Subgroups — C₃×#35336 #10653943

Order 768.

1538 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2851	C₂	1	2	yes	maximal
4	C₃×#2647	C₂	1	2	yes	maximal, complemented
5	C₃×#2629	C₂	1	2	yes	maximal
6	C₃×#2590	C₂	1	8	yes	maximal, complemented
7	C₃×#2587	C₂	1	8	yes	maximal
8	C₃×#2330	C₂	1	2	yes	maximal, complemented
9	C₃×#2314	C₂	1	2	yes	maximal
10	C₃×C₄×#125	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic
12	#35336	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂×C₃×#125	C₂²	1	2	yes	—
14	C₂×C₃×#125	C₂²	1	2	yes	—
15	C₂×C₃×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃×#117	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
18	C₃×#400	C₂²	1	8	yes	—
19	C₃×#398	C₂²	1	8	yes	—
20	C₃×#398	C₂²	1	8	yes	—
21	C₃×#396	C₂²	1	8	yes	—
22	C₃×#389	C₂²	1	4	yes	—
23	C₃×#389	C₂²	1	4	yes	—
24	C₃×#389	C₂²	1	8	yes	—
25	C₃×#389	C₂²	1	8	yes	—
26	C₃×#389	C₂²	1	8	yes	—
27	C₃×#388	C₂²	1	8	yes	—
28	C₃×#387	C₂²	1	8	yes	—
29	C₃×#387	C₂²	1	8	yes	—
30	C₃×#387	C₂²	1	8	yes	—
31	C₃×#387	C₂²	1	8	yes	—
32	C₃×#385	C₂²	1	8	yes	—
33	C₃×#384	C₂²	1	2	yes	—
34	C₃×#384	C₂²	1	2	yes	—
35	C₃×#382	C₂²	1	2	yes	—
36	C₃×#382	C₂²	1	2	yes	—
37	C₃×#380	C₂²	1	4	yes	—
38	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
39	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
40	C₃×C₄×C₄⋊C₄	C₂²	1	8	yes	—
41	C₃×C₄×C₂²⋊C₄	C₂²	1	4	yes	—
42	C₃×#376	C₂²	1	1	yes	characteristic
43	C₃×#376	C₂²	1	1	yes	characteristic
44	C₃×C₄³	C₂²	1	1	yes	characteristic
45	C₃×C₄³	C₂²	1	1	yes	characteristic
46	#2859	C₂×C₃	1	1	yes	characteristic, complemented
47	#2859	C₂×C₃	1	1	yes	characteristic, complemented
48	#2851	C₂×C₃	1	2	yes	—
49	#2647	C₂×C₃	1	2	yes	complemented
50	#2629	C₂×C₃	1	2	yes	—
51	#2590	C₂×C₃	1	8	yes	complemented
52	#2587	C₂×C₃	1	8	yes	—
53	#2330	C₂×C₃	1	2	yes	complemented
54	#2314	C₂×C₃	1	2	yes	—
55	C₄×#125	C₂×C₃	1	2	yes	complemented
56	C₄×#117	C₂×C₃	1	1	yes	characteristic
57	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
58	C₂³×C₃×C₄	C₂³	1	2	yes	—
59	C₃×#125	—	2	8	no	—
60	C₃×#125	—	2	8	no	—
61	C₃×#117	—	2	4	no	—
62	C₃×#117	Q₈	1	8	yes	—
63	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
65	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
66	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
67	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
68	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
69	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
70	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
71	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
72	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
73	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
74	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
75	C₂×C₃×C₂²⋊C₄	C₂³	1	2	yes	—
76	C₂×C₃×C₄²	—	2	4	no	—
77	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
78	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
79	C₂×C₃×C₄²	C₂³	1	2	yes	—
80	C₂×C₃×C₄²	C₂³	1	4	yes	—
81	C₂×C₃×C₄²	C₂³	1	4	yes	—
82	C₂×C₃×C₄²	C₂³	1	8	yes	—
83	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
84	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
85	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
86	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
87	C₃×#95	C₂³	1	4	yes	—
88	C₃×#95	C₂³	1	4	yes	—
89	C₃×#95	C₂³	1	8	yes	—
90	C₃×#95	C₂³	1	8	yes	—
91	C₃×#95	C₂³	1	8	yes	—
92	C₃×#95	C₂³	1	8	yes	—
93	C₃×#95	C₂³	1	8	yes	—
94	C₃×#95	C₂³	1	8	yes	—
95	C₂×#125	C₂²×C₃	1	2	yes	—
96	C₂×#125	C₂²×C₃	1	2	yes	—
97	C₂×#117	C₂²×C₃	1	1	yes	characteristic
98	C₂×#117	C₂²×C₃	1	1	yes	characteristic
99	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
100	#400	C₂²×C₃	1	8	yes	—
101	#398	C₂²×C₃	1	8	yes	—
102	#398	C₂²×C₃	1	8	yes	—
103	#396	C₂²×C₃	1	8	yes	—
104	#389	C₂²×C₃	1	4	yes	—
105	#389	C₂²×C₃	1	8	yes	—
106	#389	C₂²×C₃	1	4	yes	—
107	#389	C₂²×C₃	1	8	yes	—
108	#389	C₂²×C₃	1	8	yes	—
109	#388	C₂²×C₃	1	8	yes	—
110	#387	C₂²×C₃	1	8	yes	—
111	#387	C₂²×C₃	1	8	yes	—
112	#387	C₂²×C₃	1	8	yes	—
113	#387	C₂²×C₃	1	8	yes	—
114	#385	C₂²×C₃	1	8	yes	—
115	#384	C₂²×C₃	1	2	yes	—
116	#384	C₂²×C₃	1	2	yes	—
117	#382	C₂²×C₃	1	2	yes	—
118	#382	C₂²×C₃	1	2	yes	—
119	#380	C₂²×C₃	1	4	yes	—
120	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
121	C₄×C₄⋊C₄	C₂²×C₃	1	8	yes	—
122	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
123	C₄×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
124	#376	C₂²×C₃	1	1	yes	characteristic
125	#376	C₂²×C₃	1	1	yes	characteristic
126	C₄³	C₂²×C₃	1	1	yes	characteristic
127	C₄³	C₂²×C₃	1	1	yes	characteristic
128	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
129	C₂²×C₃×C₄	—	2	2	no	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	—	2	4	no	—
132	C₂²×C₃×C₄	—	4	4	no	—
133	C₂²×C₃×C₄	C₂×Q₈	1	4	yes	—
134	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
135	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
136	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
137	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
138	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
140	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
141	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
142	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
143	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	4	no	—
145	C₃×C₄⋊C₄	—	2	16	no	—
146	C₃×C₄⋊C₄	—	2	16	no	—
147	C₃×C₄⋊C₄	—	2	8	no	—
148	C₃×C₄⋊C₄	—	2	8	no	—
149	C₃×C₄⋊C₄	—	4	8	no	—
150	C₃×C₄⋊C₄	—	4	8	no	—
151	C₃×C₄⋊C₄	—	4	8	no	—
152	C₃×C₄⋊C₄	—	4	8	no	—
153	C₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
154	C₃×C₂²⋊C₄	—	2	4	no	—
155	C₃×C₂²⋊C₄	—	4	8	no	—
156	C₃×C₂²⋊C₄	C₂×Q₈	1	8	yes	—
157	C₃×C₄²	—	2	4	no	—
158	C₃×C₄²	—	2	2	no	—
159	C₃×C₄²	—	2	2	no	—
160	C₃×C₄²	—	4	4	no	—
161	C₃×C₄²	—	4	8	no	—
162	C₃×C₄²	—	4	4	no	—
163	C₃×C₄²	C₂×Q₈	1	4	yes	—
164	C₃×C₄²	C₂×Q₈	1	4	yes	—
165	C₃×C₄²	#41	1	4	yes	—
166	C₃×C₄²	#41	1	4	yes	—
167	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂³×C₄	C₂³×C₃	1	2	yes	—
169	#125	—	2	8	no	—
170	#125	—	2	8	no	—
171	#117	—	2	4	no	—
172	#117	C₃×Q₈	1	8	yes	—
173	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
174	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
175	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
176	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
177	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
178	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
179	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
180	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
181	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
182	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
185	C₂×C₂²⋊C₄	C₂³×C₃	1	2	yes	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₃	1	2	yes	—
190	C₂×C₄²	C₂³×C₃	1	4	yes	—
191	C₂×C₄²	C₂³×C₃	1	4	yes	—
192	C₂×C₄²	C₂³×C₃	1	8	yes	—
193	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
197	#95	C₂³×C₃	1	4	yes	—
198	#95	C₂³×C₃	1	4	yes	—
199	#95	C₂³×C₃	1	8	yes	—
200	#95	C₂³×C₃	1	8	yes	—
201	#95	C₂³×C₃	1	8	yes	—
202	#95	C₂³×C₃	1	8	yes	—
203	#95	C₂³×C₃	1	8	yes	—
204	#95	C₂³×C₃	1	8	yes	—
205	C₂³×C₃	—	2	1	no	—
206	C₂³×C₃	—	2	1	no	—
207	C₂³×C₃	—	2	4	no	—
208	C₂³×C₃	C₂²×Q₈	1	2	yes	—
209	C₂³×C₃	C₂⁵	1	1	yes	characteristic
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	1	no	—
212	C₂×C₃×C₄	—	2	4	no	—
213	C₂×C₃×C₄	—	2	4	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	8	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	8	no	—
220	C₂×C₃×C₄	—	2	8	no	—
221	C₂×C₃×C₄	—	2	8	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	8	no	—
224	C₂×C₃×C₄	—	2	8	no	—
225	C₂×C₃×C₄	—	2	8	no	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	—	4	8	no	—
228	C₂×C₃×C₄	C₂²×Q₈	1	4	yes	—
229	C₂×C₃×C₄	C₂²×Q₈	1	4	yes	—
230	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
231	C₂×C₃×C₄	C₂²×Q₈	1	2	yes	—
232	C₂×C₃×C₄	C₂×#41	1	4	yes	—
233	C₂×C₃×C₄	C₂×#41	1	4	yes	—
234	C₂×C₃×C₄	C₂×#41	1	2	yes	—
235	C₂×C₃×C₄	C₂×#41	1	2	yes	—
236	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	4	4	no	—
241	C₂²×C₄	C₂×C₃×Q₈	1	4	yes	—
242	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
244	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
245	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
246	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
248	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
249	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
250	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
251	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	16	no	—
255	C₄⋊C₄	—	2	8	no	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	4	8	no	—
258	C₄⋊C₄	—	4	8	no	—
259	C₄⋊C₄	—	4	8	no	—
260	C₄⋊C₄	—	4	8	no	—
261	C₄⋊C₄	C₂×C₃×Q₈	1	8	yes	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	4	8	no	—
264	C₂²⋊C₄	C₂×C₃×Q₈	1	8	yes	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	4	no	—
269	C₄²	—	4	8	no	—
270	C₄²	—	4	4	no	—
271	C₄²	C₂×C₃×Q₈	1	4	yes	—
272	C₄²	C₂×C₃×Q₈	1	4	yes	—
273	C₄²	C₃×#41	1	4	yes	—
274	C₄²	C₃×#41	1	4	yes	—
275	C₂²×C₃	—	2	2	no	—
276	C₂²×C₃	—	2	4	no	—
277	C₂²×C₃	—	4	2	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	C₂³×Q₈	1	1	yes	characteristic
280	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
281	C₂²×C₃	C₂²×#41	1	1	yes	characteristic
282	C₂²×C₃	#585	1	4	yes	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	2	4	no	—
286	C₃×C₄	—	2	4	no	—
287	C₃×C₄	—	2	4	no	—
288	C₃×C₄	—	4	8	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	4	2	no	—
291	C₃×C₄	—	4	8	no	—
292	C₃×C₄	#533	1	4	yes	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	4	no	—
296	C₂³	C₂²×C₃×Q₈	1	2	yes	—
297	C₂³	C₂⁵×C₃	1	1	yes	characteristic
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	8	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	8	no	—
308	C₂×C₄	—	2	8	no	—
309	C₂×C₄	—	2	8	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	8	no	—
312	C₂×C₄	—	2	8	no	—
313	C₂×C₄	—	2	8	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	4	8	no	—
316	C₂×C₄	C₂²×C₃×Q₈	1	4	yes	—
317	C₂×C₄	C₂²×C₃×Q₈	1	4	yes	—
318	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
319	C₂×C₄	C₂²×C₃×Q₈	1	2	yes	—
320	C₂×C₄	C₂×C₃×#41	1	4	yes	—
321	C₂×C₄	C₂×C₃×#41	1	4	yes	—
322	C₂×C₄	C₂×C₃×#41	1	2	yes	—
323	C₂×C₄	C₂×C₃×#41	1	2	yes	—
324	C₂×C₃	—	4	2	no	—
325	C₂×C₃	C₂×#533	1	1	yes	characteristic
326	C₂×C₃	#3455	1	2	yes	—
327	C₂×C₃	#3504	1	2	yes	—
328	C₂×C₃	#3512	1	2	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	C₂³×C₃×Q₈	1	1	yes	characteristic
334	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
335	C₂²	C₂²×C₃×#41	1	1	yes	characteristic
336	C₂²	C₃×#585	1	4	yes	—
337	C₄	—	2	2	no	—
338	C₄	—	2	4	no	—
339	C₄	—	2	4	no	—
340	C₄	—	2	4	no	—
341	C₄	—	2	4	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	8	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	8	no	—
346	C₄	C₃×#533	1	4	yes	—
347	C₃	#35336	1	1	yes	characteristic, direct product factor, Sylow
348	C₂	—	4	2	no	—
349	C₂	C₂×C₃×#533	1	1	yes	characteristic
350	C₂	C₃×#3455	1	2	yes	—
351	C₂	C₃×#3504	1	2	yes	—
352	C₂	C₃×#3512	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂⁴×C₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#2859	C₂	semidirect
2. C₃×#2859	C₂	semidirect
4. C₃×#2647	C₂	semidirect
6. C₃×#2590	C₂	semidirect
8. C₃×#2330	C₂	semidirect
10. C₃×C₄×#125	C₂	semidirect
12. #35336	C₃	direct (N × H)
46. #2859	C₂×C₃	semidirect
47. #2859	C₂×C₃	semidirect
49. #2647	C₂×C₃	semidirect
51. #2590	C₂×C₃	semidirect
53. #2330	C₂×C₃	semidirect
55. C₄×#125	C₂×C₃	semidirect
347. C₃	#35336	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#2859	2
C₃×#2851	2
C₃×#2647	2
C₃×#2629	2
C₃×#2590	2
C₃×#2587	2
C₃×#2330	2
C₃×#2314	2
C₃×C₄×#125	2
C₃×C₄×#117	2
#35336	3

Subgroups — C3×#35336 #10653943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#35336 #10653943