Subgroups — C₃×#72000 #11785461

Order 1152.

693 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	C₂²×C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#5207	C₂	1	2	yes	maximal
3	C₂×C₃×#819	C₂	1	2	yes	maximal, complemented
4	C₃×#4552	C₂	1	2	yes	maximal, complemented
5	C₃×#1811	—	3	1	no	maximal
6	#72000	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂³×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃²×M₁₆	—	2	4	no	—
9	C₂×C₃²×M₁₆	C₄	1	4	yes	—
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×#208	C₂²	1	4	yes	—
13	C₃×#830	—	2	4	no	—
14	C₃×#819	—	2	4	no	—
15	C₂²×C₃×M₁₆	—	2	1	no	—
16	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
17	C₂²×C₃×M₁₆	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×#108	—	3	2	no	—
19	C₃×#407	—	3	2	no	—
20	#5207	C₂×C₃	1	2	yes	—
21	C₂×#819	C₂×C₃	1	2	yes	complemented
22	C₃×#341	—	3	2	no	—
23	#4552	C₂×C₃	1	2	yes	complemented
24	C₂⁴×C₃²	Q₈	1	1	yes	characteristic
25	C₂²×C₃²×C₄	—	2	2	no	—
26	C₂²×C₃²×C₄	—	2	2	no	—
27	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
28	C₂²×C₃²×C₄	D₄	1	2	yes	—
29	C₂²×C₃²×C₄	Q₈	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₃²×M₁₆	—	2	8	no	—
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	D₄	1	4	yes	—
36	C₃×#208	—	2	4	no	—
37	C₃×#208	—	2	4	no	—
38	C₂×C₃×C₃⋊C₈	—	2	4	no	—
39	#1811	—	3	1	no	Sylow
40	C₂³×C₃×C₄	—	2	1	no	—
41	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
42	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₃×M₁₆	—	2	8	no	—
44	C₂×C₃×M₁₆	—	2	4	no	—
45	C₂×C₃×M₁₆	—	2	4	no	—
46	C₂×C₃×M₁₆	—	2	4	no	—
47	C₂×C₃×M₁₆	—	3	4	no	—
48	C₂×C₃×M₁₆	Dic₃	1	4	yes	—
49	C₂×C₃×M₁₆	C₃×C₄	1	4	yes	—
50	C₂²×C₃×C₈	—	2	1	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
56	C₂×#208	C₂²×C₃	1	4	yes	—
57	C₃×#108	—	6	4	no	—
58	C₃×C₂²⋊C₈	—	6	4	no	—
59	#830	—	2	4	no	—
60	#819	—	2	4	no	—
61	C₂³×C₃²	—	2	2	no	—
62	C₂³×C₃²	—	4	2	no	—
63	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
64	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	—	2	2	no	—
66	C₂×C₃²×C₄	—	2	2	no	—
67	C₂×C₃²×C₄	—	2	2	no	—
68	C₂×C₃²×C₄	—	2	4	no	—
69	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
70	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
73	C₂×C₃²×C₄	#41	1	2	yes	—
74	C₃²×C₈	—	2	2	no	—
75	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
76	C₃×C₃⋊C₈	—	4	4	no	—
77	C₂²×M₁₆	C₃×D₃	1	1	yes	characteristic
78	C₂×#108	—	3	2	no	—
79	#407	—	3	2	no	—
80	#341	—	3	2	no	—
81	C₂⁴×C₃	—	2	1	no	—
82	C₂⁴×C₃	Dic₆	1	1	yes	characteristic
83	C₂⁴×C₃	C₃×Q₈	1	1	yes	characteristic
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	—	2	4	no	—
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	—	2	2	no	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	1	no	—
91	C₂²×C₃×C₄	—	2	2	no	—
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	2	1	no	—
94	C₂²×C₃×C₄	Dic₆	1	1	yes	characteristic
95	C₂²×C₃×C₄	D₁₂	1	2	yes	—
96	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
97	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
98	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
99	C₂²×C₃×C₄	C₃×Q₈	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
102	C₃×M₁₆	—	2	16	no	—
103	C₃×M₁₆	—	2	8	no	—
104	C₃×M₁₆	—	2	8	no	—
105	C₃×M₁₆	—	6	4	no	—
106	C₃×M₁₆	—	6	4	no	—
107	C₂×C₃×C₈	—	2	4	no	—
108	C₂×C₃×C₈	—	2	2	no	—
109	C₂×C₃×C₈	—	2	2	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	—	2	2	no	—
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	6	4	no	—
114	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
115	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
116	C₂×C₃×C₈	#67	1	4	yes	—
117	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
118	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
119	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
120	#208	—	2	4	no	—
121	#208	—	2	4	no	—
122	C₂×C₃⋊C₈	—	2	4	no	—
123	C₂²×C₃²	—	2	2	no	—
124	C₂²×C₃²	—	2	2	no	—
125	C₂²×C₃²	—	2	4	no	—
126	C₂²×C₃²	—	4	4	no	—
127	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
129	C₃²×C₄	—	2	2	no	—
130	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₃²×C₄	#117	1	1	yes	characteristic
132	C₃²×C₄	#121	1	2	yes	—
133	C₂³×C₄	C₂×C₃×D₃	1	1	yes	characteristic
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	3	4	no	—
136	C₂×M₁₆	C₃×Dic₃	1	4	yes	—
137	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
139	#108	—	6	4	no	—
140	C₂²⋊C₈	—	6	4	no	—
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	—	2	2	no	—
143	C₂³×C₃	—	2	2	no	—
144	C₂³×C₃	—	2	2	no	—
145	C₂³×C₃	—	2	1	no	—
146	C₂³×C₃	—	4	2	no	—
147	C₂³×C₃	—	4	2	no	—
148	C₂³×C₃	—	4	4	no	—
149	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
150	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
151	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
152	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	4	no	—
155	C₂×C₃×C₄	—	2	4	no	—
156	C₂×C₃×C₄	—	2	8	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	2	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	4	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	4	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
171	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
172	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
173	C₂×C₃×C₄	#235	1	2	yes	—
174	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×#67	1	2	yes	—
176	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
179	C₂×C₃×C₄	C₃×#41	1	2	yes	—
180	C₃×C₈	—	2	4	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	2	4	no	—
184	C₃×C₈	—	12	4	no	—
185	C₃×C₈	#217	1	4	yes	—
186	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
187	C₃⋊C₈	—	4	4	no	—
188	C₂×C₃²	—	2	2	no	—
189	C₂×C₃²	—	4	2	no	—
190	C₂×C₃²	#380	1	1	yes	characteristic
191	C₂×C₃²	#430	1	2	yes	—
192	C₂⁴	C₃×Dic₆	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₃×Dic₆	1	1	yes	characteristic
196	C₂²×C₄	C₃×D₁₂	1	2	yes	—
197	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
198	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
199	M₁₆	—	2	8	no	—
200	M₁₆	—	6	4	no	—
201	M₁₆	—	6	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	6	4	no	—
204	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
205	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
206	C₂×C₈	C₃×#67	1	4	yes	—
207	C₂²×C₃	—	2	4	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	2	4	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	4	4	no	—
218	C₂²×C₃	—	4	8	no	—
219	C₂²×C₃	—	4	4	no	—
220	C₂²×C₃	—	4	4	no	—
221	C₂²×C₃	#924	1	2	yes	—
222	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
223	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
224	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	#926	1	1	yes	characteristic
232	C₃×C₄	#930	1	2	yes	—
233	C₃×C₄	C₂×#217	1	1	yes	characteristic
234	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
235	C₃×C₄	C₃×#117	1	1	yes	characteristic
236	C₃×C₄	C₃×#121	1	2	yes	—
237	C₃²	#1811	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	2	2	no	—
239	C₂³	—	4	2	no	—
240	C₂³	C₃×C₄⋊Dic₃	1	2	yes	—
241	C₂³	C₂×C₃×Dic₆	1	1	yes	characteristic
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
247	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
248	C₂×C₄	C₃×#235	1	2	yes	—
249	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×#67	1	2	yes	—
251	C₈	—	2	2	no	—
252	C₈	—	12	4	no	—
253	C₈	C₃×#217	1	4	yes	—
254	C₂×C₃	—	2	4	no	—
255	C₂×C₃	—	2	2	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	2	no	—
259	C₂×C₃	—	4	4	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	—	4	2	no	—
262	C₂×C₃	#5125	1	2	yes	—
263	C₂×C₃	#5212	1	1	yes	characteristic
264	C₂×C₃	C₃×#380	1	1	yes	characteristic
265	C₂×C₃	C₃×#430	1	2	yes	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	4	4	no	—
270	C₂²	C₃×#924	1	2	yes	—
271	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	C₃×#926	1	1	yes	characteristic
274	C₄	C₃×#930	1	2	yes	—
275	C₄	C₂×C₃×#217	1	1	yes	characteristic
276	C₃	—	2	1	no	—
277	C₃	#72000	1	1	yes	characteristic, direct product factor
278	C₃	C₃×#1811	1	1	yes	characteristic, complemented
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	C₃×#5125	1	2	yes	—
282	C₂	C₃×#5212	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃²×M₁₆	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂⁴×C₃²	Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂²×C₃×M₁₆	D₃
hypocenter	C₃	C₃×#1811
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ C₂²×C₃×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×#819	C₂	semidirect
4. C₃×#4552	C₂	semidirect
6. #72000	C₃	direct (N × H)
21. C₂×#819	C₂×C₃	semidirect
23. #4552	C₂×C₃	semidirect
237. C₃²	#1811	semidirect
277. C₃	#72000	direct (N × H)
278. C₃	C₃×#1811	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×M₁₆	2
C₃×#5207	2
C₂×C₃×#819	2
C₃×#4552	2
C₃×#1811	3
#72000	3

Subgroups — C3×#72000 #11785461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72000 #11785461