Subgroups — C₃×#5695 #10595792

Order 576.

504 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#898	C₂	1	2	yes	maximal, complemented
3	C₃×#895	C₂	1	2	yes	maximal, complemented
4	C₃×#895	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#895	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#894	C₂	1	2	yes	maximal, complemented
7	C₃×#893	C₂	1	2	yes	maximal, complemented
8	C₃×#892	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×C₄×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#553	—	3	1	no	maximal
13	#5695	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃×D₁₂	C₂²	1	2	yes	—
15	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
16	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
17	C₂×C₃×D₁₂	C₂²	1	2	yes	—
18	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
21	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
22	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
25	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
26	C₃²×C₄²	C₂²	1	1	yes	characteristic
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×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
32	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
33	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
34	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
35	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₃×C₄×Dic₃	C₂²	1	2	yes	—
37	C₃×#126	—	3	2	no	—
38	C₃×#125	—	2	1	no	—
39	C₃×#125	D₃	1	1	yes	characteristic, complemented
40	C₃×#125	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×#123	—	3	2	no	—
42	C₃×#121	—	3	2	no	—
43	C₃×#121	—	3	1	no	—
44	C₃×#121	—	3	1	no	—
45	C₃×C₄×D₄	—	3	2	no	—
46	C₃×C₄×D₄	—	3	1	no	—
47	C₃×C₄×D₄	—	3	1	no	—
48	C₃×#117	—	3	1	no	—
49	C₂×C₃×C₄⋊C₄	—	3	1	no	—
50	#898	C₂×C₃	1	2	yes	complemented
51	#895	C₂×C₃	1	2	yes	complemented
52	#895	C₂×C₃	1	1	yes	characteristic, complemented
53	#895	C₂×C₃	1	1	yes	characteristic, complemented
54	#894	C₂×C₃	1	2	yes	complemented
55	#893	C₂×C₃	1	2	yes	complemented
56	#892	C₂×C₃	1	1	yes	characteristic, complemented
57	D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
58	C₄×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
59	C₄×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
61	C₂²×C₃×D₃	C₂³	1	2	yes	—
62	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂³	1	2	yes	—
64	C₂×C₃²×C₄	C₂³	1	2	yes	—
65	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
69	C₂×C₃×Dic₃	C₂³	1	2	yes	—
70	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
71	C₃×D₁₂	—	2	4	no	—
72	C₃×D₁₂	—	2	2	no	—
73	C₃×D₁₂	—	2	2	no	—
74	C₃×D₁₂	—	4	2	no	—
75	C₃×C₄×D₃	—	2	2	no	—
76	C₃×C₄×D₃	—	2	2	no	—
77	C₃×C₄×D₃	—	4	2	no	—
78	C₃×C₄×D₃	—	4	2	no	—
79	#553	—	3	1	no	Sylow
80	C₂×C₃×D₄	—	3	2	no	—
81	C₂×C₃×D₄	—	3	1	no	—
82	C₂×C₃×D₄	—	3	1	no	—
83	C₂×C₃×D₄	—	3	2	no	—
84	C₂²×C₃×C₄	—	3	1	no	—
85	C₂²×C₃×C₄	—	3	1	no	—
86	C₂²×C₃×C₄	—	3	2	no	—
87	C₂²×C₃×C₄	—	3	2	no	—
88	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
89	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
90	C₂×D₁₂	C₂²×C₃	1	2	yes	—
91	C₂×D₁₂	C₂²×C₃	1	2	yes	—
92	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
94	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
95	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
96	C₃×C₄⋊C₄	—	2	1	no	—
97	C₃×C₄⋊C₄	—	2	1	no	—
98	C₃×C₄⋊C₄	—	2	2	no	—
99	C₃×C₄⋊C₄	—	2	2	no	—
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₄⋊C₄	—	3	1	no	—
102	C₃×C₄⋊C₄	—	3	1	no	—
103	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
104	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
105	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
106	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
107	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
108	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
109	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
110	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
111	C₃×C₂²⋊C₄	—	3	2	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	—	3	1	no	—
116	C₃×C₂²⋊C₄	—	3	2	no	—
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	3	2	no	—
119	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
120	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
121	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
122	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
123	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
124	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
125	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
126	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
127	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
128	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
129	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
130	C₄×Dic₃	C₂²×C₃	1	2	yes	—
131	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
132	C₂×C₃×D₃	—	2	1	no	—
133	C₂×C₃×D₃	—	2	1	no	—
134	C₂×C₃×D₃	—	2	1	no	—
135	C₂×C₃×D₃	—	2	2	no	—
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	#41	1	2	yes	—
141	C₃²×C₄	—	2	2	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	1	no	—
148	C₃×Dic₃	—	2	2	no	—
149	#126	—	3	2	no	—
150	#125	C₃×D₃	1	1	yes	characteristic, complemented
151	#123	—	3	2	no	—
152	#121	—	3	2	no	—
153	#121	—	3	1	no	—
154	#121	—	3	1	no	—
155	C₄×D₄	—	3	2	no	—
156	C₄×D₄	—	3	1	no	—
157	C₄×D₄	—	3	1	no	—
158	#117	—	3	1	no	—
159	C₂×C₄⋊C₄	—	3	1	no	—
160	C₂³×C₃	—	3	1	no	—
161	C₂³×C₃	—	3	1	no	—
162	C₂³×C₃	—	3	2	no	—
163	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
164	C₂²×D₃	C₂³×C₃	1	2	yes	—
165	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
166	C₃×D₄	—	6	4	no	—
167	C₃×D₄	—	6	2	no	—
168	C₃×D₄	—	6	2	no	—
169	C₃×D₄	—	12	2	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	2	no	—
177	C₂×C₃×C₄	—	3	1	no	—
178	C₂×C₃×C₄	—	6	2	no	—
179	C₂×C₃×C₄	—	6	2	no	—
180	C₂×C₃×C₄	—	12	2	no	—
181	C₂×C₃×C₄	—	12	2	no	—
182	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
183	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
186	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
188	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
189	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
192	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
193	C₂×Dic₃	C₂³×C₃	1	2	yes	—
194	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
195	D₁₂	—	2	2	no	—
196	D₁₂	—	2	4	no	—
197	D₁₂	—	2	2	no	—
198	D₁₂	—	4	2	no	—
199	C₄×D₃	—	2	2	no	—
200	C₄×D₃	—	2	2	no	—
201	C₄×D₃	—	4	2	no	—
202	C₄×D₃	—	4	2	no	—
203	C₂×C₃²	C₂×#41	1	1	yes	characteristic
204	C₂×C₃²	C₂×#41	1	1	yes	characteristic
205	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
206	C₃×D₃	—	2	2	no	—
207	C₃×D₃	—	4	1	no	—
208	C₃×D₃	—	4	2	no	—
209	C₂×D₄	—	3	2	no	—
210	C₂×D₄	—	3	1	no	—
211	C₂×D₄	—	3	1	no	—
212	C₂×D₄	—	3	2	no	—
213	C₂²×C₄	—	3	1	no	—
214	C₂²×C₄	—	3	1	no	—
215	C₂²×C₄	—	3	2	no	—
216	C₂²×C₄	—	3	2	no	—
217	C₄⋊C₄	—	3	1	no	—
218	C₄⋊C₄	—	3	2	no	—
219	C₄⋊C₄	—	3	1	no	—
220	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
221	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
222	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
223	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
224	C₂²⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	—	3	1	no	—
226	C₂²⋊C₄	—	3	2	no	—
227	C₂²⋊C₄	—	3	2	no	—
228	C₂²⋊C₄	—	3	1	no	—
229	C₂²⋊C₄	—	3	2	no	—
230	C₄²	—	3	2	no	—
231	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	3	2	no	—
234	C₂²×C₃	—	6	1	no	—
235	C₂²×C₃	—	6	1	no	—
236	C₂²×C₃	—	6	1	no	—
237	C₂²×C₃	—	6	2	no	—
238	C₂²×C₃	—	6	1	no	—
239	C₂²×C₃	—	6	2	no	—
240	C₂²×C₃	—	6	2	no	—
241	C₂²×C₃	—	6	1	no	—
242	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
243	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
244	C₂×D₃	—	2	1	no	—
245	C₂×D₃	—	2	1	no	—
246	C₂×D₃	—	2	1	no	—
247	C₂×D₃	—	2	2	no	—
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	—	2	2	no	—
250	C₂×D₃	—	2	2	no	—
251	C₂×D₃	—	2	1	no	—
252	C₂×D₃	C₃×#41	1	2	yes	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	2	1	no	—
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	4	2	no	—
264	C₃×C₄	—	4	1	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	6	1	no	—
267	C₃×C₄	—	6	1	no	—
268	C₃×C₄	—	6	2	no	—
269	C₃×C₄	#239	1	2	yes	—
270	C₃×C₄	C₃×#41	1	2	yes	—
271	Dic₃	—	2	1	no	—
272	Dic₃	—	2	1	no	—
273	Dic₃	—	2	2	no	—
274	C₃²	#553	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	3	1	no	—
276	C₂³	—	3	1	no	—
277	C₂³	—	3	2	no	—
278	D₄	—	6	4	no	—
279	D₄	—	6	2	no	—
280	D₄	—	6	2	no	—
281	D₄	—	12	2	no	—
282	C₂×C₄	—	3	1	no	—
283	C₂×C₄	—	3	2	no	—
284	C₂×C₄	—	3	1	no	—
285	C₂×C₄	—	6	2	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	—	12	2	no	—
288	C₂×C₄	—	12	2	no	—
289	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
290	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
293	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	6	2	no	—
298	C₂×C₃	—	12	1	no	—
299	C₂×C₃	—	12	2	no	—
300	C₂×C₃	C₂×#239	1	1	yes	characteristic
301	C₂×C₃	D₃×#41	1	1	yes	characteristic
302	C₂×C₃	#1009	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	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
306	D₃	—	2	2	no	—
307	D₃	—	4	1	no	—
308	D₃	—	4	2	no	—
309	C₂²	—	3	2	no	—
310	C₂²	—	6	1	no	—
311	C₂²	—	6	1	no	—
312	C₂²	—	6	1	no	—
313	C₂²	—	6	2	no	—
314	C₂²	—	6	1	no	—
315	C₂²	—	6	2	no	—
316	C₂²	—	6	2	no	—
317	C₂²	—	6	1	no	—
318	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	2	no	—
323	C₄	—	6	1	no	—
324	C₄	—	6	1	no	—
325	C₄	—	6	2	no	—
326	C₄	C₃×#239	1	2	yes	—
327	C₃	—	2	1	no	—
328	C₃	#5695	1	1	yes	characteristic, direct product factor
329	C₃	C₃×#553	1	1	yes	characteristic, complemented
330	C₂	—	6	2	no	—
331	C₂	—	12	1	no	—
332	C₂	—	12	2	no	—
333	C₂	C₂×C₃×#239	1	1	yes	characteristic
334	C₂	C₃×D₃×#41	1	1	yes	characteristic
335	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×#125	C₂
socle	C₂²×C₃²	C₂⁴
hypercenter	C₃×#125	D₃
hypocenter	C₃	C₃×#553
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
1. C₃²×#125	C₂	semidirect
2. C₃×#898	C₂	semidirect
3. C₃×#895	C₂	semidirect
4. C₃×#895	C₂	semidirect
5. C₃×#895	C₂	semidirect
6. C₃×#894	C₂	semidirect
7. C₃×#893	C₂	semidirect
8. C₃×#892	C₂	semidirect
9. C₃×D₃×C₄⋊C₄	C₂	semidirect
10. C₃×C₄×D₁₂	C₂	semidirect
11. C₃×C₄×D₁₂	C₂	semidirect
13. #5695	C₃	direct (N × H)
39. C₃×#125	D₃	semidirect
40. C₃×#125	C₂×C₃	semidirect
50. #898	C₂×C₃	semidirect
51. #895	C₂×C₃	semidirect
52. #895	C₂×C₃	semidirect
53. #895	C₂×C₃	semidirect
54. #894	C₂×C₃	semidirect
55. #893	C₂×C₃	semidirect
56. #892	C₂×C₃	semidirect
57. D₃×C₄⋊C₄	C₂×C₃	semidirect
58. C₄×D₁₂	C₂×C₃	semidirect
59. C₄×D₁₂	C₂×C₃	semidirect
150. #125	C₃×D₃	semidirect
274. C₃²	#553	semidirect
328. C₃	#5695	direct (N × H)
329. 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×#125	2
C₃×#898	2
C₃×#895	2
C₃×#894	2
C₃×#893	2
C₃×#892	2
C₃×D₃×C₄⋊C₄	2
C₃×C₄×D₁₂	2
C₃×#553	3
#5695	3

Subgroups — C3×#5695 #10595792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5695 #10595792