Subgroups — C₃×#33740 #10652347

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3176	C₂	1	2	yes	maximal, complemented
2	C₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2244	C₂	1	2	yes	maximal, complemented
5	C₃×#2218	C₂	1	2	yes	maximal, complemented
6	C₃×#2215	C₂	1	2	yes	maximal, complemented
7	C₃×#2177	C₂	1	2	yes	maximal, complemented
8	C₃×C₄×D₁₆	C₂	1	2	yes	maximal, complemented
9	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	#33740	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₃×#540	—	2	2	no	—
12	C₂×C₃×D₁₆	C₂²	1	2	yes	—
13	C₂×C₃×D₁₆	C₂²	1	2	yes	—
14	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×#499	C₂²	1	2	yes	—
17	C₃×#493	C₂²	1	2	yes	—
18	C₃×#490	—	2	2	no	—
19	C₃×#469	—	2	2	no	—
20	C₃×#466	C₂²	1	4	yes	—
21	C₃×#442	—	2	2	no	—
22	C₃×C₄×D₈	C₂²	1	4	yes	—
23	C₃×#405	C₂²	1	1	yes	characteristic
24	C₃×#367	C₂²	1	2	yes	—
25	C₃×#366	C₂²	1	2	yes	—
26	C₃×#357	C₂²	1	4	yes	—
27	C₃×#357	C₂²	1	4	yes	—
28	C₃×#346	C₂²	1	1	yes	characteristic
29	C₃×#346	C₂²	1	1	yes	characteristic
30	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
31	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
32	#3176	C₂×C₃	1	2	yes	complemented
33	#2253	C₂×C₃	1	1	yes	characteristic, complemented
34	#2246	C₂×C₃	1	1	yes	characteristic, complemented
35	#2244	C₂×C₃	1	2	yes	complemented
36	#2218	C₂×C₃	1	2	yes	complemented
37	#2215	C₂×C₃	1	2	yes	complemented
38	#2177	C₂×C₃	1	2	yes	complemented
39	C₄×D₁₆	C₂×C₃	1	2	yes	complemented
40	#2108	C₂×C₃	1	1	yes	characteristic, complemented
41	C₂×C₃×D₈	—	2	2	no	—
42	C₂×C₃×D₈	C₂³	1	4	yes	—
43	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
44	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
45	C₃×#127	—	2	2	no	—
46	C₃×#125	—	2	2	no	—
47	C₃×#123	—	4	4	no	—
48	C₃×#121	—	2	4	no	—
49	C₃×#121	—	4	4	no	—
50	C₃×C₄×D₄	—	2	4	no	—
51	C₃×#117	D₄	1	1	yes	characteristic
52	C₃×D₁₆	—	2	4	no	—
53	C₃×D₁₆	—	2	4	no	—
54	C₃×M₃₂	—	2	2	no	—
55	C₂×C₃×C₁₆	—	2	2	no	—
56	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₃×#107	C₂³	1	4	yes	—
61	C₃×#106	—	2	2	no	—
62	C₃×#102	—	2	4	no	—
63	C₃×#102	—	2	4	no	—
64	C₃×#97	D₄	1	1	yes	characteristic
65	C₃×#97	D₄	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
68	#540	—	2	2	no	—
69	C₂×D₁₆	C₂²×C₃	1	2	yes	—
70	C₂×D₁₆	C₂²×C₃	1	2	yes	—
71	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
72	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
73	#499	C₂²×C₃	1	2	yes	—
74	#493	C₂²×C₃	1	2	yes	—
75	#490	—	2	2	no	—
76	#469	—	2	2	no	—
77	#466	C₂²×C₃	1	4	yes	—
78	#442	—	2	2	no	—
79	C₄×D₈	C₂²×C₃	1	4	yes	—
80	#405	C₂²×C₃	1	1	yes	characteristic
81	#367	C₂²×C₃	1	2	yes	—
82	#366	C₂²×C₃	1	2	yes	—
83	#357	C₂²×C₃	1	4	yes	—
84	#357	C₂²×C₃	1	4	yes	—
85	#346	C₂²×C₃	1	1	yes	characteristic
86	#346	C₂²×C₃	1	1	yes	characteristic
87	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
88	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
89	C₂×C₃×D₄	—	2	4	no	—
90	C₂×C₃×D₄	—	4	4	no	—
91	C₂×C₃×D₄	—	4	4	no	—
92	C₂²×C₃×C₄	—	4	4	no	—
93	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
94	C₃×D₈	—	2	4	no	—
95	C₃×D₈	—	2	4	no	—
96	C₃×D₈	—	4	4	no	—
97	C₃×M₁₆	—	2	2	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	2	1	no	—
100	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
104	C₃×C₄⋊C₄	—	2	4	no	—
105	C₃×C₄⋊C₄	—	4	4	no	—
106	C₃×C₄⋊C₄	D₈	1	2	yes	—
107	C₃×C₂²⋊C₄	—	4	4	no	—
108	C₃×C₂²⋊C₄	—	4	4	no	—
109	C₃×C₂²⋊C₄	D₈	1	2	yes	—
110	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
111	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
112	C₃×C₁₆	—	2	1	no	—
113	C₃×C₁₆	—	2	1	no	—
114	C₃×C₁₆	#41	1	2	yes	—
115	C₃×C₁₆	#41	1	2	yes	—
116	C₂×D₈	—	2	2	no	—
117	C₂×D₈	C₂³×C₃	1	4	yes	—
118	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
120	#127	—	2	2	no	—
121	#125	—	2	2	no	—
122	#123	—	4	4	no	—
123	#121	—	2	4	no	—
124	#121	—	4	4	no	—
125	C₄×D₄	—	2	4	no	—
126	#117	C₃×D₄	1	1	yes	characteristic
127	D₁₆	—	2	4	no	—
128	D₁₆	—	2	4	no	—
129	M₃₂	—	2	2	no	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
132	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
133	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
134	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
135	#107	C₂³×C₃	1	4	yes	—
136	#106	—	2	2	no	—
137	#102	—	2	4	no	—
138	#102	—	2	4	no	—
139	#97	C₃×D₄	1	1	yes	characteristic
140	#97	C₃×D₄	1	1	yes	characteristic
141	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
142	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
143	C₂³×C₃	—	4	4	no	—
144	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
145	C₃×D₄	—	4	4	no	—
146	C₃×D₄	—	4	4	no	—
147	C₃×D₄	—	16	4	no	—
148	C₃×D₄	—	16	4	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	1	no	—
151	C₂×C₃×C₄	—	4	4	no	—
152	C₂×C₃×C₄	—	16	4	no	—
153	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
154	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
155	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	#135	1	2	yes	—
160	C₃×C₈	C₂×#41	1	1	yes	characteristic
161	C₃×C₈	C₂×#41	1	1	yes	characteristic
162	C₂×D₄	—	2	4	no	—
163	C₂×D₄	—	4	4	no	—
164	C₂×D₄	—	4	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
167	D₈	—	2	4	no	—
168	D₈	—	2	4	no	—
169	D₈	—	4	4	no	—
170	M₁₆	—	2	2	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	4	4	no	—
179	C₄⋊C₄	C₃×D₈	1	2	yes	—
180	C₂²⋊C₄	—	4	4	no	—
181	C₂²⋊C₄	—	4	4	no	—
182	C₂²⋊C₄	C₃×D₈	1	2	yes	—
183	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
184	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
185	C₁₆	—	2	1	no	—
186	C₁₆	—	2	1	no	—
187	C₁₆	C₃×#41	1	2	yes	—
188	C₁₆	C₃×#41	1	2	yes	—
189	C₂²×C₃	—	2	2	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	8	4	no	—
192	C₂²×C₃	—	8	4	no	—
193	C₂²×C₃	—	8	4	no	—
194	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
195	C₃×C₄	—	2	2	no	—
196	C₃×C₄	—	2	2	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	8	4	no	—
199	C₃×C₄	#532	1	1	yes	characteristic
200	C₃×C₄	C₂×#135	1	1	yes	characteristic
201	C₂³	—	4	4	no	—
202	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
203	D₄	—	4	4	no	—
204	D₄	—	4	4	no	—
205	D₄	—	16	4	no	—
206	D₄	—	16	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	16	4	no	—
211	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
212	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
213	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
214	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	C₃×#135	1	2	yes	—
218	C₈	C₂×C₃×#41	1	1	yes	characteristic
219	C₈	C₂×C₃×#41	1	1	yes	characteristic
220	C₂×C₃	—	4	1	no	—
221	C₂×C₃	—	16	4	no	—
222	C₂×C₃	#3169	1	1	yes	characteristic
223	C₂×C₃	#3415	1	2	yes	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	8	4	no	—
227	C₂²	—	8	4	no	—
228	C₂²	—	8	4	no	—
229	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	8	4	no	—
234	C₄	C₃×#532	1	1	yes	characteristic
235	C₄	C₂×C₃×#135	1	1	yes	characteristic
236	C₃	#33740	1	1	yes	characteristic, direct product factor, Sylow
237	C₂	—	4	1	no	—
238	C₂	—	16	4	no	—
239	C₂	C₃×#3169	1	1	yes	characteristic
240	C₂	C₃×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#117 ◁ C₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3176	C₂	semidirect
2. C₃×#2253	C₂	semidirect
3. C₃×#2246	C₂	semidirect
4. C₃×#2244	C₂	semidirect
5. C₃×#2218	C₂	semidirect
6. C₃×#2215	C₂	semidirect
7. C₃×#2177	C₂	semidirect
8. C₃×C₄×D₁₆	C₂	semidirect
9. C₃×#2108	C₂	semidirect
10. #33740	C₃	direct (N × H)
32. #3176	C₂×C₃	semidirect
33. #2253	C₂×C₃	semidirect
34. #2246	C₂×C₃	semidirect
35. #2244	C₂×C₃	semidirect
36. #2218	C₂×C₃	semidirect
37. #2215	C₂×C₃	semidirect
38. #2177	C₂×C₃	semidirect
39. C₄×D₁₆	C₂×C₃	semidirect
40. #2108	C₂×C₃	semidirect
236. C₃	#33740	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×#3176	2
C₃×#2253	2
C₃×#2246	2
C₃×#2244	2
C₃×#2218	2
C₃×#2215	2
C₃×#2177	2
C₃×C₄×D₁₆	2
C₃×#2108	2
#33740	3

Subgroups — C3×#33740 #10652347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33740 #10652347