Subgroups — C₃×#33463 #10652070

Order 768.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3414	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#443	C₂	1	2	yes	maximal, complemented
3	C₃×#2179	C₂	1	4	yes	maximal, complemented
4	C₃×#2178	C₂	1	4	yes	maximal, complemented
5	C₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2143	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#33463	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#135	C₂²	1	2	yes	—
10	C₂×C₃×#131	—	2	2	no	—
11	C₃×#510	C₄	1	4	yes	complemented
12	C₃×#509	C₄	1	4	yes	complemented
13	C₃×#508	C₂²	1	4	yes	—
14	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₃×#443	—	2	2	no	—
17	C₃×#443	—	2	2	no	—
18	C₃×#443	C₂²	1	4	yes	complemented
19	C₃×#443	C₂²	1	4	yes	complemented
20	C₂×C₃×#108	C₂²	1	2	yes	—
21	C₂×C₃×#104	—	2	2	no	—
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₃×#368	C₂²	1	4	yes	complemented
24	C₃×#359	C₂²	1	8	yes	complemented
25	C₃×#346	C₂²	1	2	yes	complemented
26	C₃×C₄×C₁₆	C₂²	1	2	yes	complemented
27	#3414	C₂×C₃	1	1	yes	characteristic
28	C₂×#443	C₂×C₃	1	2	yes	complemented
29	#2179	C₂×C₃	1	4	yes	complemented
30	#2178	C₂×C₃	1	4	yes	complemented
31	#2161	C₂×C₃	1	1	yes	characteristic, complemented
32	#2143	C₂×C₃	1	2	yes	complemented
33	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×C₃×#41	—	2	2	no	—
35	C₃×#135	—	2	2	no	—
36	C₃×#135	—	2	2	no	—
37	C₃×#135	C₂×C₄	1	4	yes	—
38	C₃×#135	C₂³	1	4	yes	—
39	C₂×C₃×Q₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₃×SD₁₆	—	2	2	no	—
41	C₂×C₃×D₈	C₂×C₄	1	2	yes	—
42	C₃×#131	—	2	4	no	—
43	C₃×#131	—	2	4	no	—
44	C₃×#131	—	4	2	no	—
45	C₃×#131	—	4	2	no	—
46	C₂×C₃×M₁₆	—	2	2	no	—
47	C₂×C₃×M₁₆	—	4	2	no	—
48	C₂²×C₃×C₈	—	4	2	no	—
49	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
52	C₃×Q₃₂	C₂×C₄	1	4	yes	complemented
53	C₃×SD₃₂	C₂×C₄	1	8	yes	complemented
54	C₃×D₁₆	C₂×C₄	1	4	yes	complemented
55	C₃×M₃₂	—	2	2	no	—
56	C₃×M₃₂	C₂×C₄	1	4	yes	—
57	C₂×C₃×C₁₆	C₂³	1	2	yes	—
58	C₂×C₃×C₁₆	C₂³	1	2	yes	—
59	C₃×#108	—	2	2	no	—
60	C₃×#108	C₂³	1	4	yes	—
61	C₃×#104	—	2	4	no	—
62	C₃×#104	—	2	4	no	—
63	C₃×C₄×C₈	D₄	1	1	yes	characteristic
64	C₃×C₄×C₈	D₄	1	1	yes	characteristic
65	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₂×#135	C₂²×C₃	1	2	yes	—
68	C₂×#131	—	2	2	no	—
69	#510	C₃×C₄	1	4	yes	complemented
70	#509	C₃×C₄	1	4	yes	complemented
71	#508	C₂²×C₃	1	4	yes	—
72	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
73	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
74	#443	—	2	2	no	—
75	#443	—	2	2	no	—
76	#443	C₂²×C₃	1	4	yes	complemented
77	#443	C₂²×C₃	1	4	yes	complemented
78	C₂×#108	C₂²×C₃	1	2	yes	—
79	C₂×#104	—	2	2	no	—
80	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
81	#368	C₂²×C₃	1	4	yes	complemented
82	#359	C₂²×C₃	1	8	yes	complemented
83	#346	C₂²×C₃	1	2	yes	complemented
84	C₄×C₁₆	C₂²×C₃	1	2	yes	complemented
85	C₃×#41	—	2	4	no	—
86	C₃×#41	—	2	4	no	—
87	C₃×#41	—	4	2	no	—
88	C₃×#41	—	4	2	no	—
89	C₂×C₃×Q₈	—	2	2	no	—
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂×C₃×D₄	—	4	2	no	—
92	C₂²×C₃×C₄	—	2	1	no	—
93	C₂²×C₃×C₄	—	4	2	no	—
94	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
95	C₃×Q₁₆	—	2	2	no	—
96	C₃×Q₁₆	C₂²×C₄	1	4	yes	—
97	C₃×SD₁₆	—	2	4	no	—
98	C₃×SD₁₆	—	2	4	no	—
99	C₃×D₈	—	2	2	no	—
100	C₃×D₈	C₂²×C₄	1	4	yes	—
101	C₃×M₁₆	—	2	4	no	—
102	C₃×M₁₆	—	4	2	no	—
103	C₃×M₁₆	—	4	4	no	—
104	C₃×M₁₆	—	4	4	no	—
105	C₂×C₃×C₈	—	4	2	no	—
106	C₂×C₃×C₈	—	4	4	no	—
107	C₂×C₃×C₈	—	4	4	no	—
108	C₂×C₃×C₈	D₈	1	2	yes	—
109	C₂×C₃×C₈	D₈	1	2	yes	—
110	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₃×C₈	#41	1	2	yes	—
115	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
116	C₃×C₄²	—	2	1	no	—
117	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
118	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
119	C₃×C₁₆	—	2	2	no	—
120	C₃×C₁₆	C₂²×C₄	1	4	yes	—
121	C₂×#41	—	2	2	no	—
122	#135	—	2	2	no	—
123	#135	—	2	2	no	—
124	#135	C₂×C₃×C₄	1	4	yes	—
125	#135	C₂³×C₃	1	4	yes	—
126	C₂×Q₁₆	C₂×C₃×C₄	1	2	yes	—
127	C₂×SD₁₆	—	2	2	no	—
128	C₂×D₈	C₂×C₃×C₄	1	2	yes	—
129	#131	—	2	4	no	—
130	#131	—	2	4	no	—
131	#131	—	4	2	no	—
132	#131	—	4	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	—	4	2	no	—
135	C₂²×C₈	—	4	2	no	—
136	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
138	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
139	Q₃₂	C₂×C₃×C₄	1	4	yes	complemented
140	SD₃₂	C₂×C₃×C₄	1	8	yes	complemented
141	D₁₆	C₂×C₃×C₄	1	4	yes	complemented
142	M₃₂	—	2	2	no	—
143	M₃₂	C₂×C₃×C₄	1	4	yes	—
144	C₂×C₁₆	C₂³×C₃	1	2	yes	—
145	C₂×C₁₆	C₂³×C₃	1	2	yes	—
146	#108	—	2	2	no	—
147	#108	C₂³×C₃	1	4	yes	—
148	#104	—	2	4	no	—
149	#104	—	2	4	no	—
150	C₄×C₈	C₃×D₄	1	1	yes	characteristic
151	C₄×C₈	C₃×D₄	1	1	yes	characteristic
152	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
153	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
154	C₂³×C₃	—	4	2	no	—
155	C₂³×C₃	#135	1	1	yes	characteristic
156	C₃×Q₈	—	2	4	no	—
157	C₃×Q₈	—	4	2	no	—
158	C₃×D₄	—	2	4	no	—
159	C₃×D₄	—	4	4	no	—
160	C₃×D₄	—	4	4	no	—
161	C₃×D₄	—	4	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	2	no	—
166	C₂×C₃×C₄	—	4	4	no	—
167	C₂×C₃×C₄	—	4	4	no	—
168	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
169	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
170	C₂×C₃×C₄	#135	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
172	C₃×C₈	—	4	4	no	—
173	C₃×C₈	C₄×D₄	1	2	yes	—
174	C₃×C₈	C₂×D₈	1	2	yes	—
175	C₃×C₈	C₂×D₈	1	1	yes	characteristic
176	C₃×C₈	C₂×D₈	1	1	yes	characteristic
177	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
178	C₃×C₈	C₂×#41	1	1	yes	characteristic
179	#41	—	2	4	no	—
180	#41	—	2	4	no	—
181	#41	—	4	2	no	—
182	#41	—	4	2	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
189	Q₁₆	—	2	2	no	—
190	Q₁₆	C₂²×C₃×C₄	1	4	yes	—
191	SD₁₆	—	2	4	no	—
192	SD₁₆	—	2	4	no	—
193	D₈	—	2	2	no	—
194	D₈	C₂²×C₃×C₄	1	4	yes	—
195	M₁₆	—	2	4	no	—
196	M₁₆	—	4	2	no	—
197	M₁₆	—	4	4	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	4	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₃×D₈	1	2	yes	—
203	C₂×C₈	C₃×D₈	1	2	yes	—
204	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
205	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
206	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₈	C₃×#41	1	2	yes	—
209	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
210	C₄²	—	2	1	no	—
211	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
212	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
213	C₁₆	—	2	2	no	—
214	C₁₆	C₂²×C₃×C₄	1	4	yes	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	—	4	4	no	—
217	C₂²×C₃	—	8	4	no	—
218	C₂²×C₃	C₄×D₈	1	2	yes	—
219	C₂²×C₃	C₂×#135	1	1	yes	characteristic
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	—	4	4	no	—
223	C₃×C₄	C₄×D₈	1	2	yes	—
224	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
225	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
226	C₂³	—	4	2	no	—
227	C₂³	C₃×#135	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
241	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
242	C₂×C₄	C₃×#135	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₈	C₃×C₄×D₄	1	2	yes	—
246	C₈	C₂×C₃×D₈	1	1	yes	characteristic
247	C₈	C₂×C₃×D₈	1	2	yes	—
248	C₈	C₂×C₃×D₈	1	1	yes	characteristic
249	C₈	C₂³×C₃×C₄	1	1	yes	characteristic
250	C₈	C₂×C₃×#41	1	1	yes	characteristic
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	—	8	4	no	—
254	C₂×C₃	C₂×C₄×D₈	1	1	yes	characteristic
255	C₂²	—	4	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	—	8	4	no	—
258	C₂²	C₃×C₄×D₈	1	2	yes	—
259	C₂²	C₂×C₃×#135	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	4	no	—
263	C₄	C₃×C₄×D₈	1	2	yes	—
264	C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
265	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
266	C₃	#33463	1	1	yes	characteristic, direct product factor, Sylow
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	8	4	no	—
270	C₂	C₂×C₃×C₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂×D₈
commutator	C₈	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂×C₃	C₂×C₄×D₈
prime radical	C₃×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#443	C₂	semidirect
3. C₃×#2179	C₂	semidirect
4. C₃×#2178	C₂	semidirect
5. C₃×#2161	C₂	semidirect
6. C₃×#2143	C₂	semidirect
7. C₃×C₄×M₃₂	C₂	semidirect
8. #33463	C₃	direct (N × H)
11. C₃×#510	C₄	semidirect
12. C₃×#509	C₄	semidirect
18. C₃×#443	C₂²	semidirect
19. C₃×#443	C₂²	semidirect
23. C₃×#368	C₂²	semidirect
24. C₃×#359	C₂²	semidirect
25. C₃×#346	C₂²	semidirect
26. C₃×C₄×C₁₆	C₂²	semidirect
28. C₂×#443	C₂×C₃	semidirect
29. #2179	C₂×C₃	semidirect
30. #2178	C₂×C₃	semidirect
31. #2161	C₂×C₃	semidirect
32. #2143	C₂×C₃	semidirect
33. C₄×M₃₂	C₂×C₃	semidirect
52. C₃×Q₃₂	C₂×C₄	semidirect
53. C₃×SD₃₂	C₂×C₄	semidirect
54. C₃×D₁₆	C₂×C₄	semidirect
69. #510	C₃×C₄	semidirect
70. #509	C₃×C₄	semidirect
76. #443	C₂²×C₃	semidirect
77. #443	C₂²×C₃	semidirect
81. #368	C₂²×C₃	semidirect
82. #359	C₂²×C₃	semidirect
83. #346	C₂²×C₃	semidirect
84. C₄×C₁₆	C₂²×C₃	semidirect
139. Q₃₂	C₂×C₃×C₄	semidirect
140. SD₃₂	C₂×C₃×C₄	semidirect
141. D₁₆	C₂×C₃×C₄	semidirect
266. C₃	#33463	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×#3414	2
C₂×C₃×#443	2
C₃×#2179	2
C₃×#2178	2
C₃×#2161	2
C₃×#2143	2
C₃×C₄×M₃₂	2
#33463	3

Subgroups — C3×#33463 #10652070

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33463 #10652070