Subgroups — C₃×#11671 #10630278

Order 768.

378 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	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1331	C₂	1	4	yes	maximal, complemented
5	#11671	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₃×#532	—	2	1	no	—
7	C₂²×C₃×C₁₆	—	2	1	no	—
8	C₃×#500	C₄	1	1	yes	characteristic
9	C₃×#493	C₄	1	1	yes	characteristic
10	C₃×#486	—	2	1	no	—
11	C₃×#466	—	2	1	no	—
12	C₃×#466	C₄	1	2	yes	—
13	C₃×C₄×D₈	—	2	1	no	—
14	C₃×C₄×D₈	C₂²	1	2	yes	—
15	C₃×#423	—	2	1	no	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₁₆	—	2	1	no	—
18	C₃×C₄×C₁₆	C₂²	1	2	yes	—
19	C₃×#335	—	2	1	no	—
20	C₃×#335	C₂²	1	2	yes	—
21	#3169	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	#1567	C₂×C₃	1	1	yes	characteristic, complemented
24	#1331	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×#41	—	4	1	no	—
26	C₂×C₃×D₈	—	2	1	no	—
27	C₂×C₃×D₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×M₁₆	—	4	1	no	—
29	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
31	C₃×C₄⋊Q₈	—	2	1	no	—
32	C₃×#127	—	2	1	no	—
33	C₃×#124	—	4	1	no	—
34	C₃×#121	—	2	2	no	—
35	C₃×#121	—	4	1	no	—
36	C₃×C₄×D₄	—	2	2	no	—
37	C₃×C₄×D₄	—	4	1	no	—
38	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₃×C₁₆	—	2	4	no	—
40	C₂×C₃×C₁₆	—	2	2	no	—
41	C₃×#107	—	2	1	no	—
42	C₃×#107	C₂×C₄	1	2	yes	—
43	C₃×C₄⋊C₈	—	2	2	no	—
44	C₃×C₄⋊C₈	—	4	1	no	—
45	C₃×#102	—	2	2	no	—
46	C₃×#102	—	2	2	no	—
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	D₄	1	1	yes	characteristic
49	C₃×C₄×C₈	D₄	1	1	yes	characteristic
50	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
51	#532	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	#500	C₃×C₄	1	1	yes	characteristic
54	#493	C₃×C₄	1	1	yes	characteristic
55	#486	—	2	1	no	—
56	#466	—	2	1	no	—
57	#466	C₃×C₄	1	2	yes	—
58	C₄×D₈	—	2	1	no	—
59	C₄×D₈	C₂²×C₃	1	2	yes	—
60	#423	—	2	1	no	—
61	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₄×C₁₆	—	2	1	no	—
63	C₄×C₁₆	C₂²×C₃	1	2	yes	—
64	#335	—	2	1	no	—
65	#335	C₂²×C₃	1	2	yes	—
66	C₃×#41	—	8	2	no	—
67	C₃×#41	—	8	2	no	—
68	C₂×C₃×Q₈	—	4	1	no	—
69	C₂×C₃×D₄	—	2	2	no	—
70	C₂×C₃×D₄	—	4	2	no	—
71	C₂×C₃×D₄	—	4	1	no	—
72	C₂×C₃×D₄	—	4	1	no	—
73	C₂²×C₃×C₄	—	4	2	no	—
74	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
76	C₂²×C₃×C₄	SD₁₆	1	1	yes	characteristic
77	C₃×D₈	—	2	4	no	—
78	C₃×D₈	—	4	2	no	—
79	C₃×M₁₆	—	8	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	4	2	no	—
83	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	4	1	no	—
89	C₃×C₂²⋊C₄	—	4	2	no	—
90	C₃×C₂²⋊C₄	—	4	2	no	—
91	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₄²	D₈	1	1	yes	characteristic
93	C₃×C₄²	SD₁₆	1	1	yes	characteristic
94	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
95	C₃×C₁₆	—	2	4	no	—
96	C₂×#41	—	4	1	no	—
97	C₂×D₈	—	2	1	no	—
98	C₂×D₈	C₂×C₃×C₄	1	2	yes	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
102	C₄⋊Q₈	—	2	1	no	—
103	#127	—	2	1	no	—
104	#124	—	4	1	no	—
105	#121	—	2	2	no	—
106	#121	—	4	1	no	—
107	C₄×D₄	—	2	2	no	—
108	C₄×D₄	—	4	1	no	—
109	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
110	C₂×C₁₆	—	2	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	#107	—	2	1	no	—
113	#107	C₂×C₃×C₄	1	2	yes	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	4	1	no	—
116	#102	—	2	2	no	—
117	#102	—	2	2	no	—
118	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
119	C₄×C₈	C₃×D₄	1	1	yes	characteristic
120	C₄×C₈	C₃×D₄	1	1	yes	characteristic
121	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
122	C₂³×C₃	—	4	2	no	—
123	C₂³×C₃	#102	1	1	yes	characteristic
124	C₃×Q₈	—	8	2	no	—
125	C₃×D₄	—	4	2	no	—
126	C₃×D₄	—	4	2	no	—
127	C₃×D₄	—	8	4	no	—
128	C₃×D₄	—	8	2	no	—
129	C₃×D₄	—	8	2	no	—
130	C₂×C₃×C₄	—	2	2	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	4	2	no	—
134	C₂×C₃×C₄	—	8	2	no	—
135	C₂×C₃×C₄	—	8	2	no	—
136	C₂×C₃×C₄	#102	1	1	yes	characteristic
137	C₂×C₃×C₄	#102	1	1	yes	characteristic
138	C₂×C₃×C₄	#102	1	1	yes	characteristic
139	C₂×C₃×C₄	D₁₆	1	2	yes	—
140	C₂×C₃×C₄	SD₃₂	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	8	2	no	—
147	C₃×C₈	#104	1	2	yes	—
148	C₃×C₈	#131	1	2	yes	—
149	#41	—	8	2	no	—
150	#41	—	8	2	no	—
151	C₂×Q₈	—	4	1	no	—
152	C₂×D₄	—	2	2	no	—
153	C₂×D₄	—	4	2	no	—
154	C₂×D₄	—	4	1	no	—
155	C₂×D₄	—	4	1	no	—
156	C₂²×C₄	—	4	2	no	—
157	C₂²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
159	C₂²×C₄	C₃×SD₁₆	1	1	yes	characteristic
160	D₈	—	2	4	no	—
161	D₈	—	4	2	no	—
162	M₁₆	—	8	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
169	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₂²⋊C₄	—	4	2	no	—
174	C₄²	C₃×C₂²⋊C₄	1	1	yes	characteristic
175	C₄²	C₃×D₈	1	1	yes	characteristic
176	C₄²	C₃×SD₁₆	1	1	yes	characteristic
177	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
178	C₁₆	—	2	4	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	8	2	no	—
182	C₂²×C₃	—	8	2	no	—
183	C₂²×C₃	—	8	2	no	—
184	C₂²×C₃	#357	1	2	yes	—
185	C₂²×C₃	C₂×#102	1	1	yes	characteristic
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	8	2	no	—
189	C₃×C₄	#357	1	2	yes	—
190	C₃×C₄	#359	1	2	yes	—
191	C₃×C₄	#408	1	1	yes	characteristic
192	C₃×C₄	C₂×#104	1	1	yes	characteristic
193	C₃×C₄	C₂×D₁₆	1	1	yes	characteristic
194	C₃×C₄	C₂×SD₃₂	1	1	yes	characteristic
195	C₂³	—	4	2	no	—
196	C₂³	C₃×#102	1	1	yes	characteristic
197	Q₈	—	8	2	no	—
198	D₄	—	4	2	no	—
199	D₄	—	4	2	no	—
200	D₄	—	8	4	no	—
201	D₄	—	8	2	no	—
202	D₄	—	8	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	C₃×#102	1	1	yes	characteristic
210	C₂×C₄	C₃×#102	1	1	yes	characteristic
211	C₂×C₄	C₃×#102	1	1	yes	characteristic
212	C₂×C₄	C₃×D₁₆	1	2	yes	—
213	C₂×C₄	C₃×SD₃₂	1	2	yes	—
214	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
216	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	8	2	no	—
220	C₈	C₃×#104	1	2	yes	—
221	C₈	C₃×#131	1	2	yes	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	—	16	2	no	—
224	C₂×C₃	#1492	1	1	yes	characteristic
225	C₂×C₃	C₂×#357	1	1	yes	characteristic
226	C₂×C₃	C₂×#359	1	1	yes	characteristic
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	8	2	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	C₃×#357	1	2	yes	—
233	C₂²	C₂×C₃×#102	1	1	yes	characteristic
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	8	2	no	—
237	C₄	C₃×#357	1	2	yes	—
238	C₄	C₃×#359	1	2	yes	—
239	C₄	C₃×#408	1	1	yes	characteristic
240	C₄	C₂×C₃×#104	1	1	yes	characteristic
241	C₄	C₂×C₃×D₁₆	1	1	yes	characteristic
242	C₄	C₂×C₃×SD₃₂	1	1	yes	characteristic
243	C₃	#11671	1	1	yes	characteristic, direct product factor, Sylow
244	C₂	—	2	2	no	—
245	C₂	—	16	2	no	—
246	C₂	C₃×#1492	1	1	yes	characteristic
247	C₂	C₂×C₃×#357	1	1	yes	characteristic
248	C₂	C₂×C₃×#359	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×#102
prime radical	C₃×#3169	C₂
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
2. C₂×C₃×C₄×C₁₆	C₂	semidirect
3. C₃×#1567	C₂	semidirect
4. C₃×#1331	C₂	semidirect
5. #11671	C₃	direct (N × H)
22. C₂×C₄×C₁₆	C₂×C₃	semidirect
23. #1567	C₂×C₃	semidirect
24. #1331	C₂×C₃	semidirect
243. C₃	#11671	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
C₂×C₃×C₄×C₁₆	2
C₃×#1567	2
C₃×#1331	2
#11671	3

Subgroups — C3×#11671 #10630278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#11671 #10630278