Subgroups — C₃×#9848 #10628455

Order 768.

422 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃×#1994	C₂	1	2	yes	maximal
3	C₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₃×#1567	C₂	1	1	yes	maximal, characteristic
5	C₃×#1493	C₂	1	2	yes	maximal
6	#9848	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, complemented
9	C₃×#494	C₄	1	1	yes	characteristic, complemented
10	C₃×#487	C₄	1	2	yes	complemented
11	C₃×#467	—	2	2	no	—
12	C₃×C₄×Q₁₆	—	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	C₃×C₄³	C₂²	1	1	yes	characteristic
17	C₃×#339	—	2	4	no	—
18	C₃×#336	—	2	2	no	—
19	C₃×#335	—	2	2	no	—
20	C₃×Q₈⋊C₈	—	2	4	no	—
21	#3170	C₂×C₃	1	1	yes	characteristic
22	#1994	C₂×C₃	1	2	yes	—
23	#1744	C₂×C₃	1	1	yes	characteristic
24	#1567	C₂×C₃	1	1	yes	characteristic
25	#1493	C₂×C₃	1	2	yes	—
26	C₂×C₃×Q₁₆	—	2	2	no	—
27	C₂×C₃×M₁₆	—	2	2	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	—	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×#103	—	2	4	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	#533	C₂²×C₃	1	2	yes	—
52	#500	C₃×C₄	1	1	yes	characteristic, complemented
53	#494	C₃×C₄	1	1	yes	characteristic, complemented
54	#487	C₃×C₄	1	2	yes	complemented
55	#467	—	2	2	no	—
56	C₄×Q₁₆	—	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	C₄³	C₂²×C₃	1	1	yes	characteristic
61	#339	—	2	4	no	—
62	#336	—	2	2	no	—
63	#335	—	2	2	no	—
64	Q₈⋊C₈	—	2	4	no	—
65	C₂×C₃×Q₈	—	2	2	no	—
66	C₂×C₃×Q₈	—	2	2	no	—
67	C₂²×C₃×C₄	—	2	2	no	—
68	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	#41	1	1	yes	characteristic
71	C₃×Q₁₆	—	4	4	no	—
72	C₃×M₁₆	—	4	4	no	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	—	4	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	—	2	2	no	—
87	C₃×C₄²	—	4	2	no	—
88	C₃×C₄²	—	4	2	no	—
89	C₃×C₄²	—	4	2	no	—
90	C₃×C₄²	Q₁₆	1	4	yes	complemented
91	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
92	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
93	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
94	C₃×C₄²	#41	1	1	yes	characteristic
95	C₂×Q₁₆	—	2	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	C₃×D₄	1	2	yes	—
99	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
100	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
101	#125	—	2	2	no	—
102	C₂²⋊Q₈	—	2	2	no	—
103	C₂²⋊Q₈	—	2	2	no	—
104	C₄×Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄²	—	2	1	no	—
108	C₂×C₄²	—	2	1	no	—
109	C₂×C₄²	C₃×D₄	1	2	yes	—
110	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
111	#107	—	2	2	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	2	2	no	—
114	#103	—	2	4	no	—
115	C₄×C₈	—	2	1	no	—
116	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
117	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
118	C₄×C₈	C₃×D₄	1	2	yes	—
119	C₄×C₈	C₃×D₄	1	2	yes	—
120	C₂³×C₃	#121	1	1	yes	characteristic
121	C₃×Q₈	—	4	4	no	—
122	C₃×Q₈	—	4	4	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	4	2	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	—	4	2	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	#121	1	1	yes	characteristic
134	C₂×C₃×C₄	#124	1	1	yes	characteristic
135	C₂×C₃×C₄	#127	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
137	C₂×C₃×C₄	#135	1	2	yes	—
138	C₃×C₈	—	2	2	no	—
139	C₃×C₈	—	2	2	no	—
140	C₃×C₈	—	4	2	no	—
141	C₃×C₈	—	8	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂²×C₄	—	2	2	no	—
145	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₃×#41	1	1	yes	characteristic
148	Q₁₆	—	4	4	no	—
149	M₁₆	—	4	4	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
155	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₂²⋊C₄	—	4	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	—	4	2	no	—
166	C₄²	—	4	2	no	—
167	C₄²	C₃×Q₁₆	1	4	yes	complemented
168	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
170	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
171	C₄²	C₃×#41	1	1	yes	characteristic
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	#390	1	1	yes	characteristic
175	C₂²×C₃	#467	1	2	yes	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	4	4	no	—
179	C₃×C₄	—	8	2	no	—
180	C₃×C₄	—	8	2	no	—
181	C₃×C₄	#421	1	2	yes	—
182	C₃×C₄	C₄×Q₁₆	1	2	yes	—
183	C₃×C₄	#444	1	2	yes	—
184	C₃×C₄	#494	1	1	yes	characteristic
185	C₃×C₄	#495	1	1	yes	characteristic
186	C₂³	C₃×#121	1	1	yes	characteristic
187	Q₈	—	4	4	no	—
188	Q₈	—	4	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
198	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
199	C₂×C₄	C₃×#121	1	1	yes	characteristic
200	C₂×C₄	C₃×#124	1	1	yes	characteristic
201	C₂×C₄	C₃×#127	1	1	yes	characteristic
202	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
203	C₂×C₄	C₃×#135	1	2	yes	—
204	C₈	—	2	2	no	—
205	C₈	—	2	2	no	—
206	C₈	—	4	2	no	—
207	C₈	—	8	2	no	—
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	#1963	1	1	yes	characteristic
210	C₂×C₃	#1969	1	1	yes	characteristic
211	C₂×C₃	#1975	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₃×#390	1	1	yes	characteristic
215	C₂²	C₃×#467	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	4	4	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	C₃×#421	1	2	yes	—
222	C₄	C₃×C₄×Q₁₆	1	2	yes	—
223	C₄	C₃×#444	1	2	yes	—
224	C₄	C₃×#494	1	1	yes	characteristic
225	C₄	C₃×#495	1	1	yes	characteristic
226	C₃	#9848	1	1	yes	characteristic, direct product factor, Sylow
227	C₂	—	2	2	no	—
228	C₂	C₃×#1963	1	1	yes	characteristic
229	C₂	C₃×#1969	1	1	yes	characteristic
230	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	#121
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. #9848	C₃	direct (N × H)
8. C₃×#500	C₄	semidirect
9. C₃×#494	C₄	semidirect
10. C₃×#487	C₄	semidirect
52. #500	C₃×C₄	semidirect
53. #494	C₃×C₄	semidirect
54. #487	C₃×C₄	semidirect
90. C₃×C₄²	Q₁₆	semidirect
167. C₄²	C₃×Q₁₆	semidirect
226. C₃	#9848	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×#3170	2
C₃×#1994	2
C₃×#1744	2
C₃×#1567	2
C₃×#1493	2
#9848	3

Subgroups — C3×#9848 #10628455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9848 #10628455