Subgroups — C₃×#68076 #11777265

Order 1152.

385 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#204	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4459	C₂	1	1	yes	maximal, characteristic
6	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4456	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×SD₁₆⋊C₈	—	3	1	no	maximal
9	#68076	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×#204	C₄	1	1	yes	characteristic
11	C₂×C₃×D₃×C₈	—	2	1	no	—
12	C₃×C₄×D₁₂	C₂²	1	1	yes	characteristic
13	C₃×C₄×Dic₆	C₂²	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#821	C₄	1	1	yes	characteristic
18	C₃×(C₂×D₃)⋊C₈	—	2	1	no	—
19	C₃×#817	C₄	1	1	yes	characteristic
20	C₃×#816	C₄	1	1	yes	characteristic
21	C₃×Dic₃⋊C₈	—	2	1	no	—
22	C₃×C₈×Dic₃	—	2	1	no	—
23	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₃×C₈×Q₈	—	3	1	no	—
26	C₃×C₄×SD₁₆	—	3	1	no	—
27	C₃×C₈×D₄	—	3	1	no	—
28	C₄×#204	C₂×C₃	1	1	yes	characteristic
29	C₈×D₁₂	C₂×C₃	1	1	yes	characteristic
30	C₈×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
31	C₃×#335	—	3	1	no	—
32	C₃×Q₈⋊C₈	—	3	1	no	—
33	C₃×D₄⋊C₈	—	3	1	no	—
34	C₃×#322	—	2	1	no	—
35	C₃×#322	D₃	1	1	yes	characteristic, complemented
36	C₃×#322	C₂×C₃	1	1	yes	characteristic, complemented
37	#4459	C₂×C₃	1	1	yes	characteristic
38	#4458	C₂×C₃	1	1	yes	characteristic, complemented
39	#4456	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃×D₁₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄×D₃	—	2	1	no	—
42	C₂×C₃×Dic₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
48	C₃²×C₄²	C₂³	1	1	yes	characteristic
49	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
50	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₃×Dic₃⋊C₄	—	2	1	no	—
53	C₃×C₄×Dic₃	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	—	2	1	no	—
56	C₃×#204	C₈	1	4	yes	complemented
57	C₃×D₃×C₈	—	4	2	no	—
58	SD₁₆⋊C₈	—	3	1	no	Sylow
59	C₂×C₃×SD₁₆	—	3	1	no	—
60	C₂²×C₃×C₈	—	6	1	no	—
61	C₃×C₄×Q₈	—	3	1	no	—
62	C₃×C₄×D₄	—	3	1	no	—
63	C₂×#204	C₃×C₄	1	1	yes	characteristic
64	C₂×D₃×C₈	—	2	1	no	—
65	C₄×D₁₂	C₂²×C₃	1	1	yes	characteristic
66	C₄×Dic₆	C₂²×C₃	1	1	yes	characteristic
67	C₃×#106	—	3	1	no	—
68	C₃×C₄⋊C₈	—	3	1	no	—
69	C₃×C₄⋊C₈	—	3	1	no	—
70	C₃×C₄⋊C₈	—	6	1	no	—
71	C₃×#103	—	3	1	no	—
72	C₃×#102	—	3	1	no	—
73	C₃×C₂²⋊C₈	—	6	1	no	—
74	C₃×C₄×C₈	—	2	1	no	—
75	C₃×C₄×C₈	—	2	1	no	—
76	C₃×C₄×C₈	—	2	1	no	—
77	C₃×C₄×C₈	—	6	1	no	—
78	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
79	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
82	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
84	#821	C₃×C₄	1	1	yes	characteristic
85	(C₂×D₃)⋊C₈	—	2	1	no	—
86	#817	C₃×C₄	1	1	yes	characteristic
87	#816	C₃×C₄	1	1	yes	characteristic
88	Dic₃⋊C₈	—	2	1	no	—
89	C₈×Dic₃	—	2	1	no	—
90	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
91	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
92	C₂²×C₃×D₃	—	2	1	no	—
93	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	#41	1	1	yes	characteristic
96	C₂×C₃×Dic₃	—	2	1	no	—
97	C₂×C₃×Dic₃	—	2	1	no	—
98	C₂×C₃×Dic₃	—	2	1	no	—
99	C₃×D₁₂	—	2	1	no	—
100	C₃×D₁₂	C₂×C₈	1	2	yes	—
101	C₃×C₄×D₃	—	4	1	no	—
102	C₃×C₄×D₃	—	4	1	no	—
103	C₃×Dic₆	—	2	1	no	—
104	C₃×Dic₆	C₂×C₈	1	2	yes	complemented
105	C₃²×C₈	—	2	1	no	—
106	C₃²×C₈	—	2	1	no	—
107	C₃²×C₈	—	2	1	no	—
108	C₃²×C₈	—	4	1	no	—
109	C₃²×C₈	C₂×C₈	1	2	yes	complemented
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	—	4	1	no	—
112	C₈×Q₈	—	3	1	no	—
113	C₄×SD₁₆	—	3	1	no	—
114	C₈×D₄	—	3	1	no	—
115	#335	—	3	1	no	—
116	Q₈⋊C₈	—	3	1	no	—
117	D₄⋊C₈	—	3	1	no	—
118	#322	C₃×D₃	1	1	yes	characteristic, complemented
119	C₂×C₃×Q₈	—	3	1	no	—
120	C₂×C₃×D₄	—	3	1	no	—
121	C₂²×C₃×C₄	—	6	1	no	—
122	C₂×D₁₂	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×C₄×D₃	—	2	1	no	—
124	C₂×Dic₆	C₂×C₃×C₄	1	1	yes	characteristic
125	C₃×SD₁₆	—	3	4	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	1	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	6	1	no	—
134	C₂×C₃×C₈	—	6	1	no	—
135	C₂×C₃×C₈	—	12	2	no	—
136	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
137	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
138	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
139	C₂×C₃×C₈	D₁₂	1	1	yes	characteristic
140	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
142	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	3	1	no	—
145	C₃×C₄⋊C₄	—	3	1	no	—
146	C₃×C₄⋊C₄	—	6	1	no	—
147	C₃×C₂²⋊C₄	—	6	1	no	—
148	C₃×C₄²	—	2	1	no	—
149	C₃×C₄²	—	6	1	no	—
150	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
151	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
152	(C₂×D₃)⋊C₄	—	2	1	no	—
153	C₄⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
154	C₄⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
155	Dic₃⋊C₄	—	2	1	no	—
156	C₄×Dic₃	—	2	1	no	—
157	C₂×C₃⋊C₈	—	2	1	no	—
158	C₂×C₃⋊C₈	—	2	1	no	—
159	#204	C₃×C₈	1	4	yes	complemented
160	D₃×C₈	—	4	2	no	—
161	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
162	C₂×C₃×D₃	—	2	2	no	—
163	C₂×C₃×D₃	—	4	1	no	—
164	C₂×C₃×D₃	—	4	1	no	—
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
167	C₃²×C₄	#131	1	1	yes	characteristic
168	C₃²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₃²×C₄	#137	1	1	yes	characteristic
170	C₃×Dic₃	—	2	2	no	—
171	C₃×Dic₃	—	4	1	no	—
172	C₃×Dic₃	—	4	1	no	—
173	C₂×SD₁₆	—	3	1	no	—
174	C₂²×C₈	—	6	1	no	—
175	C₄×Q₈	—	3	1	no	—
176	C₄×D₄	—	3	1	no	—
177	#106	—	3	1	no	—
178	C₄⋊C₈	—	3	1	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₄⋊C₈	—	6	1	no	—
181	#103	—	3	1	no	—
182	#102	—	3	1	no	—
183	C₂²⋊C₈	—	6	1	no	—
184	C₄×C₈	—	6	1	no	—
185	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
186	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
187	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
188	C₂³×C₃	—	6	1	no	—
189	C₂²×D₃	—	2	1	no	—
190	C₃×Q₈	—	3	2	no	—
191	C₃×Q₈	—	6	1	no	—
192	C₃×D₄	—	3	2	no	—
193	C₃×D₄	—	6	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	6	1	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	6	1	no	—
200	C₂×C₃×C₄	—	12	1	no	—
201	C₂×C₃×C₄	—	12	1	no	—
202	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
204	C₂×C₃×C₄	#235	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
208	C₂×Dic₃	—	2	1	no	—
209	C₂×Dic₃	—	2	1	no	—
210	C₂×Dic₃	—	2	1	no	—
211	D₁₂	—	2	1	no	—
212	D₁₂	C₂×C₃×C₈	1	2	yes	—
213	C₄×D₃	—	4	1	no	—
214	C₄×D₃	—	4	1	no	—
215	Dic₆	—	2	1	no	—
216	Dic₆	C₂×C₃×C₈	1	2	yes	complemented
217	C₃×C₈	—	2	2	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	2	1	no	—
220	C₃×C₈	—	2	1	no	—
221	C₃×C₈	—	2	1	no	—
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	—	2	2	no	—
225	C₃×C₈	—	4	1	no	—
226	C₃×C₈	—	4	1	no	—
227	C₃×C₈	—	4	2	no	—
228	C₃×C₈	—	4	1	no	—
229	C₃×C₈	—	4	1	no	—
230	C₃×C₈	—	12	1	no	—
231	C₃×C₈	—	12	1	no	—
232	C₃×C₈	D₃×C₈	1	2	yes	complemented
233	C₃×C₈	C₂×C₃×C₈	1	2	yes	complemented
234	C₃⋊C₈	—	4	1	no	—
235	C₃⋊C₈	—	4	1	no	—
236	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
237	C₂×C₃²	#440	1	1	yes	characteristic
238	C₂×C₃²	#444	1	1	yes	characteristic
239	C₃×D₃	—	4	2	no	—
240	C₂×Q₈	—	3	1	no	—
241	C₂×D₄	—	3	1	no	—
242	C₂²×C₄	—	6	1	no	—
243	SD₁₆	—	3	4	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	6	1	no	—
246	C₂×C₈	—	6	1	no	—
247	C₂×C₈	—	12	2	no	—
248	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
249	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
250	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
251	C₂×C₈	C₃×D₁₂	1	1	yes	characteristic
252	C₄⋊C₄	—	3	1	no	—
253	C₄⋊C₄	—	3	1	no	—
254	C₄⋊C₄	—	6	1	no	—
255	C₂²⋊C₄	—	6	1	no	—
256	C₄²	—	6	1	no	—
257	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	6	2	no	—
260	C₂²×C₃	—	12	1	no	—
261	C₂²×C₃	—	12	1	no	—
262	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
263	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
264	C₂×D₃	—	2	2	no	—
265	C₂×D₃	—	4	1	no	—
266	C₂×D₃	—	4	1	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	6	2	no	—
275	C₃×C₄	—	12	1	no	—
276	C₃×C₄	—	12	1	no	—
277	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
278	C₃×C₄	#901	1	1	yes	characteristic
279	C₃×C₄	#908	1	1	yes	characteristic
280	C₃×C₄	#909	1	1	yes	characteristic
281	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
282	C₃×C₄	C₃×#131	1	1	yes	characteristic
283	C₃×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
284	C₃×C₄	C₃×#137	1	1	yes	characteristic
285	Dic₃	—	2	2	no	—
286	Dic₃	—	4	1	no	—
287	Dic₃	—	4	1	no	—
288	C₃²	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Sylow
289	C₂³	—	6	1	no	—
290	Q₈	—	3	2	no	—
291	Q₈	—	6	1	no	—
292	D₄	—	3	2	no	—
293	D₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	6	1	no	—
296	C₂×C₄	—	6	1	no	—
297	C₂×C₄	—	12	1	no	—
298	C₂×C₄	—	12	1	no	—
299	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
300	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
301	C₂×C₄	C₃×#235	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	4	1	no	—
306	C₈	—	12	1	no	—
307	C₈	—	12	1	no	—
308	C₈	C₃×D₃×C₈	1	2	yes	complemented
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	12	2	no	—
313	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
314	C₂×C₃	#4710	1	1	yes	characteristic
315	C₂×C₃	#4717	1	1	yes	characteristic
316	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
317	C₂×C₃	C₃×#440	1	1	yes	characteristic
318	C₂×C₃	C₃×#444	1	1	yes	characteristic
319	D₃	—	4	2	no	—
320	C₂²	—	6	2	no	—
321	C₂²	—	12	1	no	—
322	C₂²	—	12	1	no	—
323	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	6	2	no	—
326	C₄	—	12	1	no	—
327	C₄	—	12	1	no	—
328	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
329	C₄	C₃×#901	1	1	yes	characteristic
330	C₄	C₃×#908	1	1	yes	characteristic
331	C₄	C₃×#909	1	1	yes	characteristic
332	C₃	—	2	1	no	—
333	C₃	#68076	1	1	yes	characteristic, direct product factor
334	C₃	C₃×SD₁₆⋊C₈	1	1	yes	characteristic, complemented
335	C₂	—	12	2	no	—
336	C₂	C₃×C₈×D₁₂	1	1	yes	characteristic
337	C₂	C₃×#4710	1	1	yes	characteristic
338	C₂	C₃×#4717	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#322	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₁₂	C₂×C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#322	D₃
hypocenter	C₃	C₃×SD₁₆⋊C₈
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
3. C₃×C₈×Dic₆	C₂	semidirect
4. C₃²×#322	C₂	semidirect
6. C₃×#4458	C₂	semidirect
7. C₃×#4456	C₂	semidirect
9. #68076	C₃	direct (N × H)
30. C₈×Dic₆	C₂×C₃	semidirect
35. C₃×#322	D₃	semidirect
36. C₃×#322	C₂×C₃	semidirect
38. #4458	C₂×C₃	semidirect
39. #4456	C₂×C₃	semidirect
56. C₃×#204	C₈	semidirect
104. C₃×Dic₆	C₂×C₈	semidirect
109. C₃²×C₈	C₂×C₈	semidirect
118. #322	C₃×D₃	semidirect
159. #204	C₃×C₈	semidirect
216. Dic₆	C₂×C₃×C₈	semidirect
232. C₃×C₈	D₃×C₈	semidirect
233. C₃×C₈	C₂×C₃×C₈	semidirect
288. C₃²	SD₁₆⋊C₈	semidirect
308. C₈	C₃×D₃×C₈	semidirect
333. C₃	#68076	direct (N × H)
334. C₃	C₃×SD₁₆⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#204	2
C₃×C₈×D₁₂	2
C₃×C₈×Dic₆	2
C₃²×#322	2
C₃×#4459	2
C₃×#4458	2
C₃×#4456	2
C₃×SD₁₆⋊C₈	3
#68076	3

Subgroups — C3×#68076 #11777265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68076 #11777265