Subgroups — C₃×#13407 #10632014

Order 768.

566 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#357	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#357	C₂	1	2	yes	maximal, complemented
4	C₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	#13407	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂²×C₃×D₈	—	2	1	no	—
8	C₂²×C₃×D₈	C₂²	1	2	yes	—
9	C₂×C₃×#127	—	2	1	no	—
10	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
11	C₃×#493	—	2	4	no	—
12	C₂×C₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×#389	—	2	1	no	—
16	C₃×#357	—	2	4	no	—
17	C₃×#357	—	2	4	no	—
18	C₃×#340	—	2	1	no	—
19	C₂×#493	C₂×C₃	1	1	yes	characteristic
20	C₂×#357	C₂×C₃	1	2	yes	complemented
21	C₂×#357	C₂×C₃	1	2	yes	complemented
22	#1942	C₂×C₃	1	1	yes	characteristic, complemented
23	#1374	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂²×C₃×D₄	—	2	2	no	—
25	C₂²×C₃×D₄	—	4	1	no	—
26	C₂²×C₃×D₄	—	4	1	no	—
27	C₂×C₃×D₈	—	2	4	no	—
28	C₂×C₃×D₈	—	2	2	no	—
29	C₂×C₃×D₈	—	2	2	no	—
30	C₂×C₃×D₈	—	2	2	no	—
31	C₂×C₃×D₈	—	2	2	no	—
32	C₂×C₃×D₈	—	2	2	no	—
33	C₂×C₃×D₈	—	4	2	no	—
34	C₂×C₃×D₈	—	4	2	no	—
35	C₂×C₃×D₈	D₄	1	4	yes	—
36	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
37	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
38	C₃×#127	—	2	4	no	—
39	C₃×#127	—	4	2	no	—
40	C₂×C₃×C₄⋊C₄	—	2	1	no	—
41	C₂×C₃×C₄⋊C₄	—	2	1	no	—
42	C₂×C₃×C₄⋊C₄	—	4	1	no	—
43	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
44	C₂×C₃×C₁₆	—	2	2	no	—
45	C₂×C₃×C₁₆	—	2	2	no	—
46	C₂×C₃×C₁₆	—	2	2	no	—
47	C₃×#107	—	2	1	no	—
48	C₃×#107	—	2	2	no	—
49	C₃×#107	D₄	1	2	yes	complemented
50	C₃×C₄×C₈	—	2	2	no	—
51	C₃×#95	—	4	1	no	—
52	C₂²×D₈	—	2	1	no	—
53	C₂²×D₈	C₂²×C₃	1	2	yes	—
54	C₂×#127	—	2	1	no	—
55	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
56	#493	—	2	4	no	—
57	C₂×#107	C₂²×C₃	1	1	yes	characteristic
58	C₂×#107	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	#389	—	2	1	no	—
61	#357	—	2	4	no	—
62	#357	—	2	4	no	—
63	#340	—	2	1	no	—
64	C₂⁴×C₃	—	4	2	no	—
65	C₂×C₃×D₄	—	2	4	no	—
66	C₂×C₃×D₄	—	4	4	no	—
67	C₂×C₃×D₄	—	4	4	no	—
68	C₂×C₃×D₄	—	4	2	no	—
69	C₂×C₃×D₄	—	4	2	no	—
70	C₂×C₃×D₄	—	4	2	no	—
71	C₂×C₃×D₄	—	4	2	no	—
72	C₂×C₃×D₄	—	4	2	no	—
73	C₂×C₃×D₄	—	8	2	no	—
74	C₂×C₃×D₄	—	8	2	no	—
75	C₂×C₃×D₄	—	8	2	no	—
76	C₂×C₃×D₄	—	8	2	no	—
77	C₂²×C₃×C₄	—	4	1	no	—
78	C₂²×C₃×C₄	—	4	1	no	—
79	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
80	C₂²×C₃×C₄	D₈	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
82	C₃×D₈	—	2	4	no	—
83	C₃×D₈	—	2	4	no	—
84	C₃×D₈	—	4	8	no	—
85	C₃×D₈	—	4	2	no	—
86	C₃×D₈	—	4	2	no	—
87	C₂×C₃×C₈	—	2	1	no	—
88	C₂×C₃×C₈	—	2	1	no	—
89	C₂×C₃×C₈	SD₁₆	1	2	yes	—
90	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₈	#41	1	2	yes	—
93	C₂×C₃×C₈	#41	1	1	yes	characteristic
94	C₃×C₄⋊C₄	—	2	2	no	—
95	C₃×C₄⋊C₄	—	4	1	no	—
96	C₃×C₄⋊C₄	—	4	1	no	—
97	C₃×C₄⋊C₄	—	4	1	no	—
98	C₃×C₄⋊C₄	—	8	2	no	—
99	C₃×C₄²	—	2	1	no	—
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₁₆	—	4	2	no	—
102	C₂²×D₄	—	2	2	no	—
103	C₂²×D₄	—	4	1	no	—
104	C₂²×D₄	—	4	1	no	—
105	C₂×D₈	—	2	4	no	—
106	C₂×D₈	—	2	2	no	—
107	C₂×D₈	—	2	2	no	—
108	C₂×D₈	—	2	2	no	—
109	C₂×D₈	—	2	2	no	—
110	C₂×D₈	—	2	2	no	—
111	C₂×D₈	—	4	2	no	—
112	C₂×D₈	—	4	2	no	—
113	C₂×D₈	C₃×D₄	1	4	yes	—
114	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
115	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
116	#127	—	2	4	no	—
117	#127	—	4	2	no	—
118	C₂×C₄⋊C₄	—	2	1	no	—
119	C₂×C₄⋊C₄	—	2	1	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
122	C₂×C₁₆	—	2	2	no	—
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	#107	—	2	1	no	—
126	#107	—	2	2	no	—
127	#107	C₃×D₄	1	2	yes	complemented
128	C₄×C₈	—	2	2	no	—
129	#95	—	4	1	no	—
130	C₂³×C₃	—	4	4	no	—
131	C₂³×C₃	—	8	2	no	—
132	C₂³×C₃	—	8	2	no	—
133	C₂³×C₃	—	8	2	no	—
134	C₂³×C₃	—	8	2	no	—
135	C₂³×C₃	—	8	2	no	—
136	C₂³×C₃	—	8	2	no	—
137	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
138	C₃×D₄	—	4	4	no	—
139	C₃×D₄	—	4	4	no	—
140	C₃×D₄	—	8	8	no	—
141	C₃×D₄	—	8	8	no	—
142	C₃×D₄	—	8	2	no	—
143	C₃×D₄	—	8	2	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	—	8	1	no	—
150	C₂×C₃×C₄	—	8	1	no	—
151	C₂×C₃×C₄	—	8	1	no	—
152	C₂×C₃×C₄	—	8	1	no	—
153	C₂×C₃×C₄	—	8	1	no	—
154	C₂×C₃×C₄	D₁₆	1	2	yes	—
155	C₂×C₃×C₄	SD₃₂	1	2	yes	—
156	C₂×C₃×C₄	#120	1	1	yes	characteristic
157	C₂×C₃×C₄	#121	1	1	yes	characteristic
158	C₂×C₃×C₄	#124	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
160	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₂×C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₃×C₈	—	2	2	no	—
163	C₃×C₈	#121	1	2	yes	—
164	C₃×C₈	#123	1	2	yes	—
165	C₂⁴	—	4	2	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	2	no	—
172	C₂×D₄	—	4	2	no	—
173	C₂×D₄	—	4	2	no	—
174	C₂×D₄	—	8	2	no	—
175	C₂×D₄	—	8	2	no	—
176	C₂×D₄	—	8	2	no	—
177	C₂×D₄	—	8	2	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
181	C₂²×C₄	C₃×D₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
183	D₈	—	2	4	no	—
184	D₈	—	2	4	no	—
185	D₈	—	4	8	no	—
186	D₈	—	4	2	no	—
187	D₈	—	4	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	C₃×SD₁₆	1	2	yes	—
191	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
193	C₂×C₈	C₃×#41	1	2	yes	—
194	C₂×C₈	C₃×#41	1	1	yes	characteristic
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	8	2	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	2	1	no	—
202	C₁₆	—	4	2	no	—
203	C₂²×C₃	—	8	4	no	—
204	C₂²×C₃	—	8	4	no	—
205	C₂²×C₃	—	8	4	no	—
206	C₂²×C₃	—	16	2	no	—
207	C₂²×C₃	—	16	2	no	—
208	C₂²×C₃	—	16	2	no	—
209	C₂²×C₃	—	16	2	no	—
210	C₂²×C₃	#447	1	1	yes	characteristic
211	C₂²×C₃	#459	1	1	yes	characteristic
212	C₂²×C₃	#486	1	1	yes	characteristic
213	C₂²×C₃	C₂×D₁₆	1	1	yes	characteristic
214	C₂²×C₃	C₂×SD₃₂	1	1	yes	characteristic
215	C₂²×C₃	#509	1	1	yes	characteristic
216	C₂²×C₃	#509	1	1	yes	characteristic
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	8	2	no	—
220	C₃×C₄	—	16	1	no	—
221	C₃×C₄	#394	1	1	yes	characteristic
222	C₃×C₄	#450	1	1	yes	characteristic
223	C₃×C₄	#460	1	1	yes	characteristic
224	C₃×C₄	#490	1	1	yes	characteristic
225	C₂³	—	4	4	no	—
226	C₂³	—	8	2	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	—	8	2	no	—
232	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
233	D₄	—	4	4	no	—
234	D₄	—	4	4	no	—
235	D₄	—	8	8	no	—
236	D₄	—	8	8	no	—
237	D₄	—	8	2	no	—
238	D₄	—	8	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	8	1	no	—
245	C₂×C₄	—	8	1	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	—	8	1	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	C₃×D₁₆	1	2	yes	—
250	C₂×C₄	C₃×SD₃₂	1	2	yes	—
251	C₂×C₄	C₃×#120	1	1	yes	characteristic
252	C₂×C₄	C₃×#121	1	1	yes	characteristic
253	C₂×C₄	C₃×#124	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
255	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₂×C₄	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	C₃×#121	1	2	yes	—
259	C₈	C₃×#123	1	2	yes	—
260	C₂×C₃	—	16	4	no	—
261	C₂×C₃	#2041	1	1	yes	characteristic
262	C₂×C₃	#2184	1	1	yes	characteristic
263	C₂×C₃	#2187	1	1	yes	characteristic
264	C₂×C₃	#2206	1	1	yes	characteristic
265	C₂×C₃	#2207	1	1	yes	characteristic
266	C₂×C₃	#2240	1	1	yes	characteristic
267	C₂×C₃	#2244	1	1	yes	characteristic
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	—	16	2	no	—
272	C₂²	—	16	2	no	—
273	C₂²	—	16	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	C₃×#447	1	1	yes	characteristic
276	C₂²	C₃×#459	1	1	yes	characteristic
277	C₂²	C₃×#486	1	1	yes	characteristic
278	C₂²	C₂×C₃×D₁₆	1	1	yes	characteristic
279	C₂²	C₂×C₃×SD₃₂	1	1	yes	characteristic
280	C₂²	C₃×#509	1	1	yes	characteristic
281	C₂²	C₃×#509	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	8	2	no	—
285	C₄	—	16	1	no	—
286	C₄	C₃×#394	1	1	yes	characteristic
287	C₄	C₃×#450	1	1	yes	characteristic
288	C₄	C₃×#460	1	1	yes	characteristic
289	C₄	C₃×#490	1	1	yes	characteristic
290	C₃	#13407	1	1	yes	characteristic, direct product factor, Sylow
291	C₂	—	16	4	no	—
292	C₂	C₃×#2041	1	1	yes	characteristic
293	C₂	C₃×#2184	1	1	yes	characteristic
294	C₂	C₃×#2187	1	1	yes	characteristic
295	C₂	C₃×#2206	1	1	yes	characteristic
296	C₂	C₃×#2207	1	1	yes	characteristic
297	C₂	C₃×#2240	1	1	yes	characteristic
298	C₂	C₃×#2244	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂×C₃×#493	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ 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×#357	C₂	semidirect
3. C₂×C₃×#357	C₂	semidirect
4. C₃×#1942	C₂	semidirect
5. C₃×#1374	C₂	semidirect
6. #13407	C₃	direct (N × H)
20. C₂×#357	C₂×C₃	semidirect
21. C₂×#357	C₂×C₃	semidirect
22. #1942	C₂×C₃	semidirect
23. #1374	C₂×C₃	semidirect
49. C₃×#107	D₄	semidirect
127. #107	C₃×D₄	semidirect
290. C₃	#13407	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#493	2
C₂×C₃×#357	2
C₃×#1942	2
C₃×#1374	2
#13407	3

Subgroups — C3×#13407 #10632014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13407 #10632014