Subgroups — C₃×#33735 #10652342

Order 768.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3168	C₂	1	2	yes	maximal, complemented
2	C₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2242	C₂	1	2	yes	maximal, complemented
5	C₃×#2217	C₂	1	2	yes	maximal, complemented
6	C₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#2215	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄×SD₃₂	C₂	1	2	yes	maximal, complemented
10	C₃×C₄×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#33735	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₃×#529	—	2	2	no	—
14	C₂×C₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₃×D₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
19	C₃×#499	C₂²	1	2	yes	—
20	C₃×#495	C₂²	1	2	yes	—
21	C₃×#488	—	2	2	no	—
22	C₃×#467	C₂²	1	2	yes	—
23	C₃×#466	C₂²	1	2	yes	—
24	C₃×#465	—	2	2	no	—
25	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
26	C₃×C₄×SD₁₆	—	2	2	no	—
27	C₃×C₄×D₈	C₂²	1	2	yes	—
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×#367	C₂²	1	2	yes	—
30	C₃×#366	C₂²	1	1	yes	characteristic
31	C₃×#366	C₂²	1	1	yes	characteristic
32	C₃×#358	C₂²	1	2	yes	—
33	C₃×#358	C₂²	1	2	yes	—
34	C₃×#357	C₂²	1	2	yes	—
35	C₃×#357	C₂²	1	2	yes	—
36	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
37	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
38	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
39	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
40	#3168	C₂×C₃	1	2	yes	complemented
41	#2253	C₂×C₃	1	1	yes	characteristic, complemented
42	#2249	C₂×C₃	1	1	yes	characteristic, complemented
43	#2242	C₂×C₃	1	2	yes	complemented
44	#2217	C₂×C₃	1	2	yes	complemented
45	#2216	C₂×C₃	1	1	yes	characteristic, complemented
46	#2215	C₂×C₃	1	1	yes	characteristic
47	C₄×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
48	C₄×SD₃₂	C₂×C₃	1	2	yes	complemented
49	C₄×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
51	C₂×C₃×Q₁₆	C₂³	1	2	yes	—
52	C₂×C₃×SD₁₆	—	2	2	no	—
53	C₂×C₃×D₈	C₂³	1	2	yes	—
54	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₃×#126	—	4	2	no	—
57	C₃×#125	—	2	2	no	—
58	C₃×#124	—	2	2	no	—
59	C₃×#123	—	4	2	no	—
60	C₃×C₂²⋊Q₈	—	2	2	no	—
61	C₃×#121	—	2	2	no	—
62	C₃×C₄×Q₈	—	2	2	no	—
63	C₃×C₄×D₄	—	2	2	no	—
64	C₃×C₄×D₄	—	4	2	no	—
65	C₃×#117	—	4	2	no	—
66	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
67	C₃×Q₃₂	—	2	2	no	—
68	C₃×SD₃₂	—	2	4	no	—
69	C₃×D₁₆	—	2	2	no	—
70	C₂×C₃×C₁₆	—	2	2	no	—
71	C₂×C₃×C₁₆	—	2	2	no	—
72	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
75	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
76	C₃×#107	C₂³	1	2	yes	—
77	C₃×#107	C₂³	1	2	yes	—
78	C₃×#106	—	2	2	no	—
79	C₃×#103	—	2	2	no	—
80	C₃×#103	—	2	2	no	—
81	C₃×#102	—	2	2	no	—
82	C₃×#102	—	2	2	no	—
83	C₃×C₄×C₈	D₄	1	1	yes	characteristic
84	C₃×C₄×C₈	D₄	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
86	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
87	#529	—	2	2	no	—
88	C₂×Q₃₂	C₂²×C₃	1	1	yes	characteristic
89	C₂×SD₃₂	C₂²×C₃	1	2	yes	—
90	C₂×D₁₆	C₂²×C₃	1	1	yes	characteristic
91	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
92	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
93	#499	C₂²×C₃	1	2	yes	—
94	#495	C₂²×C₃	1	2	yes	—
95	#488	—	2	2	no	—
96	#467	C₂²×C₃	1	2	yes	—
97	#466	C₂²×C₃	1	2	yes	—
98	#465	—	2	2	no	—
99	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
100	C₄×SD₁₆	—	2	2	no	—
101	C₄×D₈	C₂²×C₃	1	2	yes	—
102	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
103	#367	C₂²×C₃	1	2	yes	—
104	#366	C₂²×C₃	1	1	yes	characteristic
105	#366	C₂²×C₃	1	1	yes	characteristic
106	#358	C₂²×C₃	1	2	yes	—
107	#358	C₂²×C₃	1	2	yes	—
108	#357	C₂²×C₃	1	2	yes	—
109	#357	C₂²×C₃	1	2	yes	—
110	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
111	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
112	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
113	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	—	4	2	no	—
117	C₂²×C₃×C₄	—	4	2	no	—
118	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
121	C₃×Q₁₆	—	2	2	no	—
122	C₃×Q₁₆	—	2	2	no	—
123	C₃×SD₁₆	—	4	4	no	—
124	C₃×D₈	—	2	2	no	—
125	C₃×D₈	—	2	2	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	C₂×D₄	1	1	yes	characteristic
130	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
133	C₃×C₄⋊C₄	—	2	2	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	4	2	no	—
136	C₃×C₄⋊C₄	—	4	2	no	—
137	C₃×C₄⋊C₄	—	4	2	no	—
138	C₃×C₄⋊C₄	—	4	2	no	—
139	C₃×C₂²⋊C₄	—	4	2	no	—
140	C₃×C₂²⋊C₄	—	4	2	no	—
141	C₃×C₂²⋊C₄	—	4	2	no	—
142	C₃×C₂²⋊C₄	—	4	2	no	—
143	C₃×C₂²⋊C₄	—	4	2	no	—
144	C₃×C₄²	—	4	2	no	—
145	C₃×C₄²	D₈	1	1	yes	characteristic
146	C₃×C₄²	D₈	1	1	yes	characteristic
147	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
148	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
149	C₃×C₁₆	—	2	1	no	—
150	C₃×C₁₆	—	2	1	no	—
151	C₃×C₁₆	#41	1	2	yes	—
152	C₃×C₁₆	#41	1	2	yes	—
153	C₂×Q₁₆	C₂³×C₃	1	2	yes	—
154	C₂×SD₁₆	—	2	2	no	—
155	C₂×D₈	C₂³×C₃	1	2	yes	—
156	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
158	#126	—	4	2	no	—
159	#125	—	2	2	no	—
160	#124	—	2	2	no	—
161	#123	—	4	2	no	—
162	C₂²⋊Q₈	—	2	2	no	—
163	#121	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	4	2	no	—
167	#117	—	4	2	no	—
168	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
169	Q₃₂	—	2	2	no	—
170	SD₃₂	—	2	4	no	—
171	D₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
176	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
178	#107	C₂³×C₃	1	2	yes	—
179	#107	C₂³×C₃	1	2	yes	—
180	#106	—	2	2	no	—
181	#103	—	2	2	no	—
182	#103	—	2	2	no	—
183	#102	—	2	2	no	—
184	#102	—	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	—	4	2	no	—
190	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
191	C₃×Q₈	—	4	2	no	—
192	C₃×Q₈	—	4	2	no	—
193	C₃×D₄	—	4	2	no	—
194	C₃×D₄	—	4	2	no	—
195	C₃×D₄	—	8	4	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	4	2	no	—
201	C₂×C₃×C₄	—	4	2	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	—	8	4	no	—
204	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
210	C₃×C₈	—	2	1	no	—
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	#135	1	2	yes	—
213	C₃×C₈	C₂×#41	1	1	yes	characteristic
214	C₃×C₈	C₂×#41	1	1	yes	characteristic
215	C₂×Q₈	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
220	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	Q₁₆	—	2	2	no	—
223	Q₁₆	—	2	2	no	—
224	SD₁₆	—	4	4	no	—
225	D₈	—	2	2	no	—
226	D₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
231	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
232	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	C₃×D₈	1	1	yes	characteristic
247	C₄²	C₃×D₈	1	1	yes	characteristic
248	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
249	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
250	C₁₆	—	2	1	no	—
251	C₁₆	—	2	1	no	—
252	C₁₆	C₃×#41	1	2	yes	—
253	C₁₆	C₃×#41	1	2	yes	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	8	2	no	—
257	C₂²×C₃	—	8	2	no	—
258	C₂²×C₃	—	8	2	no	—
259	C₂²×C₃	#508	1	2	yes	—
260	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	2	1	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	8	2	no	—
266	C₃×C₄	—	8	2	no	—
267	C₃×C₄	—	8	2	no	—
268	C₃×C₄	#508	1	2	yes	—
269	C₃×C₄	#532	1	1	yes	characteristic
270	C₃×C₄	C₂×#135	1	1	yes	characteristic
271	C₂³	—	4	2	no	—
272	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
287	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
288	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
289	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
290	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	C₃×#135	1	2	yes	—
295	C₈	C₂×C₃×#41	1	1	yes	characteristic
296	C₈	C₂×C₃×#41	1	1	yes	characteristic
297	C₂×C₃	—	2	2	no	—
298	C₂×C₃	—	16	2	no	—
299	C₂×C₃	#3169	1	1	yes	characteristic
300	C₂×C₃	C₂×#508	1	1	yes	characteristic
301	C₂×C₃	C₂×#508	1	1	yes	characteristic
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	C₃×#508	1	2	yes	—
308	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	C₃×#508	1	2	yes	—
317	C₄	C₃×#532	1	1	yes	characteristic
318	C₄	C₂×C₃×#135	1	1	yes	characteristic
319	C₃	#33735	1	1	yes	characteristic, direct product factor, Sylow
320	C₂	—	2	2	no	—
321	C₂	—	16	2	no	—
322	C₂	C₃×#3169	1	1	yes	characteristic
323	C₂	C₂×C₃×#508	1	1	yes	characteristic
324	C₂	C₂×C₃×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₈
prime radical	C₃×#2215	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3168	C₂	semidirect
2. C₃×#2253	C₂	semidirect
3. C₃×#2249	C₂	semidirect
4. C₃×#2242	C₂	semidirect
5. C₃×#2217	C₂	semidirect
6. C₃×#2216	C₂	semidirect
8. C₃×C₄×Q₃₂	C₂	semidirect
9. C₃×C₄×SD₃₂	C₂	semidirect
10. C₃×C₄×D₁₆	C₂	semidirect
11. C₂×C₃×C₄×C₁₆	C₂	semidirect
12. #33735	C₃	direct (N × H)
40. #3168	C₂×C₃	semidirect
41. #2253	C₂×C₃	semidirect
42. #2249	C₂×C₃	semidirect
43. #2242	C₂×C₃	semidirect
44. #2217	C₂×C₃	semidirect
45. #2216	C₂×C₃	semidirect
47. C₄×Q₃₂	C₂×C₃	semidirect
48. C₄×SD₃₂	C₂×C₃	semidirect
49. C₄×D₁₆	C₂×C₃	semidirect
50. C₂×C₄×C₁₆	C₂×C₃	semidirect
319. C₃	#33735	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×#3168	2
C₃×#2253	2
C₃×#2249	2
C₃×#2242	2
C₃×#2217	2
C₃×#2216	2
C₃×#2215	2
C₃×C₄×Q₃₂	2
C₃×C₄×SD₃₂	2
C₃×C₄×D₁₆	2
C₂×C₃×C₄×C₁₆	2
#33735	3

Subgroups — C3×#33735 #10652342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33735 #10652342