Subgroups — C₃×#33465 #10652072

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₃×#2957	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	C₂×C₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2143	C₂	1	2	yes	maximal, complemented
7	C₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#33465	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#135	C₂²	1	2	yes	—
10	C₃×#568	—	2	2	no	—
11	C₃×#508	C₄	1	2	yes	complemented
12	C₃×#508	C₄	1	2	yes	complemented
13	C₃×#508	C₂²	1	2	yes	—
14	C₃×#508	C₂²	1	2	yes	—
15	C₂×C₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
16	C₂×C₃×SD₃₂	C₄	1	2	yes	complemented
17	C₂×C₃×D₁₆	C₄	1	1	yes	characteristic, complemented
18	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
19	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
20	C₃×#444	—	2	4	no	—
21	C₃×#443	C₂²	1	8	yes	complemented
22	C₂×C₃×#108	C₂²	1	2	yes	—
23	C₃×#421	—	2	2	no	—
24	C₃×#405	C₂²	1	1	yes	characteristic
25	C₃×#368	C₂²	1	2	yes	complemented
26	C₃×#368	C₂²	1	2	yes	complemented
27	C₃×#359	C₂²	1	8	yes	complemented
28	C₃×#346	C₂²	1	2	yes	complemented
29	C₃×C₄×C₁₆	C₂²	1	2	yes	complemented
30	C₂×#508	C₂×C₃	1	1	yes	characteristic
31	#2957	C₂×C₃	1	2	yes	complemented
32	#2179	C₂×C₃	1	4	yes	complemented
33	#2178	C₂×C₃	1	4	yes	complemented
34	C₂×#368	C₂×C₃	1	1	yes	characteristic, complemented
35	#2143	C₂×C₃	1	2	yes	complemented
36	#2108	C₂×C₃	1	1	yes	characteristic, complemented
37	C₂×C₃×#41	—	2	2	no	—
38	C₃×#135	—	2	2	no	—
39	C₃×#135	—	2	2	no	—
40	C₃×#135	C₂×C₄	1	4	yes	—
41	C₃×#135	C₂³	1	4	yes	—
42	C₂×C₃×Q₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₃×SD₁₆	—	2	2	no	—
44	C₂×C₃×D₈	C₂×C₄	1	2	yes	—
45	C₃×#131	—	2	8	no	—
46	C₃×#131	—	4	4	no	—
47	C₂×C₃×M₁₆	—	2	2	no	—
48	C₂×C₃×M₁₆	—	4	4	no	—
49	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
51	C₃×#117	D₄	1	1	yes	characteristic
52	C₃×Q₃₂	C₂×C₄	1	2	yes	—
53	C₃×Q₃₂	C₂×C₄	1	2	yes	—
54	C₃×SD₃₂	C₂×C₄	1	4	yes	—
55	C₃×SD₃₂	C₂×C₄	1	4	yes	—
56	C₃×D₁₆	C₂×C₄	1	2	yes	—
57	C₃×D₁₆	C₂×C₄	1	2	yes	—
58	C₃×M₃₂	—	2	1	no	—
59	C₃×M₃₂	—	2	1	no	—
60	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
61	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
62	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
63	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
64	C₂×C₃×C₁₆	C₂³	1	2	yes	—
65	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
66	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
67	C₃×#108	—	2	2	no	—
68	C₃×#108	C₂³	1	4	yes	—
69	C₃×#104	—	2	8	no	—
70	C₃×#97	D₄	1	2	yes	—
71	C₃×C₄×C₈	C₂³	1	2	yes	—
72	C₂×#135	C₂²×C₃	1	2	yes	—
73	#568	—	2	2	no	—
74	#508	C₃×C₄	1	2	yes	complemented
75	#508	C₃×C₄	1	2	yes	complemented
76	#508	C₂²×C₃	1	2	yes	—
77	#508	C₂²×C₃	1	2	yes	—
78	C₂×Q₃₂	C₃×C₄	1	1	yes	characteristic, complemented
79	C₂×SD₃₂	C₃×C₄	1	2	yes	complemented
80	C₂×D₁₆	C₃×C₄	1	1	yes	characteristic, complemented
81	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
82	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
83	#444	—	2	4	no	—
84	#443	C₂²×C₃	1	8	yes	complemented
85	C₂×#108	C₂²×C₃	1	2	yes	—
86	#421	—	2	2	no	—
87	#405	C₂²×C₃	1	1	yes	characteristic
88	#368	C₂²×C₃	1	2	yes	complemented
89	#368	C₂²×C₃	1	2	yes	complemented
90	#359	C₂²×C₃	1	8	yes	complemented
91	#346	C₂²×C₃	1	2	yes	complemented
92	C₄×C₁₆	C₂²×C₃	1	2	yes	complemented
93	C₃×#41	—	2	4	no	—
94	C₃×#41	—	2	4	no	—
95	C₃×#41	—	4	2	no	—
96	C₃×#41	—	4	2	no	—
97	C₂×C₃×Q₈	—	2	2	no	—
98	C₂×C₃×D₄	—	2	2	no	—
99	C₂×C₃×D₄	—	4	2	no	—
100	C₂²×C₃×C₄	—	4	2	no	—
101	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
102	C₃×Q₁₆	—	2	2	no	—
103	C₃×Q₁₆	C₂²×C₄	1	4	yes	—
104	C₃×SD₁₆	—	2	4	no	—
105	C₃×SD₁₆	—	2	4	no	—
106	C₃×D₈	—	2	2	no	—
107	C₃×D₈	C₂²×C₄	1	4	yes	—
108	C₃×M₁₆	—	2	4	no	—
109	C₃×M₁₆	—	4	2	no	—
110	C₃×M₁₆	—	4	8	no	—
111	C₃×M₁₆	D₈	1	2	yes	—
112	C₃×M₁₆	D₈	1	2	yes	—
113	C₂×C₃×C₈	—	4	2	no	—
114	C₂×C₃×C₈	—	4	8	no	—
115	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
119	C₂×C₃×C₈	#41	1	1	yes	characteristic
120	C₂×C₃×C₈	#41	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₂²⋊C₄	—	2	1	no	—
124	C₃×C₄²	C₂×D₄	1	2	yes	—
125	C₃×C₁₆	—	2	2	no	—
126	C₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
127	C₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
128	C₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
129	C₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
130	C₂×#41	—	2	2	no	—
131	#135	—	2	2	no	—
132	#135	—	2	2	no	—
133	#135	C₂×C₃×C₄	1	4	yes	—
134	#135	C₂³×C₃	1	4	yes	—
135	C₂×Q₁₆	C₂×C₃×C₄	1	2	yes	—
136	C₂×SD₁₆	—	2	2	no	—
137	C₂×D₈	C₂×C₃×C₄	1	2	yes	—
138	#131	—	2	8	no	—
139	#131	—	4	4	no	—
140	C₂×M₁₆	—	2	2	no	—
141	C₂×M₁₆	—	4	4	no	—
142	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
144	#117	C₃×D₄	1	1	yes	characteristic
145	Q₃₂	C₂×C₃×C₄	1	2	yes	—
146	Q₃₂	C₂×C₃×C₄	1	2	yes	—
147	SD₃₂	C₂×C₃×C₄	1	4	yes	—
148	SD₃₂	C₂×C₃×C₄	1	4	yes	—
149	D₁₆	C₂×C₃×C₄	1	2	yes	—
150	D₁₆	C₂×C₃×C₄	1	2	yes	—
151	M₃₂	—	2	1	no	—
152	M₃₂	—	2	1	no	—
153	C₂×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
154	C₂×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
155	C₂×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
156	C₂×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
157	C₂×C₁₆	C₂³×C₃	1	2	yes	—
158	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
159	C₂×C₁₆	C₂³×C₃	1	1	yes	characteristic
160	#108	—	2	2	no	—
161	#108	C₂³×C₃	1	4	yes	—
162	#104	—	2	8	no	—
163	#97	C₃×D₄	1	2	yes	—
164	C₄×C₈	C₂³×C₃	1	2	yes	—
165	C₂³×C₃	—	4	2	no	—
166	C₂³×C₃	#135	1	1	yes	characteristic
167	C₃×Q₈	—	2	4	no	—
168	C₃×Q₈	—	4	2	no	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	4	4	no	—
171	C₃×D₄	—	4	4	no	—
172	C₃×D₄	—	4	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	—	4	4	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
181	C₂×C₃×C₄	#135	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
183	C₃×C₈	—	4	4	no	—
184	C₃×C₈	C₄×D₄	1	1	yes	characteristic
185	C₃×C₈	C₄×D₄	1	1	yes	characteristic
186	C₃×C₈	C₂×D₈	1	4	yes	—
187	C₃×C₈	C₂³×C₄	1	1	yes	characteristic
188	C₃×C₈	C₂×#41	1	1	yes	characteristic
189	#41	—	2	4	no	—
190	#41	—	2	4	no	—
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	Q₁₆	—	2	2	no	—
199	Q₁₆	C₂²×C₃×C₄	1	4	yes	—
200	SD₁₆	—	2	4	no	—
201	SD₁₆	—	2	4	no	—
202	D₈	—	2	2	no	—
203	D₈	C₂²×C₃×C₄	1	4	yes	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	4	2	no	—
206	M₁₆	—	4	8	no	—
207	M₁₆	C₃×D₈	1	2	yes	—
208	M₁₆	C₃×D₈	1	2	yes	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	8	no	—
211	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
213	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
215	C₂×C₈	C₃×#41	1	1	yes	characteristic
216	C₂×C₈	C₃×#41	1	1	yes	characteristic
217	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
218	C₄⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₄²	C₂×C₃×D₄	1	2	yes	—
221	C₁₆	—	2	2	no	—
222	C₁₆	C₂²×C₃×C₄	1	1	yes	characteristic
223	C₁₆	C₂²×C₃×C₄	1	1	yes	characteristic
224	C₁₆	C₂²×C₃×C₄	1	1	yes	characteristic
225	C₁₆	C₂²×C₃×C₄	1	1	yes	characteristic
226	C₂²×C₃	—	4	1	no	—
227	C₂²×C₃	—	4	4	no	—
228	C₂²×C₃	—	16	2	no	—
229	C₂²×C₃	C₄×D₈	1	1	yes	characteristic
230	C₂²×C₃	C₄×D₈	1	1	yes	characteristic
231	C₂²×C₃	C₂×#135	1	1	yes	characteristic
232	C₃×C₄	—	4	2	no	—
233	C₃×C₄	—	4	4	no	—
234	C₃×C₄	C₄×D₈	1	1	yes	characteristic
235	C₃×C₄	C₄×D₈	1	1	yes	characteristic
236	C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
237	C₃×C₄	C₂²×D₈	1	1	yes	characteristic
238	C₂³	—	4	2	no	—
239	C₂³	C₃×#135	1	1	yes	characteristic
240	Q₈	—	2	4	no	—
241	Q₈	—	4	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	4	4	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
251	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
253	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
254	C₂×C₄	C₃×#135	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
256	C₈	—	4	4	no	—
257	C₈	C₃×C₄×D₄	1	1	yes	characteristic
258	C₈	C₃×C₄×D₄	1	1	yes	characteristic
259	C₈	C₂×C₃×D₈	1	4	yes	—
260	C₈	C₂³×C₃×C₄	1	1	yes	characteristic
261	C₈	C₂×C₃×#41	1	1	yes	characteristic
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	8	4	no	—
266	C₂×C₃	C₂×C₄×D₈	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	4	4	no	—
269	C₂²	—	16	2	no	—
270	C₂²	C₃×C₄×D₈	1	1	yes	characteristic
271	C₂²	C₃×C₄×D₈	1	1	yes	characteristic
272	C₂²	C₂×C₃×#135	1	1	yes	characteristic
273	C₄	—	4	2	no	—
274	C₄	—	4	4	no	—
275	C₄	C₃×C₄×D₈	1	1	yes	characteristic
276	C₄	C₃×C₄×D₈	1	1	yes	characteristic
277	C₄	C₂×C₃×C₄×D₄	1	1	yes	characteristic
278	C₄	C₂²×C₃×D₈	1	1	yes	characteristic
279	C₃	#33465	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×C₃×C₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂²×D₈
commutator	C₈	C₂³×C₃×C₄
Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂×C₃	C₂×C₄×D₈
prime radical	C₂×C₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆ ◁ C₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2957	C₂	semidirect
3. C₃×#2179	C₂	semidirect
4. C₃×#2178	C₂	semidirect
5. C₂×C₃×#368	C₂	semidirect
6. C₃×#2143	C₂	semidirect
7. C₃×#2108	C₂	semidirect
8. #33465	C₃	direct (N × H)
11. C₃×#508	C₄	semidirect
12. C₃×#508	C₄	semidirect
15. C₂×C₃×Q₃₂	C₄	semidirect
16. C₂×C₃×SD₃₂	C₄	semidirect
17. C₂×C₃×D₁₆	C₄	semidirect
21. C₃×#443	C₂²	semidirect
25. C₃×#368	C₂²	semidirect
26. C₃×#368	C₂²	semidirect
27. C₃×#359	C₂²	semidirect
28. C₃×#346	C₂²	semidirect
29. C₃×C₄×C₁₆	C₂²	semidirect
31. #2957	C₂×C₃	semidirect
32. #2179	C₂×C₃	semidirect
33. #2178	C₂×C₃	semidirect
34. C₂×#368	C₂×C₃	semidirect
35. #2143	C₂×C₃	semidirect
36. #2108	C₂×C₃	semidirect
74. #508	C₃×C₄	semidirect
75. #508	C₃×C₄	semidirect
78. C₂×Q₃₂	C₃×C₄	semidirect
79. C₂×SD₃₂	C₃×C₄	semidirect
80. C₂×D₁₆	C₃×C₄	semidirect
84. #443	C₂²×C₃	semidirect
88. #368	C₂²×C₃	semidirect
89. #368	C₂²×C₃	semidirect
90. #359	C₂²×C₃	semidirect
91. #346	C₂²×C₃	semidirect
92. C₄×C₁₆	C₂²×C₃	semidirect
279. C₃	#33465	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×#508	2
C₃×#2957	2
C₃×#2179	2
C₃×#2178	2
C₂×C₃×#368	2
C₃×#2143	2
C₃×#2108	2
#33465	3

Subgroups — C3×#33465 #10652072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33465 #10652072