Subgroups — C₃×#70702 #11784372

Order 1152.

517 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#108	C₂	1	1	yes	maximal, characteristic
2	C₃×#5122	C₂	1	2	yes	maximal
3	C₃×#5106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
5	C₃×#4555	C₂	1	2	yes	maximal
6	C₃×#1772	—	3	1	no	maximal
7	#70702	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₃×#926	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃²×#108	C₄	1	4	yes	complemented
14	C₃×#818	C₄	1	2	yes	complemented
15	C₃×#817	C₄	1	2	yes	complemented
16	C₃×#815	—	2	2	no	—
17	C₃×C₈×Dic₃	—	2	2	no	—
18	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
19	C₂×C₃×#108	—	2	1	no	—
20	C₂×C₃×#108	D₃	1	1	yes	characteristic
21	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
22	C₃×#427	—	3	1	no	—
23	C₃×#405	—	3	2	no	—
24	C₂×C₃×C₄×C₈	—	3	1	no	—
25	#5122	C₂×C₃	1	2	yes	—
26	#5106	C₂×C₃	1	1	yes	characteristic
27	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
28	C₃×#339	—	3	2	no	—
29	#4555	C₂×C₃	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	—	2	1	no	—
32	C₃²×M₁₆	—	2	2	no	—
33	C₃²×M₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₃²×C₈	—	2	2	no	—
35	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
39	C₃×#217	—	2	2	no	—
40	C₃×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
41	C₃×C₄×Dic₃	—	2	2	no	—
42	C₃×C₄×Dic₃	—	2	1	no	—
43	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	Q₈	1	2	yes	—
48	#1772	—	3	1	no	Sylow
49	C₂×C₃×M₁₆	—	2	2	no	—
50	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
51	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	—	3	1	no	—
54	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₃×#117	—	3	2	no	—
57	C₂×C₃×C₄²	—	3	1	no	—
58	#926	C₂²×C₃	1	2	yes	—
59	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₃×#108	—	2	4	no	—
62	C₃×#108	Dic₃	1	4	yes	complemented
63	C₃×#108	C₃×C₄	1	4	yes	complemented
64	C₃×#107	—	3	2	no	—
65	C₃×#106	—	3	2	no	—
66	C₃×#97	—	6	2	no	—
67	C₃×C₄×C₈	—	3	4	no	—
68	C₃×C₄×C₈	—	6	2	no	—
69	#818	C₃×C₄	1	2	yes	complemented
70	#817	C₃×C₄	1	2	yes	complemented
71	#815	—	2	2	no	—
72	C₈×Dic₃	—	2	2	no	—
73	C₈×Dic₃	C₃×C₄	1	4	yes	complemented
74	C₂³×C₃²	#41	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	2	4	no	—
82	C₂×C₃×Dic₃	—	2	2	no	—
83	C₂×C₃×Dic₃	—	2	4	no	—
84	C₃²×C₈	—	2	4	no	—
85	C₃²×C₈	C₄²	1	2	yes	—
86	C₃²×C₈	C₄²	1	2	yes	—
87	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
88	C₃×C₃⋊C₈	C₄⋊C₄	1	2	yes	—
89	C₂×#108	C₃×D₃	1	1	yes	characteristic
90	#427	—	3	1	no	—
91	#405	—	3	2	no	—
92	C₂×C₄×C₈	—	3	1	no	—
93	#339	—	3	2	no	—
94	C₂²×C₃×C₄	—	2	1	no	—
95	C₂²×C₃×C₄	—	6	1	no	—
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂²×Dic₃	—	2	1	no	—
99	C₃×M₁₆	—	2	2	no	—
100	C₃×M₁₆	—	2	2	no	—
101	C₃×M₁₆	—	2	4	no	—
102	C₃×M₁₆	—	4	2	no	—
103	C₃×M₁₆	C₂×Dic₃	1	4	yes	—
104	C₃×M₁₆	C₂×C₃×C₄	1	4	yes	—
105	C₂×C₃×C₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	1	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	3	2	no	—
112	C₂×C₃×C₈	—	3	1	no	—
113	C₂×C₃×C₈	—	3	1	no	—
114	C₂×C₃×C₈	—	3	2	no	—
115	C₂×C₃×C₈	—	4	2	no	—
116	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
117	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
118	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
119	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
120	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
121	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
124	C₃×C₄⋊C₄	—	3	4	no	—
125	C₃×C₂²⋊C₄	—	6	2	no	—
126	C₃×C₄²	—	3	2	no	—
127	C₃×C₄²	—	6	2	no	—
128	C₃×C₄²	—	6	1	no	—
129	#217	—	2	2	no	—
130	C₄⋊Dic₃	C₂×C₃×C₄	1	4	yes	—
131	C₄×Dic₃	—	2	2	no	—
132	C₄×Dic₃	—	2	1	no	—
133	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
134	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
135	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
136	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
137	C₂×C₃⋊C₈	C₃×Q₈	1	2	yes	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	#117	1	1	yes	characteristic
140	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
141	C₂²×C₃²	C₄×Q₈	1	1	yes	characteristic
142	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
143	C₃²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
144	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
145	C₃²×C₄	C₄×Q₈	1	1	yes	characteristic
146	C₃×Dic₃	—	2	4	no	—
147	C₃×Dic₃	—	4	4	no	—
148	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
149	C₂²×C₈	—	3	1	no	—
150	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
151	#117	—	3	2	no	—
152	C₂×C₄²	—	3	1	no	—
153	#108	C₃×Dic₃	1	4	yes	complemented
154	#107	—	3	2	no	—
155	#106	—	3	2	no	—
156	#97	—	6	2	no	—
157	C₄×C₈	—	3	4	no	—
158	C₄×C₈	—	6	2	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	#239	1	1	yes	characteristic
161	C₂³×C₃	C₃×#41	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	6	4	no	—
169	C₂×C₃×C₄	—	6	2	no	—
170	C₂×C₃×C₄	—	6	4	no	—
171	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	#237	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
183	C₂×Dic₃	—	2	4	no	—
184	C₂×Dic₃	—	2	2	no	—
185	C₂×Dic₃	—	2	4	no	—
186	C₃×C₈	—	2	4	no	—
187	C₃×C₈	—	2	4	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	3	2	no	—
191	C₃×C₈	—	3	2	no	—
192	C₃×C₈	—	4	4	no	—
193	C₃×C₈	C₄×Dic₃	1	2	yes	—
194	C₃×C₈	C₄×Dic₃	1	2	yes	—
195	C₃×C₈	C₃×C₄²	1	2	yes	—
196	C₃×C₈	C₃×C₄²	1	2	yes	—
197	C₃⋊C₈	C₃×C₄⋊C₄	1	2	yes	—
198	C₃⋊C₈	C₃×C₄⋊C₄	1	2	yes	—
199	C₂×C₃²	—	2	1	no	—
200	C₂×C₃²	—	2	1	no	—
201	C₂×C₃²	C₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₃²	#443	1	2	yes	—
203	C₂²×C₄	—	6	1	no	—
204	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
205	M₁₆	—	2	2	no	—
206	M₁₆	C₂×C₃×Dic₃	1	4	yes	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	—	3	1	no	—
210	C₂×C₈	—	3	1	no	—
211	C₂×C₈	—	3	2	no	—
212	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
213	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
214	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
216	C₄⋊C₄	—	3	4	no	—
217	C₂²⋊C₄	—	6	2	no	—
218	C₄²	—	3	2	no	—
219	C₄²	—	6	2	no	—
220	C₄²	—	6	1	no	—
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	—	2	1	no	—
225	C₂²×C₃	—	2	1	no	—
226	C₂²×C₃	—	4	2	no	—
227	C₂²×C₃	#892	1	1	yes	characteristic
228	C₂²×C₃	#893	1	1	yes	characteristic
229	C₂²×C₃	Q₈×Dic₃	1	1	yes	characteristic
230	C₂²×C₃	C₃×#117	1	1	yes	characteristic
231	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
232	C₂²×C₃	C₃×C₄×Q₈	1	1	yes	characteristic
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	1	no	—
235	C₃×C₄	—	2	1	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	6	4	no	—
238	C₃×C₄	—	12	4	no	—
239	C₃×C₄	#887	1	1	yes	characteristic
240	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
241	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
242	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
243	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
244	C₃×C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
245	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
246	C₃×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
247	Dic₃	—	2	4	no	—
248	Dic₃	—	4	4	no	—
249	C₃²	#1772	1	1	yes	characteristic, complemented, Sylow
250	C₂³	C₃×#239	1	1	yes	characteristic
251	C₂×C₄	—	6	4	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
256	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
257	C₂×C₄	C₃×#237	1	1	yes	characteristic
258	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	3	2	no	—
262	C₈	—	3	2	no	—
263	C₈	C₃×C₄×Dic₃	1	2	yes	—
264	C₈	C₃×C₄×Dic₃	1	2	yes	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	2	no	—
271	C₂×C₃	—	4	1	no	—
272	C₂×C₃	—	4	1	no	—
273	C₂×C₃	#4895	1	2	yes	—
274	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
275	C₂×C₃	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
276	C₂×C₃	C₃×#443	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	C₃×#892	1	1	yes	characteristic
279	C₂²	C₃×#893	1	1	yes	characteristic
280	C₂²	C₃×Q₈×Dic₃	1	1	yes	characteristic
281	C₄	—	6	4	no	—
282	C₄	—	12	4	no	—
283	C₄	C₃×#887	1	1	yes	characteristic
284	C₄	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
285	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
286	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
287	C₃	—	2	1	no	—
288	C₃	#70702	1	1	yes	characteristic, direct product factor
289	C₃	C₃×#1772	1	1	yes	characteristic, complemented
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	C₃×#4895	1	2	yes	—
293	C₂	C₃×Dic₃×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₃×C₄	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#108	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#108	D₃
hypocenter	C₃	C₃×#1772
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#108

Complements

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

N	H ≅ G/N	type
7. #70702	C₃	direct (N × H)
13. C₃²×#108	C₄	semidirect
14. C₃×#818	C₄	semidirect
15. C₃×#817	C₄	semidirect
18. C₃×C₈×Dic₃	C₄	semidirect
62. C₃×#108	Dic₃	semidirect
63. C₃×#108	C₃×C₄	semidirect
69. #818	C₃×C₄	semidirect
70. #817	C₃×C₄	semidirect
73. C₈×Dic₃	C₃×C₄	semidirect
153. #108	C₃×Dic₃	semidirect
249. C₃²	#1772	semidirect
288. C₃	#70702	direct (N × H)
289. C₃	C₃×#1772	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#108	2
C₃×#5122	2
C₃×#5106	2
C₂×C₃×C₈×Dic₃	2
C₃×#4555	2
C₃×#1772	3
#70702	3

Subgroups — C3×#70702 #11784372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70702 #11784372