Subgroups — C₃×#72012 #11785487

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#131	C₂	1	1	yes	maximal, characteristic
2	C₃×#5240	C₂	1	2	yes	maximal
3	C₃×#5106	C₂	1	2	yes	maximal, complemented
4	C₃×#4557	C₂	1	2	yes	maximal, complemented
5	C₃×#1812	—	3	1	no	maximal
6	#72012	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
8	C₃²×#131	—	2	2	no	—
9	C₃²×#131	—	2	2	no	—
10	C₂×C₃²×M₁₆	C₄	1	2	yes	complemented
11	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃²×C₈	C₄	1	2	yes	complemented
13	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃×#926	C₂²	1	4	yes	—
15	C₃×#834	—	2	4	no	—
16	C₃×#818	—	2	2	no	—
17	C₃×#817	—	2	2	no	—
18	C₂×C₃×#131	—	2	1	no	—
19	C₂×C₃×#131	D₃	1	1	yes	characteristic
20	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
21	C₃×#427	—	3	2	no	—
22	C₃×#410	—	3	2	no	—
23	#5240	C₂×C₃	1	2	yes	—
24	#5106	C₂×C₃	1	2	yes	complemented
25	C₃×H(3,ℤ/4)	—	3	2	no	—
26	#4557	C₂×C₃	1	2	yes	complemented
27	C₃²×#41	—	2	2	no	—
28	C₃²×#41	—	2	2	no	—
29	C₂×C₃²×Q₈	Q₈	1	1	yes	characteristic
30	C₂×C₃²×D₄	D₄	1	2	yes	—
31	C₂×C₃²×D₄	Q₈	1	1	yes	characteristic
32	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₃²×M₁₆	—	2	2	no	—
35	C₃²×M₁₆	—	2	4	no	—
36	C₂×C₃²×C₈	—	2	2	no	—
37	C₂×C₃²×C₈	—	2	2	no	—
38	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	D₄	1	2	yes	—
42	C₂×C₃²×C₈	D₄	1	2	yes	—
43	C₃×#217	—	2	4	no	—
44	C₃×C₄⋊Dic₃	—	2	4	no	—
45	C₃×C₄×Dic₃	—	2	4	no	—
46	#1812	—	3	1	no	Sylow
47	C₂×C₃×#41	—	2	1	no	—
48	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
49	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
50	C₃×#131	—	2	4	no	—
51	C₃×#131	—	2	4	no	—
52	C₃×#131	—	2	2	no	—
53	C₃×#131	—	2	2	no	—
54	C₃×#131	—	2	2	no	—
55	C₃×#131	—	2	2	no	—
56	C₂×C₃×M₁₆	—	2	2	no	—
57	C₂×C₃×M₁₆	—	2	1	no	—
58	C₂×C₃×M₁₆	Dic₃	1	2	yes	complemented
59	C₂×C₃×M₁₆	C₃×C₄	1	2	yes	complemented
60	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	—	2	2	no	—
63	C₂²×C₃×C₈	—	2	1	no	—
64	C₂²×C₃×C₈	Dic₃	1	2	yes	complemented
65	C₂²×C₃×C₈	C₃×C₄	1	2	yes	complemented
66	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
68	C₃×#117	—	3	4	no	—
69	#926	C₂²×C₃	1	4	yes	—
70	C₃×#107	—	6	2	no	—
71	C₃×#106	—	6	2	no	—
72	C₃×#99	—	6	4	no	—
73	#834	—	2	4	no	—
74	#818	—	2	2	no	—
75	#817	—	2	2	no	—
76	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
77	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
78	C₃²×Q₈	—	2	2	no	—
79	C₃²×D₄	—	2	4	no	—
80	C₃²×D₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
84	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
86	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₃²×C₄	#41	1	2	yes	—
88	C₂×C₃×Dic₃	—	4	4	no	—
89	C₃²×C₈	—	2	2	no	—
90	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
93	C₂×#131	C₃×D₃	1	1	yes	characteristic
94	#427	—	3	2	no	—
95	#410	—	3	2	no	—
96	H(3,ℤ/4)	—	3	2	no	—
97	C₃×#41	—	2	4	no	—
98	C₃×#41	—	2	4	no	—
99	C₃×#41	—	2	2	no	—
100	C₃×#41	—	2	2	no	—
101	C₃×#41	—	2	2	no	—
102	C₃×#41	—	2	2	no	—
103	C₂×C₃×Q₈	—	2	1	no	—
104	C₂×C₃×Q₈	Dic₆	1	1	yes	characteristic
105	C₂×C₃×Q₈	C₃×Q₈	1	1	yes	characteristic
106	C₂×C₃×D₄	—	2	1	no	—
107	C₂×C₃×D₄	—	2	2	no	—
108	C₂×C₃×D₄	Dic₆	1	1	yes	characteristic
109	C₂×C₃×D₄	D₁₂	1	2	yes	—
110	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
111	C₂×C₃×D₄	C₃×Q₈	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
116	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₃×M₁₆	—	2	4	no	—
119	C₃×M₁₆	—	2	8	no	—
120	C₃×M₁₆	—	2	2	no	—
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	2	4	no	—
123	C₃×M₁₆	—	2	4	no	—
124	C₂×C₃×C₈	—	2	4	no	—
125	C₂×C₃×C₈	—	2	4	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
136	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
137	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
138	C₂×C₃×C₈	#67	1	2	yes	—
139	C₂×C₃×C₈	#67	1	2	yes	—
140	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
141	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
142	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
144	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
145	C₃×C₄⋊C₄	—	6	4	no	—
146	C₃×C₂²⋊C₄	—	6	4	no	—
147	C₃×C₄²	—	6	4	no	—
148	#217	—	2	4	no	—
149	C₄⋊Dic₃	—	2	4	no	—
150	C₄×Dic₃	—	2	4	no	—
151	C₂²×C₃²	—	2	2	no	—
152	C₂²×C₃²	—	4	1	no	—
153	C₂²×C₃²	—	4	2	no	—
154	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₃²	#121	1	2	yes	—
156	C₃²×C₄	—	2	2	no	—
157	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₃²×C₄	#117	1	1	yes	characteristic
159	C₃²×C₄	C₂²⋊Q₈	1	2	yes	—
160	C₃×Dic₃	—	8	4	no	—
161	C₂×#41	C₂×C₃×D₃	1	1	yes	characteristic
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	C₃×Dic₃	1	2	yes	complemented
165	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
166	C₂²×C₈	C₃×Dic₃	1	2	yes	complemented
167	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
168	#117	—	3	4	no	—
169	#107	—	6	2	no	—
170	#106	—	6	2	no	—
171	#99	—	6	4	no	—
172	C₂³×C₃	—	2	2	no	—
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
175	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
176	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
177	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
178	C₃×Q₈	—	2	4	no	—
179	C₃×Q₈	—	2	2	no	—
180	C₃×Q₈	—	2	2	no	—
181	C₃×D₄	—	2	8	no	—
182	C₃×D₄	—	2	4	no	—
183	C₃×D₄	—	2	4	no	—
184	C₃×D₄	—	2	4	no	—
185	C₃×D₄	—	2	2	no	—
186	C₃×D₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	12	4	no	—
199	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
200	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
201	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
202	C₂×C₃×C₄	#235	1	2	yes	—
203	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×#67	1	2	yes	—
205	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
208	C₂×C₃×C₄	C₃×#41	1	2	yes	—
209	C₂×Dic₃	—	4	4	no	—
210	C₃×C₈	—	2	4	no	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	—	2	2	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	#217	1	1	yes	characteristic
217	C₃×C₈	#217	1	1	yes	characteristic
218	C₃×C₈	#217	1	2	yes	—
219	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
220	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
222	C₂×C₃²	—	2	1	no	—
223	C₂×C₃²	—	2	2	no	—
224	C₂×C₃²	—	4	2	no	—
225	C₂×C₃²	#380	1	1	yes	characteristic
226	#41	—	2	2	no	—
227	#41	—	2	2	no	—
228	C₂×Q₈	C₃×Dic₆	1	1	yes	characteristic
229	C₂×D₄	C₃×Dic₆	1	1	yes	characteristic
230	C₂×D₄	C₃×D₁₂	1	2	yes	—
231	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
232	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
238	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
239	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
240	C₂×C₈	C₃×#67	1	2	yes	—
241	C₂×C₈	C₃×#67	1	2	yes	—
242	C₄⋊C₄	—	6	4	no	—
243	C₂²⋊C₄	—	6	4	no	—
244	C₄²	—	6	4	no	—
245	C₂²×C₃	—	2	4	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	4	1	no	—
251	C₂²×C₃	—	4	1	no	—
252	C₂²×C₃	—	4	2	no	—
253	C₂²×C₃	—	4	2	no	—
254	C₂²×C₃	—	4	2	no	—
255	C₂²×C₃	—	4	4	no	—
256	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
257	C₂²×C₃	#930	1	2	yes	—
258	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
259	C₂²×C₃	C₃×#121	1	2	yes	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	2	2	no	—
262	C₃×C₄	—	2	2	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	24	4	no	—
267	C₃×C₄	#924	1	2	yes	—
268	C₃×C₄	#926	1	1	yes	characteristic
269	C₃×C₄	C₂×#217	1	1	yes	characteristic
270	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
271	C₃×C₄	C₃×#117	1	1	yes	characteristic
272	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
273	Dic₃	—	8	4	no	—
274	C₃²	#1812	1	1	yes	characteristic, complemented, Sylow
275	C₂³	C₃×C₄⋊Dic₃	1	2	yes	—
276	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	D₄	—	2	4	no	—
279	D₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	12	4	no	—
283	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
284	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
285	C₂×C₄	C₃×#235	1	2	yes	—
286	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₃×#67	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	C₃×#217	1	1	yes	characteristic
290	C₈	C₃×#217	1	1	yes	characteristic
291	C₈	C₃×#217	1	2	yes	—
292	C₂×C₃	—	2	2	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	2	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	4	2	no	—
299	C₂×C₃	—	4	4	no	—
300	C₂×C₃	—	4	2	no	—
301	C₂×C₃	—	4	2	no	—
302	C₂×C₃	#5212	1	1	yes	characteristic
303	C₂×C₃	C₃×#380	1	1	yes	characteristic
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
308	C₂²	C₃×#930	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	24	4	no	—
311	C₄	C₃×#924	1	2	yes	—
312	C₄	C₃×#926	1	1	yes	characteristic
313	C₄	C₂×C₃×#217	1	1	yes	characteristic
314	C₃	—	2	1	no	—
315	C₃	#72012	1	1	yes	characteristic, direct product factor
316	C₃	C₃×#1812	1	1	yes	characteristic, complemented
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	—	4	2	no	—
320	C₂	C₃×#5212	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	#926
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂×C₃²	#380
prime radical	C₂×C₃²×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#131	D₃
hypocenter	C₃	C₃×#1812
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×M₁₆ ◁ C₂×C₃×#131

Complements

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

N	H ≅ G/N	type
3. C₃×#5106	C₂	semidirect
4. C₃×#4557	C₂	semidirect
6. #72012	C₃	direct (N × H)
10. C₂×C₃²×M₁₆	C₄	semidirect
12. C₂²×C₃²×C₈	C₄	semidirect
24. #5106	C₂×C₃	semidirect
26. #4557	C₂×C₃	semidirect
58. C₂×C₃×M₁₆	Dic₃	semidirect
59. C₂×C₃×M₁₆	C₃×C₄	semidirect
64. C₂²×C₃×C₈	Dic₃	semidirect
65. C₂²×C₃×C₈	C₃×C₄	semidirect
164. C₂×M₁₆	C₃×Dic₃	semidirect
166. C₂²×C₈	C₃×Dic₃	semidirect
274. C₃²	#1812	semidirect
315. C₃	#72012	direct (N × H)
316. C₃	C₃×#1812	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×#131	2
C₃×#5240	2
C₃×#5106	2
C₃×#4557	2
C₃×#1812	3
#72012	3

Subgroups — C3×#72012 #11785487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72012 #11785487