Subgroups — C₃×#69769 #11790097

Order 1152.

577 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
4	C₃×#4928	C₂	1	2	yes	maximal
5	C₃×#4551	C₂	1	2	yes	maximal
6	C₃×#1941	—	3	1	no	maximal
7	#69769	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₃×Dic₃⋊C₄	—	2	2	no	—
14	C₃²×C₄×C₈	—	2	2	no	—
15	C₃×C₃⋊#95	—	2	2	no	—
16	C₃×#817	—	2	2	no	—
17	C₃×#817	C₄	1	4	yes	complemented
18	C₂×C₃×#106	—	3	1	no	—
19	C₂×C₃×#106	—	3	1	no	—
20	C₂×C₃×C₄×C₈	—	2	1	no	—
21	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
23	C₃×#389	—	3	2	no	—
24	C₂×#817	C₂×C₃	1	1	yes	characteristic
25	C₂×#817	C₂×C₃	1	1	yes	characteristic
26	#4928	C₂×C₃	1	2	yes	—
27	C₃×#340	—	3	2	no	—
28	#4551	C₂×C₃	1	2	yes	—
29	C₂²×C₃²×C₄	D₄	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	2	no	—
32	C₂²×C₃×Dic₃	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	1	no	—
34	C₂×C₃²×C₈	—	2	1	no	—
35	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	C₂×C₃²×C₈	Q₈	1	2	yes	—
38	C₂×C₃²×C₈	Q₈	1	2	yes	—
39	C₃²×C₄²	—	2	1	no	—
40	C₃²×C₄²	—	2	1	no	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	—	2	2	no	—
43	C₃×C₄⋊Dic₃	—	2	2	no	—
44	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
45	C₃×Dic₃⋊C₄	—	4	4	no	—
46	#1941	—	3	1	no	Sylow
47	C₂²×C₃×C₈	—	2	1	no	—
48	C₂²×C₃×C₈	—	2	1	no	—
49	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×C₄⋊C₄	—	3	2	no	—
54	C₂×C₃×C₄⋊C₄	—	3	2	no	—
55	C₂×C₃×C₄⋊C₄	—	6	2	no	—
56	C₂×C₃×C₄²	—	2	1	no	—
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
60	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
61	C₂×Dic₃⋊C₄	—	2	2	no	—
62	C₃×#106	—	3	4	no	—
63	C₃×#106	—	6	2	no	—
64	C₃×C₄×C₈	—	2	4	no	—
65	C₃×C₄×C₈	—	2	2	no	—
66	C₃×C₄×C₈	—	2	2	no	—
67	C₃×#95	—	6	2	no	—
68	C₃⋊#95	—	2	2	no	—
69	#817	—	2	2	no	—
70	#817	C₃×C₄	1	4	yes	complemented
71	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	—	2	2	no	—
73	C₂×C₃²×C₄	—	2	2	no	—
74	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	2	4	no	—
82	C₂×C₃×Dic₃	—	4	2	no	—
83	C₂×C₃×Dic₃	—	4	2	no	—
84	C₂×C₃×Dic₃	—	4	2	no	—
85	C₂×C₃×Dic₃	—	4	2	no	—
86	C₂×C₃×Dic₃	—	4	2	no	—
87	C₃²×C₈	—	2	2	no	—
88	C₃²×C₈	C₄⋊C₄	1	4	yes	complemented
89	C₂×#106	—	3	1	no	—
90	C₂×#106	—	3	1	no	—
91	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
92	#389	—	3	2	no	—
93	#340	—	3	2	no	—
94	C₂²×C₃×C₄	—	2	2	no	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	6	2	no	—
97	C₂²×C₃×C₄	—	6	2	no	—
98	C₂²×C₃×C₄	D₁₂	1	2	yes	—
99	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₂²×Dic₃	—	2	2	no	—
103	C₂²×Dic₃	—	2	2	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	Dic₆	1	2	yes	—
115	C₂×C₃×C₈	Dic₆	1	2	yes	—
116	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
117	C₂×C₃×C₈	#67	1	2	yes	—
118	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
119	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
120	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
121	C₂×C₃×C₈	C₃×Q₈	1	2	yes	—
122	C₃×C₄⋊C₄	—	3	4	no	—
123	C₃×C₄⋊C₄	—	6	2	no	—
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₄⋊C₄	—	6	2	no	—
126	C₃×C₄⋊C₄	—	12	4	no	—
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₄⋊Dic₃	—	2	2	no	—
134	C₄⋊Dic₃	—	2	2	no	—
135	C₄⋊Dic₃	—	2	2	no	—
136	C₄⋊Dic₃	C₂×C₃×C₄	1	4	yes	—
137	Dic₃⋊C₄	—	4	4	no	—
138	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
139	C₂²×C₃²	#121	1	1	yes	characteristic
140	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
142	C₂²×C₃²	#135	1	2	yes	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
146	C₃²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
147	C₃²×C₄	#125	1	1	yes	characteristic
148	C₃×Dic₃	—	4	4	no	—
149	C₃×Dic₃	—	8	2	no	—
150	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	3	2	no	—
153	C₂×C₄⋊C₄	—	3	2	no	—
154	C₂×C₄⋊C₄	—	6	2	no	—
155	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
156	#106	—	3	4	no	—
157	#106	—	6	2	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	6	2	no	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
162	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	—	2	4	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	4	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	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	6	4	no	—
177	C₂×C₃×C₄	—	12	2	no	—
178	C₂×C₃×C₄	—	12	2	no	—
179	C₂×C₃×C₄	—	12	2	no	—
180	C₂×C₃×C₄	—	12	2	no	—
181	C₂×C₃×C₄	—	12	2	no	—
182	C₂×C₃×C₄	#204	1	4	yes	—
183	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
184	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
187	C₂×C₃×C₄	#235	1	1	yes	characteristic
188	C₂×C₃×C₄	#235	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
195	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
196	C₂×Dic₃	—	2	4	no	—
197	C₂×Dic₃	—	4	2	no	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	4	2	no	—
200	C₂×Dic₃	—	4	2	no	—
201	C₂×Dic₃	—	4	2	no	—
202	C₃×C₈	—	2	4	no	—
203	C₃×C₈	—	2	2	no	—
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	2	4	no	—
206	C₃×C₈	Dic₃⋊C₄	1	4	yes	complemented
207	C₃×C₈	C₃×C₄⋊C₄	1	4	yes	complemented
208	C₂×C₃²	#389	1	1	yes	characteristic
209	C₂×C₃²	C₄×SD₁₆	1	2	yes	—
210	C₂×C₃²	#465	1	2	yes	—
211	C₂×C₃²	#498	1	1	yes	characteristic
212	C₂×C₃²	#499	1	1	yes	characteristic
213	C₂²×C₄	—	6	2	no	—
214	C₂²×C₄	—	6	2	no	—
215	C₂²×C₄	C₃×D₁₂	1	2	yes	—
216	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	C₃×Dic₆	1	2	yes	—
220	C₂×C₈	C₃×Dic₆	1	2	yes	—
221	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
222	C₂×C₈	C₃×#67	1	2	yes	—
223	C₄⋊C₄	—	3	4	no	—
224	C₄⋊C₄	—	6	2	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₄⋊C₄	—	6	2	no	—
227	C₄⋊C₄	—	12	4	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	#869	1	1	yes	characteristic
236	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
237	C₂²×C₃	C₂×#204	1	2	yes	—
238	C₂²×C₃	#904	1	2	yes	—
239	C₂²×C₃	#930	1	1	yes	characteristic
240	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
241	C₂²×C₃	C₃×#121	1	1	yes	characteristic
242	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
243	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
244	C₂²×C₃	C₃×#135	1	2	yes	—
245	C₃×C₄	—	2	8	no	—
246	C₃×C₄	—	2	4	no	—
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	1	no	—
252	C₃×C₄	—	12	4	no	—
253	C₃×C₄	—	24	2	no	—
254	C₃×C₄	C₄×Dic₆	1	1	yes	characteristic
255	C₃×C₄	#870	1	1	yes	characteristic
256	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
257	C₃×C₄	#924	1	1	yes	characteristic
258	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
259	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
260	C₃×C₄	C₃×C₂²⋊Q₈	1	1	yes	characteristic
261	C₃×C₄	C₃×#125	1	1	yes	characteristic
262	Dic₃	—	4	4	no	—
263	Dic₃	—	8	2	no	—
264	C₃²	#1941	1	1	yes	characteristic, complemented, Sylow
265	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	6	4	no	—
269	C₂×C₄	—	12	2	no	—
270	C₂×C₄	—	12	2	no	—
271	C₂×C₄	—	12	2	no	—
272	C₂×C₄	—	12	2	no	—
273	C₂×C₄	—	12	2	no	—
274	C₂×C₄	C₃×#204	1	4	yes	—
275	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
276	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
278	C₂×C₄	C₃×#235	1	1	yes	characteristic
279	C₂×C₄	C₃×#235	1	1	yes	characteristic
280	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	C₃×Dic₃⋊C₄	1	4	yes	complemented
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	2	2	no	—
287	C₂×C₃	—	2	2	no	—
288	C₂×C₃	#4680	1	1	yes	characteristic
289	C₂×C₃	#4683	1	1	yes	characteristic
290	C₂×C₃	C₄×#204	1	2	yes	—
291	C₂×C₃	#4928	1	1	yes	characteristic
292	C₂×C₃	#5114	1	2	yes	—
293	C₂×C₃	C₃×#389	1	1	yes	characteristic
294	C₂×C₃	C₃×C₄×SD₁₆	1	2	yes	—
295	C₂×C₃	C₃×#465	1	2	yes	—
296	C₂×C₃	C₃×#498	1	1	yes	characteristic
297	C₂×C₃	C₃×#499	1	1	yes	characteristic
298	C₂²	C₃×#869	1	1	yes	characteristic
299	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
300	C₂²	C₂×C₃×#204	1	2	yes	—
301	C₂²	C₃×#904	1	2	yes	—
302	C₂²	C₃×#930	1	1	yes	characteristic
303	C₄	—	2	4	no	—
304	C₄	—	12	4	no	—
305	C₄	—	24	2	no	—
306	C₄	C₃×C₄×Dic₆	1	1	yes	characteristic
307	C₄	C₃×#870	1	1	yes	characteristic
308	C₄	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
309	C₄	C₃×#924	1	1	yes	characteristic
310	C₃	—	2	1	no	—
311	C₃	#69769	1	1	yes	characteristic, direct product factor
312	C₃	C₃×#1941	1	1	yes	characteristic, complemented
313	C₂	C₃×#4680	1	1	yes	characteristic
314	C₂	C₃×#4683	1	1	yes	characteristic
315	C₂	C₃×C₄×#204	1	2	yes	—
316	C₂	C₃×#4928	1	1	yes	characteristic
317	C₂	C₃×#5114	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#1941
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #69769	C₃	direct (N × H)
17. C₃×#817	C₄	semidirect
70. #817	C₃×C₄	semidirect
88. C₃²×C₈	C₄⋊C₄	semidirect
206. C₃×C₈	Dic₃⋊C₄	semidirect
207. C₃×C₈	C₃×C₄⋊C₄	semidirect
264. C₃²	#1941	semidirect
282. C₈	C₃×Dic₃⋊C₄	semidirect
311. C₃	#69769	direct (N × H)
312. C₃	C₃×#1941	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×C₃×#817	2
C₃×#4928	2
C₃×#4551	2
C₃×#1941	3
#69769	3

Subgroups — C3×#69769 #11790097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69769 #11790097