Subgroups — C₃×#35322 #10653929

Order 768.

2074 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#533	C₂	1	2	yes	maximal
2	C₃×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2862	C₂	1	2	yes	maximal, complemented
4	C₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#2621	C₂	1	4	yes	maximal
6	C₃×#2565	C₂	1	8	yes	maximal, complemented
7	C₃×#2499	C₂	1	8	yes	maximal
8	C₃×#2327	C₂	1	4	yes	maximal, complemented
9	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic
10	#35322	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#533	—	2	16	no	—
12	C₂×C₃×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₃×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₃×#125	C₂²	1	4	yes	—
16	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
17	C₂×C₃×C₂²⋊Q₈	C₂²	1	4	yes	—
18	C₂×C₃×C₄×Q₈	C₂²	1	4	yes	—
19	C₂×C₃×C₄×Q₈	C₂²	1	4	yes	—
20	C₂×C₃×#117	C₂²	1	2	yes	—
21	C₂×C₃×#117	C₂²	1	4	yes	—
22	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#398	C₂²	1	16	yes	—
24	C₃×#397	C₂²	1	16	yes	—
25	C₃×#391	C₂²	1	8	yes	—
26	C₃×#391	C₂²	1	8	yes	—
27	C₃×#391	C₂²	1	8	yes	—
28	C₃×#387	C₂²	1	32	yes	—
29	C₃×#384	C₂²	1	8	yes	—
30	C₃×#382	C₂²	1	4	yes	—
31	C₃×#382	C₂²	1	8	yes	—
32	C₃×C₄×C₄⋊C₄	C₂²	1	4	yes	—
33	C₃×C₄×C₂²⋊C₄	C₂²	1	4	yes	—
34	C₃×#376	C₂²	1	2	yes	—
35	C₃×C₄³	C₂²	1	1	yes	characteristic
36	C₃×C₄³	C₂²	1	1	yes	characteristic
37	C₂×#533	C₂×C₃	1	2	yes	—
38	#2868	C₂×C₃	1	1	yes	characteristic, complemented
39	#2862	C₂×C₃	1	1	yes	characteristic, complemented
40	#2862	C₂×C₃	1	2	yes	complemented
41	#2621	C₂×C₃	1	4	yes	—
42	#2565	C₂×C₃	1	8	yes	complemented
43	#2499	C₂×C₃	1	8	yes	—
44	#2327	C₂×C₃	1	4	yes	complemented
45	C₄×#117	C₂×C₃	1	1	yes	characteristic
46	C₂²×C₃×Q₈	C₂³	1	4	yes	—
47	C₂²×C₃×Q₈	C₂³	1	4	yes	—
48	C₂³×C₃×C₄	C₂³	1	2	yes	—
49	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
50	C₃×C₄⋊Q₈	—	2	8	no	—
51	C₃×C₄⋊Q₈	—	2	8	no	—
52	C₃×C₄⋊Q₈	—	2	16	no	—
53	C₃×#125	—	2	16	no	—
54	C₃×C₂²⋊Q₈	—	2	16	no	—
55	C₃×C₂²⋊Q₈	—	2	16	no	—
56	C₃×C₄×Q₈	—	2	16	no	—
57	C₃×C₄×Q₈	—	2	16	no	—
58	C₃×#117	—	2	16	no	—
59	C₃×#117	—	2	8	no	—
60	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
61	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
62	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
63	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
64	C₂×C₃×C₄⋊C₄	C₂³	1	16	yes	—
65	C₂×C₃×C₄⋊C₄	C₂³	1	4	yes	—
66	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
67	C₂×C₃×C₄⋊C₄	C₂³	1	8	yes	—
68	C₂×C₃×C₂²⋊C₄	C₂³	1	4	yes	—
69	C₂×C₃×C₂²⋊C₄	C₂³	1	8	yes	—
70	C₂×C₃×C₄²	D₄	1	8	yes	—
71	C₂×C₃×C₄²	C₂³	1	2	yes	—
72	C₂×C₃×C₄²	C₂³	1	4	yes	—
73	C₂×C₃×C₄²	C₂³	1	4	yes	—
74	C₂×C₃×C₄²	C₂³	1	8	yes	—
75	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
76	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
77	C₂×C₃×C₄²	C₂³	1	2	yes	—
78	C₂×C₃×C₄²	C₂³	1	2	yes	—
79	C₃×#95	C₂³	1	8	yes	—
80	C₃×#95	C₂³	1	32	yes	—
81	C₃×#95	C₂³	1	16	yes	—
82	#533	—	2	16	no	—
83	C₂×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
84	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
85	C₂×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
86	C₂×#125	C₂²×C₃	1	4	yes	—
87	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
88	C₂×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
89	C₂×C₄×Q₈	C₂²×C₃	1	4	yes	—
90	C₂×C₄×Q₈	C₂²×C₃	1	4	yes	—
91	C₂×#117	C₂²×C₃	1	2	yes	—
92	C₂×#117	C₂²×C₃	1	4	yes	—
93	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
94	#398	C₂²×C₃	1	16	yes	—
95	#397	C₂²×C₃	1	16	yes	—
96	#391	C₂²×C₃	1	8	yes	—
97	#391	C₂²×C₃	1	8	yes	—
98	#391	C₂²×C₃	1	8	yes	—
99	#387	C₂²×C₃	1	32	yes	—
100	#384	C₂²×C₃	1	8	yes	—
101	#382	C₂²×C₃	1	8	yes	—
102	#382	C₂²×C₃	1	4	yes	—
103	C₄×C₄⋊C₄	C₂²×C₃	1	4	yes	—
104	C₄×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
105	#376	C₂²×C₃	1	2	yes	—
106	C₄³	C₂²×C₃	1	1	yes	characteristic
107	C₄³	C₂²×C₃	1	1	yes	characteristic
108	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
109	C₂×C₃×Q₈	—	2	8	no	—
110	C₂×C₃×Q₈	—	2	8	no	—
111	C₂×C₃×Q₈	—	2	16	no	—
112	C₂×C₃×Q₈	—	2	16	no	—
113	C₂²×C₃×C₄	—	2	4	no	—
114	C₂²×C₃×C₄	—	2	8	no	—
115	C₂²×C₃×C₄	C₂×D₄	1	8	yes	—
116	C₂²×C₃×C₄	C₂×D₄	1	4	yes	—
117	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
118	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
119	C₂²×C₃×C₄	C₂⁴	1	8	yes	—
120	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
121	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂⁴	1	4	yes	—
123	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂⁴	1	2	yes	—
125	C₃×C₄⋊C₄	—	2	8	no	—
126	C₃×C₄⋊C₄	—	2	8	no	—
127	C₃×C₄⋊C₄	—	2	16	no	—
128	C₃×C₄⋊C₄	—	2	16	no	—
129	C₃×C₄⋊C₄	—	2	8	no	—
130	C₃×C₄⋊C₄	—	2	8	no	—
131	C₃×C₄⋊C₄	—	2	8	no	—
132	C₃×C₄⋊C₄	—	2	32	no	—
133	C₃×C₂²⋊C₄	—	2	16	no	—
134	C₃×C₂²⋊C₄	—	2	8	no	—
135	C₃×C₄²	—	2	16	no	—
136	C₃×C₄²	—	2	4	no	—
137	C₃×C₄²	—	2	8	no	—
138	C₃×C₄²	—	2	8	no	—
139	C₃×C₄²	C₂×D₄	1	4	yes	—
140	C₃×C₄²	C₂×D₄	1	8	yes	—
141	C₃×C₄²	C₂×D₄	1	4	yes	—
142	C₃×C₄²	#41	1	8	yes	—
143	C₂²×Q₈	C₂³×C₃	1	4	yes	—
144	C₂²×Q₈	C₂³×C₃	1	4	yes	—
145	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
146	C₂³×C₄	C₂³×C₃	1	2	yes	—
147	C₄⋊Q₈	—	2	8	no	—
148	C₄⋊Q₈	—	2	8	no	—
149	C₄⋊Q₈	—	2	16	no	—
150	#125	—	2	16	no	—
151	C₂²⋊Q₈	—	2	16	no	—
152	C₂²⋊Q₈	—	2	16	no	—
153	C₄×Q₈	—	2	16	no	—
154	C₄×Q₈	—	2	16	no	—
155	#117	—	2	16	no	—
156	#117	—	2	8	no	—
157	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
158	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
159	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
161	C₂×C₄⋊C₄	C₂³×C₃	1	8	yes	—
162	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
163	C₂×C₄⋊C₄	C₂³×C₃	1	4	yes	—
164	C₂×C₄⋊C₄	C₂³×C₃	1	16	yes	—
165	C₂×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
166	C₂×C₂²⋊C₄	C₂³×C₃	1	4	yes	—
167	C₂×C₄²	C₃×D₄	1	8	yes	—
168	C₂×C₄²	C₂³×C₃	1	8	yes	—
169	C₂×C₄²	C₂³×C₃	1	2	yes	—
170	C₂×C₄²	C₂³×C₃	1	4	yes	—
171	C₂×C₄²	C₂³×C₃	1	4	yes	—
172	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₃	1	2	yes	—
174	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₃	1	2	yes	—
176	#95	C₂³×C₃	1	8	yes	—
177	#95	C₂³×C₃	1	32	yes	—
178	#95	C₂³×C₃	1	16	yes	—
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	—	2	4	no	—
181	C₂³×C₃	C₂²×D₄	1	2	yes	—
182	C₂³×C₃	C₂⁵	1	1	yes	characteristic
183	C₃×Q₈	—	4	16	no	—
184	C₃×Q₈	—	4	16	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	8	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	16	no	—
190	C₂×C₃×C₄	—	2	8	no	—
191	C₂×C₃×C₄	—	2	8	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	16	no	—
194	C₂×C₃×C₄	—	2	8	no	—
195	C₂×C₃×C₄	—	2	16	no	—
196	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
197	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
198	C₂×C₃×C₄	C₂²×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₂²×D₄	1	4	yes	—
200	C₂×C₃×C₄	C₂×#41	1	8	yes	—
201	C₂×C₃×C₄	C₂×#41	1	4	yes	—
202	C₂×C₃×C₄	2₋¹⁺⁴	1	8	yes	—
203	C₂⁴	C₂⁴×C₃	1	1	yes	characteristic
204	C₂×Q₈	—	2	8	no	—
205	C₂×Q₈	—	2	8	no	—
206	C₂×Q₈	—	2	16	no	—
207	C₂×Q₈	—	2	16	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	C₂×C₃×D₄	1	8	yes	—
211	C₂²×C₄	C₂×C₃×D₄	1	4	yes	—
212	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
213	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
214	C₂²×C₄	C₂⁴×C₃	1	8	yes	—
215	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
217	C₂²×C₄	C₂⁴×C₃	1	4	yes	—
218	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
219	C₂²×C₄	C₂⁴×C₃	1	2	yes	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	2	16	no	—
223	C₄⋊C₄	—	2	16	no	—
224	C₄⋊C₄	—	2	8	no	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	32	no	—
228	C₂²⋊C₄	—	2	16	no	—
229	C₂²⋊C₄	—	2	8	no	—
230	C₄²	—	2	16	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	2	8	no	—
233	C₄²	—	2	8	no	—
234	C₄²	C₂×C₃×D₄	1	4	yes	—
235	C₄²	C₂×C₃×D₄	1	8	yes	—
236	C₄²	C₂×C₃×D₄	1	4	yes	—
237	C₄²	C₃×#41	1	8	yes	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	2	4	no	—
240	C₂²×C₃	—	4	4	no	—
241	C₂²×C₃	C₂³×D₄	1	1	yes	characteristic
242	C₂²×C₃	C₂²×#41	1	2	yes	—
243	C₂²×C₃	C₂×2₋¹⁺⁴	1	4	yes	—
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	4	no	—
246	C₃×C₄	—	2	8	no	—
247	C₃×C₄	—	4	8	no	—
248	C₃×C₄	—	4	8	no	—
249	C₃×C₄	—	4	4	no	—
250	C₃×C₄	#532	1	4	yes	—
251	C₃×C₄	#536	1	8	yes	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	4	no	—
254	C₂³	C₂²×C₃×D₄	1	2	yes	—
255	C₂³	C₂⁵×C₃	1	1	yes	characteristic
256	Q₈	—	4	16	no	—
257	Q₈	—	4	16	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	16	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	16	no	—
269	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
270	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
271	C₂×C₄	C₂²×C₃×D₄	1	2	yes	—
272	C₂×C₄	C₂²×C₃×D₄	1	4	yes	—
273	C₂×C₄	C₂×C₃×#41	1	8	yes	—
274	C₂×C₄	C₂×C₃×#41	1	4	yes	—
275	C₂×C₄	C₃×2₋¹⁺⁴	1	8	yes	—
276	C₂×C₃	—	4	2	no	—
277	C₂×C₃	C₂×#532	1	1	yes	characteristic
278	C₂×C₃	C₂×#536	1	2	yes	—
279	C₂×C₃	#3509	1	4	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₂³×C₃×D₄	1	1	yes	characteristic
284	C₂²	C₂²×C₃×#41	1	2	yes	—
285	C₂²	C₂×C₃×2₋¹⁺⁴	1	4	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	2	4	no	—
288	C₄	—	2	8	no	—
289	C₄	—	4	8	no	—
290	C₄	—	4	8	no	—
291	C₄	—	4	4	no	—
292	C₄	C₃×#532	1	4	yes	—
293	C₄	C₃×#536	1	8	yes	—
294	C₃	#35322	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	4	2	no	—
296	C₂	C₂×C₃×#532	1	1	yes	characteristic
297	C₂	C₂×C₃×#536	1	2	yes	—
298	C₂	C₃×#3509	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃
prime radical	C₂⁴×C₃	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2868	C₂	semidirect
3. C₃×#2862	C₂	semidirect
4. C₃×#2862	C₂	semidirect
6. C₃×#2565	C₂	semidirect
8. C₃×#2327	C₂	semidirect
10. #35322	C₃	direct (N × H)
38. #2868	C₂×C₃	semidirect
39. #2862	C₂×C₃	semidirect
40. #2862	C₂×C₃	semidirect
42. #2565	C₂×C₃	semidirect
44. #2327	C₂×C₃	semidirect
294. C₃	#35322	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#533	2
C₃×#2868	2
C₃×#2862	2
C₃×#2621	2
C₃×#2565	2
C₃×#2499	2
C₃×#2327	2
C₃×C₄×#117	2
#35322	3

Subgroups — C3×#35322 #10653929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#35322 #10653929