Subgroups — C₃×#9849 #10628456

Order 768.

530 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₃×#1962	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1492	C₂	1	2	yes	maximal, complemented
6	#9849	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#532	C₂²	1	2	yes	—
8	C₃×#500	C₄	1	1	yes	characteristic, complemented
9	C₃×#493	C₄	1	1	yes	characteristic, complemented
10	C₃×#486	C₄	1	2	yes	complemented
11	C₃×#466	—	2	2	no	—
12	C₃×C₄×D₈	—	2	2	no	—
13	C₃×#423	C₂²	1	2	yes	—
14	C₂×C₃×#104	—	2	4	no	—
15	C₂×C₃×#97	—	2	2	no	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄³	C₂²	1	1	yes	characteristic
19	C₃×#334	—	2	2	no	—
20	C₃×D₄⋊C₈	—	2	4	no	—
21	#3169	C₂×C₃	1	1	yes	characteristic
22	#1962	C₂×C₃	1	2	yes	complemented
23	C₄×#97	C₂×C₃	1	1	yes	characteristic, complemented
24	#1566	C₂×C₃	1	1	yes	characteristic, complemented
25	#1492	C₂×C₃	1	2	yes	complemented
26	C₂×C₃×#41	—	2	2	no	—
27	C₂×C₃×D₈	—	2	2	no	—
28	C₂×C₃×M₁₆	—	2	2	no	—
29	C₂²×C₃×C₈	—	2	1	no	—
30	C₂²×C₃×C₈	D₄	1	2	yes	—
31	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₃×#127	C₂×C₄	1	2	yes	—
33	C₃×#124	—	2	2	no	—
34	C₃×#121	—	2	2	no	—
35	C₃×#121	—	2	2	no	—
36	C₃×C₄×D₄	—	2	2	no	—
37	C₃×C₄×D₄	—	2	2	no	—
38	C₂×C₃×C₄²	—	2	1	no	—
39	C₂×C₃×C₄²	—	2	1	no	—
40	C₂×C₃×C₄²	D₄	1	2	yes	—
41	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
42	C₃×#107	—	2	2	no	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄⋊C₈	—	2	2	no	—
45	C₃×#104	—	4	8	no	—
46	C₃×#102	—	2	4	no	—
47	C₃×#97	—	2	8	no	—
48	C₃×C₄×C₈	—	2	1	no	—
49	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄×C₈	D₄	1	2	yes	—
52	C₃×C₄×C₈	D₄	1	2	yes	—
53	#532	C₂²×C₃	1	2	yes	—
54	#500	C₃×C₄	1	1	yes	characteristic, complemented
55	#493	C₃×C₄	1	1	yes	characteristic, complemented
56	#486	C₃×C₄	1	2	yes	complemented
57	#466	—	2	2	no	—
58	C₄×D₈	—	2	2	no	—
59	#423	C₂²×C₃	1	2	yes	—
60	C₂×#104	—	2	4	no	—
61	C₂×#97	—	2	2	no	—
62	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₄³	C₂²×C₃	1	1	yes	characteristic
65	#334	—	2	2	no	—
66	D₄⋊C₈	—	2	4	no	—
67	C₃×#41	—	4	4	no	—
68	C₃×#41	—	4	4	no	—
69	C₂×C₃×Q₈	—	2	2	no	—
70	C₂×C₃×D₄	—	2	2	no	—
71	C₂×C₃×D₄	—	2	2	no	—
72	C₂×C₃×D₄	—	2	2	no	—
73	C₂×C₃×D₄	—	4	2	no	—
74	C₂²×C₃×C₄	—	2	2	no	—
75	C₂²×C₃×C₄	—	4	2	no	—
76	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₃×C₄	#41	1	1	yes	characteristic
79	C₃×D₈	—	4	4	no	—
80	C₃×M₁₆	—	4	4	no	—
81	C₂×C₃×C₈	—	2	4	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	—	4	2	no	—
85	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₂²⋊C₄	—	4	2	no	—
90	C₃×C₂²⋊C₄	—	4	2	no	—
91	C₃×C₄²	—	2	4	no	—
92	C₃×C₄²	—	2	4	no	—
93	C₃×C₄²	—	4	2	no	—
94	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
97	C₃×C₄²	#41	1	1	yes	characteristic
98	C₂×#41	—	2	2	no	—
99	C₂×D₈	—	2	2	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	C₃×D₄	1	2	yes	—
103	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
104	#127	C₂×C₃×C₄	1	2	yes	—
105	#124	—	2	2	no	—
106	#121	—	2	2	no	—
107	#121	—	2	2	no	—
108	C₄×D₄	—	2	2	no	—
109	C₄×D₄	—	2	2	no	—
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	—	2	1	no	—
112	C₂×C₄²	C₃×D₄	1	2	yes	—
113	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
114	#107	—	2	2	no	—
115	C₄⋊C₈	—	2	2	no	—
116	C₄⋊C₈	—	2	2	no	—
117	#104	—	4	8	no	—
118	#102	—	2	4	no	—
119	#97	—	2	8	no	—
120	C₄×C₈	—	2	1	no	—
121	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
122	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
123	C₄×C₈	C₃×D₄	1	2	yes	—
124	C₄×C₈	C₃×D₄	1	2	yes	—
125	C₂³×C₃	—	4	2	no	—
126	C₂³×C₃	#121	1	1	yes	characteristic
127	C₃×Q₈	—	4	4	no	—
128	C₃×D₄	—	4	4	no	—
129	C₃×D₄	—	4	4	no	—
130	C₃×D₄	—	4	4	no	—
131	C₃×D₄	—	8	4	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	4	2	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	—	8	2	no	—
139	C₂×C₃×C₄	—	8	2	no	—
140	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	#121	1	1	yes	characteristic
144	C₂×C₃×C₄	#124	1	1	yes	characteristic
145	C₂×C₃×C₄	#127	1	1	yes	characteristic
146	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
147	C₃×C₈	—	2	4	no	—
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	—	8	2	no	—
150	C₃×C₈	#104	1	4	yes	complemented
151	#41	—	4	4	no	—
152	#41	—	4	4	no	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	C₂×D₄	—	2	2	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂×D₄	—	4	2	no	—
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂²×C₄	C₃×#41	1	1	yes	characteristic
163	D₈	—	4	4	no	—
164	M₁₆	—	4	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
170	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₄²	—	2	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	4	2	no	—
178	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
180	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
181	C₄²	C₃×#41	1	1	yes	characteristic
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	8	2	no	—
185	C₂²×C₃	—	8	2	no	—
186	C₂²×C₃	—	8	2	no	—
187	C₂²×C₃	#390	1	1	yes	characteristic
188	C₂²×C₃	#469	1	2	yes	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	4	4	no	—
192	C₃×C₄	—	8	2	no	—
193	C₃×C₄	C₂×#104	1	2	yes	—
194	C₃×C₄	#442	1	2	yes	—
195	C₃×C₄	#444	1	2	yes	—
196	C₃×C₄	#496	1	2	yes	—
197	C₂³	—	4	2	no	—
198	C₂³	C₃×#121	1	1	yes	characteristic
199	Q₈	—	4	4	no	—
200	D₄	—	4	4	no	—
201	D₄	—	4	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	8	4	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	2	no	—
212	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
214	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
215	C₂×C₄	C₃×#121	1	1	yes	characteristic
216	C₂×C₄	C₃×#124	1	1	yes	characteristic
217	C₂×C₄	C₃×#127	1	1	yes	characteristic
218	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
219	C₈	—	2	4	no	—
220	C₈	—	2	2	no	—
221	C₈	—	8	2	no	—
222	C₈	C₃×#104	1	4	yes	complemented
223	C₂×C₃	—	2	2	no	—
224	C₂×C₃	—	16	2	no	—
225	C₂×C₃	#1962	1	1	yes	characteristic
226	C₂×C₃	#1972	1	1	yes	characteristic
227	C₂×C₃	#1975	1	1	yes	characteristic
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	—	8	2	no	—
233	C₂²	C₃×#390	1	1	yes	characteristic
234	C₂²	C₃×#469	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	4	4	no	—
238	C₄	—	8	2	no	—
239	C₄	C₂×C₃×#104	1	2	yes	—
240	C₄	C₃×#442	1	2	yes	—
241	C₄	C₃×#444	1	2	yes	—
242	C₄	C₃×#496	1	2	yes	—
243	C₃	#9849	1	1	yes	characteristic, direct product factor, Sylow
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	C₃×#1962	1	1	yes	characteristic
247	C₂	C₃×#1972	1	1	yes	characteristic
248	C₂	C₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#390
prime radical	C₃×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#1962	C₂	semidirect
3. C₃×C₄×#97	C₂	semidirect
4. C₃×#1566	C₂	semidirect
5. C₃×#1492	C₂	semidirect
6. #9849	C₃	direct (N × H)
8. C₃×#500	C₄	semidirect
9. C₃×#493	C₄	semidirect
10. C₃×#486	C₄	semidirect
22. #1962	C₂×C₃	semidirect
23. C₄×#97	C₂×C₃	semidirect
24. #1566	C₂×C₃	semidirect
25. #1492	C₂×C₃	semidirect
54. #500	C₃×C₄	semidirect
55. #493	C₃×C₄	semidirect
56. #486	C₃×C₄	semidirect
150. C₃×C₈	#104	semidirect
222. C₈	C₃×#104	semidirect
243. C₃	#9849	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×#3169	2
C₃×#1962	2
C₃×C₄×#97	2
C₃×#1566	2
C₃×#1492	2
#9849	3

Subgroups — C3×#9849 #10628456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9849 #10628456