Subgroups — C₃×#12983 #10631590

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#2929	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#350	C₂	1	2	yes	maximal, complemented
4	C₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1379	C₂	1	2	yes	maximal, complemented
6	#12983	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#131	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₃×#525	C₄	1	1	yes	characteristic
10	C₂×C₃×M₃₂	C₂²	1	2	yes	—
11	C₂²×C₃×C₁₆	—	2	2	no	—
12	C₂²×C₃×C₁₆	—	2	4	no	—
13	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
14	C₃×#435	—	2	2	no	—
15	C₃×C₈×D₄	—	2	2	no	—
16	C₃×#424	C₄	1	1	yes	characteristic
17	C₃×#408	C₄	1	1	yes	characteristic
18	C₃×#407	C₄	1	1	yes	characteristic
19	C₃×#405	C₂²	1	1	yes	characteristic
20	C₃×#350	—	2	4	no	—
21	C₃×C₂²⋊C₁₆	—	2	2	no	—
22	C₂³×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	#2929	C₂×C₃	1	1	yes	characteristic
24	C₂×#350	C₂×C₃	1	2	yes	complemented
25	#2112	C₂×C₃	1	1	yes	characteristic, complemented
26	#1379	C₂×C₃	1	2	yes	complemented
27	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
29	C₃×#131	—	2	2	no	—
30	C₃×#131	—	2	2	no	—
31	C₂×C₃×M₁₆	—	2	1	no	—
32	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₈	—	2	2	no	—
35	C₂²×C₃×C₈	—	2	2	no	—
36	C₂²×C₃×C₈	—	2	1	no	—
37	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₈	D₄	1	2	yes	—
39	C₂²×C₃×C₈	D₄	1	2	yes	—
40	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
41	C₃×#123	—	2	1	no	—
42	C₃×C₂²⋊Q₈	C₈	1	2	yes	—
43	C₃×#121	C₈	1	2	yes	—
44	C₃×#120	—	2	1	no	—
45	C₃×C₄×D₄	—	2	2	no	—
46	C₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₃×M₃₂	—	2	2	no	—
48	C₃×M₃₂	—	2	2	no	—
49	C₂×C₃×C₁₆	—	2	8	no	—
50	C₂×C₃×C₁₆	—	2	2	no	—
51	C₂×C₃×C₁₆	—	2	2	no	—
52	C₂×C₃×C₁₆	—	2	2	no	—
53	C₂×C₃×C₁₆	D₄	1	4	yes	complemented
54	C₃×C₄⋊C₈	—	2	2	no	—
55	C₃×C₂²⋊C₈	—	2	2	no	—
56	C₃×C₂²⋊C₈	—	2	2	no	—
57	C₃×#97	—	2	1	no	—
58	C₃×C₄×C₈	—	2	1	no	—
59	C₂×#131	C₂²×C₃	1	1	yes	characteristic
60	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
61	#525	C₃×C₄	1	1	yes	characteristic
62	C₂×M₃₂	C₂²×C₃	1	2	yes	—
63	C₂²×C₁₆	—	2	2	no	—
64	C₂²×C₁₆	—	2	4	no	—
65	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
66	#435	—	2	2	no	—
67	C₈×D₄	—	2	2	no	—
68	#424	C₃×C₄	1	1	yes	characteristic
69	#408	C₃×C₄	1	1	yes	characteristic
70	#407	C₃×C₄	1	1	yes	characteristic
71	#405	C₂²×C₃	1	1	yes	characteristic
72	#350	—	2	4	no	—
73	C₂²⋊C₁₆	—	2	2	no	—
74	C₂⁴×C₃	M₁₆	1	1	yes	characteristic
75	C₃×#41	—	2	2	no	—
76	C₃×#41	—	2	2	no	—
77	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×D₄	—	2	2	no	—
79	C₂×C₃×D₄	—	2	1	no	—
80	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	2	2	no	—
82	C₂²×C₃×C₄	—	2	2	no	—
83	C₂²×C₃×C₄	—	2	1	no	—
84	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
86	C₂²×C₃×C₄	M₁₆	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
88	C₃×M₁₆	—	2	2	no	—
89	C₃×M₁₆	—	2	2	no	—
90	C₃×M₁₆	—	4	1	no	—
91	C₃×M₁₆	—	4	1	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	4	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	1	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	4	2	no	—
99	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
100	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
101	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	#41	1	2	yes	—
104	C₃×C₄⋊C₄	—	2	2	no	—
105	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
106	C₃×C₂²⋊C₄	—	2	2	no	—
107	C₃×C₂²⋊C₄	—	2	1	no	—
108	C₃×C₂²⋊C₄	—	2	2	no	—
109	C₃×C₄²	—	2	1	no	—
110	C₃×C₁₆	—	2	4	no	—
111	C₃×C₁₆	—	4	2	no	—
112	C₂×#41	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
114	#131	—	2	2	no	—
115	#131	—	2	2	no	—
116	C₂×M₁₆	—	2	1	no	—
117	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂²×C₈	—	2	2	no	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂²×C₈	C₃×D₄	1	2	yes	—
124	C₂²×C₈	C₃×D₄	1	2	yes	—
125	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
126	#123	—	2	1	no	—
127	C₂²⋊Q₈	C₃×C₈	1	2	yes	—
128	#121	C₃×C₈	1	2	yes	—
129	#120	—	2	1	no	—
130	C₄×D₄	—	2	2	no	—
131	#117	C₂×C₃×C₄	1	1	yes	characteristic
132	M₃₂	—	2	2	no	—
133	M₃₂	—	2	2	no	—
134	C₂×C₁₆	—	2	8	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	C₃×D₄	1	4	yes	complemented
139	C₄⋊C₈	—	2	2	no	—
140	C₂²⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	#97	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₂³×C₃	—	2	4	no	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	C₂²⋊C₈	1	2	yes	—
148	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
149	C₃×Q₈	—	2	2	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	—	4	4	no	—
152	C₃×D₄	—	4	2	no	—
153	C₂×C₃×C₄	—	2	2	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	4	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
162	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
164	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
165	C₂×C₃×C₄	#131	1	2	yes	—
166	C₃×C₈	—	2	2	no	—
167	C₃×C₈	—	4	1	no	—
168	C₃×C₈	—	4	1	no	—
169	C₃×C₈	#120	1	1	yes	characteristic
170	C₃×C₈	#121	1	2	yes	—
171	C₃×C₈	#123	1	1	yes	characteristic
172	C₂⁴	C₃×M₁₆	1	1	yes	characteristic
173	#41	—	2	2	no	—
174	#41	—	2	2	no	—
175	C₂×Q₈	C₂×C₃×C₈	1	1	yes	characteristic
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	C₂×C₃×C₈	1	1	yes	characteristic
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
183	C₂²×C₄	C₂×C₃×C₈	1	2	yes	—
184	C₂²×C₄	C₃×M₁₆	1	1	yes	characteristic
185	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	2	no	—
188	M₁₆	—	4	1	no	—
189	M₁₆	—	4	1	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
198	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
199	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
200	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
201	C₂×C₈	C₃×#41	1	2	yes	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	C₂×C₃×C₈	1	2	yes	—
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₄²	—	2	1	no	—
208	C₁₆	—	2	4	no	—
209	C₁₆	—	4	2	no	—
210	C₂²×C₃	—	2	8	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	4	2	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	#350	1	4	yes	complemented
217	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
218	C₂²×C₃	#435	1	2	yes	—
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	4	2	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	#386	1	1	yes	characteristic
223	C₃×C₄	#408	1	1	yes	characteristic
224	C₃×C₄	C₈×D₄	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	1	no	—
228	C₂³	C₃×C₂²⋊C₈	1	2	yes	—
229	C₂³	C₂×C₃×M₁₆	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	4	2	no	—
233	D₄	—	4	4	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	C₃×C₂²⋊C₈	1	2	yes	—
243	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
245	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
246	C₂×C₄	C₃×#131	1	2	yes	—
247	C₈	—	2	2	no	—
248	C₈	—	4	1	no	—
249	C₈	—	4	1	no	—
250	C₈	C₃×#120	1	1	yes	characteristic
251	C₈	C₃×#121	1	2	yes	—
252	C₈	C₃×#123	1	1	yes	characteristic
253	C₂×C₃	—	2	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	—	8	1	no	—
256	C₂×C₃	#1869	1	1	yes	characteristic
257	C₂×C₃	C₂×#350	1	2	yes	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	1	no	—
264	C₂²	C₃×#350	1	4	yes	complemented
265	C₂²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
266	C₂²	C₃×#435	1	2	yes	—
267	C₄	—	2	2	no	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	C₃×#386	1	1	yes	characteristic
271	C₄	C₃×#408	1	1	yes	characteristic
272	C₄	C₃×C₈×D₄	1	2	yes	—
273	C₃	#12983	1	1	yes	characteristic, direct product factor, Sylow
274	C₂	—	2	2	no	—
275	C₂	—	2	4	no	—
276	C₂	—	8	1	no	—
277	C₂	C₃×#1869	1	1	yes	characteristic
278	C₂	C₂×C₃×#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₃×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂³×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₁₆	C₂	semidirect
3. C₂×C₃×#350	C₂	semidirect
4. C₃×#2112	C₂	semidirect
5. C₃×#1379	C₂	semidirect
6. #12983	C₃	direct (N × H)
22. C₂³×C₁₆	C₂×C₃	semidirect
24. C₂×#350	C₂×C₃	semidirect
25. #2112	C₂×C₃	semidirect
26. #1379	C₂×C₃	semidirect
53. C₂×C₃×C₁₆	D₄	semidirect
138. C₂×C₁₆	C₃×D₄	semidirect
216. C₂²×C₃	#350	semidirect
264. C₂²	C₃×#350	semidirect
273. C₃	#12983	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₁₆	2
C₃×#2929	2
C₂×C₃×#350	2
C₃×#2112	2
C₃×#1379	2
#12983	3

Subgroups — C3×#12983 #10631590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12983 #10631590