Subgroups — C₃×#9859 #10628466

Order 768.

390 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₃×#1837	C₂	1	1	yes	maximal, characteristic
3	C₃×#1493	C₂	1	2	yes	maximal
4	C₃×#1493	C₂	1	2	yes	maximal
5	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic
6	#9859	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₃×#533	C₂²	1	2	yes	—
8	C₃×#500	C₄	1	1	yes	characteristic
9	C₃×#500	C₄	1	1	yes	characteristic
10	C₃×#499	—	2	2	no	—
11	C₃×#498	C₄	1	2	yes	—
12	C₃×#427	—	2	2	no	—
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×Q₈⋊C₈	—	2	4	no	—
19	C₃×Q₈⋊C₈	—	2	4	no	—
20	C₃×#322	—	2	2	no	—
21	#3160	C₂×C₃	1	1	yes	characteristic
22	#1837	C₂×C₃	1	1	yes	characteristic
23	#1493	C₂×C₃	1	2	yes	—
24	#1493	C₂×C₃	1	2	yes	—
25	C₂×#322	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×M₁₆	—	2	2	no	—
27	C₂²×C₃×C₈	—	2	1	no	—
28	C₂²×C₃×C₈	—	2	1	no	—
29	C₂²×C₃×C₈	D₄	1	2	yes	—
30	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₃×#125	—	2	2	no	—
33	C₃×C₂²⋊Q₈	—	2	2	no	—
34	C₃×C₂²⋊Q₈	—	2	2	no	—
35	C₃×C₄×Q₈	—	2	2	no	—
36	C₃×C₄×Q₈	—	2	2	no	—
37	C₃×#117	—	2	2	no	—
38	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
39	C₃×#107	—	2	2	no	—
40	C₃×#107	—	2	2	no	—
41	C₃×#106	—	2	4	no	—
42	C₃×C₄⋊C₈	—	2	2	no	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄×C₈	—	2	1	no	—
45	C₃×C₄×C₈	—	2	1	no	—
46	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	D₄	1	2	yes	—
49	C₃×C₄×C₈	D₄	1	2	yes	—
50	C₃×C₄×C₈	D₄	1	2	yes	—
51	#533	C₂²×C₃	1	2	yes	—
52	#500	C₃×C₄	1	1	yes	characteristic
53	#500	C₃×C₄	1	1	yes	characteristic
54	#499	—	2	2	no	—
55	#498	C₃×C₄	1	2	yes	—
56	#427	—	2	2	no	—
57	#423	C₂²×C₃	1	2	yes	—
58	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	#341	—	2	2	no	—
62	Q₈⋊C₈	—	2	4	no	—
63	Q₈⋊C₈	—	2	4	no	—
64	#322	—	2	2	no	—
65	C₂×C₃×Q₈	—	2	2	no	—
66	C₂×C₃×Q₈	—	2	2	no	—
67	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₃×C₄	#41	1	1	yes	characteristic
69	C₂²×C₃×C₄	#41	1	1	yes	characteristic
70	C₃×M₁₆	—	4	4	no	—
71	C₂×C₃×C₈	—	2	4	no	—
72	C₂×C₃×C₈	—	2	2	no	—
73	C₂×C₃×C₈	—	2	4	no	—
74	C₂×C₃×C₈	—	2	2	no	—
75	C₂×C₃×C₈	—	4	2	no	—
76	C₂×C₃×C₈	—	4	2	no	—
77	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₄⋊C₄	—	2	2	no	—
81	C₃×C₄⋊C₄	—	2	2	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	—	4	2	no	—
84	C₃×C₄⋊C₄	—	4	2	no	—
85	C₃×C₂²⋊C₄	—	4	2	no	—
86	C₃×C₄²	—	4	2	no	—
87	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
89	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
91	C₂×M₁₆	—	2	2	no	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	C₃×D₄	1	2	yes	—
95	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
96	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
97	#125	—	2	2	no	—
98	C₂²⋊Q₈	—	2	2	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	C₄×Q₈	—	2	2	no	—
102	#117	—	2	2	no	—
103	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
104	#107	—	2	2	no	—
105	#107	—	2	2	no	—
106	#106	—	2	4	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	2	2	no	—
109	C₄×C₈	—	2	1	no	—
110	C₄×C₈	—	2	1	no	—
111	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
112	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
113	C₄×C₈	C₃×D₄	1	2	yes	—
114	C₄×C₈	C₃×D₄	1	2	yes	—
115	C₄×C₈	C₃×D₄	1	2	yes	—
116	C₂³×C₃	#124	1	1	yes	characteristic
117	C₃×Q₈	—	4	4	no	—
118	C₃×Q₈	—	4	4	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	4	2	no	—
122	C₂×C₃×C₄	—	4	2	no	—
123	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
126	C₂×C₃×C₄	#121	1	1	yes	characteristic
127	C₂×C₃×C₄	#121	1	1	yes	characteristic
128	C₂×C₃×C₄	#127	1	1	yes	characteristic
129	C₂×C₃×C₄	#137	1	4	yes	—
130	C₃×C₈	—	2	4	no	—
131	C₃×C₈	—	2	2	no	—
132	C₃×C₈	—	2	2	no	—
133	C₃×C₈	—	4	2	no	—
134	C₃×C₈	—	8	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	—	2	2	no	—
137	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂²×C₄	C₃×#41	1	1	yes	characteristic
139	C₂²×C₄	C₃×#41	1	1	yes	characteristic
140	M₁₆	—	4	4	no	—
141	C₂×C₈	—	2	4	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	—	4	2	no	—
147	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
148	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄⋊C₄	—	4	2	no	—
155	C₂²⋊C₄	—	4	2	no	—
156	C₄²	—	4	2	no	—
157	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
159	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
160	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	#390	1	1	yes	characteristic
164	C₂²×C₃	#491	1	2	yes	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	8	2	no	—
168	C₃×C₄	—	8	2	no	—
169	C₃×C₄	#419	1	2	yes	—
170	C₃×C₄	#443	1	2	yes	—
171	C₃×C₄	#444	1	2	yes	—
172	C₃×C₄	#497	1	2	yes	—
173	C₂³	C₃×#124	1	1	yes	characteristic
174	Q₈	—	4	4	no	—
175	Q₈	—	4	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
182	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
183	C₂×C₄	C₃×#121	1	1	yes	characteristic
184	C₂×C₄	C₃×#121	1	1	yes	characteristic
185	C₂×C₄	C₃×#127	1	1	yes	characteristic
186	C₂×C₄	C₃×#137	1	4	yes	—
187	C₈	—	2	4	no	—
188	C₈	—	2	2	no	—
189	C₈	—	2	2	no	—
190	C₈	—	4	2	no	—
191	C₈	—	8	2	no	—
192	C₂×C₃	—	2	2	no	—
193	C₂×C₃	#1960	1	1	yes	characteristic
194	C₂×C₃	#1974	1	1	yes	characteristic
195	C₂×C₃	#1975	1	1	yes	characteristic
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	C₃×#390	1	1	yes	characteristic
199	C₂²	C₃×#491	1	2	yes	—
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	2	no	—
203	C₄	—	8	2	no	—
204	C₄	C₃×#419	1	2	yes	—
205	C₄	C₃×#443	1	2	yes	—
206	C₄	C₃×#444	1	2	yes	—
207	C₄	C₃×#497	1	2	yes	—
208	C₃	#9859	1	1	yes	characteristic, direct product factor, Sylow
209	C₂	—	2	2	no	—
210	C₂	C₃×#1960	1	1	yes	characteristic
211	C₂	C₃×#1974	1	1	yes	characteristic
212	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	C₂³×C₃	#124
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
6. #9859	C₃	direct (N × H)
208. C₃	#9859	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×#3160	2
C₃×#1837	2
C₃×#1493	2
C₂×C₃×#322	2
#9859	3

Subgroups — C3×#9859 #10628466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9859 #10628466