Subgroups — C₃×#9860 #10628467

Order 768.

498 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3147	C₂	1	1	yes	maximal, characteristic
2	C₃×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#1492	C₂	1	2	yes	maximal, complemented
4	C₃×#1492	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	#9860	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#135	—	2	2	no	—
8	C₃×#532	C₂²	1	2	yes	—
9	C₃×#495	—	2	2	no	—
10	C₃×#494	C₄	1	1	yes	characteristic
11	C₃×#493	C₄	1	1	yes	characteristic
12	C₃×#492	C₄	1	2	yes	—
13	C₃×#423	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#341	—	2	2	no	—
18	C₃×D₄⋊C₈	—	2	4	no	—
19	C₃×D₄⋊C₈	—	2	4	no	—
20	C₃×#322	—	2	2	no	—
21	#3147	C₂×C₃	1	1	yes	characteristic
22	#1837	C₂×C₃	1	1	yes	characteristic, complemented
23	#1492	C₂×C₃	1	2	yes	complemented
24	#1492	C₂×C₃	1	2	yes	complemented
25	C₂×#322	C₂×C₃	1	1	yes	characteristic, complemented
26	C₂×C₃×#41	—	2	2	no	—
27	C₃×#135	—	4	8	no	—
28	C₂×C₃×Q₁₆	—	2	2	no	—
29	C₂×C₃×SD₁₆	—	2	4	no	—
30	C₂×C₃×D₈	—	2	2	no	—
31	C₂×C₃×M₁₆	—	2	2	no	—
32	C₂²×C₃×C₈	—	2	1	no	—
33	C₂²×C₃×C₈	—	2	1	no	—
34	C₂²×C₃×C₈	D₄	1	2	yes	—
35	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₃×#127	C₂×C₄	1	2	yes	—
37	C₃×#124	—	2	2	no	—
38	C₃×#121	—	2	2	no	—
39	C₃×#121	—	2	2	no	—
40	C₃×C₄×D₄	—	2	2	no	—
41	C₃×C₄×D₄	—	2	2	no	—
42	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄⋊C₈	—	2	2	no	—
45	C₃×C₄×C₈	—	2	1	no	—
46	C₃×C₄×C₈	—	2	1	no	—
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×C₈	D₄	1	2	yes	—
50	C₃×C₄×C₈	D₄	1	2	yes	—
51	C₃×C₄×C₈	D₄	1	2	yes	—
52	C₂×#135	—	2	2	no	—
53	#532	C₂²×C₃	1	2	yes	—
54	#495	—	2	2	no	—
55	#494	C₃×C₄	1	1	yes	characteristic
56	#493	C₃×C₄	1	1	yes	characteristic
57	#492	C₃×C₄	1	2	yes	—
58	#423	C₂²×C₃	1	2	yes	—
59	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
62	#341	—	2	2	no	—
63	D₄⋊C₈	—	2	4	no	—
64	D₄⋊C₈	—	2	4	no	—
65	#322	—	2	2	no	—
66	C₃×#41	—	4	8	no	—
67	C₂×C₃×Q₈	—	2	2	no	—
68	C₂×C₃×D₄	—	2	2	no	—
69	C₂×C₃×D₄	—	2	2	no	—
70	C₂×C₃×D₄	—	2	2	no	—
71	C₂×C₃×D₄	—	4	2	no	—
72	C₂²×C₃×C₄	—	4	2	no	—
73	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₃×C₄	#41	1	1	yes	characteristic
75	C₂²×C₃×C₄	#41	1	1	yes	characteristic
76	C₃×Q₁₆	—	4	4	no	—
77	C₃×SD₁₆	—	4	8	no	—
78	C₃×D₈	—	4	4	no	—
79	C₃×M₁₆	—	4	4	no	—
80	C₂×C₃×C₈	—	2	4	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	4	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	—	4	2	no	—
85	C₂×C₃×C₈	—	4	2	no	—
86	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₂²⋊C₄	—	4	2	no	—
91	C₃×C₂²⋊C₄	—	4	2	no	—
92	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
96	C₂×#41	—	2	2	no	—
97	#135	—	4	8	no	—
98	C₂×Q₁₆	—	2	2	no	—
99	C₂×SD₁₆	—	2	4	no	—
100	C₂×D₈	—	2	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	C₃×D₄	1	2	yes	—
105	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
106	#127	C₂×C₃×C₄	1	2	yes	—
107	#124	—	2	2	no	—
108	#121	—	2	2	no	—
109	#121	—	2	2	no	—
110	C₄×D₄	—	2	2	no	—
111	C₄×D₄	—	2	2	no	—
112	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	—	2	1	no	—
117	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
118	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
119	C₄×C₈	C₃×D₄	1	2	yes	—
120	C₄×C₈	C₃×D₄	1	2	yes	—
121	C₄×C₈	C₃×D₄	1	2	yes	—
122	C₂³×C₃	—	4	2	no	—
123	C₂³×C₃	#124	1	1	yes	characteristic
124	C₃×Q₈	—	4	4	no	—
125	C₃×D₄	—	4	4	no	—
126	C₃×D₄	—	4	4	no	—
127	C₃×D₄	—	4	4	no	—
128	C₃×D₄	—	8	4	no	—
129	C₂×C₃×C₄	—	2	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	—	8	4	no	—
133	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
136	C₂×C₃×C₄	#121	1	1	yes	characteristic
137	C₂×C₃×C₄	#121	1	1	yes	characteristic
138	C₂×C₃×C₄	#127	1	1	yes	characteristic
139	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
140	C₃×C₈	—	2	4	no	—
141	C₃×C₈	—	2	2	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	4	2	no	—
144	C₃×C₈	—	8	2	no	—
145	#41	—	4	8	no	—
146	C₂×Q₈	—	2	2	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	—	4	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂²×C₄	C₃×#41	1	1	yes	characteristic
154	C₂²×C₄	C₃×#41	1	1	yes	characteristic
155	Q₁₆	—	4	4	no	—
156	SD₁₆	—	4	8	no	—
157	D₈	—	4	4	no	—
158	M₁₆	—	4	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
166	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	2	no	—
170	C₂²⋊C₄	—	4	2	no	—
171	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
172	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
173	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
174	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	8	2	no	—
178	C₂²×C₃	—	8	2	no	—
179	C₂²×C₃	—	8	2	no	—
180	C₂²×C₃	#390	1	1	yes	characteristic
181	C₂²×C₃	#490	1	2	yes	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	8	2	no	—
185	C₃×C₄	#418	1	2	yes	—
186	C₃×C₄	#443	1	2	yes	—
187	C₃×C₄	#444	1	2	yes	—
188	C₃×C₄	#496	1	2	yes	—
189	C₂³	—	4	2	no	—
190	C₂³	C₃×#124	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	D₄	—	4	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	4	4	no	—
195	D₄	—	8	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	8	4	no	—
200	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
202	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
203	C₂×C₄	C₃×#121	1	1	yes	characteristic
204	C₂×C₄	C₃×#121	1	1	yes	characteristic
205	C₂×C₄	C₃×#127	1	1	yes	characteristic
206	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
207	C₈	—	2	4	no	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	4	2	no	—
211	C₈	—	8	2	no	—
212	C₂×C₃	—	2	2	no	—
213	C₂×C₃	—	16	2	no	—
214	C₂×C₃	#1961	1	1	yes	characteristic
215	C₂×C₃	#1974	1	1	yes	characteristic
216	C₂×C₃	#1975	1	1	yes	characteristic
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₃×#390	1	1	yes	characteristic
223	C₂²	C₃×#490	1	2	yes	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	C₃×#418	1	2	yes	—
228	C₄	C₃×#443	1	2	yes	—
229	C₄	C₃×#444	1	2	yes	—
230	C₄	C₃×#496	1	2	yes	—
231	C₃	#9860	1	1	yes	characteristic, direct product factor, Sylow
232	C₂	—	2	2	no	—
233	C₂	—	16	2	no	—
234	C₂	C₃×#1961	1	1	yes	characteristic
235	C₂	C₃×#1974	1	1	yes	characteristic
236	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×#3147	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×#1837	C₂	semidirect
3. C₃×#1492	C₂	semidirect
4. C₃×#1492	C₂	semidirect
5. C₂×C₃×#322	C₂	semidirect
6. #9860	C₃	direct (N × H)
22. #1837	C₂×C₃	semidirect
23. #1492	C₂×C₃	semidirect
24. #1492	C₂×C₃	semidirect
25. C₂×#322	C₂×C₃	semidirect
231. C₃	#9860	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×#3147	2
C₃×#1837	2
C₃×#1492	2
C₂×C₃×#322	2
#9860	3

Subgroups — C3×#9860 #10628467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9860 #10628467