Subgroups — C₃×#72007 #11796052

Order 1152.

573 conjugacy classes of subgroups.

The subgroup lattice has 298 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	C₂×C₃×#812	C₂	1	2	yes	maximal, complemented
3	C₃×#4553	C₂	1	4	yes	maximal, complemented
4	C₃×#2115	—	3	1	no	maximal
5	#72007	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×#41	C₄	1	1	yes	characteristic
7	C₃²×#131	—	2	2	no	—
8	C₃²×#131	—	2	2	no	—
9	C₂×C₃²×M₁₆	C₄	1	1	yes	characteristic
10	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
11	C₂²×C₃²×C₈	C₄	1	2	yes	—
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃×#812	—	2	2	no	—
14	C₃×#812	C₂²	1	4	yes	—
15	C₂×C₃×C₃⋊C₁₆	—	2	2	no	—
16	C₂×C₃×#131	—	2	1	no	—
17	C₂×C₃×#131	D₃	1	1	yes	characteristic
18	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×M₃₂	—	3	2	no	—
20	C₂×#812	C₂×C₃	1	2	yes	complemented
21	C₃×#349	—	3	4	no	—
22	#4553	C₂×C₃	1	4	yes	complemented
23	C₃²×#41	—	2	2	no	—
24	C₃²×#41	—	2	2	no	—
25	C₂×C₃²×Q₈	C₈	1	1	yes	characteristic
26	C₂×C₃²×D₄	C₈	1	2	yes	—
27	C₂×C₃²×D₄	C₈	1	1	yes	characteristic
28	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₃²×M₁₆	—	2	2	no	—
31	C₃²×M₁₆	—	2	4	no	—
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	—	2	2	no	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃²×C₈	D₄	1	2	yes	—
37	C₂×C₃²×C₈	D₄	1	2	yes	—
38	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
39	C₃×C₃⋊C₁₆	—	2	4	no	—
40	#2115	—	3	1	no	Sylow
41	C₂×C₃×#41	—	2	1	no	—
42	C₂×C₃×#41	Dic₃	1	1	yes	characteristic
43	C₂×C₃×#41	C₃×C₄	1	1	yes	characteristic
44	C₃×#131	—	2	4	no	—
45	C₃×#131	—	2	4	no	—
46	C₃×#131	—	2	2	no	—
47	C₃×#131	—	2	2	no	—
48	C₃×#131	—	2	2	no	—
49	C₃×#131	—	2	2	no	—
50	C₂×C₃×M₁₆	—	2	1	no	—
51	C₂×C₃×M₁₆	—	2	2	no	—
52	C₂×C₃×M₁₆	Dic₃	1	1	yes	characteristic
53	C₂×C₃×M₁₆	C₃×C₄	1	1	yes	characteristic
54	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
55	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
56	C₂²×C₃×C₈	—	2	2	no	—
57	C₂²×C₃×C₈	—	2	1	no	—
58	C₂²×C₃×C₈	Dic₃	1	2	yes	—
59	C₂²×C₃×C₈	C₃×C₄	1	2	yes	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₃×M₃₂	—	3	4	no	—
63	C₃×M₃₂	—	6	2	no	—
64	C₂×C₃×C₁₆	—	6	2	no	—
65	#812	—	2	2	no	—
66	#812	C₂²×C₃	1	4	yes	—
67	C₂×C₃⋊C₁₆	—	2	2	no	—
68	C₂³×C₃²	C₂×C₈	1	2	yes	—
69	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
70	C₃²×Q₈	—	2	2	no	—
71	C₃²×D₄	—	2	4	no	—
72	C₃²×D₄	—	2	2	no	—
73	C₂×C₃²×C₄	—	2	2	no	—
74	C₂×C₃²×C₄	—	2	2	no	—
75	C₂×C₃²×C₄	C₂²⋊C₄	1	2	yes	—
76	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
77	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃²×C₄	M₁₆	1	2	yes	—
79	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
80	C₃²×C₈	—	2	2	no	—
81	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₃²×C₈	C₂²⋊C₄	1	1	yes	characteristic
83	C₃²×C₈	C₂×D₄	1	2	yes	—
84	C₂×#131	C₃×D₃	1	1	yes	characteristic
85	C₂×M₃₂	—	3	2	no	—
86	#349	—	3	4	no	—
87	C₃×#41	—	2	4	no	—
88	C₃×#41	—	2	4	no	—
89	C₃×#41	—	2	2	no	—
90	C₃×#41	—	2	2	no	—
91	C₃×#41	—	2	2	no	—
92	C₃×#41	—	2	2	no	—
93	C₂×C₃×Q₈	—	2	1	no	—
94	C₂×C₃×Q₈	C₃⋊C₈	1	1	yes	characteristic
95	C₂×C₃×Q₈	C₃×C₈	1	1	yes	characteristic
96	C₂×C₃×D₄	—	2	2	no	—
97	C₂×C₃×D₄	—	2	1	no	—
98	C₂×C₃×D₄	C₃⋊C₈	1	2	yes	—
99	C₂×C₃×D₄	C₃⋊C₈	1	1	yes	characteristic
100	C₂×C₃×D₄	C₃×C₈	1	2	yes	—
101	C₂×C₃×D₄	C₃×C₈	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	—	2	1	no	—
104	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
105	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
107	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
108	C₃×M₁₆	—	2	4	no	—
109	C₃×M₁₆	—	2	8	no	—
110	C₃×M₁₆	—	2	2	no	—
111	C₃×M₁₆	—	2	2	no	—
112	C₃×M₁₆	—	2	4	no	—
113	C₃×M₁₆	—	2	4	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	—	2	2	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
126	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
127	C₂×C₃×C₈	#67	1	2	yes	—
128	C₂×C₃×C₈	#67	1	2	yes	—
129	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
130	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
132	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
133	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
135	C₃×C₁₆	—	6	4	no	—
136	C₃⋊C₁₆	—	2	4	no	—
137	C₂²×C₃²	—	2	2	no	—
138	C₂²×C₃²	—	4	1	no	—
139	C₂²×C₃²	—	4	2	no	—
140	C₂²×C₃²	C₂²⋊C₈	1	2	yes	—
141	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
142	C₃²×C₄	—	2	2	no	—
143	C₃²×C₄	C₂²⋊C₈	1	2	yes	—
144	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
146	C₂×#41	C₃×Dic₃	1	1	yes	characteristic
147	#131	—	2	2	no	—
148	#131	—	2	2	no	—
149	C₂×M₁₆	C₃×Dic₃	1	1	yes	characteristic
150	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
151	C₂²×C₈	C₃×Dic₃	1	2	yes	—
152	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
153	M₃₂	—	3	4	no	—
154	M₃₂	—	6	2	no	—
155	C₂×C₁₆	—	6	2	no	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	—	2	1	no	—
158	C₂³×C₃	C₂×C₃⋊C₈	1	2	yes	—
159	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
160	C₂³×C₃	C₂×C₃×C₈	1	2	yes	—
161	C₂³×C₃	C₂×C₃×C₈	1	1	yes	characteristic
162	C₃×Q₈	—	2	4	no	—
163	C₃×Q₈	—	2	2	no	—
164	C₃×Q₈	—	2	2	no	—
165	C₃×D₄	—	2	8	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	4	no	—
168	C₃×D₄	—	2	4	no	—
169	C₃×D₄	—	2	2	no	—
170	C₃×D₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
183	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
184	C₂×C₃×C₄	#208	1	2	yes	—
185	C₂×C₃×C₄	#217	1	2	yes	—
186	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
187	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
188	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
189	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
190	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
192	C₃×C₈	—	2	4	no	—
193	C₃×C₈	—	2	2	no	—
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	—	2	1	no	—
196	C₃×C₈	—	2	1	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	#217	1	1	yes	characteristic
199	C₃×C₈	#217	1	1	yes	characteristic
200	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
201	C₃×C₈	C₃×C₂²⋊C₄	1	1	yes	characteristic
202	C₃×C₈	C₂×#67	1	2	yes	—
203	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
204	C₂×C₃²	—	2	1	no	—
205	C₂×C₃²	—	2	2	no	—
206	C₂×C₃²	—	4	2	no	—
207	C₂×C₃²	C₂×C₂²⋊C₈	1	1	yes	characteristic
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	C₂×Q₈	C₃×C₃⋊C₈	1	1	yes	characteristic
211	C₂×D₄	C₃×C₃⋊C₈	1	2	yes	—
212	C₂×D₄	C₃×C₃⋊C₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
214	C₂²×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
220	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
221	C₂×C₈	C₃×#67	1	2	yes	—
222	C₂×C₈	C₃×#67	1	2	yes	—
223	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
224	C₁₆	—	6	4	no	—
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	4	1	no	—
231	C₂²×C₃	—	4	1	no	—
232	C₂²×C₃	—	4	2	no	—
233	C₂²×C₃	—	4	2	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	—	4	4	no	—
236	C₂²×C₃	#830	1	2	yes	—
237	C₂²×C₃	C₃×C₂²⋊C₈	1	2	yes	—
238	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
239	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	#830	1	2	yes	—
247	C₃×C₄	C₃×C₂²⋊C₈	1	2	yes	—
248	C₃×C₄	C₂×#208	1	1	yes	characteristic
249	C₃×C₄	C₂×#217	1	1	yes	characteristic
250	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
251	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
252	C₃²	#2115	1	1	yes	characteristic, complemented, Sylow
253	C₂³	C₂×C₃×C₃⋊C₈	1	2	yes	—
254	C₂³	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	C₂×C₃×C₃⋊C₈	1	2	yes	—
261	C₂×C₄	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
262	C₂×C₄	C₃×#208	1	2	yes	—
263	C₂×C₄	C₃×#217	1	2	yes	—
264	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	C₃×#217	1	1	yes	characteristic
267	C₈	C₃×#217	1	1	yes	characteristic
268	C₈	C₂×C₃×#67	1	2	yes	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	2	1	no	—
273	C₂×C₃	—	2	2	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	4	2	no	—
276	C₂×C₃	—	4	4	no	—
277	C₂×C₃	—	4	2	no	—
278	C₂×C₃	—	4	2	no	—
279	C₂×C₃	C₂×#830	1	1	yes	characteristic
280	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₃×#830	1	2	yes	—
285	C₂²	C₂²×C₃×C₃⋊C₈	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	C₃×#830	1	2	yes	—
288	C₄	C₂×C₃×#208	1	1	yes	characteristic
289	C₄	C₂×C₃×#217	1	1	yes	characteristic
290	C₃	—	2	1	no	—
291	C₃	#72007	1	1	yes	characteristic, direct product factor
292	C₃	C₃×#2115	1	1	yes	characteristic, complemented
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×C₃×#830	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	#217
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₂×C₃²×#131	C₂
socle	C₂×C₃²	C₂×C₂²⋊C₈
prime radical	C₂×C₃²×#41	C₄
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₂×C₃×#131	D₃
hypocenter	C₃	C₃×#2115
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₈ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#131

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#812	C₂	semidirect
3. C₃×#4553	C₂	semidirect
5. #72007	C₃	direct (N × H)
20. C₂×#812	C₂×C₃	semidirect
22. #4553	C₂×C₃	semidirect
252. C₃²	#2115	semidirect
291. C₃	#72007	direct (N × H)
292. C₃	C₃×#2115	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
C₂×C₃×#812	2
C₃×#4553	2
C₃×#2115	3
#72007	3

Subgroups — C3×#72007 #11796052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72007 #11796052