Subgroups — C₃×#19188 #10637795

Order 768.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3226	C₂	1	2	yes	maximal
2	C₂×C₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2931	C₂	1	1	yes	maximal, characteristic
4	C₃×#1689	C₂	1	4	yes	maximal, complemented
5	C₃×#1669	C₂	1	4	yes	maximal, complemented
6	C₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#1546	C₂	1	2	yes	maximal, complemented
8	#19188	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂²×C₃×Q₁₆	—	2	1	no	—
10	C₂²×C₃×Q₁₆	—	2	1	no	—
11	C₂×C₃×#131	—	2	1	no	—
12	C₃×#532	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₃×#500	—	2	2	no	—
15	C₃×#494	—	2	1	no	—
16	C₃×#494	—	2	1	no	—
17	C₃×#494	C₂²	1	2	yes	—
18	C₃×#494	C₂²	1	2	yes	—
19	C₃×#486	—	2	2	no	—
20	C₃×#480	—	2	4	no	—
21	C₃×#474	C₂²	1	8	yes	—
22	C₃×#436	—	2	1	no	—
23	C₃×#436	C₂²	1	2	yes	—
24	C₃×C₈×D₄	—	2	1	no	—
25	C₃×C₈×D₄	C₂²	1	2	yes	—
26	C₃×#431	C₂²	1	2	yes	—
27	C₃×#423	C₂²	1	1	yes	characteristic
28	C₃×#408	—	2	1	no	—
29	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
30	C₃×#335	C₂²	1	2	yes	—
31	C₃×#334	C₂²	1	2	yes	—
32	C₃×#332	C₂²	1	8	yes	—
33	#3226	C₂×C₃	1	2	yes	—
34	C₂×#494	C₂×C₃	1	1	yes	characteristic, complemented
35	#2931	C₂×C₃	1	1	yes	characteristic
36	#1689	C₂×C₃	1	4	yes	complemented
37	#1669	C₂×C₃	1	4	yes	complemented
38	#1564	C₂×C₃	1	1	yes	characteristic, complemented
39	#1546	C₂×C₃	1	2	yes	complemented
40	C₂×C₃×#41	—	2	1	no	—
41	C₂²×C₃×Q₈	—	2	2	no	—
42	C₂×C₃×Q₁₆	—	2	2	no	—
43	C₂×C₃×Q₁₆	—	2	2	no	—
44	C₂×C₃×Q₁₆	—	2	2	no	—
45	C₂×C₃×Q₁₆	—	2	2	no	—
46	C₂×C₃×Q₁₆	—	4	4	no	—
47	C₂×C₃×Q₁₆	—	4	4	no	—
48	C₃×#131	—	4	4	no	—
49	C₂×C₃×M₁₆	—	2	1	no	—
50	C₂×C₃×M₁₆	—	2	2	no	—
51	C₂²×C₃×C₈	—	2	2	no	—
52	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
53	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
54	C₃×C₄⋊Q₈	—	2	2	no	—
55	C₃×C₄⋊Q₈	—	2	2	no	—
56	C₃×C₄⋊Q₈	—	2	2	no	—
57	C₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₃×C₄⋊Q₈	C₂³	1	4	yes	—
59	C₃×#127	D₄	1	1	yes	characteristic
60	C₃×#124	—	2	1	no	—
61	C₃×#121	—	2	1	no	—
62	C₃×#121	D₄	1	2	yes	—
63	C₃×C₄×D₄	—	2	1	no	—
64	C₃×C₄×D₄	C₂³	1	2	yes	—
65	C₂×C₃×C₄⋊C₄	—	2	2	no	—
66	C₂×C₃×C₄⋊C₄	—	2	2	no	—
67	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
68	C₃×#107	—	2	8	no	—
69	C₃×C₄⋊C₈	—	2	1	no	—
70	C₃×C₄⋊C₈	C₂³	1	2	yes	—
71	C₃×C₄⋊C₈	C₂³	1	4	yes	—
72	C₃×#102	—	2	8	no	—
73	C₃×C₂²⋊C₈	—	2	2	no	—
74	C₃×C₂²⋊C₈	—	2	2	no	—
75	C₃×C₂²⋊C₈	—	2	2	no	—
76	C₃×C₄×C₈	—	2	2	no	—
77	C₃×C₄×C₈	D₄	1	1	yes	characteristic
78	C₃×C₄×C₈	D₄	1	1	yes	characteristic
79	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
81	C₂²×Q₁₆	—	2	1	no	—
82	C₂²×Q₁₆	—	2	1	no	—
83	C₂×#131	—	2	1	no	—
84	#532	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
86	#500	—	2	2	no	—
87	#494	—	2	1	no	—
88	#494	—	2	1	no	—
89	#494	C₂²×C₃	1	2	yes	—
90	#494	C₂²×C₃	1	2	yes	—
91	#486	—	2	2	no	—
92	#480	—	2	4	no	—
93	#474	C₂²×C₃	1	8	yes	—
94	#436	—	2	1	no	—
95	#436	C₂²×C₃	1	2	yes	—
96	C₈×D₄	—	2	1	no	—
97	C₈×D₄	C₂²×C₃	1	2	yes	—
98	#431	C₂²×C₃	1	2	yes	—
99	#423	C₂²×C₃	1	1	yes	characteristic
100	#408	—	2	1	no	—
101	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
102	#335	C₂²×C₃	1	2	yes	—
103	#334	C₂²×C₃	1	2	yes	—
104	#332	C₂²×C₃	1	8	yes	—
105	C₃×#41	—	4	2	no	—
106	C₃×#41	—	4	2	no	—
107	C₂×C₃×Q₈	—	2	1	no	—
108	C₂×C₃×Q₈	—	2	4	no	—
109	C₂×C₃×Q₈	—	4	2	no	—
110	C₂×C₃×Q₈	—	4	4	no	—
111	C₂×C₃×Q₈	—	4	4	no	—
112	C₂×C₃×D₄	—	2	2	no	—
113	C₂×C₃×D₄	—	2	1	no	—
114	C₂×C₃×D₄	—	2	1	no	—
115	C₂×C₃×D₄	C₂×D₄	1	2	yes	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	—	4	2	no	—
118	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₃×C₄	#41	1	1	yes	characteristic
121	C₃×Q₁₆	—	4	4	no	—
122	C₃×Q₁₆	—	4	4	no	—
123	C₃×Q₁₆	—	4	4	no	—
124	C₃×Q₁₆	—	4	4	no	—
125	C₃×M₁₆	—	4	2	no	—
126	C₃×M₁₆	—	4	4	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	—	2	4	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	4	4	no	—
131	C₂×C₃×C₈	D₈	1	4	yes	—
132	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	2	4	no	—
138	C₃×C₄⋊C₄	—	2	4	no	—
139	C₃×C₄⋊C₄	—	4	2	no	—
140	C₃×C₄⋊C₄	—	4	2	no	—
141	C₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
142	C₃×C₂²⋊C₄	—	2	2	no	—
143	C₃×C₂²⋊C₄	—	2	2	no	—
144	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
145	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
146	C₃×C₄²	#41	1	1	yes	characteristic
147	C₃×C₄²	C₂⁴	1	1	yes	characteristic
148	C₂×#41	—	2	1	no	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂×Q₁₆	—	2	2	no	—
151	C₂×Q₁₆	—	2	2	no	—
152	C₂×Q₁₆	—	2	2	no	—
153	C₂×Q₁₆	—	2	2	no	—
154	C₂×Q₁₆	—	4	4	no	—
155	C₂×Q₁₆	—	4	4	no	—
156	#131	—	4	4	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
161	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	2	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	C₃×D₄	1	1	yes	characteristic
166	C₄⋊Q₈	C₂³×C₃	1	4	yes	—
167	#127	C₃×D₄	1	1	yes	characteristic
168	#124	—	2	1	no	—
169	#121	—	2	1	no	—
170	#121	C₃×D₄	1	2	yes	—
171	C₄×D₄	—	2	1	no	—
172	C₄×D₄	C₂³×C₃	1	2	yes	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
176	#107	—	2	8	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	C₂³×C₃	1	2	yes	—
179	C₄⋊C₈	C₂³×C₃	1	4	yes	—
180	#102	—	2	8	no	—
181	C₂²⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₃×D₄	1	1	yes	characteristic
186	C₄×C₈	C₃×D₄	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
189	C₂³×C₃	—	2	2	no	—
190	C₂³×C₃	#121	1	1	yes	characteristic
191	C₃×Q₈	—	4	2	no	—
192	C₃×Q₈	—	4	4	no	—
193	C₃×Q₈	—	4	4	no	—
194	C₃×Q₈	—	8	4	no	—
195	C₃×D₄	—	2	4	no	—
196	C₃×D₄	—	4	4	no	—
197	C₃×D₄	—	4	2	no	—
198	C₃×D₄	—	4	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	4	4	no	—
205	C₂×C₃×C₄	—	8	4	no	—
206	C₂×C₃×C₄	#121	1	1	yes	characteristic
207	C₂×C₃×C₄	#121	1	1	yes	characteristic
208	C₂×C₃×C₄	#121	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
210	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
211	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	4	2	no	—
217	C₃×C₈	—	4	4	no	—
218	C₃×C₈	C₂×D₈	1	2	yes	—
219	C₃×C₈	C₂×D₈	1	2	yes	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	C₂×Q₈	—	2	1	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	4	2	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	C₂×C₃×D₄	1	2	yes	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂²×C₄	C₃×#41	1	1	yes	characteristic
236	Q₁₆	—	4	4	no	—
237	Q₁₆	—	4	4	no	—
238	Q₁₆	—	4	4	no	—
239	Q₁₆	—	4	4	no	—
240	M₁₆	—	4	2	no	—
241	M₁₆	—	4	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	4	no	—
246	C₂×C₈	C₃×D₈	1	4	yes	—
247	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
248	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
249	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
250	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₂×C₃×D₄	1	2	yes	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
260	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
261	C₄²	C₃×#41	1	1	yes	characteristic
262	C₄²	C₂⁴×C₃	1	1	yes	characteristic
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	4	2	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	#459	1	2	yes	—
269	C₂²×C₃	C₂×#121	1	1	yes	characteristic
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	—	8	4	no	—
274	C₃×C₄	#459	1	2	yes	—
275	C₃×C₄	#473	1	2	yes	—
276	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
277	C₃×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₃×C₄	#578	1	2	yes	—
279	C₂³	—	2	2	no	—
280	C₂³	C₃×#121	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	4	4	no	—
283	Q₈	—	4	4	no	—
284	Q₈	—	8	4	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	C₃×#121	1	1	yes	characteristic
297	C₂×C₄	C₃×#121	1	1	yes	characteristic
298	C₂×C₄	C₃×#121	1	1	yes	characteristic
299	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
300	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
301	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	4	no	—
308	C₈	C₂×C₃×D₈	1	2	yes	—
309	C₈	C₂×C₃×D₈	1	2	yes	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	—	8	2	no	—
312	C₂×C₃	C₂×#459	1	1	yes	characteristic
313	C₂×C₃	#3059	1	1	yes	characteristic
314	C₂×C₃	C₂×#473	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₃×#459	1	2	yes	—
321	C₂²	C₂×C₃×#121	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	4	no	—
326	C₄	C₃×#459	1	2	yes	—
327	C₄	C₃×#473	1	2	yes	—
328	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
329	C₄	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
330	C₄	C₃×#578	1	2	yes	—
331	C₃	#19188	1	1	yes	characteristic, direct product factor, Sylow
332	C₂	—	2	2	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₃×#459	1	1	yes	characteristic
335	C₂	C₃×#3059	1	1	yes	characteristic
336	C₂	C₂×C₃×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#494	C₂	semidirect
4. C₃×#1689	C₂	semidirect
5. C₃×#1669	C₂	semidirect
6. C₃×#1564	C₂	semidirect
7. C₃×#1546	C₂	semidirect
8. #19188	C₃	direct (N × H)
34. C₂×#494	C₂×C₃	semidirect
36. #1689	C₂×C₃	semidirect
37. #1669	C₂×C₃	semidirect
38. #1564	C₂×C₃	semidirect
39. #1546	C₂×C₃	semidirect
331. C₃	#19188	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#3226	2
C₂×C₃×#494	2
C₃×#2931	2
C₃×#1689	2
C₃×#1669	2
C₃×#1564	2
C₃×#1546	2
#19188	3

Subgroups — C3×#19188 #10637795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#19188 #10637795