C₄⋊C₄	—	3	2	no	—
99	C₃×C₄⋊C₄	—	3	2	no	—
100	C₃×C₄⋊C₄	—	3	1	no	—
101	C₃×C₄⋊C₄	—	3	2	no	—
102	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
103	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
104	C₃×C₂²⋊C₄	—	2	2	no	—
105	C₃×C₂²⋊C₄	—	3	2	no	—
106	C₃×C₂²⋊C₄	—	3	2	no	—
107	C₃×C₂²⋊C₄	—	3	2	no	—
108	C₃×C₂²⋊C₄	—	3	2	no	—
109	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
110	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
111	C₃×C₄²	—	3	1	no	—
112	C₃×C₄²	—	3	1	no	—
113	C₃×C₄²	—	3	1	no	—
114	C₃×C₄²	—	3	1	no	—
115	C₃×C₄²	—	3	2	no	—
116	#217	C₂²×C₃	1	2	yes	—
117	#217	C₂²×C₃	1	2	yes	—
118	#217	C₂²×C₃	1	2	yes	—
119	#217	C₂²×C₃	1	2	yes	—
120	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
121	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
122	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
123	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
124	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
125	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
126	C₄×Dic₃	C₂²×C₃	1	2	yes	—
127	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
128	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
129	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
130	C₂²×C₃²	—	2	1	no	—
131	C₂²×C₃²	—	2	1	no	—
132	C₂²×C₃²	—	2	2	no	—
133	C₂²×C₃²	—	2	2	no	—
134	C₂²×C₃²	—	2	2	no	—
135	C₂²×C₃²	#41	1	2	yes	—
136	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
137	C₃²×C₄	—	2	1	no	—
138	C₃²×C₄	—	2	2	no	—
139	C₃²×C₄	#41	1	2	yes	—
140	C₃×Dic₃	—	2	2	no	—
141	C₃×Dic₃	—	2	1	no	—
142	C₃×Dic₃	—	2	1	no	—
143	C₃×Dic₃	—	2	1	no	—
144	C₃×Dic₃	—	2	2	no	—
145	C₃×Dic₃	#41	1	2	yes	—
146	#126	—	3	2	no	—
147	#125	—	3	1	no	—
148	#124	—	3	1	no	—
149	#123	—	3	2	no	—
150	C₂²⋊Q₈	—	3	1	no	—
151	#121	C₃×D₃	1	1	yes	characteristic
152	C₄×Q₈	—	3	1	no	—
153	C₄×D₄	—	3	1	no	—
154	C₄×D₄	—	3	2	no	—
155	#117	—	3	2	no	—
156	C₂×C₄²	—	3	1	no	—
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	—	2	2	no	—
159	C₂³×C₃	C₂²×D₃	1	2	yes	—
160	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
161	C₂³×C₃	C₂³×C₃	1	2	yes	—
162	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
163	C₃×Q₈	—	6	2	no	—
164	C₃×D₄	—	2	4	no	—
165	C₃×D₄	—	2	4	no	—
166	C₃×D₄	—	2	2	no	—
167	C₃×D₄	—	2	2	no	—
168	C₃×D₄	—	4	2	no	—
169	C₃×D₄	—	4	4	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	3	1	no	—
179	C₂×C₃×C₄	—	3	2	no	—
180	C₂×C₃×C₄	—	3	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	6	2	no	—
183	C₂×C₃×C₄	—	6	4	no	—
184	C₂×C₃×C₄	—	6	2	no	—
185	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
187	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
190	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
191	C₂×Dic₃	—	2	2	no	—
192	C₂×Dic₃	—	2	2	no	—
193	C₂×Dic₃	—	2	4	no	—
194	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
195	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
196	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
197	C₂×Dic₃	C₂³×C₃	1	2	yes	—
198	C₂×Dic₃	C₂³×C₃	1	2	yes	—
199	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
200	Dic₆	—	2	2	no	—
201	C₂×C₃²	—	2	2	no	—
202	C₂×C₃²	—	4	2	no	—
203	C₂×C₃²	C₂×#41	1	1	yes	characteristic
204	C₂×C₃²	C₂×#41	1	1	yes	characteristic
205	C₂×C₃²	C₂×#41	1	1	yes	characteristic
206	C₂×Q₈	—	3	1	no	—
207	C₂×D₄	C₂×C₃×D₃	1	2	yes	—
208	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	3	1	no	—
211	C₂²×C₄	—	3	2	no	—
212	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
213	C₄⋊C₄	—	3	2	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₄⋊C₄	—	3	1	no	—
216	C₄⋊C₄	—	3	2	no	—
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
219	C₂²⋊C₄	—	3	2	no	—
220	C₂²⋊C₄	—	3	2	no	—
221	C₂²⋊C₄	—	3	2	no	—
222	C₂²⋊C₄	—	3	2	no	—
223	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
224	C₄²	—	3	1	no	—
225	C₄²	—	3	1	no	—
226	C₄²	—	3	1	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	2	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	1	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	4	1	no	—
242	C₂²×C₃	—	4	1	no	—
243	C₂²×C₃	—	4	2	no	—
244	C₂²×C₃	—	4	2	no	—
245	C₂²×C₃	—	4	2	no	—
246	C₂²×C₃	#237	1	2	yes	—
247	C₂²×C₃	C₃×#41	1	2	yes	—
248	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
249	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	2	no	—
255	C₃×C₄	—	3	2	no	—
256	C₃×C₄	—	4	1	no	—
257	C₃×C₄	—	4	2	no	—
258	C₃×C₄	—	6	1	no	—
259	C₃×C₄	—	6	1	no	—
260	C₃×C₄	—	6	2	no	—
261	C₃×C₄	—	6	1	no	—
262	C₃×C₄	—	6	2	no	—
263	C₃×C₄	#237	1	2	yes	—
264	C₃×C₄	C₃×#41	1	2	yes	—
265	Dic₃	—	2	2	no	—
266	Dic₃	—	2	1	no	—
267	Dic₃	—	2	1	no	—
268	Dic₃	—	2	1	no	—
269	Dic₃	—	2	2	no	—
270	Dic₃	C₃×#41	1	2	yes	—
271	C₃²	#529	1	1	yes	characteristic, complemented, Sylow
272	C₂³	C₂²×C₃×D₃	1	2	yes	—
273	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
274	Q₈	—	6	2	no	—
275	D₄	—	2	4	no	—
276	D₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	3	1	no	—
280	C₂×C₄	—	3	1	no	—
281	C₂×C₄	—	3	1	no	—
282	C₂×C₄	—	3	2	no	—
283	C₂×C₄	—	3	2	no	—
284	C₂×C₄	—	6	2	no	—
285	C₂×C₄	—	6	4	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
289	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	2	2	no	—
292	C₂×C₃	—	2	1	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	4	2	no	—
296	C₂×C₃	—	4	2	no	—
297	C₂×C₃	—	4	2	no	—
298	C₂×C₃	—	8	2	no	—
299	C₂×C₃	C₂×#237	1	1	yes	characteristic
300	C₂×C₃	C₂×#237	1	1	yes	characteristic
301	C₂×C₃	D₃×#41	1	1	yes	characteristic
302	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
303	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
304	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	C₃×#237	1	2	yes	—
311	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	3	2	no	—
315	C₄	—	6	1	no	—
316	C₄	—	6	1	no	—
317	C₄	—	6	2	no	—
318	C₄	—	6	1	no	—
319	C₄	—	6	2	no	—
320	C₄	C₃×#237	1	2	yes	—
321	C₃	—	2	1	no	—
322	C₃	#5600	1	1	yes	characteristic, direct product factor
323	C₃	C₃×#529	1	1	yes	characteristic, complemented
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	C₂×C₃×#237	1	1	yes	characteristic
327	C₂	C₂×C₃×#237	1	1	yes	characteristic
328	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, prime radical	C₃²×#121	C₂
socle	C₂²×C₃²	C₂⁴
hypercenter	C₃×#121	D₃
hypocenter	C₃	C₃×#529
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
2. C₃×#936	C₂	semidirect
3. C₃×#935	C₂	semidirect
4. C₃×D₄×Dic₃	C₂	semidirect
5. C₃×D₄×Dic₃	C₂	semidirect
6. C₃×#924	C₂	semidirect
7. C₂×C₃×C₄×Dic₃	C₂	semidirect
8. C₃×#890	C₂	semidirect
9. C₃×#887	C₂	semidirect
10. C₃×#879	C₂	semidirect
11. C₃×#877	C₂	semidirect
13. #5600	C₃	direct (N × H)
50. #936	C₂×C₃	semidirect
51. #935	C₂×C₃	semidirect
52. D₄×Dic₃	C₂×C₃	semidirect
53. D₄×Dic₃	C₂×C₃	semidirect
54. #924	C₂×C₃	semidirect
55. C₂×C₄×Dic₃	C₂×C₃	semidirect
56. #890	C₂×C₃	semidirect
57. #887	C₂×C₃	semidirect
58. #879	C₂×C₃	semidirect
59. #877	C₂×C₃	semidirect
271. C₃²	#529	semidirect
322. C₃	#5600	direct (N × H)
323. C₃	C₃×#529	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×#936	2
C₃×#935	2
C₃×D₄×Dic₃	2
C₃×#924	2
C₂×C₃×C₄×Dic₃	2
C₃×#890	2
C₃×#887	2
C₃×#879	2
C₃×#877	2
C₃×#529	3
#5600	3

Subgroups — C3×#5600 #10595697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5600 #10595697