Subgroups — C₃×#5635 #10595732

Order 576.

493 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#946	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×#67	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#897	C₂	1	2	yes	maximal, complemented
7	C₃×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#889	C₂	1	2	yes	maximal, complemented
9	C₃×#888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#67⋊C₄	C₂	1	2	yes	maximal
11	C₃×#878	C₂	1	2	yes	maximal, complemented
12	C₃×#554	—	3	1	no	maximal
13	#5635	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×#67	C₂²	1	1	yes	characteristic
17	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
18	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
20	C₂×C₃×Dic₆	C₂²	1	2	yes	—
21	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
23	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₃×#217	C₂²	1	1	yes	characteristic
25	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₂²	1	2	yes	—
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	3	1	no	—
38	C₃×#125	—	3	2	no	—
39	C₃×C₂²⋊Q₈	—	2	1	no	—
40	C₃×C₂²⋊Q₈	—	3	1	no	—
41	C₃×C₂²⋊Q₈	—	3	2	no	—
42	C₃×C₂²⋊Q₈	—	3	2	no	—
43	C₃×C₂²⋊Q₈	D₃	1	1	yes	characteristic, complemented
44	C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
45	C₃×C₄×Q₈	—	3	1	no	—
46	C₃×C₄×D₄	—	3	1	no	—
47	C₃×C₄×D₄	—	3	2	no	—
48	C₂×C₃×C₄⋊C₄	—	3	1	no	—
49	C₂×C₃×C₄⋊C₄	—	3	1	no	—
50	#946	C₂×C₃	1	1	yes	characteristic, complemented
51	Q₈×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
52	C₄×#67	C₂×C₃	1	1	yes	characteristic
53	C₂×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
54	#897	C₂×C₃	1	2	yes	complemented
55	D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
56	#889	C₂×C₃	1	2	yes	complemented
57	#888	C₂×C₃	1	1	yes	characteristic, complemented
58	#67⋊C₄	C₂×C₃	1	2	yes	—
59	#878	C₂×C₃	1	2	yes	complemented
60	C₂³×C₃²	C₂³	1	1	yes	characteristic
61	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
62	C₃²×Q₈	—	2	2	no	—
63	C₂×C₃²×C₄	—	2	2	no	—
64	C₂×C₃²×C₄	C₂³	1	2	yes	—
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	C₃×#67	Q₈	1	4	yes	complemented
69	C₂×C₃×Dic₃	—	4	2	no	—
70	C₂×C₃×Dic₃	C₂³	1	2	yes	—
71	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
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	C₃×C₄×D₃	—	2	2	no	—
76	C₃×C₄×D₃	—	4	2	no	—
77	C₃×Dic₆	—	4	2	no	—
78	#554	—	3	1	no	Sylow
79	C₂×C₃×Q₈	—	2	1	no	—
80	C₂×C₃×Q₈	—	3	2	no	—
81	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
82	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×D₄	—	3	1	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	3	2	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂²×C₃×C₄	—	3	2	no	—
88	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
90	C₂×#67	C₂²×C₃	1	1	yes	characteristic
91	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
92	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
93	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
94	C₂×Dic₆	C₂²×C₃	1	2	yes	—
95	C₃×C₄⋊C₄	—	2	1	no	—
96	C₃×C₄⋊C₄	—	2	2	no	—
97	C₃×C₄⋊C₄	—	3	2	no	—
98	C₃×C₄⋊C₄	—	3	1	no	—
99	C₃×C₄⋊C₄	—	3	2	no	—
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₄⋊C₄	—	3	1	no	—
102	C₃×C₄⋊C₄	—	3	2	no	—
103	C₃×C₄⋊C₄	—	3	1	no	—
104	C₃×C₄⋊C₄	—	3	2	no	—
105	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
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	2	yes	—
109	C₃×C₂²⋊C₄	—	2	2	no	—
110	C₃×C₂²⋊C₄	—	3	1	no	—
111	C₃×C₂²⋊C₄	—	3	1	no	—
112	C₃×C₂²⋊C₄	—	3	2	no	—
113	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
114	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
115	C₃×C₄²	—	3	2	no	—
116	C₃×C₄²	—	3	1	no	—
117	#217	C₂²×C₃	1	1	yes	characteristic
118	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
119	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
120	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
121	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
122	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
123	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
124	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
125	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
126	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
127	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
128	C₄×Dic₃	C₂²×C₃	1	2	yes	—
129	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
130	C₂²×C₃²	—	2	1	no	—
131	C₂²×C₃²	—	2	1	no	—
132	C₂²×C₃²	C₂×Q₈	1	2	yes	—
133	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
134	C₂×C₃×D₃	—	2	1	no	—
135	C₂×C₃×D₃	—	2	1	no	—
136	C₂×C₃×D₃	C₂×Q₈	1	2	yes	—
137	C₃²×C₄	—	2	1	no	—
138	C₃²×C₄	—	2	2	no	—
139	C₃²×C₄	—	2	2	no	—
140	C₃²×C₄	#41	1	2	yes	—
141	C₃×Dic₃	—	2	1	no	—
142	C₃×Dic₃	—	2	1	no	—
143	C₃×Dic₃	—	2	2	no	—
144	C₃×Dic₃	—	2	2	no	—
145	C₃×Dic₃	C₂×Q₈	1	2	yes	—
146	C₄⋊Q₈	—	3	1	no	—
147	#125	—	3	2	no	—
148	C₂²⋊Q₈	—	3	1	no	—
149	C₂²⋊Q₈	—	3	2	no	—
150	C₂²⋊Q₈	—	3	2	no	—
151	C₂²⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
152	C₄×Q₈	—	3	1	no	—
153	C₄×D₄	—	3	1	no	—
154	C₄×D₄	—	3	2	no	—
155	C₂×C₄⋊C₄	—	3	1	no	—
156	C₂×C₄⋊C₄	—	3	1	no	—
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	—	3	1	no	—
159	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
160	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
161	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
162	C₃×Q₈	—	2	2	no	—
163	C₃×Q₈	—	2	2	no	—
164	C₃×Q₈	—	4	2	no	—
165	C₃×Q₈	—	12	2	no	—
166	C₃×D₄	—	3	4	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	1	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	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	3	2	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	3	2	no	—
178	C₂×C₃×C₄	—	4	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	2	yes	—
185	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
187	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
188	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
190	#67	C₃×Q₈	1	4	yes	complemented
191	C₂×Dic₃	—	4	2	no	—
192	C₂×Dic₃	C₂³×C₃	1	2	yes	—
193	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
194	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
195	C₂×Dic₃	C₂³×C₃	1	2	yes	—
196	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
197	C₄×D₃	—	2	2	no	—
198	C₄×D₃	—	4	2	no	—
199	Dic₆	—	4	2	no	—
200	C₂×C₃²	—	2	2	no	—
201	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₃²	C₂×#41	1	1	yes	characteristic
203	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
204	C₃×D₃	—	2	2	no	—
205	C₂×Q₈	—	3	2	no	—
206	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
207	C₂×D₄	—	3	1	no	—
208	C₂²×C₄	—	3	2	no	—
209	C₂²×C₄	—	3	1	no	—
210	C₂²×C₄	—	3	2	no	—
211	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	3	2	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₄⋊C₄	—	3	1	no	—
216	C₄⋊C₄	—	3	1	no	—
217	C₄⋊C₄	—	3	2	no	—
218	C₄⋊C₄	—	3	2	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
221	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
222	C₂²⋊C₄	—	3	1	no	—
223	C₂²⋊C₄	—	3	1	no	—
224	C₂²⋊C₄	—	3	2	no	—
225	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
226	C₄²	—	3	2	no	—
227	C₄²	—	3	1	no	—
228	C₂²×C₃	—	2	1	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	—	3	2	no	—
235	C₂²×C₃	—	4	1	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	—	6	1	no	—
238	C₂²×C₃	—	6	1	no	—
239	C₂²×C₃	D₃×Q₈	1	2	yes	—
240	C₂²×C₃	C₂×C₃×Q₈	1	2	yes	—
241	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
242	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
243	C₂×D₃	—	2	1	no	—
244	C₂×D₃	—	2	1	no	—
245	C₂×D₃	C₂×C₃×Q₈	1	2	yes	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	2	no	—
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	3	2	no	—
254	C₃×C₄	—	4	1	no	—
255	C₃×C₄	—	4	2	no	—
256	C₃×C₄	—	4	2	no	—
257	C₃×C₄	—	6	1	no	—
258	C₃×C₄	—	6	2	no	—
259	C₃×C₄	—	6	2	no	—
260	C₃×C₄	—	6	1	no	—
261	C₃×C₄	#237	1	2	yes	—
262	C₃×C₄	C₃×#41	1	2	yes	—
263	Dic₃	—	2	1	no	—
264	Dic₃	—	2	1	no	—
265	Dic₃	—	2	2	no	—
266	Dic₃	—	2	2	no	—
267	Dic₃	C₂×C₃×Q₈	1	2	yes	—
268	C₃²	#554	1	1	yes	characteristic, complemented, Sylow
269	C₂³	—	3	1	no	—
270	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
271	Q₈	—	2	2	no	—
272	Q₈	—	12	2	no	—
273	D₄	—	3	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	3	1	no	—
277	C₂×C₄	—	3	2	no	—
278	C₂×C₄	—	3	1	no	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	6	2	no	—
281	C₂×C₄	—	12	2	no	—
282	C₂×C₄	—	12	2	no	—
283	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
284	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
286	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	—	2	2	no	—
289	C₂×C₃	—	2	1	no	—
290	C₂×C₃	—	2	1	no	—
291	C₂×C₃	—	2	1	no	—
292	C₂×C₃	—	4	2	no	—
293	C₂×C₃	—	6	2	no	—
294	C₂×C₃	C₂×#237	1	1	yes	characteristic
295	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
296	C₂×C₃	#1009	1	1	yes	characteristic
297	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
298	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
299	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
300	D₃	—	2	2	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	3	2	no	—
304	C₂²	—	6	1	no	—
305	C₂²	—	6	1	no	—
306	C₂²	C₃×D₃×Q₈	1	2	yes	—
307	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	3	2	no	—
312	C₄	—	6	1	no	—
313	C₄	—	6	2	no	—
314	C₄	—	6	2	no	—
315	C₄	—	6	1	no	—
316	C₄	C₃×#237	1	2	yes	—
317	C₃	—	2	1	no	—
318	C₃	#5635	1	1	yes	characteristic, direct product factor
319	C₃	C₃×#554	1	1	yes	characteristic, complemented
320	C₂	—	2	2	no	—
321	C₂	—	6	2	no	—
322	C₂	C₂×C₃×#237	1	1	yes	characteristic
323	C₂	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
324	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×C₂²⋊Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×#67	C₂²
hypercenter	C₃×C₂²⋊Q₈	D₃
hypocenter	C₃	C₃×#554
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃²×C₂²⋊Q₈	C₂	semidirect
2. C₃×#946	C₂	semidirect
3. C₃×Q₈×Dic₃	C₂	semidirect
5. C₂×C₃×C₄⋊Dic₃	C₂	semidirect
6. C₃×#897	C₂	semidirect
7. C₃×D₃×C₄⋊C₄	C₂	semidirect
8. C₃×#889	C₂	semidirect
9. C₃×#888	C₂	semidirect
11. C₃×#878	C₂	semidirect
13. #5635	C₃	direct (N × H)
43. C₃×C₂²⋊Q₈	D₃	semidirect
44. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
50. #946	C₂×C₃	semidirect
51. Q₈×Dic₃	C₂×C₃	semidirect
53. C₂×C₄⋊Dic₃	C₂×C₃	semidirect
54. #897	C₂×C₃	semidirect
55. D₃×C₄⋊C₄	C₂×C₃	semidirect
56. #889	C₂×C₃	semidirect
57. #888	C₂×C₃	semidirect
59. #878	C₂×C₃	semidirect
68. C₃×#67	Q₈	semidirect
151. C₂²⋊Q₈	C₃×D₃	semidirect
190. #67	C₃×Q₈	semidirect
268. C₃²	#554	semidirect
318. C₃	#5635	direct (N × H)
319. C₃	C₃×#554	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂²⋊Q₈	2
C₃×#946	2
C₃×Q₈×Dic₃	2
C₃×C₄×#67	2
C₂×C₃×C₄⋊Dic₃	2
C₃×#897	2
C₃×D₃×C₄⋊C₄	2
C₃×#889	2
C₃×#888	2
C₃×#67⋊C₄	2
C₃×#878	2
C₃×#554	3
#5635	3

Subgroups — C3×#5635 #10595732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5635 #10595732