Subgroups — C₃×#69103 #11797480

Order 1152.

361 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×#67	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4502	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2168	—	3	1	no	maximal
9	#69103	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×#67	C₄	1	1	yes	characteristic
12	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
15	C₃×#830	C₄	1	1	yes	characteristic
16	C₃×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
17	C₃×Dic₃⋊C₈	C₄	1	1	yes	characteristic
18	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
19	C₃×#812	—	2	2	no	—
20	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₃×D₃×C₁₆	—	2	2	no	—
23	C₂×C₃×M₃₂	—	3	1	no	—
24	C₂²×C₃×C₁₆	—	3	1	no	—
25	C₃×C₈×D₄	—	3	1	no	—
26	C₈×#67	C₂×C₃	1	1	yes	characteristic
27	C₂×#812	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×D₃×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×C₄⋊C₁₆	—	3	1	no	—
30	C₃×C₂²⋊C₁₆	—	2	1	no	—
31	C₃×C₂²⋊C₁₆	—	3	1	no	—
32	C₃×C₂²⋊C₁₆	D₃	1	1	yes	characteristic, complemented
33	C₃×C₂²⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×#346	—	3	1	no	—
35	(C₂×D₃)⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
36	#4502	C₂×C₃	1	1	yes	characteristic, complemented
37	Dic₃⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#67	C₈	1	1	yes	characteristic
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	—	2	1	no	—
42	C₂×C₃²×C₈	—	2	1	no	—
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
45	C₃×#217	C₈	1	1	yes	characteristic
46	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
47	C₃×Dic₃⋊C₄	C₈	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₃×D₃×C₈	—	2	1	no	—
52	C₃×D₃×C₈	—	2	1	no	—
53	C₃²×C₁₆	—	2	1	no	—
54	C₃²×C₁₆	—	2	1	no	—
55	C₃×C₃⋊C₁₆	—	2	1	no	—
56	C₃×C₃⋊C₁₆	D₄	1	2	yes	complemented
57	#2168	—	3	1	no	Sylow
58	C₂²×C₃×C₈	—	2	1	no	—
59	C₂²×C₃×C₈	—	3	1	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×C₄×D₄	—	3	1	no	—
63	C₄×#67	C₃×C₄	1	1	yes	characteristic
64	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₃×M₃₂	—	6	2	no	—
66	C₂×C₃×C₁₆	—	2	1	no	—
67	C₂×C₃×C₁₆	—	2	1	no	—
68	C₂×C₃×C₁₆	—	3	1	no	—
69	C₂×C₃×C₁₆	—	3	1	no	—
70	C₂×C₃×C₁₆	—	6	2	no	—
71	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
73	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄⋊C₈	—	3	1	no	—
76	C₃×C₂²⋊C₈	—	3	1	no	—
77	C₃×C₂²⋊C₈	—	3	1	no	—
78	C₃×C₄×C₈	—	3	1	no	—
79	#830	C₃×C₄	1	1	yes	characteristic
80	(C₂×D₃)⋊C₈	C₃×C₄	1	1	yes	characteristic
81	Dic₃⋊C₈	C₃×C₄	1	1	yes	characteristic
82	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic
83	#812	—	2	2	no	—
84	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
86	D₃×C₁₆	—	2	2	no	—
87	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
89	C₂×C₃²×C₄	—	2	1	no	—
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₃×#67	—	2	2	no	—
94	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
96	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
97	C₃×C₄×D₃	—	2	1	no	—
98	C₃×C₄×D₃	—	2	1	no	—
99	C₃²×C₈	—	2	1	no	—
100	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₃²×C₈	#41	1	1	yes	characteristic
102	C₃×C₃⋊C₈	—	2	1	no	—
103	C₃×C₃⋊C₈	—	2	1	no	—
104	C₂×M₃₂	—	3	1	no	—
105	C₂²×C₁₆	—	3	1	no	—
106	C₈×D₄	—	3	1	no	—
107	C₄⋊C₁₆	—	3	1	no	—
108	C₂²⋊C₁₆	—	3	1	no	—
109	C₂²⋊C₁₆	C₃×D₃	1	1	yes	characteristic, complemented
110	#346	—	3	1	no	—
111	C₂×C₃×D₄	—	3	1	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	—	3	1	no	—
114	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂×#67	C₃×C₈	1	1	yes	characteristic
117	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	—	3	1	no	—
125	C₂×C₃×C₈	—	3	1	no	—
126	C₂×C₃×C₈	—	4	1	no	—
127	C₂×C₃×C₈	—	4	1	no	—
128	C₂×C₃×C₈	—	6	1	no	—
129	C₂×C₃×C₈	—	6	1	no	—
130	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	3	1	no	—
135	C₃×C₂²⋊C₄	—	3	1	no	—
136	C₃×C₂²⋊C₄	—	3	1	no	—
137	C₃×C₄²	—	3	1	no	—
138	#217	C₃×C₈	1	1	yes	characteristic
139	(C₂×D₃)⋊C₄	C₃×C₈	1	1	yes	characteristic
140	Dic₃⋊C₄	C₃×C₈	1	1	yes	characteristic
141	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
142	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
144	D₃×C₈	—	2	1	no	—
145	D₃×C₈	—	2	1	no	—
146	C₃×C₁₆	—	2	1	no	—
147	C₃×C₁₆	—	2	1	no	—
148	C₃×C₁₆	—	2	1	no	—
149	C₃×C₁₆	—	2	1	no	—
150	C₃×C₁₆	—	3	2	no	—
151	C₃×C₁₆	—	4	1	no	—
152	C₃×C₁₆	—	4	1	no	—
153	C₃×C₁₆	—	6	1	no	—
154	C₃⋊C₁₆	—	2	1	no	—
155	C₃⋊C₁₆	C₃×D₄	1	2	yes	complemented
156	C₂²×C₃²	—	2	1	no	—
157	C₂²×C₃²	—	2	1	no	—
158	C₂²×C₃²	—	2	1	no	—
159	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
160	C₂×C₃×D₃	—	2	1	no	—
161	C₂×C₃×D₃	—	2	1	no	—
162	C₂×C₃×D₃	M₃₂	1	2	yes	complemented
163	C₃²×C₄	—	2	1	no	—
164	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
165	C₃²×C₄	#131	1	1	yes	characteristic
166	C₃×Dic₃	—	2	1	no	—
167	C₃×Dic₃	—	2	1	no	—
168	C₃×Dic₃	—	2	1	no	—
169	C₂²×C₈	—	3	1	no	—
170	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
171	C₄×D₄	—	3	1	no	—
172	M₃₂	—	6	2	no	—
173	C₂×C₁₆	—	3	1	no	—
174	C₂×C₁₆	—	3	1	no	—
175	C₂×C₁₆	—	6	2	no	—
176	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
178	C₄⋊C₈	—	3	1	no	—
179	C₂²⋊C₈	—	3	1	no	—
180	C₂²⋊C₈	—	3	1	no	—
181	C₄×C₈	—	3	1	no	—
182	C₂³×C₃	—	2	1	no	—
183	C₂³×C₃	—	3	1	no	—
184	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
185	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
186	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
187	C₃×D₄	—	6	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	3	1	no	—
195	C₂×C₃×C₄	—	3	1	no	—
196	C₂×C₃×C₄	—	3	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	—	4	1	no	—
199	C₂×C₃×C₄	—	6	1	no	—
200	C₂×C₃×C₄	—	6	1	no	—
201	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
205	#67	—	2	2	no	—
206	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
207	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
208	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
209	C₄×D₃	—	2	1	no	—
210	C₄×D₃	—	2	1	no	—
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	4	1	no	—
216	C₃×C₈	—	6	1	no	—
217	C₃×C₈	—	6	1	no	—
218	C₃×C₈	D₃×D₄	1	1	yes	characteristic
219	C₃×C₈	#237	1	1	yes	characteristic
220	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
221	C₃×C₈	C₃×#41	1	1	yes	characteristic
222	C₃⋊C₈	—	2	1	no	—
223	C₃⋊C₈	—	2	1	no	—
224	C₂×C₃²	—	4	1	no	—
225	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
226	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
227	C₂×C₃²	#504	1	1	yes	characteristic
228	C₃×D₃	—	2	2	no	—
229	C₂×D₄	—	3	1	no	—
230	C₂²×C₄	—	3	1	no	—
231	C₂²×C₄	C₃×C₄×D₃	1	1	yes	characteristic
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	3	1	no	—
235	C₂×C₈	—	3	1	no	—
236	C₂×C₈	—	6	1	no	—
237	C₂×C₈	—	6	1	no	—
238	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
239	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
240	C₄⋊C₄	—	3	1	no	—
241	C₂²⋊C₄	—	3	1	no	—
242	C₂²⋊C₄	—	3	1	no	—
243	C₄²	—	3	1	no	—
244	C₁₆	—	2	1	no	—
245	C₁₆	—	2	1	no	—
246	C₁₆	—	3	2	no	—
247	C₁₆	—	6	1	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	3	2	no	—
257	C₂²×C₃	—	4	1	no	—
258	C₂²×C₃	—	4	1	no	—
259	C₂²×C₃	—	6	1	no	—
260	C₂²×C₃	—	6	1	no	—
261	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
262	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
263	C₂×D₃	—	2	1	no	—
264	C₂×D₃	—	2	1	no	—
265	C₂×D₃	C₃×M₃₂	1	2	yes	complemented
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	4	1	no	—
271	C₃×C₄	—	6	1	no	—
272	C₃×C₄	—	6	1	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
275	C₃×C₄	#907	1	1	yes	characteristic
276	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
277	C₃×C₄	C₃×#131	1	1	yes	characteristic
278	Dic₃	—	2	1	no	—
279	Dic₃	—	2	1	no	—
280	Dic₃	—	2	1	no	—
281	C₃²	#2168	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	3	1	no	—
283	C₂³	C₃×D₃×C₈	1	1	yes	characteristic
284	D₄	—	6	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	3	1	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	6	1	no	—
296	C₈	—	6	1	no	—
297	C₈	C₃×D₃×D₄	1	1	yes	characteristic
298	C₈	C₃×#237	1	1	yes	characteristic
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	4	2	no	—
303	C₂×C₃	—	4	1	no	—
304	C₂×C₃	—	4	1	no	—
305	C₂×C₃	—	6	2	no	—
306	C₂×C₃	#67⋊C₈	1	1	yes	characteristic
307	C₂×C₃	#4901	1	1	yes	characteristic
308	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
309	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
310	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
311	C₂×C₃	C₃×#504	1	1	yes	characteristic
312	D₃	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	3	2	no	—
317	C₂²	—	6	1	no	—
318	C₂²	—	6	1	no	—
319	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	6	1	no	—
322	C₄	—	6	1	no	—
323	C₄	—	6	1	no	—
324	C₄	C₃×#67⋊C₄	1	1	yes	characteristic
325	C₄	C₃×#907	1	1	yes	characteristic
326	C₃	—	2	1	no	—
327	C₃	#69103	1	1	yes	characteristic, direct product factor
328	C₃	C₃×#2168	1	1	yes	characteristic, complemented
329	C₂	—	4	1	no	—
330	C₂	—	6	2	no	—
331	C₂	C₃×#67⋊C₈	1	1	yes	characteristic
332	C₂	C₃×#4901	1	1	yes	characteristic
333	C₂	C₃×D₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×C₂²⋊C₁₆	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃×#67	C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×C₂²⋊C₁₆	D₃
hypocenter	C₃	C₃×#2168
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#812	C₂	semidirect
3. C₂×C₃×D₃×C₁₆	C₂	semidirect
4. C₃²×C₂²⋊C₁₆	C₂	semidirect
5. C₃×(C₂×D₃)⋊C₁₆	C₂	semidirect
6. C₃×#4502	C₂	semidirect
7. C₃×Dic₃⋊C₁₆	C₂	semidirect
9. #69103	C₃	direct (N × H)
27. C₂×#812	C₂×C₃	semidirect
28. C₂×D₃×C₁₆	C₂×C₃	semidirect
32. C₃×C₂²⋊C₁₆	D₃	semidirect
33. C₃×C₂²⋊C₁₆	C₂×C₃	semidirect
35. (C₂×D₃)⋊C₁₆	C₂×C₃	semidirect
36. #4502	C₂×C₃	semidirect
37. Dic₃⋊C₁₆	C₂×C₃	semidirect
56. C₃×C₃⋊C₁₆	D₄	semidirect
109. C₂²⋊C₁₆	C₃×D₃	semidirect
155. C₃⋊C₁₆	C₃×D₄	semidirect
162. C₂×C₃×D₃	M₃₂	semidirect
265. C₂×D₃	C₃×M₃₂	semidirect
281. C₃²	#2168	semidirect
327. C₃	#69103	direct (N × H)
328. C₃	C₃×#2168	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈×#67	2
C₂×C₃×#812	2
C₂×C₃×D₃×C₁₆	2
C₃²×C₂²⋊C₁₆	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×#4502	2
C₃×Dic₃⋊C₁₆	2
C₃×#2168	3
#69103	3

Subgroups — C3×#69103 #11797480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69103 #11797480