Subgroups — C₃×#12982 #10631589

Order 768.

474 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#2931	C₂	1	1	yes	maximal, characteristic
3	C₃×#2117	C₂	1	2	yes	maximal, complemented
4	C₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1376	C₂	1	2	yes	maximal, complemented
6	#12982	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×#131	C₂²	1	1	yes	characteristic
8	C₂×C₃×#131	C₂²	1	1	yes	characteristic
9	C₃×#532	C₄	1	1	yes	characteristic
10	C₃×#504	—	2	4	no	—
11	C₂×C₃×M₃₂	—	2	2	no	—
12	C₂×C₃×M₃₂	C₂²	1	2	yes	—
13	C₂×C₃×M₃₂	C₂²	1	2	yes	—
14	C₃×#436	—	2	2	no	—
15	C₃×C₈×D₄	—	2	2	no	—
16	C₃×#423	C₄	1	1	yes	characteristic
17	C₃×#408	C₄	1	1	yes	characteristic
18	C₃×#408	C₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₃×#350	—	2	4	no	—
21	C₃×#349	—	2	2	no	—
22	#3407	C₂×C₃	1	1	yes	characteristic, complemented
23	#2931	C₂×C₃	1	1	yes	characteristic
24	#2117	C₂×C₃	1	2	yes	complemented
25	#2115	C₂×C₃	1	1	yes	characteristic, complemented
26	#1376	C₂×C₃	1	2	yes	complemented
27	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×#41	C₂×C₄	1	1	yes	characteristic
29	C₃×#131	—	2	2	no	—
30	C₃×#131	—	2	2	no	—
31	C₃×#131	—	2	2	no	—
32	C₃×#131	—	2	2	no	—
33	C₂×C₃×M₁₆	—	2	1	no	—
34	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×M₁₆	D₄	1	2	yes	—
37	C₂²×C₃×C₈	—	2	1	no	—
38	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₈	D₄	1	2	yes	—
40	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
42	C₃×#127	C₈	1	1	yes	characteristic
43	C₃×#124	C₈	1	1	yes	characteristic
44	C₃×#124	C₈	1	1	yes	characteristic
45	C₃×#121	—	2	2	no	—
46	C₃×C₄×D₄	—	2	2	no	—
47	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
48	C₃×M₃₂	—	2	4	no	—
49	C₃×M₃₂	—	2	2	no	—
50	C₃×M₃₂	—	2	2	no	—
51	C₃×M₃₂	—	2	2	no	—
52	C₃×M₃₂	D₄	1	4	yes	complemented
53	C₂×C₃×C₁₆	—	2	4	no	—
54	C₂×C₃×C₁₆	—	2	2	no	—
55	C₂×C₃×C₁₆	—	2	2	no	—
56	C₃×C₄⋊C₈	—	2	2	no	—
57	C₃×C₂²⋊C₈	—	2	2	no	—
58	C₃×C₂²⋊C₈	—	2	2	no	—
59	C₃×C₄×C₈	—	2	1	no	—
60	C₃×C₄×C₈	—	2	1	no	—
61	C₂×#131	C₂²×C₃	1	1	yes	characteristic
62	C₂×#131	C₂²×C₃	1	1	yes	characteristic
63	#532	C₃×C₄	1	1	yes	characteristic
64	#504	—	2	4	no	—
65	C₂×M₃₂	—	2	2	no	—
66	C₂×M₃₂	C₂²×C₃	1	2	yes	—
67	C₂×M₃₂	C₂²×C₃	1	2	yes	—
68	#436	—	2	2	no	—
69	C₈×D₄	—	2	2	no	—
70	#423	C₃×C₄	1	1	yes	characteristic
71	#408	C₃×C₄	1	1	yes	characteristic
72	#408	C₃×C₄	1	1	yes	characteristic
73	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	#350	—	2	4	no	—
75	#349	—	2	2	no	—
76	C₃×#41	—	2	2	no	—
77	C₃×#41	—	2	2	no	—
78	C₃×#41	—	2	2	no	—
79	C₃×#41	—	2	2	no	—
80	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
81	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
82	C₂×C₃×D₄	—	2	2	no	—
83	C₂×C₃×D₄	—	2	1	no	—
84	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
86	C₂×C₃×D₄	M₁₆	1	2	yes	—
87	C₂²×C₃×C₄	—	2	1	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₃×M₁₆	—	2	2	no	—
92	C₃×M₁₆	—	2	4	no	—
93	C₃×M₁₆	—	2	2	no	—
94	C₃×M₁₆	—	4	2	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	2	1	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	4	1	no	—
101	C₂×C₃×C₈	—	4	1	no	—
102	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
103	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	#41	1	2	yes	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₂²⋊C₄	—	2	2	no	—
109	C₃×C₂²⋊C₄	—	2	2	no	—
110	C₃×C₄²	—	2	1	no	—
111	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
112	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
113	C₃×C₁₆	—	2	4	no	—
114	C₃×C₁₆	—	4	2	no	—
115	C₂×#41	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂×#41	C₂×C₃×C₄	1	1	yes	characteristic
117	#131	—	2	2	no	—
118	#131	—	2	2	no	—
119	#131	—	2	2	no	—
120	#131	—	2	2	no	—
121	C₂×M₁₆	—	2	1	no	—
122	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
124	C₂×M₁₆	C₃×D₄	1	2	yes	—
125	C₂²×C₈	—	2	1	no	—
126	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
127	C₂²×C₈	C₃×D₄	1	2	yes	—
128	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
129	C₄⋊Q₈	C₃×C₈	1	1	yes	characteristic
130	#127	C₃×C₈	1	1	yes	characteristic
131	#124	C₃×C₈	1	1	yes	characteristic
132	#124	C₃×C₈	1	1	yes	characteristic
133	#121	—	2	2	no	—
134	C₄×D₄	—	2	2	no	—
135	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
136	M₃₂	—	2	4	no	—
137	M₃₂	—	2	2	no	—
138	M₃₂	—	2	2	no	—
139	M₃₂	—	2	2	no	—
140	M₃₂	C₃×D₄	1	4	yes	complemented
141	C₂×C₁₆	—	2	4	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₄⋊C₈	—	2	2	no	—
145	C₂²⋊C₈	—	2	2	no	—
146	C₂²⋊C₈	—	2	2	no	—
147	C₄×C₈	—	2	1	no	—
148	C₄×C₈	—	2	1	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	C₂²⋊C₈	1	2	yes	—
151	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
152	C₃×Q₈	—	2	2	no	—
153	C₃×Q₈	—	2	2	no	—
154	C₃×D₄	—	2	4	no	—
155	C₃×D₄	—	2	2	no	—
156	C₃×D₄	—	2	2	no	—
157	C₃×D₄	—	4	4	no	—
158	C₃×D₄	—	4	2	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
170	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
171	C₂×C₃×C₄	#131	1	2	yes	—
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	4	1	no	—
174	C₃×C₈	—	4	1	no	—
175	C₃×C₈	#120	1	1	yes	characteristic
176	C₃×C₈	#121	1	2	yes	—
177	C₃×C₈	#123	1	1	yes	characteristic
178	#41	—	2	2	no	—
179	#41	—	2	2	no	—
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	C₂×C₃×C₈	1	1	yes	characteristic
183	C₂×Q₈	C₂×C₃×C₈	1	1	yes	characteristic
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂×C₃×C₈	1	1	yes	characteristic
187	C₂×D₄	C₂×C₃×C₈	1	1	yes	characteristic
188	C₂×D₄	C₃×M₁₆	1	2	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
192	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
193	M₁₆	—	2	2	no	—
194	M₁₆	—	2	4	no	—
195	M₁₆	—	2	2	no	—
196	M₁₆	—	4	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
205	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
206	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
207	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
208	C₂×C₈	C₃×#41	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂×C₃×C₈	1	1	yes	characteristic
214	C₄²	C₂×C₃×C₈	1	1	yes	characteristic
215	C₁₆	—	2	4	no	—
216	C₁₆	—	4	2	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	—	4	1	no	—
219	C₂²×C₃	—	4	2	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	4	1	no	—
222	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
223	C₂²×C₃	#435	1	2	yes	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	4	1	no	—
227	C₃×C₄	—	4	1	no	—
228	C₃×C₄	#386	1	1	yes	characteristic
229	C₃×C₄	#408	1	1	yes	characteristic
230	C₃×C₄	C₈×D₄	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	C₃×C₂²⋊C₈	1	2	yes	—
233	C₂³	C₂×C₃×M₁₆	1	1	yes	characteristic
234	Q₈	—	2	2	no	—
235	Q₈	—	2	2	no	—
236	D₄	—	2	4	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	2	no	—
239	D₄	—	4	4	no	—
240	D₄	—	4	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₃×C₂²⋊C₈	1	2	yes	—
250	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
252	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
253	C₂×C₄	C₃×#131	1	2	yes	—
254	C₈	—	2	2	no	—
255	C₈	—	4	1	no	—
256	C₈	—	4	1	no	—
257	C₈	C₃×#120	1	1	yes	characteristic
258	C₈	C₃×#121	1	2	yes	—
259	C₈	C₃×#123	1	1	yes	characteristic
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	—	4	2	no	—
263	C₂×C₃	—	8	1	no	—
264	C₂×C₃	#1869	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
271	C₂²	C₃×#435	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	C₃×#386	1	1	yes	characteristic
277	C₄	C₃×#408	1	1	yes	characteristic
278	C₄	C₃×C₈×D₄	1	2	yes	—
279	C₃	#12982	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	—	2	1	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₃×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	#120
commutator	C₂×C₄	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1869
prime radical	C₃×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#3407	C₂	semidirect
3. C₃×#2117	C₂	semidirect
4. C₃×#2115	C₂	semidirect
5. C₃×#1376	C₂	semidirect
6. #12982	C₃	direct (N × H)
22. #3407	C₂×C₃	semidirect
24. #2117	C₂×C₃	semidirect
25. #2115	C₂×C₃	semidirect
26. #1376	C₂×C₃	semidirect
52. C₃×M₃₂	D₄	semidirect
140. M₃₂	C₃×D₄	semidirect
279. C₃	#12982	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×#3407	2
C₃×#2931	2
C₃×#2117	2
C₃×#2115	2
C₃×#1376	2
#12982	3

Subgroups — C3×#12982 #10631589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12982 #10631589