Subgroups — C₃×#12981 #10631588

Order 768.

488 conjugacy classes of subgroups.

The subgroup lattice has 284 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×#2929	C₂	1	1	yes	maximal, characteristic
3	C₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#2112	C₂	1	2	yes	maximal, complemented
5	C₃×#1378	C₂	1	2	yes	maximal, complemented
6	#12981	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	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×#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×#349	—	2	2	no	—
21	C₃×C₂²⋊C₁₆	—	2	4	no	—
22	#3407	C₂×C₃	1	1	yes	characteristic, complemented
23	#2929	C₂×C₃	1	1	yes	characteristic
24	#2115	C₂×C₃	1	1	yes	characteristic, complemented
25	#2112	C₂×C₃	1	2	yes	complemented
26	#1378	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	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×M₁₆	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×M₁₆	D₄	1	2	yes	—
34	C₂²×C₃×C₈	—	2	1	no	—
35	C₂²×C₃×C₈	—	2	1	no	—
36	C₂²×C₃×C₈	—	2	2	no	—
37	C₂²×C₃×C₈	—	2	2	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×#123	C₈	1	2	yes	—
42	C₃×C₂²⋊Q₈	—	2	1	no	—
43	C₃×#121	—	2	1	no	—
44	C₃×#120	C₈	1	2	yes	—
45	C₃×C₄×D₄	—	2	2	no	—
46	C₃×#117	C₂×C₄	1	1	yes	characteristic
47	C₃×M₃₂	—	2	4	no	—
48	C₃×M₃₂	—	2	2	no	—
49	C₃×M₃₂	—	2	2	no	—
50	C₃×M₃₂	—	2	2	no	—
51	C₃×M₃₂	—	2	2	no	—
52	C₂×C₃×C₁₆	—	2	4	no	—
53	C₂×C₃×C₁₆	—	2	2	no	—
54	C₂×C₃×C₁₆	D₄	1	4	yes	complemented
55	C₃×C₄⋊C₈	—	2	2	no	—
56	C₃×C₂²⋊C₈	—	2	2	no	—
57	C₃×C₂²⋊C₈	—	2	2	no	—
58	C₃×#97	—	2	1	no	—
59	C₃×C₄×C₈	—	2	1	no	—
60	C₂×#131	C₂²×C₃	1	1	yes	characteristic
61	C₂³×C₈	C₂²×C₃	1	1	yes	characteristic
62	#525	C₃×C₄	1	1	yes	characteristic
63	#504	—	2	4	no	—
64	C₂×M₃₂	—	2	2	no	—
65	C₂×M₃₂	C₂²×C₃	1	2	yes	—
66	C₂×M₃₂	C₂²×C₃	1	2	yes	—
67	#435	—	2	2	no	—
68	C₈×D₄	—	2	2	no	—
69	#424	C₃×C₄	1	1	yes	characteristic
70	#408	C₃×C₄	1	1	yes	characteristic
71	#407	C₃×C₄	1	1	yes	characteristic
72	#405	C₂²×C₃	1	1	yes	characteristic
73	#349	—	2	2	no	—
74	C₂²⋊C₁₆	—	2	4	no	—
75	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
76	C₃×#41	—	2	2	no	—
77	C₃×#41	—	2	2	no	—
78	C₂×C₃×Q₈	M₁₆	1	1	yes	characteristic
79	C₂×C₃×D₄	—	2	2	no	—
80	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
81	C₂×C₃×D₄	M₁₆	1	1	yes	characteristic
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	—	2	2	no	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
87	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
89	C₃×M₁₆	—	2	2	no	—
90	C₃×M₁₆	—	2	4	no	—
91	C₃×M₁₆	—	2	2	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	—	2	2	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	4	1	no	—
98	C₂×C₃×C₈	—	4	2	no	—
99	C₂×C₃×C₈	—	4	1	no	—
100	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
102	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	#41	1	2	yes	—
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₄⋊C₄	—	2	1	no	—
107	C₃×C₂²⋊C₄	—	2	2	no	—
108	C₃×C₂²⋊C₄	—	2	2	no	—
109	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
110	C₃×C₄²	—	2	1	no	—
111	C₃×C₁₆	—	2	4	no	—
112	C₃×C₁₆	—	4	2	no	—
113	C₂×#41	C₂×C₃×C₄	1	1	yes	characteristic
114	C₂³×C₄	C₂×C₃×C₄	1	1	yes	characteristic
115	#131	—	2	2	no	—
116	#131	—	2	2	no	—
117	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
118	C₂×M₁₆	C₂×C₃×C₄	1	1	yes	characteristic
119	C₂×M₁₆	C₃×D₄	1	2	yes	—
120	C₂²×C₈	—	2	2	no	—
121	C₂²×C₈	—	2	1	no	—
122	C₂²×C₈	—	2	1	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂²×C₈	C₃×D₄	1	2	yes	—
126	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
127	#123	C₃×C₈	1	2	yes	—
128	C₂²⋊Q₈	—	2	1	no	—
129	#121	—	2	1	no	—
130	#120	C₃×C₈	1	2	yes	—
131	C₄×D₄	—	2	2	no	—
132	#117	C₂×C₃×C₄	1	1	yes	characteristic
133	M₃₂	—	2	4	no	—
134	M₃₂	—	2	2	no	—
135	M₃₂	—	2	2	no	—
136	M₃₂	—	2	2	no	—
137	M₃₂	—	2	2	no	—
138	C₂×C₁₆	—	2	4	no	—
139	C₂×C₁₆	—	2	2	no	—
140	C₂×C₁₆	C₃×D₄	1	4	yes	complemented
141	C₄⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	2	2	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	#97	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	—	2	2	no	—
148	C₂³×C₃	—	4	2	no	—
149	C₂³×C₃	C₂²⋊C₈	1	2	yes	—
150	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
151	C₃×Q₈	—	2	2	no	—
152	C₃×D₄	—	2	4	no	—
153	C₃×D₄	—	2	2	no	—
154	C₃×D₄	—	4	4	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
164	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
166	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
167	C₂×C₃×C₄	#131	1	2	yes	—
168	C₃×C₈	—	2	2	no	—
169	C₃×C₈	—	4	1	no	—
170	C₃×C₈	—	4	1	no	—
171	C₃×C₈	#120	1	1	yes	characteristic
172	C₃×C₈	#121	1	2	yes	—
173	C₃×C₈	#123	1	1	yes	characteristic
174	C₂⁴	C₂×C₃×C₈	1	1	yes	characteristic
175	#41	—	2	2	no	—
176	#41	—	2	2	no	—
177	C₂×Q₈	C₃×M₁₆	1	1	yes	characteristic
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	C₂×C₃×C₈	1	2	yes	—
180	C₂×D₄	C₃×M₁₆	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₃×C₂²⋊C₄	1	2	yes	—
186	C₂²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
187	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
188	M₁₆	—	2	2	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
200	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
201	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
202	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
203	C₂×C₈	C₃×#41	1	2	yes	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	C₂×C₃×C₈	1	2	yes	—
209	C₄²	—	2	1	no	—
210	C₁₆	—	2	4	no	—
211	C₁₆	—	4	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	—	4	2	no	—
217	C₂²×C₃	—	4	1	no	—
218	C₂²×C₃	—	4	2	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
221	C₂²×C₃	C₈×D₄	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	4	2	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	#386	1	1	yes	characteristic
226	C₃×C₄	#408	1	1	yes	characteristic
227	C₃×C₄	#435	1	2	yes	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₃×C₂²⋊C₈	1	2	yes	—
232	C₂³	C₂²×C₃×C₈	1	1	yes	characteristic
233	Q₈	—	2	2	no	—
234	D₄	—	2	4	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	C₃×C₂²⋊C₈	1	2	yes	—
246	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
248	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
249	C₂×C₄	C₃×#131	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	4	1	no	—
252	C₈	—	4	1	no	—
253	C₈	C₃×#120	1	1	yes	characteristic
254	C₈	C₃×#121	1	2	yes	—
255	C₈	C₃×#123	1	1	yes	characteristic
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	4	2	no	—
260	C₂×C₃	#1869	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
270	C₂²	C₃×C₈×D₄	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	C₃×#386	1	1	yes	characteristic
275	C₄	C₃×#408	1	1	yes	characteristic
276	C₄	C₃×#435	1	2	yes	—
277	C₃	#12981	1	1	yes	characteristic, direct product factor, Sylow
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	4	2	no	—
282	C₂	C₃×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	#120
commutator	C₂³	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1869
prime radical	C₃×#525	C₄
LCS	G ▷ C₂³ ▷ 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×#2115	C₂	semidirect
4. C₃×#2112	C₂	semidirect
5. C₃×#1378	C₂	semidirect
6. #12981	C₃	direct (N × H)
22. #3407	C₂×C₃	semidirect
24. #2115	C₂×C₃	semidirect
25. #2112	C₂×C₃	semidirect
26. #1378	C₂×C₃	semidirect
54. C₂×C₃×C₁₆	D₄	semidirect
140. C₂×C₁₆	C₃×D₄	semidirect
277. C₃	#12981	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×#2929	2
C₃×#2115	2
C₃×#2112	2
C₃×#1378	2
#12981	3

Subgroups — C3×#12981 #10631588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12981 #10631588