Subgroups — C₃×#70853 #11781174

Order 1152.

387 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	C₃²×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₃×#5008	C₂	1	2	yes	maximal
3	C₃×#4682	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4481	C₂	1	2	yes	maximal, complemented
5	C₃×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1675	—	3	1	no	maximal
7	#70853	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	—	2	1	no	—
9	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
10	C₃×#869	C₂²	1	2	yes	—
11	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₂²⋊C₈	—	2	1	no	—
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#832	—	2	2	no	—
15	C₃×#818	—	2	1	no	—
16	C₃×#818	—	2	1	no	—
17	C₃×#808	—	2	2	no	—
18	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₃×#500	—	3	1	no	—
20	C₃×#476	—	3	2	no	—
21	C₃×C₈×D₄	—	2	1	no	—
22	C₃×C₈×D₄	D₃	1	1	yes	characteristic
23	C₃×C₈×D₄	C₂×C₃	1	1	yes	characteristic
24	#5008	C₂×C₃	1	2	yes	—
25	#4682	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×#335	—	3	1	no	—
27	C₃×#333	—	3	2	no	—
28	#4481	C₂×C₃	1	2	yes	complemented
29	#4458	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
31	C₂²×C₃²×C₄	—	2	1	no	—
32	C₂×C₃×Dic₆	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	1	no	—
34	C₂×C₃²×C₈	—	4	2	no	—
35	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
37	C₃²×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₃²×C₂²⋊C₄	—	2	1	no	—
39	C₃²×C₄²	C₂³	1	1	yes	characteristic
40	C₃×C₄⋊Dic₃	—	2	2	no	—
41	C₃×C₄⋊Dic₃	—	2	2	no	—
42	C₂×C₃×C₃⋊C₈	—	2	2	no	—
43	#1675	—	3	1	no	Sylow
44	C₂²×C₃×C₈	—	2	2	no	—
45	C₂²×C₃×C₈	—	2	1	no	—
46	C₂²×C₃×C₈	—	2	1	no	—
47	C₃×C₄⋊Q₈	—	3	2	no	—
48	C₃×C₄×D₄	—	2	1	no	—
49	C₃×C₄×D₄	C₂×D₃	1	1	yes	characteristic
50	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
51	#869	C₂²×C₃	1	2	yes	—
52	C₃×#107	—	6	1	no	—
53	C₃×#107	—	6	1	no	—
54	C₃×#106	—	6	2	no	—
55	C₃×C₄⋊C₈	—	2	1	no	—
56	C₃×C₄⋊C₈	—	3	2	no	—
57	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃×#102	—	6	2	no	—
60	C₃×C₂²⋊C₈	—	2	2	no	—
61	C₃×C₂²⋊C₈	—	2	1	no	—
62	C₃×C₂²⋊C₈	—	2	1	no	—
63	C₃×C₄×C₈	—	2	1	no	—
64	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
65	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
66	#832	—	2	2	no	—
67	#818	—	2	1	no	—
68	#818	—	2	1	no	—
69	#808	—	2	2	no	—
70	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
71	C₂³×C₃²	—	2	1	no	—
72	C₃²×D₄	—	2	1	no	—
73	C₃²×D₄	Q₁₆	1	2	yes	—
74	C₂×C₃²×C₄	—	2	1	no	—
75	C₂×C₃²×C₄	—	4	1	no	—
76	C₂×C₃²×C₄	—	4	1	no	—
77	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	1	yes	characteristic
80	C₂×C₃×Dic₃	—	4	2	no	—
81	C₃×Dic₆	—	8	2	no	—
82	C₃²×C₈	—	2	1	no	—
83	C₃²×C₈	—	4	1	no	—
84	C₃²×C₈	SD₁₆	1	2	yes	—
85	C₃×C₃⋊C₈	—	4	2	no	—
86	#500	—	3	1	no	—
87	#476	—	3	2	no	—
88	C₈×D₄	C₃×D₃	1	1	yes	characteristic
89	#335	—	3	1	no	—
90	#333	—	3	2	no	—
91	C₂×C₃×Q₈	—	6	2	no	—
92	C₂×C₃×D₄	—	2	1	no	—
93	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
94	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	—	2	2	no	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	—	2	1	no	—
98	C₂×Dic₆	—	2	2	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	1	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	—	4	4	no	—
105	C₂×C₃×C₈	—	4	2	no	—
106	C₂×C₃×C₈	—	4	2	no	—
107	C₂×C₃×C₈	—	6	2	no	—
108	C₂×C₃×C₈	#67	1	1	yes	characteristic
109	C₂×C₃×C₈	#67	1	1	yes	characteristic
110	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
111	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
112	C₃×C₄⋊C₄	—	2	1	no	—
113	C₃×C₄⋊C₄	—	6	2	no	—
114	C₃×C₄⋊C₄	—	6	2	no	—
115	C₃×C₄⋊C₄	D₁₂	1	1	yes	characteristic
116	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
117	C₃×C₂²⋊C₄	—	2	2	no	—
118	C₃×C₂²⋊C₄	—	2	1	no	—
119	C₃×C₂²⋊C₄	—	2	1	no	—
120	C₃×C₄²	—	2	1	no	—
121	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
122	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
123	C₄⋊Dic₃	—	2	2	no	—
124	C₄⋊Dic₃	—	2	2	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂²×C₃²	—	2	2	no	—
127	C₂²×C₃²	—	4	1	no	—
128	C₂²×C₃²	—	4	1	no	—
129	C₂²×C₃²	#121	1	1	yes	characteristic
130	C₃²×C₄	—	2	1	no	—
131	C₃²×C₄	—	4	1	no	—
132	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
133	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
134	C₃²×C₄	#135	1	1	yes	characteristic
135	C₃²×C₄	#137	1	1	yes	characteristic
136	C₃×Dic₃	—	8	2	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₄⋊Q₈	—	3	2	no	—
139	C₄×D₄	C₂×C₃×D₃	1	1	yes	characteristic
140	#107	—	6	1	no	—
141	#107	—	6	1	no	—
142	#106	—	6	2	no	—
143	C₄⋊C₈	—	3	2	no	—
144	C₄⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
145	#102	—	6	2	no	—
146	C₂²⋊C₈	—	2	1	no	—
147	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
148	C₂³×C₃	—	2	2	no	—
149	C₂³×C₃	—	2	1	no	—
150	C₂³×C₃	—	2	1	no	—
151	C₃×Q₈	—	24	2	no	—
152	C₃×D₄	—	2	2	no	—
153	C₃×D₄	—	2	1	no	—
154	C₃×D₄	—	2	1	no	—
155	C₃×D₄	—	2	2	no	—
156	C₃×D₄	Dic₁₂	1	2	yes	—
157	C₃×D₄	C₃×Q₁₆	1	2	yes	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	1	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	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	2	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	—	4	1	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	—	12	2	no	—
171	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
172	C₂×C₃×C₄	#235	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
177	C₂×Dic₃	—	4	2	no	—
178	Dic₆	—	8	2	no	—
179	C₃×C₈	—	2	1	no	—
180	C₃×C₈	—	2	1	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	—	4	1	no	—
183	C₃×C₈	—	4	2	no	—
184	C₃×C₈	—	4	1	no	—
185	C₃×C₈	—	4	1	no	—
186	C₃×C₈	—	12	2	no	—
187	C₃×C₈	#214	1	2	yes	—
188	C₃×C₈	C₃×SD₁₆	1	2	yes	—
189	C₃⋊C₈	—	4	2	no	—
190	C₂×C₃²	—	4	2	no	—
191	C₂×C₃²	#461	1	1	yes	characteristic
192	C₂×C₃²	#467	1	1	yes	characteristic
193	C₂×C₃²	#472	1	1	yes	characteristic
194	C₂×Q₈	—	6	2	no	—
195	C₂×D₄	C₃×D₁₂	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	C₃×#67	1	1	yes	characteristic
201	C₂×C₈	C₃×#67	1	1	yes	characteristic
202	C₄⋊C₄	—	6	2	no	—
203	C₄⋊C₄	—	6	2	no	—
204	C₄⋊C₄	C₃×D₁₂	1	1	yes	characteristic
205	C₂²⋊C₄	—	2	1	no	—
206	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	1	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	#930	1	1	yes	characteristic
218	C₂²×C₃	C₃×#121	1	1	yes	characteristic
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	1	no	—
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	4	1	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	—	4	1	no	—
228	C₃×C₄	—	4	1	no	—
229	C₃×C₄	—	24	2	no	—
230	C₃×C₄	#904	1	1	yes	characteristic
231	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
232	C₃×C₄	C₂×#214	1	1	yes	characteristic
233	C₃×C₄	#951	1	1	yes	characteristic
234	C₃×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
235	C₃×C₄	C₂×C₃×Q₁₆	1	1	yes	characteristic
236	C₃×C₄	C₃×#135	1	1	yes	characteristic
237	C₃×C₄	C₃×#137	1	1	yes	characteristic
238	Dic₃	—	8	2	no	—
239	C₃²	#1675	1	1	yes	characteristic, complemented, Sylow
240	C₂³	—	2	1	no	—
241	Q₈	—	24	2	no	—
242	D₄	—	2	1	no	—
243	D₄	C₃×Dic₁₂	1	2	yes	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	12	2	no	—
248	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
249	C₂×C₄	C₃×#235	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	4	1	no	—
253	C₈	—	12	2	no	—
254	C₈	C₃×#214	1	2	yes	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	4	4	no	—
259	C₂×C₃	—	4	2	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	#5019	1	1	yes	characteristic
262	C₂×C₃	#5116	1	1	yes	characteristic
263	C₂×C₃	#5142	1	1	yes	characteristic
264	C₂×C₃	C₃×#461	1	1	yes	characteristic
265	C₂×C₃	C₃×#467	1	1	yes	characteristic
266	C₂×C₃	C₃×#472	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₃×#930	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	24	2	no	—
274	C₄	C₃×#904	1	1	yes	characteristic
275	C₄	C₂×C₃×Dic₁₂	1	1	yes	characteristic
276	C₄	C₂×C₃×#214	1	1	yes	characteristic
277	C₄	C₃×#951	1	1	yes	characteristic
278	C₃	—	2	1	no	—
279	C₃	#70853	1	1	yes	characteristic, direct product factor
280	C₃	C₃×#1675	1	1	yes	characteristic, complemented
281	C₂	—	4	2	no	—
282	C₂	C₃×#5019	1	1	yes	characteristic
283	C₂	C₃×#5116	1	1	yes	characteristic
284	C₂	C₃×#5142	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#930
commutator, cosocle	C₃×C₄²	C₂³×C₃
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₈×D₄	C₂
socle	C₂²×C₃²	#121
prime radical	C₂×C₃²×D₄	D₄
hypercenter	C₃×C₈×D₄	D₃
hypocenter	C₃	C₃×#1675
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈×D₄

Complements

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

N	H ≅ G/N	type
3. C₃×#4682	C₂	semidirect
4. C₃×#4481	C₂	semidirect
5. C₃×#4458	C₂	semidirect
7. #70853	C₃	direct (N × H)
25. #4682	C₂×C₃	semidirect
28. #4481	C₂×C₃	semidirect
29. #4458	C₂×C₃	semidirect
239. C₃²	#1675	semidirect
279. C₃	#70853	direct (N × H)
280. C₃	C₃×#1675	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₈×D₄	2
C₃×#5008	2
C₃×#4682	2
C₃×#4481	2
C₃×#4458	2
C₃×#1675	3
#70853	3

Subgroups — C3×#70853 #11781174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70853 #11781174