Subgroups — C₃×#17962 #10636569

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 272 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	C₃×#2931	C₂	1	2	yes	maximal, complemented
3	C₃×#1594	C₂	1	4	yes	maximal, complemented
4	C₃×#1580	C₂	1	4	yes	maximal, complemented
5	C₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1492	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	#17962	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
9	C₂×C₃×#131	—	2	2	no	—
10	C₃×#532	C₂²	1	2	yes	—
11	C₃×#500	C₄	1	1	yes	characteristic
12	C₃×#493	C₄	1	1	yes	characteristic
13	C₃×#486	C₄	1	2	yes	—
14	C₃×#466	C₄	1	4	yes	—
15	C₃×C₄×D₈	C₂²	1	4	yes	—
16	C₃×#436	—	2	2	no	—
17	C₃×#436	C₂²	1	4	yes	—
18	C₃×C₈×D₄	—	2	2	no	—
19	C₃×C₈×D₄	C₂²	1	4	yes	—
20	C₃×#423	C₂²	1	2	yes	—
21	C₃×#408	—	2	2	no	—
22	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₃×#334	C₂²	1	4	yes	—
26	C₃×D₄⋊C₈	C₂²	1	8	yes	—
27	C₃×#322	C₂²	1	2	yes	—
28	C₃×#322	C₂²	1	2	yes	—
29	#3169	C₂×C₃	1	1	yes	characteristic
30	#2931	C₂×C₃	1	2	yes	complemented
31	#1594	C₂×C₃	1	4	yes	complemented
32	#1580	C₂×C₃	1	4	yes	complemented
33	#1566	C₂×C₃	1	1	yes	characteristic, complemented
34	#1492	C₂×C₃	1	2	yes	complemented
35	C₂×#322	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×C₃×#41	—	2	2	no	—
37	C₂×C₃×D₈	C₂×C₄	1	4	yes	—
38	C₃×#131	—	4	8	no	—
39	C₂×C₃×M₁₆	—	2	2	no	—
40	C₂×C₃×M₁₆	—	2	4	no	—
41	C₂²×C₃×C₈	—	2	1	no	—
42	C₂²×C₃×C₈	—	2	4	no	—
43	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₃×C₈	D₄	1	2	yes	—
45	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
46	C₃×#127	C₂×C₄	1	2	yes	—
47	C₃×#124	—	2	2	no	—
48	C₃×#121	—	2	2	no	—
49	C₃×#121	C₂×C₄	1	4	yes	—
50	C₃×C₄×D₄	—	2	2	no	—
51	C₃×C₄×D₄	C₂³	1	4	yes	—
52	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₃×#107	C₂×C₄	1	4	yes	—
54	C₃×C₄⋊C₈	—	2	2	no	—
55	C₃×C₄⋊C₈	C₂³	1	4	yes	—
56	C₃×#102	C₂×C₄	1	8	yes	—
57	C₃×C₂²⋊C₈	—	2	4	no	—
58	C₃×C₂²⋊C₈	—	2	4	no	—
59	C₃×C₄×C₈	—	2	1	no	—
60	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
61	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
62	C₃×C₄×C₈	D₄	1	2	yes	—
63	C₃×C₄×C₈	C₂³	1	2	yes	—
64	C₃×C₄×C₈	C₂³	1	2	yes	—
65	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
67	C₂×#131	—	2	2	no	—
68	#532	C₂²×C₃	1	2	yes	—
69	#500	C₃×C₄	1	1	yes	characteristic
70	#493	C₃×C₄	1	1	yes	characteristic
71	#486	C₃×C₄	1	2	yes	—
72	#466	C₃×C₄	1	4	yes	—
73	C₄×D₈	C₂²×C₃	1	4	yes	—
74	#436	—	2	2	no	—
75	#436	C₂²×C₃	1	4	yes	—
76	C₈×D₄	—	2	2	no	—
77	C₈×D₄	C₂²×C₃	1	4	yes	—
78	#423	C₂²×C₃	1	2	yes	—
79	#408	—	2	2	no	—
80	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
83	#334	C₂²×C₃	1	4	yes	—
84	D₄⋊C₈	C₂²×C₃	1	8	yes	—
85	#322	C₂²×C₃	1	2	yes	—
86	#322	C₂²×C₃	1	2	yes	—
87	C₃×#41	—	4	4	no	—
88	C₃×#41	—	4	4	no	—
89	C₂×C₃×Q₈	—	2	2	no	—
90	C₂×C₃×D₄	—	2	4	no	—
91	C₂×C₃×D₄	—	2	2	no	—
92	C₂×C₃×D₄	—	2	2	no	—
93	C₂×C₃×D₄	C₂²×C₄	1	4	yes	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	#41	1	1	yes	characteristic
98	C₃×D₈	—	2	8	no	—
99	C₃×M₁₆	—	4	8	no	—
100	C₃×M₁₆	—	4	4	no	—
101	C₂×C₃×C₈	—	2	4	no	—
102	C₂×C₃×C₈	—	2	4	no	—
103	C₂×C₃×C₈	—	2	4	no	—
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	4	8	no	—
106	C₂×C₃×C₈	—	4	2	no	—
107	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
113	C₃×C₂²⋊C₄	—	2	4	no	—
114	C₃×C₂²⋊C₄	—	2	4	no	—
115	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
116	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
117	C₃×C₄²	#41	1	1	yes	characteristic
118	C₃×C₄²	C₂⁴	1	1	yes	characteristic
119	C₂×#41	—	2	2	no	—
120	C₂×D₈	C₂×C₃×C₄	1	4	yes	—
121	#131	—	4	8	no	—
122	C₂×M₁₆	—	2	2	no	—
123	C₂×M₁₆	—	2	4	no	—
124	C₂²×C₈	—	2	1	no	—
125	C₂²×C₈	—	2	4	no	—
126	C₂²×C₈	C₂×C₃×C₄	1	2	yes	—
127	C₂²×C₈	C₃×D₄	1	2	yes	—
128	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
129	#127	C₂×C₃×C₄	1	2	yes	—
130	#124	—	2	2	no	—
131	#121	—	2	2	no	—
132	#121	C₂×C₃×C₄	1	4	yes	—
133	C₄×D₄	—	2	2	no	—
134	C₄×D₄	C₂³×C₃	1	4	yes	—
135	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
136	#107	C₂×C₃×C₄	1	4	yes	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	C₂³×C₃	1	4	yes	—
139	#102	C₂×C₃×C₄	1	8	yes	—
140	C₂²⋊C₈	—	2	4	no	—
141	C₂²⋊C₈	—	2	4	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
144	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
145	C₄×C₈	C₃×D₄	1	2	yes	—
146	C₄×C₈	C₂³×C₃	1	2	yes	—
147	C₄×C₈	C₂³×C₃	1	2	yes	—
148	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
149	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
150	C₂³×C₃	—	2	4	no	—
151	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
152	C₃×Q₈	—	4	4	no	—
153	C₃×D₄	—	2	8	no	—
154	C₃×D₄	—	4	8	no	—
155	C₃×D₄	—	4	4	no	—
156	C₃×D₄	—	4	4	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	4	no	—
160	C₂×C₃×C₄	—	4	4	no	—
161	C₂×C₃×C₄	—	4	4	no	—
162	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄	AGL(1,ℤ/8)	1	4	yes	—
166	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
169	C₃×C₈	—	2	2	no	—
170	C₃×C₈	—	2	2	no	—
171	C₃×C₈	—	2	2	no	—
172	C₃×C₈	—	4	2	no	—
173	C₃×C₈	—	4	4	no	—
174	C₃×C₈	#131	1	4	yes	complemented
175	#41	—	4	4	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×D₄	—	2	4	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	C₂²×C₃×C₄	1	4	yes	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
185	C₂²×C₄	C₃×#41	1	1	yes	characteristic
186	D₈	—	2	8	no	—
187	M₁₆	—	4	8	no	—
188	M₁₆	—	4	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	8	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
196	C₂×C₈	C₂²×C₃×C₄	1	2	yes	—
197	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
198	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	C₂²×C₃×C₄	1	4	yes	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
204	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
205	C₄²	C₃×#41	1	1	yes	characteristic
206	C₄²	C₂⁴×C₃	1	1	yes	characteristic
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	4	4	no	—
210	C₂²×C₃	—	4	4	no	—
211	C₂²×C₃	—	4	4	no	—
212	C₂²×C₃	#442	1	2	yes	—
213	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	—	4	4	no	—
217	C₃×C₄	#442	1	2	yes	—
218	C₃×C₄	#444	1	2	yes	—
219	C₃×C₄	C₂×#131	1	2	yes	—
220	C₃×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
221	C₂³	—	2	4	no	—
222	C₂³	C₃×C₄×D₄	1	1	yes	characteristic
223	Q₈	—	4	4	no	—
224	D₄	—	2	8	no	—
225	D₄	—	4	8	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
234	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
235	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
236	C₂×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
237	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
238	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	4	4	no	—
245	C₈	C₃×#131	1	4	yes	complemented
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	8	4	no	—
248	C₂×C₃	#2931	1	1	yes	characteristic
249	C₂×C₃	C₂×#442	1	1	yes	characteristic
250	C₂×C₃	C₂×#444	1	1	yes	characteristic
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₃×#442	1	2	yes	—
257	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	4	no	—
261	C₄	C₃×#442	1	2	yes	—
262	C₄	C₃×#444	1	2	yes	—
263	C₄	C₂×C₃×#131	1	2	yes	—
264	C₄	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
265	C₃	#17962	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	2	2	no	—
267	C₂	—	8	4	no	—
268	C₂	C₃×#2931	1	1	yes	characteristic
269	C₂	C₂×C₃×#442	1	1	yes	characteristic
270	C₂	C₂×C₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₄²	C₂⁴×C₃
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#2931	C₂	semidirect
3. C₃×#1594	C₂	semidirect
4. C₃×#1580	C₂	semidirect
5. C₃×#1566	C₂	semidirect
6. C₃×#1492	C₂	semidirect
7. C₂×C₃×#322	C₂	semidirect
8. #17962	C₃	direct (N × H)
30. #2931	C₂×C₃	semidirect
31. #1594	C₂×C₃	semidirect
32. #1580	C₂×C₃	semidirect
33. #1566	C₂×C₃	semidirect
34. #1492	C₂×C₃	semidirect
35. C₂×#322	C₂×C₃	semidirect
174. C₃×C₈	#131	semidirect
245. C₈	C₃×#131	semidirect
265. C₃	#17962	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
C₃×#2931	2
C₃×#1594	2
C₃×#1580	2
C₃×#1566	2
C₃×#1492	2
C₂×C₃×#322	2
#17962	3

Subgroups — C3×#17962 #10636569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#17962 #10636569