Subgroups — C₃×#29219 #10647826

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#2099	C₂	1	4	yes	maximal
4	C₃×#1990	C₂	1	2	yes	maximal
5	C₃×#1988	C₂	1	4	yes	maximal
6	C₃×#1834	C₂	1	2	yes	maximal
7	C₃×C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#29219	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄×Q₈	—	2	1	no	—
14	C₂×C₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
16	C₃×#391	—	2	1	no	—
17	C₃×#389	—	2	1	no	—
18	C₃×#389	C₂²	1	2	yes	—
19	C₃×#384	—	2	1	no	—
20	C₃×#384	—	2	1	no	—
21	C₃×#384	C₂²	1	2	yes	—
22	C₃×#384	C₂²	1	1	yes	characteristic
23	C₃×#384	C₂²	1	2	yes	—
24	C₃×C₄×C₄⋊C₄	—	2	1	no	—
25	C₃×C₄×C₄⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄³	C₂²	1	1	yes	characteristic
27	C₃×#340	C₂²	1	8	yes	—
28	C₃×#340	C₂²	1	8	yes	—
29	C₃×#336	C₂²	1	2	yes	—
30	#2868	C₂×C₃	1	1	yes	characteristic
31	C₄×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
32	#2099	C₂×C₃	1	4	yes	—
33	#1990	C₂×C₃	1	2	yes	—
34	#1988	C₂×C₃	1	4	yes	—
35	#1834	C₂×C₃	1	2	yes	—
36	C₄×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
37	C₂²×C₃×Q₈	—	2	1	no	—
38	C₂²×C₃×Q₈	—	2	2	no	—
39	C₂²×C₃×C₈	C₂³	1	4	yes	—
40	C₃×C₄⋊Q₈	—	2	2	no	—
41	C₃×C₄⋊Q₈	—	2	8	no	—
42	C₃×C₄⋊Q₈	—	4	4	no	—
43	C₃×C₄⋊Q₈	—	4	4	no	—
44	C₃×C₄⋊Q₈	Q₈	1	4	yes	complemented
45	C₃×C₄×Q₈	—	4	4	no	—
46	C₂×C₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₃×C₄⋊C₄	—	2	1	no	—
48	C₂×C₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₃×C₄⋊C₄	—	2	2	no	—
52	C₂×C₃×C₄⋊C₄	—	2	2	no	—
53	C₂×C₃×C₄⋊C₄	—	2	1	no	—
54	C₂×C₃×C₄⋊C₄	—	2	2	no	—
55	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
57	C₂×C₃×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₃×C₄²	—	2	2	no	—
59	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
60	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
61	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
62	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
63	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
65	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
66	C₃×C₄⋊C₈	—	2	4	no	—
67	C₃×C₄×C₈	—	2	4	no	—
68	C₃×#95	—	2	2	no	—
69	C₃×#95	—	2	2	no	—
70	C₂×C₄⋊Q₈	—	2	1	no	—
71	C₂×C₄⋊Q₈	—	2	1	no	—
72	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
73	C₂×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₄×Q₈	—	2	1	no	—
75	C₂×C₄⋊C₈	C₂²×C₃	1	2	yes	—
76	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
77	#391	—	2	1	no	—
78	#389	—	2	1	no	—
79	#389	C₂²×C₃	1	2	yes	—
80	#384	—	2	1	no	—
81	#384	—	2	1	no	—
82	#384	C₂²×C₃	1	1	yes	characteristic
83	#384	C₂²×C₃	1	2	yes	—
84	#384	C₂²×C₃	1	2	yes	—
85	C₄×C₄⋊C₄	—	2	1	no	—
86	C₄×C₄⋊C₄	C₂²×C₃	1	2	yes	—
87	C₄³	C₂²×C₃	1	1	yes	characteristic
88	#340	C₂²×C₃	1	8	yes	—
89	#340	C₂²×C₃	1	8	yes	—
90	#336	C₂²×C₃	1	2	yes	—
91	C₂×C₃×Q₈	—	2	4	no	—
92	C₂×C₃×Q₈	—	4	4	no	—
93	C₂×C₃×Q₈	—	4	2	no	—
94	C₂×C₃×Q₈	—	4	4	no	—
95	C₂×C₃×Q₈	—	4	4	no	—
96	C₂×C₃×Q₈	—	4	2	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	2	no	—
99	C₂²×C₃×C₄	—	2	4	no	—
100	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	—	2	4	no	—
109	C₂×C₃×C₈	—	2	4	no	—
110	C₃×C₄⋊C₄	—	2	4	no	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	8	no	—
113	C₃×C₄⋊C₄	—	2	4	no	—
114	C₃×C₄⋊C₄	—	4	4	no	—
115	C₃×C₄⋊C₄	—	4	4	no	—
116	C₃×C₄⋊C₄	—	4	2	no	—
117	C₃×C₄⋊C₄	—	4	4	no	—
118	C₃×C₄⋊C₄	—	4	4	no	—
119	C₃×C₄⋊C₄	—	4	4	no	—
120	C₃×C₄⋊C₄	—	4	4	no	—
121	C₃×C₄⋊C₄	—	4	2	no	—
122	C₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
123	C₃×C₄²	—	2	2	no	—
124	C₃×C₄²	—	2	2	no	—
125	C₃×C₄²	—	2	2	no	—
126	C₃×C₄²	—	4	4	no	—
127	C₃×C₄²	D₈	1	4	yes	—
128	C₃×C₄²	SD₁₆	1	4	yes	—
129	C₃×C₄²	C₂×Q₈	1	2	yes	—
130	C₃×C₄²	#41	1	2	yes	—
131	C₃×C₄²	#41	1	4	yes	—
132	C₂²×Q₈	—	2	1	no	—
133	C₂²×Q₈	—	2	2	no	—
134	C₂²×C₈	C₂³×C₃	1	4	yes	—
135	C₄⋊Q₈	—	2	2	no	—
136	C₄⋊Q₈	—	2	8	no	—
137	C₄⋊Q₈	—	4	4	no	—
138	C₄⋊Q₈	—	4	4	no	—
139	C₄⋊Q₈	C₃×Q₈	1	4	yes	complemented
140	C₄×Q₈	—	4	4	no	—
141	C₂×C₄⋊C₄	—	2	1	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
151	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂³×C₃	1	2	yes	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
155	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
156	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
157	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
159	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
160	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
161	C₄⋊C₈	—	2	4	no	—
162	C₄×C₈	—	2	4	no	—
163	#95	—	2	2	no	—
164	#95	—	2	2	no	—
165	C₂³×C₃	C₂²×D₄	1	1	yes	characteristic
166	C₃×Q₈	—	4	8	no	—
167	C₃×Q₈	—	8	8	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	4	2	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	4	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	—	4	4	no	—
181	C₂×C₃×C₄	—	4	4	no	—
182	C₂×C₃×C₄	—	4	2	no	—
183	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
184	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
185	C₂×C₃×C₄	#124	1	2	yes	—
186	C₂×C₃×C₄	#124	1	2	yes	—
187	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
188	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
189	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
190	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
191	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
192	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
193	C₂×C₃×C₄	#137	1	2	yes	—
194	C₂×C₃×C₄	#137	1	2	yes	—
195	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×#41	1	2	yes	—
198	C₂×C₃×C₄	2₋¹⁺⁴	1	2	yes	—
199	C₃×C₈	—	4	4	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	4	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	C₂×C₃×Q₈	1	4	yes	—
232	C₄²	—	2	2	no	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	4	4	no	—
236	C₄²	C₃×D₈	1	4	yes	—
237	C₄²	C₃×SD₁₆	1	4	yes	—
238	C₄²	C₂×C₃×Q₈	1	2	yes	—
239	C₄²	C₃×#41	1	2	yes	—
240	C₄²	C₃×#41	1	4	yes	—
241	C₂²×C₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂²×C₃	C₂×#124	1	1	yes	characteristic
243	C₂²×C₃	#536	1	1	yes	characteristic
244	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
245	C₂²×C₃	C₂²×SD₁₆	1	1	yes	characteristic
246	C₂²×C₃	C₂×#137	1	1	yes	characteristic
247	C₂²×C₃	C₂×#137	1	1	yes	characteristic
248	C₃×C₄	—	2	2	no	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	—	4	4	no	—
252	C₃×C₄	—	8	4	no	—
253	C₃×C₄	—	8	2	no	—
254	C₃×C₄	#474	1	4	yes	—
255	C₃×C₄	#475	1	4	yes	—
256	C₃×C₄	#486	1	4	yes	—
257	C₃×C₄	#556	1	2	yes	—
258	C₃×C₄	#557	1	2	yes	—
259	C₂³	C₂²×C₃×D₄	1	1	yes	characteristic
260	Q₈	—	4	8	no	—
261	Q₈	—	8	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
278	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
279	C₂×C₄	C₃×#124	1	2	yes	—
280	C₂×C₄	C₃×#124	1	2	yes	—
281	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
282	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
283	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
284	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
285	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
286	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
287	C₂×C₄	C₃×#137	1	2	yes	—
288	C₂×C₄	C₃×#137	1	2	yes	—
289	C₂×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
291	C₂×C₄	C₂×C₃×#41	1	2	yes	—
292	C₂×C₄	C₃×2₋¹⁺⁴	1	2	yes	—
293	C₈	—	4	4	no	—
294	C₂×C₃	#2568	1	1	yes	characteristic
295	C₂×C₃	C₂×#474	1	1	yes	characteristic
296	C₂×C₃	C₂×#475	1	1	yes	characteristic
297	C₂×C₃	C₂×#486	1	1	yes	characteristic
298	C₂×C₃	#3139	1	1	yes	characteristic
299	C₂×C₃	#3226	1	1	yes	characteristic
300	C₂×C₃	#3229	1	1	yes	characteristic
301	C₂²	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
302	C₂²	C₂×C₃×#124	1	1	yes	characteristic
303	C₂²	C₃×#536	1	1	yes	characteristic
304	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
305	C₂²	C₂²×C₃×SD₁₆	1	1	yes	characteristic
306	C₂²	C₂×C₃×#137	1	1	yes	characteristic
307	C₂²	C₂×C₃×#137	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	8	4	no	—
313	C₄	—	8	2	no	—
314	C₄	C₃×#474	1	4	yes	—
315	C₄	C₃×#475	1	4	yes	—
316	C₄	C₃×#486	1	4	yes	—
317	C₄	C₃×#556	1	2	yes	—
318	C₄	C₃×#557	1	2	yes	—
319	C₃	#29219	1	1	yes	characteristic, direct product factor, Sylow
320	C₂	C₃×#2568	1	1	yes	characteristic
321	C₂	C₂×C₃×#474	1	1	yes	characteristic
322	C₂	C₂×C₃×#475	1	1	yes	characteristic
323	C₂	C₂×C₃×#486	1	1	yes	characteristic
324	C₂	C₃×#3139	1	1	yes	characteristic
325	C₂	C₃×#3226	1	1	yes	characteristic
326	C₂	C₃×#3229	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×C₄³ ◁ G

Complements

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

N	H ≅ G/N	type
8. #29219	C₃	direct (N × H)
44. C₃×C₄⋊Q₈	Q₈	semidirect
139. C₄⋊Q₈	C₃×Q₈	semidirect
319. C₃	#29219	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×#2868	2
C₃×C₄×C₄⋊Q₈	2
C₃×#2099	2
C₃×#1990	2
C₃×#1988	2
C₃×#1834	2
C₃×C₄×C₄⋊C₈	2
#29219	3

Subgroups — C3×#29219 #10647826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#29219 #10647826