Subgroups — C₃×#71778 #11791428

Order 1152.

793 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#835	C₂	1	4	yes	maximal
3	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₃×#1981	—	3	1	no	maximal
6	#71778	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
8	C₃²×C₄⋊Q₈	—	2	2	no	—
9	C₃²×C₄⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
11	C₂×C₃²×C₄⋊C₄	—	2	1	no	—
12	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₄×Dic₃	—	2	1	no	—
15	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×#835	—	2	8	no	—
17	C₃×C₃⋊#95	—	2	1	no	—
18	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
19	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
20	C₂×C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
21	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×C₄⋊C₈	—	3	1	no	—
23	C₂×C₃×#103	—	3	4	no	—
24	C₃×C₄×C₄⋊C₄	—	3	1	no	—
25	C₂×#835	C₂×C₃	1	4	yes	—
26	C₄×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
27	C₂×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃²×Q₈	—	2	2	no	—
29	C₂×C₃²×Q₈	—	2	4	no	—
30	C₂×C₃²×Q₈	—	2	4	no	—
31	C₂×C₃²×Q₈	C₂×C₄	1	4	yes	—
32	C₂²×C₃²×C₄	—	2	2	no	—
33	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
34	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
35	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃×Dic₃	—	2	2	no	—
37	C₃²×C₄⋊C₄	—	2	4	no	—
38	C₃²×C₄⋊C₄	—	2	4	no	—
39	C₃²×C₄²	C₂×C₄	1	2	yes	—
40	C₃²×C₄²	D₄	1	2	yes	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₃×C₄⋊Dic₃	D₄	1	4	yes	—
43	C₃×C₄×Dic₃	—	4	2	no	—
44	C₂×C₃×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃×C₃⋊C₈	—	2	4	no	—
46	#1981	—	3	1	no	Sylow
47	C₂²×C₃×Q₈	—	2	2	no	—
48	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
49	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
50	C₂²×C₃×C₈	—	3	2	no	—
51	C₃×C₄⋊Q₈	—	2	4	no	—
52	C₃×C₄⋊Q₈	—	2	2	no	—
53	C₃×C₄⋊Q₈	—	2	2	no	—
54	C₃×C₄⋊Q₈	—	2	4	no	—
55	C₃×C₄⋊Q₈	Dic₃	1	4	yes	complemented
56	C₃×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
57	C₂×C₃×C₄⋊C₄	—	2	2	no	—
58	C₂×C₃×C₄⋊C₄	—	2	2	no	—
59	C₂×C₃×C₄⋊C₄	—	2	1	no	—
60	C₂×C₃×C₄⋊C₄	—	2	1	no	—
61	C₂×C₃×C₄⋊C₄	—	2	1	no	—
62	C₂×C₃×C₄⋊C₄	—	2	1	no	—
63	C₂×C₃×C₄⋊C₄	—	3	2	no	—
64	C₂×C₃×C₄²	—	2	1	no	—
65	C₂×C₃×C₄²	—	6	1	no	—
66	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
69	C₂×C₄×Dic₃	—	2	1	no	—
70	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
71	C₃×C₄⋊C₈	—	6	2	no	—
72	C₃×#103	—	6	8	no	—
73	C₃×#95	—	6	1	no	—
74	#835	—	2	8	no	—
75	C₃⋊#95	—	2	1	no	—
76	C₄⋊(C₃⋊C₈)	—	2	2	no	—
77	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
78	C₃²×Q₈	—	2	8	no	—
79	C₃²×Q₈	—	4	2	no	—
80	C₃²×Q₈	—	4	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	4	no	—
83	C₂×C₃²×C₄	—	4	4	no	—
84	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
87	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
88	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
90	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	#41	1	2	yes	—
92	C₂×C₃×Dic₃	—	2	4	no	—
93	C₂×C₃×Dic₃	—	4	2	no	—
94	C₂×C₃×Dic₃	—	4	2	no	—
95	C₃×C₃⋊C₈	—	4	2	no	—
96	C₂×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
97	C₂×C₄⋊C₈	—	3	1	no	—
98	C₂×#103	—	3	4	no	—
99	C₄×C₄⋊C₄	—	3	1	no	—
100	C₂×C₃×Q₈	—	2	4	no	—
101	C₂×C₃×Q₈	—	2	2	no	—
102	C₂×C₃×Q₈	—	2	2	no	—
103	C₂×C₃×Q₈	—	2	4	no	—
104	C₂×C₃×Q₈	—	2	4	no	—
105	C₂×C₃×Q₈	—	2	4	no	—
106	C₂×C₃×Q₈	—	2	4	no	—
107	C₂×C₃×Q₈	—	2	4	no	—
108	C₂×C₃×Q₈	—	4	4	no	—
109	C₂×C₃×Q₈	—	4	4	no	—
110	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
111	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
112	C₂²×C₃×C₄	—	2	4	no	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	2	1	no	—
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	6	2	no	—
119	C₂²×C₃×C₄	#67	1	1	yes	characteristic
120	C₂²×C₃×C₄	#67	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
125	C₂²×Dic₃	—	2	2	no	—
126	C₂×C₃×C₈	—	6	4	no	—
127	C₂×C₃×C₈	—	6	2	no	—
128	C₃×C₄⋊C₄	—	2	8	no	—
129	C₃×C₄⋊C₄	—	2	8	no	—
130	C₃×C₄⋊C₄	—	2	4	no	—
131	C₃×C₄⋊C₄	—	2	4	no	—
132	C₃×C₄⋊C₄	—	2	4	no	—
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	3	4	no	—
135	C₃×C₄⋊C₄	—	6	2	no	—
136	C₃×C₄²	—	2	2	no	—
137	C₃×C₄²	—	2	2	no	—
138	C₃×C₄²	—	12	2	no	—
139	C₃×C₄²	C₂×Dic₃	1	2	yes	—
140	C₃×C₄²	#67	1	2	yes	—
141	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
142	C₃×C₄²	C₃×D₄	1	2	yes	—
143	C₄⋊Dic₃	—	2	2	no	—
144	C₄⋊Dic₃	C₃×D₄	1	4	yes	—
145	C₄×Dic₃	—	4	2	no	—
146	C₂×C₃⋊C₈	—	2	2	no	—
147	C₂×C₃⋊C₈	—	2	4	no	—
148	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₃²	#121	1	2	yes	—
150	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
151	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
152	C₂²×C₃²	#137	1	2	yes	—
153	C₃²×C₄	—	2	2	no	—
154	C₃²×C₄	—	4	4	no	—
155	C₃²×C₄	#103	1	4	yes	complemented
156	C₃²×C₄	C₄×D₄	1	2	yes	—
157	C₃²×C₄	#124	1	1	yes	characteristic
158	C₃²×C₄	#127	1	1	yes	characteristic
159	C₃×Dic₃	—	4	4	no	—
160	C₂²×Q₈	C₂×C₃×D₃	1	2	yes	—
161	C₂²×C₈	—	3	2	no	—
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	C₃×Dic₃	1	4	yes	complemented
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	3	2	no	—
167	C₂×C₄²	—	6	1	no	—
168	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
169	C₄⋊C₈	—	6	2	no	—
170	#103	—	6	8	no	—
171	#95	—	6	1	no	—
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	C₂×#67	1	1	yes	characteristic
174	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
175	C₃×Q₈	—	2	8	no	—
176	C₃×Q₈	—	2	8	no	—
177	C₃×Q₈	—	4	8	no	—
178	C₃×Q₈	—	4	4	no	—
179	C₃×Q₈	—	4	4	no	—
180	C₃×Q₈	—	4	2	no	—
181	C₃×Q₈	—	4	2	no	—
182	C₃×Q₈	—	4	2	no	—
183	C₃×Q₈	—	4	2	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	8	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	4	8	no	—
199	C₂×C₃×C₄	—	4	4	no	—
200	C₂×C₃×C₄	—	4	4	no	—
201	C₂×C₃×C₄	—	6	4	no	—
202	C₂×C₃×C₄	—	12	2	no	—
203	C₂×C₃×C₄	—	12	2	no	—
204	C₂×C₃×C₄	#215	1	2	yes	—
205	C₂×C₃×C₄	#216	1	2	yes	—
206	C₂×C₃×C₄	#217	1	2	yes	—
207	C₂×C₃×C₄	#217	1	2	yes	—
208	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
209	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
210	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
211	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
212	C₂×C₃×C₄	D₃×D₄	1	2	yes	—
213	C₂×C₃×C₄	#237	1	2	yes	—
214	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
218	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₃×#41	1	2	yes	—
220	C₂×Dic₃	—	2	4	no	—
221	C₂×Dic₃	—	4	2	no	—
222	C₂×Dic₃	—	4	2	no	—
223	C₃×C₈	—	12	2	no	—
224	C₃⋊C₈	—	4	2	no	—
225	C₂×C₃²	#390	1	1	yes	characteristic
226	C₂×C₃²	C₂×#103	1	1	yes	characteristic
227	C₂×C₃²	#419	1	1	yes	characteristic
228	C₂×C₃²	#461	1	2	yes	—
229	C₂×C₃²	#462	1	2	yes	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	C₂×C₃×Dic₃	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	6	2	no	—
236	C₂²×C₄	C₃×#67	1	1	yes	characteristic
237	C₂²×C₄	C₃×#67	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
239	C₂×C₈	—	6	4	no	—
240	C₂×C₈	—	6	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	3	4	no	—
244	C₄⋊C₄	—	6	2	no	—
245	C₄²	—	12	2	no	—
246	C₄²	C₂×C₃×Dic₃	1	2	yes	—
247	C₄²	C₃×#67	1	2	yes	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	#938	1	2	yes	—
254	C₂²×C₃	C₂×#215	1	1	yes	characteristic
255	C₂²×C₃	#942	1	2	yes	—
256	C₂²×C₃	C₂×#216	1	1	yes	characteristic
257	C₂²×C₃	C₂×#217	1	1	yes	characteristic
258	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
259	C₂²×C₃	C₃×#121	1	2	yes	—
260	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
261	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
262	C₂²×C₃	C₃×#137	1	2	yes	—
263	C₃×C₄	—	2	4	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	2	2	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	4	no	—
270	C₃×C₄	—	4	8	no	—
271	C₃×C₄	—	4	4	no	—
272	C₃×C₄	—	4	4	no	—
273	C₃×C₄	—	12	4	no	—
274	C₃×C₄	#835	1	4	yes	complemented
275	C₃×C₄	C₃×#103	1	4	yes	complemented
276	C₃×C₄	D₄×Dic₃	1	2	yes	—
277	C₃×C₄	#936	1	1	yes	characteristic
278	C₃×C₄	#940	1	1	yes	characteristic
279	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
280	C₃×C₄	C₃×#124	1	1	yes	characteristic
281	C₃×C₄	C₃×#127	1	1	yes	characteristic
282	Dic₃	—	4	4	no	—
283	C₃²	#1981	1	1	yes	characteristic, complemented, Sylow
284	C₂³	C₂×C₃×#67	1	1	yes	characteristic
285	Q₈	—	2	8	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	4	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	6	4	no	—
292	C₂×C₄	—	12	2	no	—
293	C₂×C₄	—	12	2	no	—
294	C₂×C₄	C₃×#215	1	2	yes	—
295	C₂×C₄	C₃×#216	1	2	yes	—
296	C₂×C₄	C₃×#217	1	2	yes	—
297	C₂×C₄	C₃×#217	1	2	yes	—
298	C₂×C₄	C₃×D₃×D₄	1	2	yes	—
299	C₂×C₄	C₃×#237	1	2	yes	—
300	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
302	C₈	—	12	2	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	2	2	no	—
308	C₂×C₃	#5083	1	2	yes	—
309	C₂×C₃	#5090	1	2	yes	—
310	C₂×C₃	#5221	1	1	yes	characteristic
311	C₂×C₃	C₂×#835	1	1	yes	characteristic
312	C₂×C₃	#5225	1	1	yes	characteristic
313	C₂×C₃	C₃×#390	1	1	yes	characteristic
314	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
315	C₂×C₃	C₃×#419	1	1	yes	characteristic
316	C₂×C₃	C₃×#461	1	2	yes	—
317	C₂×C₃	C₃×#462	1	2	yes	—
318	C₂²	C₃×#938	1	2	yes	—
319	C₂²	C₂×C₃×#215	1	1	yes	characteristic
320	C₂²	C₃×#942	1	2	yes	—
321	C₂²	C₂×C₃×#216	1	1	yes	characteristic
322	C₂²	C₂×C₃×#217	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	4	4	no	—
325	C₄	—	12	4	no	—
326	C₄	C₃×#835	1	4	yes	complemented
327	C₄	C₃×D₄×Dic₃	1	2	yes	—
328	C₄	C₃×#936	1	1	yes	characteristic
329	C₄	C₃×#940	1	1	yes	characteristic
330	C₃	—	2	1	no	—
331	C₃	#71778	1	1	yes	characteristic, direct product factor
332	C₃	C₃×#1981	1	1	yes	characteristic, complemented
333	C₂	C₃×#5083	1	2	yes	—
334	C₂	C₃×#5090	1	2	yes	—
335	C₂	C₃×#5221	1	1	yes	characteristic
336	C₂	C₂×C₃×#835	1	1	yes	characteristic
337	C₂	C₃×#5225	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#1981
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. #71778	C₃	direct (N × H)
9. C₃²×C₄⋊Q₈	C₄	semidirect
55. C₃×C₄⋊Q₈	Dic₃	semidirect
56. C₃×C₄⋊Q₈	C₃×C₄	semidirect
155. C₃²×C₄	#103	semidirect
163. C₄⋊Q₈	C₃×Dic₃	semidirect
274. C₃×C₄	#835	semidirect
275. C₃×C₄	C₃×#103	semidirect
283. C₃²	#1981	semidirect
326. C₄	C₃×#835	semidirect
331. C₃	#71778	direct (N × H)
332. C₃	C₃×#1981	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⋊Q₈	2
C₂×C₃×#835	2
C₃×C₄×C₄⋊Dic₃	2
C₂×C₃×C₄⋊(C₃⋊C₈)	2
C₃×#1981	3
#71778	3

Subgroups — C3×#71778 #11791428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71778 #11791428