Subgroups — C₃×#69771 #11790612

Order 1152.

745 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	C₂×C₃×#869	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#816	C₂	1	4	yes	maximal
4	C₃×#4935	C₂	1	1	yes	maximal, characteristic
5	C₃×#1956	—	3	1	no	maximal
6	#69771	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
8	C₂²×C₃²×C₈	C₂²	1	2	yes	—
9	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₃×#869	—	2	2	no	—
14	C₃×#869	C₄	1	4	yes	complemented
15	C₃²×C₄×C₈	—	2	2	no	—
16	C₃×#816	—	2	8	no	—
17	C₂×C₃×C₄⋊Q₈	—	3	1	no	—
18	C₂×#869	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×#103	—	3	4	no	—
20	C₂×C₃×C₄×C₈	—	2	1	no	—
21	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
23	C₃×#384	—	3	1	no	—
24	C₂×#816	C₂×C₃	1	4	yes	—
25	#4935	C₂×C₃	1	1	yes	characteristic
26	C₂²×C₃²×C₄	D₄	1	2	yes	—
27	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₂²×C₃×Dic₃	—	2	2	no	—
29	C₂²×C₃×Dic₃	—	2	2	no	—
30	C₂×C₃×Dic₆	—	2	4	no	—
31	C₂×C₃×Dic₆	—	2	4	no	—
32	C₂×C₃×Dic₆	—	2	2	no	—
33	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
34	C₂×C₃²×C₈	—	2	2	no	—
35	C₂×C₃²×C₈	—	2	2	no	—
36	C₂×C₃²×C₈	D₄	1	4	yes	—
37	C₃²×C₄²	C₂×C₄	1	2	yes	—
38	C₃²×C₄²	D₄	1	2	yes	—
39	C₃×C₄⋊Dic₃	—	2	8	no	—
40	C₃×C₄⋊Dic₃	—	2	4	no	—
41	C₃×C₄⋊Dic₃	—	4	4	no	—
42	#1956	—	3	1	no	Sylow
43	C₂²×C₃×Q₈	—	3	2	no	—
44	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
45	C₂²×C₃×C₈	—	2	2	no	—
46	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
47	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
48	C₃×C₄⋊Q₈	—	3	4	no	—
49	C₃×C₄⋊Q₈	—	6	2	no	—
50	C₂×C₃×C₄⋊C₄	—	3	2	no	—
51	C₂×C₃×C₄⋊C₄	—	6	2	no	—
52	C₂×C₃×C₄⋊C₄	—	6	2	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
56	C₂×C₄⋊Dic₃	—	2	2	no	—
57	C₂×C₄⋊Dic₃	—	2	2	no	—
58	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
59	#869	—	2	2	no	—
60	#869	C₃×C₄	1	4	yes	complemented
61	C₃×#103	—	6	8	no	—
62	C₃×C₄×C₈	—	2	4	no	—
63	C₃×C₄×C₈	—	2	2	no	—
64	C₃×C₄×C₈	—	2	2	no	—
65	#816	—	2	8	no	—
66	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²⋊C₄	1	4	yes	—
68	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
69	C₂×C₃²×C₄	Q₁₆	1	4	yes	—
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
72	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	#41	1	2	yes	—
74	C₂×C₃×Dic₃	—	2	4	no	—
75	C₂×C₃×Dic₃	—	4	2	no	—
76	C₂×C₃×Dic₃	—	4	2	no	—
77	C₂×C₃×Dic₃	—	4	2	no	—
78	C₂×C₃×Dic₃	—	4	4	no	—
79	C₃×Dic₆	—	2	8	no	—
80	C₃×Dic₆	—	4	4	no	—
81	C₃²×C₈	—	2	4	no	—
82	C₂×C₄⋊Q₈	—	3	1	no	—
83	C₂×#103	—	3	4	no	—
84	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
85	#384	—	3	1	no	—
86	C₂×C₃×Q₈	—	3	4	no	—
87	C₂×C₃×Q₈	—	6	2	no	—
88	C₂×C₃×Q₈	—	6	4	no	—
89	C₂×C₃×Q₈	—	6	4	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	2	1	no	—
92	C₂²×C₃×C₄	—	6	2	no	—
93	C₂²×C₃×C₄	—	6	2	no	—
94	C₂²×C₃×C₄	D₁₂	1	2	yes	—
95	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
96	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₂²×Dic₃	—	2	2	no	—
99	C₂²×Dic₃	—	2	2	no	—
100	C₂×Dic₆	—	2	4	no	—
101	C₂×Dic₆	—	2	4	no	—
102	C₂×Dic₆	—	2	2	no	—
103	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
104	C₂×C₃×C₈	—	2	4	no	—
105	C₂×C₃×C₈	—	2	4	no	—
106	C₂×C₃×C₈	—	2	2	no	—
107	C₂×C₃×C₈	—	2	2	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	4	no	—
111	C₂×C₃×C₈	D₁₂	1	4	yes	—
112	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
113	C₃×C₄⋊C₄	—	6	8	no	—
114	C₃×C₄⋊C₄	—	6	4	no	—
115	C₃×C₄⋊C₄	—	12	4	no	—
116	C₃×C₄²	—	2	2	no	—
117	C₃×C₄²	—	2	2	no	—
118	C₃×C₄²	C₄×D₃	1	2	yes	—
119	C₃×C₄²	#67	1	2	yes	—
120	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
121	C₃×C₄²	C₃×D₄	1	2	yes	—
122	C₄⋊Dic₃	—	2	8	no	—
123	C₄⋊Dic₃	—	2	4	no	—
124	C₄⋊Dic₃	—	4	4	no	—
125	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂²×C₃²	#124	1	1	yes	characteristic
127	C₂²×C₃²	#127	1	1	yes	characteristic
128	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
129	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
130	C₃²×C₄	#103	1	8	yes	complemented
131	C₃²×C₄	C₄×D₄	1	2	yes	—
132	C₃²×C₄	#121	1	2	yes	—
133	C₃×Dic₃	—	4	4	no	—
134	C₃×Dic₃	—	8	2	no	—
135	C₂²×Q₈	—	3	2	no	—
136	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
137	C₄⋊Q₈	—	3	4	no	—
138	C₄⋊Q₈	—	6	2	no	—
139	C₂×C₄⋊C₄	—	3	2	no	—
140	C₂×C₄⋊C₄	—	6	2	no	—
141	C₂×C₄⋊C₄	—	6	2	no	—
142	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
143	#103	—	6	8	no	—
144	C₄×C₈	—	2	2	no	—
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
147	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
148	C₃×Q₈	—	6	8	no	—
149	C₃×Q₈	—	12	4	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	6	4	no	—
158	C₂×C₃×C₄	—	12	2	no	—
159	C₂×C₃×C₄	—	12	2	no	—
160	C₂×C₃×C₄	—	12	2	no	—
161	C₂×C₃×C₄	—	12	4	no	—
162	C₂×C₃×C₄	#204	1	4	yes	—
163	C₂×C₃×C₄	Dic₁₂	1	4	yes	—
164	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
165	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	4	yes	—
166	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
167	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
168	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
170	C₂×C₃×C₄	#235	1	2	yes	—
171	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₃×#41	1	2	yes	—
176	C₂×Dic₃	—	2	4	no	—
177	C₂×Dic₃	—	4	2	no	—
178	C₂×Dic₃	—	4	2	no	—
179	C₂×Dic₃	—	4	2	no	—
180	C₂×Dic₃	—	4	4	no	—
181	Dic₆	—	2	8	no	—
182	Dic₆	—	4	4	no	—
183	C₃×C₈	—	2	8	no	—
184	C₃×C₈	—	2	4	no	—
185	C₃×C₈	—	2	4	no	—
186	C₂×C₃²	#390	1	1	yes	characteristic
187	C₂×C₃²	C₂×#103	1	2	yes	—
188	C₂×C₃²	#487	1	2	yes	—
189	C₂×C₃²	#492	1	1	yes	characteristic
190	C₂×C₃²	#494	1	1	yes	characteristic
191	C₂×Q₈	—	3	4	no	—
192	C₂×Q₈	—	6	2	no	—
193	C₂×Q₈	—	6	4	no	—
194	C₂×Q₈	—	6	4	no	—
195	C₂²×C₄	—	6	2	no	—
196	C₂²×C₄	—	6	2	no	—
197	C₂²×C₄	C₃×D₁₂	1	2	yes	—
198	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₃×D₁₂	1	4	yes	—
202	C₄⋊C₄	—	6	8	no	—
203	C₄⋊C₄	—	6	4	no	—
204	C₄⋊C₄	—	12	4	no	—
205	C₄²	C₃×C₄×D₃	1	2	yes	—
206	C₄²	C₃×#67	1	2	yes	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	#874	1	1	yes	characteristic
213	C₂²×C₃	#875	1	1	yes	characteristic
214	C₂²×C₃	C₂×#204	1	2	yes	—
215	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
216	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
217	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
218	C₂²×C₃	C₃×#124	1	1	yes	characteristic
219	C₂²×C₃	C₃×#127	1	1	yes	characteristic
220	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
221	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	8	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	12	4	no	—
226	C₃×C₄	—	24	2	no	—
227	C₃×C₄	#816	1	8	yes	complemented
228	C₃×C₄	C₃×#103	1	8	yes	complemented
229	C₃×C₄	C₄×D₁₂	1	2	yes	—
230	C₃×C₄	#930	1	2	yes	—
231	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
232	C₃×C₄	C₃×#121	1	2	yes	—
233	Dic₃	—	4	4	no	—
234	Dic₃	—	8	2	no	—
235	C₃²	#1956	1	1	yes	characteristic, complemented, Sylow
236	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
237	Q₈	—	6	8	no	—
238	Q₈	—	12	4	no	—
239	C₂×C₄	—	6	4	no	—
240	C₂×C₄	—	12	2	no	—
241	C₂×C₄	—	12	2	no	—
242	C₂×C₄	—	12	4	no	—
243	C₂×C₄	—	12	2	no	—
244	C₂×C₄	C₃×#204	1	4	yes	—
245	C₂×C₄	C₃×Dic₁₂	1	4	yes	—
246	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
247	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
249	C₂×C₄	C₃×#235	1	2	yes	—
250	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
251	C₈	—	2	4	no	—
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	#4681	1	2	yes	—
258	C₂×C₃	#4693	1	1	yes	characteristic
259	C₂×C₃	#4699	1	1	yes	characteristic
260	C₂×C₃	#4939	1	1	yes	characteristic
261	C₂×C₃	C₂×#816	1	2	yes	—
262	C₂×C₃	C₃×#390	1	1	yes	characteristic
263	C₂×C₃	C₂×C₃×#103	1	2	yes	—
264	C₂×C₃	C₃×#487	1	2	yes	—
265	C₂×C₃	C₃×#492	1	1	yes	characteristic
266	C₂×C₃	C₃×#494	1	1	yes	characteristic
267	C₂²	C₃×#874	1	1	yes	characteristic
268	C₂²	C₃×#875	1	1	yes	characteristic
269	C₂²	C₂×C₃×#204	1	2	yes	—
270	C₂²	C₂×C₃×Dic₁₂	1	2	yes	—
271	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
272	C₄	—	12	4	no	—
273	C₄	—	24	2	no	—
274	C₄	C₃×#816	1	8	yes	complemented
275	C₄	C₃×C₄×D₁₂	1	2	yes	—
276	C₄	C₃×#930	1	2	yes	—
277	C₃	—	2	1	no	—
278	C₃	#69771	1	1	yes	characteristic, direct product factor
279	C₃	C₃×#1956	1	1	yes	characteristic, complemented
280	C₂	C₃×#4681	1	2	yes	—
281	C₂	C₃×#4693	1	1	yes	characteristic
282	C₂	C₃×#4699	1	1	yes	characteristic
283	C₂	C₃×#4939	1	1	yes	characteristic
284	C₂	C₂×C₃×#816	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#1956
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #69771	C₃	direct (N × H)
14. C₃×#869	C₄	semidirect
60. #869	C₃×C₄	semidirect
130. C₃²×C₄	#103	semidirect
227. C₃×C₄	#816	semidirect
228. C₃×C₄	C₃×#103	semidirect
235. C₃²	#1956	semidirect
274. C₄	C₃×#816	semidirect
278. C₃	#69771	direct (N × H)
279. C₃	C₃×#1956	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#869	2
C₂×C₃²×C₄×C₈	2
C₂×C₃×#816	2
C₃×#4935	2
C₃×#1956	3
#69771	3

Subgroups — C3×#69771 #11790612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69771 #11790612