Subgroups — C₃×#13274 #10631881

Order 768.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 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	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#108	C₂	1	1	yes	maximal, characteristic
4	C₃×#1382	C₂	1	4	yes	maximal
5	#13274	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₃×#533	—	2	1	no	—
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₃×#500	C₄	1	1	yes	characteristic
9	C₃×#500	C₄	1	1	yes	characteristic
10	C₃×#499	—	2	1	no	—
11	C₃×#499	C₄	1	2	yes	—
12	C₃×#498	—	2	1	no	—
13	C₂×C₃×#108	C₂²	1	2	yes	—
14	C₃×#427	—	2	1	no	—
15	C₃×#427	C₂²	1	2	yes	—
16	C₃×C₄×M₁₆	—	2	1	no	—
17	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊C₁₆	—	2	1	no	—
19	C₃×C₄⋊C₁₆	—	2	1	no	—
20	C₃×#341	—	2	1	no	—
21	#3160	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
23	C₄×#108	C₂×C₃	1	1	yes	characteristic
24	#1382	C₂×C₃	1	4	yes	—
25	C₂×C₃×M₁₆	—	2	2	no	—
26	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×C₄⋊Q₈	—	2	1	no	—
29	C₃×C₄⋊Q₈	—	2	1	no	—
30	C₃×#125	—	2	2	no	—
31	C₃×C₂²⋊Q₈	—	4	1	no	—
32	C₃×C₂²⋊Q₈	—	4	1	no	—
33	C₃×C₄×Q₈	—	4	1	no	—
34	C₃×C₄×Q₈	—	4	1	no	—
35	C₃×#117	—	2	2	no	—
36	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
37	C₂×C₃×C₁₆	—	2	2	no	—
38	C₂×C₃×C₁₆	—	2	4	no	—
39	C₃×#108	—	2	2	no	—
40	C₃×#108	Q₈	1	4	yes	complemented
41	C₃×#107	—	2	1	no	—
42	C₃×#107	—	2	1	no	—
43	C₃×#107	C₂×C₄	1	2	yes	—
44	C₃×#107	C₂×C₄	1	2	yes	—
45	C₃×#106	—	2	2	no	—
46	C₃×#106	—	2	2	no	—
47	C₃×#97	—	4	1	no	—
48	C₃×C₄×C₈	—	4	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	1	yes	characteristic
52	C₃×C₄×C₈	D₄	1	1	yes	characteristic
53	#533	—	2	1	no	—
54	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
55	#500	C₃×C₄	1	1	yes	characteristic
56	#500	C₃×C₄	1	1	yes	characteristic
57	#499	—	2	1	no	—
58	#499	C₃×C₄	1	2	yes	—
59	#498	—	2	1	no	—
60	C₂×#108	C₂²×C₃	1	2	yes	—
61	#427	—	2	1	no	—
62	#427	C₂²×C₃	1	2	yes	—
63	C₄×M₁₆	—	2	1	no	—
64	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₄⋊C₁₆	—	2	1	no	—
66	C₄⋊C₁₆	—	2	1	no	—
67	#341	—	2	1	no	—
68	C₂×C₃×Q₈	—	4	1	no	—
69	C₂×C₃×Q₈	—	4	1	no	—
70	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
71	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
73	C₃×M₁₆	—	2	4	no	—
74	C₃×M₁₆	—	4	2	no	—
75	C₂×C₃×C₈	—	2	2	no	—
76	C₂×C₃×C₈	—	4	2	no	—
77	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
82	C₂×C₃×C₈	#41	1	2	yes	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	—	2	2	no	—
85	C₃×C₄⋊C₄	—	4	2	no	—
86	C₃×C₄⋊C₄	—	4	2	no	—
87	C₃×C₄⋊C₄	—	4	1	no	—
88	C₃×C₄⋊C₄	—	4	1	no	—
89	C₃×C₂²⋊C₄	—	4	2	no	—
90	C₃×C₄²	—	4	2	no	—
91	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₄²	D₈	1	1	yes	characteristic
94	C₃×C₄²	D₈	1	1	yes	characteristic
95	C₃×C₁₆	—	4	2	no	—
96	C₂×M₁₆	—	2	2	no	—
97	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
99	C₄⋊Q₈	—	2	1	no	—
100	C₄⋊Q₈	—	2	1	no	—
101	#125	—	2	2	no	—
102	C₂²⋊Q₈	—	4	1	no	—
103	C₂²⋊Q₈	—	4	1	no	—
104	C₄×Q₈	—	4	1	no	—
105	C₄×Q₈	—	4	1	no	—
106	#117	—	2	2	no	—
107	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
108	C₂×C₁₆	—	2	2	no	—
109	C₂×C₁₆	—	2	4	no	—
110	#108	—	2	2	no	—
111	#108	C₃×Q₈	1	4	yes	complemented
112	#107	—	2	1	no	—
113	#107	—	2	1	no	—
114	#107	C₂×C₃×C₄	1	2	yes	—
115	#107	C₂×C₃×C₄	1	2	yes	—
116	#106	—	2	2	no	—
117	#106	—	2	2	no	—
118	#97	—	4	1	no	—
119	C₄×C₈	—	4	1	no	—
120	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
121	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
122	C₄×C₈	C₃×D₄	1	1	yes	characteristic
123	C₄×C₈	C₃×D₄	1	1	yes	characteristic
124	C₂³×C₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₃×Q₈	—	8	2	no	—
126	C₃×Q₈	—	8	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	4	1	no	—
129	C₂×C₃×C₄	—	4	2	no	—
130	C₂×C₃×C₄	—	4	2	no	—
131	C₂×C₃×C₄	#102	1	1	yes	characteristic
132	C₂×C₃×C₄	#102	1	1	yes	characteristic
133	C₂×C₃×C₄	#102	1	1	yes	characteristic
134	C₂×C₃×C₄	#102	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²⋊Q₈	1	2	yes	—
137	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
138	C₂×C₃×C₄	#137	1	2	yes	—
139	C₃×C₈	—	2	1	no	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	4	4	no	—
142	C₃×C₈	C₄×Q₈	1	2	yes	—
143	C₃×C₈	#124	1	1	yes	characteristic
144	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
145	C₂×Q₈	—	4	1	no	—
146	C₂×Q₈	—	4	1	no	—
147	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
148	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
150	M₁₆	—	2	4	no	—
151	M₁₆	—	4	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
156	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
157	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
159	C₂×C₈	C₃×#41	1	2	yes	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₄⋊C₄	—	4	1	no	—
165	C₄⋊C₄	—	4	1	no	—
166	C₂²⋊C₄	—	4	2	no	—
167	C₄²	—	4	2	no	—
168	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
169	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₄²	C₃×D₈	1	1	yes	characteristic
171	C₄²	C₃×D₈	1	1	yes	characteristic
172	C₁₆	—	4	2	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	C₂×#102	1	1	yes	characteristic
176	C₂²×C₃	#475	1	2	yes	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	8	2	no	—
181	C₃×C₄	—	8	2	no	—
182	C₃×C₄	#359	1	2	yes	—
183	C₃×C₄	#391	1	1	yes	characteristic
184	C₃×C₄	#419	1	1	yes	characteristic
185	C₃×C₄	#474	1	2	yes	—
186	C₂³	C₂×C₃×SD₁₆	1	1	yes	characteristic
187	Q₈	—	8	2	no	—
188	Q₈	—	8	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	4	1	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	C₃×#102	1	1	yes	characteristic
194	C₂×C₄	C₃×#102	1	1	yes	characteristic
195	C₂×C₄	C₃×#102	1	1	yes	characteristic
196	C₂×C₄	C₃×#102	1	1	yes	characteristic
197	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
199	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
200	C₂×C₄	C₃×#137	1	2	yes	—
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	4	no	—
204	C₈	C₃×C₄×Q₈	1	2	yes	—
205	C₈	C₃×#124	1	1	yes	characteristic
206	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
207	C₂×C₃	—	2	2	no	—
208	C₂×C₃	#1988	1	1	yes	characteristic
209	C₂×C₃	C₂×#359	1	1	yes	characteristic
210	C₂×C₃	#2143	1	1	yes	characteristic
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	C₂×C₃×#102	1	1	yes	characteristic
214	C₂²	C₃×#475	1	2	yes	—
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	2	no	—
220	C₄	C₃×#359	1	2	yes	—
221	C₄	C₃×#391	1	1	yes	characteristic
222	C₄	C₃×#419	1	1	yes	characteristic
223	C₄	C₃×#474	1	2	yes	—
224	C₃	#13274	1	1	yes	characteristic, direct product factor, Sylow
225	C₂	—	2	2	no	—
226	C₂	C₃×#1988	1	1	yes	characteristic
227	C₂	C₂×C₃×#359	1	1	yes	characteristic
228	C₂	C₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×#102
prime radical	C₂³×C₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
5. #13274	C₃	direct (N × H)
40. C₃×#108	Q₈	semidirect
111. #108	C₃×Q₈	semidirect
224. C₃	#13274	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
C₂×C₃×C₄⋊C₁₆	2
C₃×C₄×#108	2
C₃×#1382	2
#13274	3

Subgroups — C3×#13274 #10631881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13274 #10631881