Subgroups — C₃×#78535 #11876976

Order 1152.

963 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5477	C₂	1	2	yes	maximal, complemented
2	C₃²×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#5135	C₂	1	4	yes	maximal, complemented
4	C₃×#4711	C₂	1	4	yes	maximal, complemented
5	C₃×#4695	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4694	C₂	1	2	yes	maximal, complemented
7	C₃×#4680	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3184	—	3	1	no	maximal
9	#78535	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#235	—	2	2	no	—
11	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₃×#930	—	2	2	no	—
14	C₃×#930	C₂²	1	4	yes	—
15	C₂×C₃×D₂₄	C₂²	1	4	yes	—
16	C₃×#875	—	2	2	no	—
17	C₃×#874	C₂²	1	2	yes	—
18	C₃×C₄×D₁₂	—	2	2	no	—
19	C₃×C₄×D₁₂	C₂²	1	4	yes	—
20	C₃×#869	C₂²	1	2	yes	—
21	C₃²×#97	C₂²	1	2	yes	—
22	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₃×#821	C₂²	1	8	yes	—
25	C₃×#817	C₂²	1	4	yes	—
26	C₃×#532	—	3	2	no	—
27	#5477	C₂×C₃	1	2	yes	complemented
28	C₃×#498	—	3	1	no	—
29	C₃×#493	—	3	1	no	—
30	C₃×#486	—	3	2	no	—
31	C₃×#469	—	3	4	no	—
32	C₃×#442	—	3	4	no	—
33	C₃×C₄×M₁₆	—	2	1	no	—
34	C₃×C₄×M₁₆	D₃	1	1	yes	characteristic, complemented
35	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
36	#5135	C₂×C₃	1	4	yes	complemented
37	#4711	C₂×C₃	1	4	yes	complemented
38	#4695	C₂×C₃	1	1	yes	characteristic, complemented
39	#4694	C₂×C₃	1	2	yes	complemented
40	#4680	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂²×C₃²×C₄	D₄	1	2	yes	—
42	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
43	C₂×C₃×#67	—	2	4	no	—
44	C₃×#235	—	4	8	no	—
45	C₂×C₃×D₁₂	—	2	2	no	—
46	C₂×C₃×D₁₂	—	2	2	no	—
47	C₂×C₃×D₁₂	C₂³	1	4	yes	—
48	C₂×C₃×C₄×D₃	—	2	4	no	—
49	C₂×C₃×Dic₆	—	2	2	no	—
50	C₃²×M₁₆	—	2	4	no	—
51	C₂×C₃²×C₈	C₂³	1	2	yes	—
52	C₂×C₃²×C₈	C₂³	1	2	yes	—
53	C₃²×C₄²	D₄	1	2	yes	—
54	C₃²×C₄²	C₂³	1	1	yes	characteristic
55	C₃²×C₄²	C₂³	1	1	yes	characteristic
56	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
57	C₃×(C₂×D₃)⋊C₄	—	2	4	no	—
58	C₃×C₄⋊Dic₃	—	2	2	no	—
59	C₃×C₄⋊Dic₃	C₂³	1	4	yes	—
60	C₃×D₂₄	—	2	8	no	—
61	#3184	—	3	1	no	Sylow
62	C₂×C₃×#41	—	6	2	no	—
63	C₂×#235	—	2	2	no	—
64	C₂×C₃×D₈	—	3	4	no	—
65	C₂×C₃×M₁₆	—	2	2	no	—
66	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
67	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
68	C₃×C₄⋊Q₈	—	3	2	no	—
69	C₃×#127	—	3	2	no	—
70	C₃×#124	—	6	2	no	—
71	C₃×#121	—	3	4	no	—
72	C₃×#121	—	6	2	no	—
73	C₃×C₄×D₄	—	3	4	no	—
74	C₃×C₄×D₄	—	6	2	no	—
75	C₂×C₃×C₄²	—	2	1	no	—
76	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
77	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
78	#930	—	2	2	no	—
79	#930	C₂²×C₃	1	4	yes	—
80	C₂×D₂₄	C₂²×C₃	1	4	yes	—
81	#875	—	2	2	no	—
82	#874	C₂²×C₃	1	2	yes	—
83	C₄×D₁₂	—	2	2	no	—
84	C₄×D₁₂	C₂²×C₃	1	4	yes	—
85	#869	C₂²×C₃	1	2	yes	—
86	C₃×#106	—	3	4	no	—
87	C₃×#102	—	3	8	no	—
88	C₃×#97	—	2	2	no	—
89	C₃×#97	C₂×D₃	1	2	yes	—
90	C₃×#97	C₂²×C₃	1	2	yes	—
91	C₃×C₄×C₈	—	2	1	no	—
92	C₃×C₄×C₈	—	2	1	no	—
93	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
94	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
95	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
96	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
97	#821	C₂²×C₃	1	8	yes	—
98	#817	C₂²×C₃	1	4	yes	—
99	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₃×D₃	—	2	4	no	—
101	C₂×C₃²×C₄	—	2	2	no	—
102	C₂×C₃²×C₄	—	2	4	no	—
103	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
105	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
106	C₂×C₃²×C₄	C₂⁴	1	1	yes	characteristic
107	C₃×#67	—	4	8	no	—
108	C₂×C₃×Dic₃	—	2	4	no	—
109	C₃×D₁₂	—	2	8	no	—
110	C₃×D₁₂	—	4	4	no	—
111	C₃×D₁₂	—	4	4	no	—
112	C₃×C₄×D₃	—	4	8	no	—
113	C₃×Dic₆	—	4	4	no	—
114	C₃²×C₈	—	2	2	no	—
115	C₃²×C₈	#41	1	4	yes	complemented
116	#532	—	3	2	no	—
117	#498	—	3	1	no	—
118	#493	—	3	1	no	—
119	#486	—	3	2	no	—
120	#469	—	3	4	no	—
121	#442	—	3	4	no	—
122	C₄×M₁₆	C₃×D₃	1	1	yes	characteristic, complemented
123	C₃×#41	—	12	8	no	—
124	C₂×C₃×Q₈	—	6	2	no	—
125	C₂×C₃×D₄	—	3	4	no	—
126	C₂×C₃×D₄	—	6	4	no	—
127	C₂×C₃×D₄	—	6	2	no	—
128	C₂×C₃×D₄	—	6	2	no	—
129	C₂²×C₃×C₄	—	2	2	no	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	—	6	4	no	—
132	C₂²×C₃×C₄	D₁₂	1	2	yes	—
133	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
134	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
136	C₂×#67	—	2	4	no	—
137	#235	—	4	8	no	—
138	C₂×D₁₂	—	2	2	no	—
139	C₂×D₁₂	—	2	2	no	—
140	C₂×D₁₂	C₂³×C₃	1	4	yes	—
141	C₂×C₄×D₃	—	2	4	no	—
142	C₂×Dic₆	—	2	2	no	—
143	C₃×D₈	—	6	8	no	—
144	C₃×M₁₆	—	2	8	no	—
145	C₃×M₁₆	—	2	4	no	—
146	C₃×M₁₆	—	2	4	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
150	C₂×C₃×C₈	C₂²×D₃	1	2	yes	—
151	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
152	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
153	C₃×C₄⋊C₄	—	3	4	no	—
154	C₃×C₄⋊C₄	—	6	2	no	—
155	C₃×C₂²⋊C₄	—	6	4	no	—
156	C₃×C₂²⋊C₄	—	6	4	no	—
157	C₃×C₄²	—	2	2	no	—
158	C₃×C₄²	—	2	1	no	—
159	C₃×C₄²	—	2	1	no	—
160	C₃×C₄²	D₁₂	1	2	yes	—
161	C₃×C₄²	C₃×D₄	1	2	yes	—
162	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
163	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
164	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
165	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
166	(C₂×D₃)⋊C₄	—	2	4	no	—
167	(C₂×D₃)⋊C₄	—	2	4	no	—
168	C₄⋊Dic₃	—	2	2	no	—
169	C₄⋊Dic₃	C₂³×C₃	1	4	yes	—
170	D₂₄	—	2	8	no	—
171	C₂²×C₃²	—	2	1	no	—
172	C₂²×C₃²	—	2	2	no	—
173	C₂²×C₃²	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₃×D₃	—	4	4	no	—
175	C₂×C₃×D₃	—	4	4	no	—
176	C₂×C₃×D₃	—	4	4	no	—
177	C₃²×C₄	—	2	1	no	—
178	C₃²×C₄	—	2	2	no	—
179	C₃²×C₄	AGL(1,ℤ/8)	1	4	yes	—
180	C₃²×C₄	C₂×#41	1	2	yes	—
181	C₃×Dic₃	—	4	4	no	—
182	C₂×#41	—	6	2	no	—
183	C₂×D₈	—	3	4	no	—
184	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
185	C₄⋊Q₈	—	3	2	no	—
186	#127	—	3	2	no	—
187	#124	—	6	2	no	—
188	#121	—	3	4	no	—
189	#121	—	6	2	no	—
190	C₄×D₄	—	3	4	no	—
191	C₄×D₄	—	6	2	no	—
192	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
193	#106	—	3	4	no	—
194	#102	—	3	8	no	—
195	#97	C₂×C₃×D₃	1	2	yes	—
196	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
197	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
198	C₂³×C₃	—	2	1	no	—
199	C₂³×C₃	—	6	4	no	—
200	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
201	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂²×D₃	—	2	4	no	—
203	C₃×Q₈	—	12	4	no	—
204	C₃×D₄	—	6	8	no	—
205	C₃×D₄	—	12	8	no	—
206	C₃×D₄	—	12	4	no	—
207	C₃×D₄	—	12	4	no	—
208	C₂×C₃×C₄	—	2	4	no	—
209	C₂×C₃×C₄	—	2	4	no	—
210	C₂×C₃×C₄	—	2	4	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	—	4	4	no	—
218	C₂×C₃×C₄	—	6	4	no	—
219	C₂×C₃×C₄	—	12	8	no	—
220	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
222	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
223	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
225	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
226	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
228	#67	—	4	8	no	—
229	C₂×Dic₃	—	2	4	no	—
230	D₁₂	—	2	8	no	—
231	D₁₂	—	4	4	no	—
232	D₁₂	—	4	4	no	—
233	C₄×D₃	—	4	8	no	—
234	Dic₆	—	4	4	no	—
235	C₃×C₈	—	2	4	no	—
236	C₃×C₈	—	2	2	no	—
237	C₃×C₈	—	2	2	no	—
238	C₃×C₈	—	2	4	no	—
239	C₃×C₈	#235	1	4	yes	complemented
240	C₃×C₈	C₃×#41	1	4	yes	complemented
241	C₂×C₃²	—	4	1	no	—
242	C₂×C₃²	#532	1	1	yes	characteristic
243	C₂×C₃²	C₂×AGL(1,ℤ/8)	1	2	yes	—
244	C₃×D₃	—	8	4	no	—
245	#41	—	12	8	no	—
246	C₂×Q₈	—	6	2	no	—
247	C₂×D₄	—	3	4	no	—
248	C₂×D₄	—	6	4	no	—
249	C₂×D₄	—	6	2	no	—
250	C₂×D₄	—	6	2	no	—
251	C₂²×C₄	—	6	4	no	—
252	C₂²×C₄	C₃×D₁₂	1	2	yes	—
253	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
254	D₈	—	6	8	no	—
255	M₁₆	—	2	4	no	—
256	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
257	C₂×C₈	C₂²×C₃×D₃	1	2	yes	—
258	C₄⋊C₄	—	3	4	no	—
259	C₄⋊C₄	—	6	2	no	—
260	C₂²⋊C₄	—	6	4	no	—
261	C₂²⋊C₄	—	6	4	no	—
262	C₄²	C₃×D₁₂	1	2	yes	—
263	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
264	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	2	2	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	2	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	4	2	no	—
272	C₂²×C₃	—	12	4	no	—
273	C₂²×C₃	—	12	4	no	—
274	C₂²×C₃	—	12	4	no	—
275	C₂²×C₃	C₂²×D₁₂	1	1	yes	characteristic
276	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
277	C₂×D₃	—	4	4	no	—
278	C₂×D₃	—	4	4	no	—
279	C₂×D₃	—	4	4	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	4	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	4	2	no	—
288	C₃×C₄	—	12	4	no	—
289	C₃×C₄	#908	1	4	yes	—
290	C₃×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
291	C₃×C₄	C₂×#235	1	2	yes	—
292	C₃×C₄	C₂×C₃×#41	1	2	yes	—
293	Dic₃	—	4	4	no	—
294	C₃²	#3184	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	6	4	no	—
296	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
297	Q₈	—	12	4	no	—
298	D₄	—	6	8	no	—
299	D₄	—	12	8	no	—
300	D₄	—	12	4	no	—
301	D₄	—	12	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	6	4	no	—
305	C₂×C₄	—	12	8	no	—
306	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
307	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
308	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
309	C₂×C₄	C₂³×C₃×D₃	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	C₃×#235	1	4	yes	complemented
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	4	2	no	—
315	C₂×C₃	—	4	1	no	—
316	C₂×C₃	—	4	1	no	—
317	C₂×C₃	—	24	4	no	—
318	C₂×C₃	#5477	1	1	yes	characteristic
319	C₂×C₃	C₂×#908	1	2	yes	—
320	C₂×C₃	C₃×#532	1	1	yes	characteristic
321	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
322	D₃	—	8	4	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	12	4	no	—
326	C₂²	—	12	4	no	—
327	C₂²	—	12	4	no	—
328	C₂²	C₂²×C₃×D₁₂	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	12	4	no	—
332	C₄	C₃×#908	1	4	yes	—
333	C₄	C₂×C₃×#235	1	2	yes	—
334	C₃	—	2	1	no	—
335	C₃	#78535	1	1	yes	characteristic, direct product factor
336	C₃	C₃×#3184	1	1	yes	characteristic, complemented
337	C₂	—	4	1	no	—
338	C₂	—	24	4	no	—
339	C₂	C₃×#5477	1	1	yes	characteristic
340	C₂	C₂×C₃×#908	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₃
Fitting	C₃²×C₄×M₁₆	C₂
socle	C₂²×C₃²	C₂²×D₄
hypercenter	C₃×C₄×M₁₆	D₃
hypocenter	C₃	C₃×#3184
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₃×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₃×#5477	C₂	semidirect
2. C₃²×C₄×M₁₆	C₂	semidirect
3. C₃×#5135	C₂	semidirect
4. C₃×#4711	C₂	semidirect
5. C₃×#4695	C₂	semidirect
6. C₃×#4694	C₂	semidirect
7. C₃×#4680	C₂	semidirect
9. #78535	C₃	direct (N × H)
27. #5477	C₂×C₃	semidirect
34. C₃×C₄×M₁₆	D₃	semidirect
35. C₃×C₄×M₁₆	C₂×C₃	semidirect
36. #5135	C₂×C₃	semidirect
37. #4711	C₂×C₃	semidirect
38. #4695	C₂×C₃	semidirect
39. #4694	C₂×C₃	semidirect
40. #4680	C₂×C₃	semidirect
115. C₃²×C₈	#41	semidirect
122. C₄×M₁₆	C₃×D₃	semidirect
239. C₃×C₈	#235	semidirect
240. C₃×C₈	C₃×#41	semidirect
294. C₃²	#3184	semidirect
311. C₈	C₃×#235	semidirect
335. C₃	#78535	direct (N × H)
336. C₃	C₃×#3184	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5477	2
C₃²×C₄×M₁₆	2
C₃×#5135	2
C₃×#4711	2
C₃×#4695	2
C₃×#4694	2
C₃×#4680	2
C₃×#3184	3
#78535	3

Subgroups — C3×#78535 #11876976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#78535 #11876976