Subgroups — C₃×#72014 #11785515

Order 1152.

637 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	C₃×#5237	C₂	1	2	yes	maximal
3	C₂×C₃×#819	C₂	1	2	yes	maximal, complemented
4	C₃×C₃⋊#338	C₂	1	2	yes	maximal, complemented
5	C₃×#1813	—	3	1	no	maximal
6	#72014	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×#41	C₂²	1	1	yes	characteristic
8	C₃²×#131	—	2	4	no	—
9	C₂×C₃²×M₁₆	C₄	1	2	yes	—
10	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃²×C₈	C₄	1	2	yes	—
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×#208	C₂²	1	4	yes	—
14	C₃×#836	—	2	2	no	—
15	C₃×#833	—	2	2	no	—
16	C₃×#819	—	2	4	no	—
17	C₂×C₃×#131	—	2	1	no	—
18	C₂×C₃×#131	D₃	1	1	yes	characteristic
19	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#108	—	3	2	no	—
21	C₃×#413	—	3	2	no	—
22	#5237	C₂×C₃	1	2	yes	—
23	C₂×#819	C₂×C₃	1	2	yes	complemented
24	C₃×#338	—	3	2	no	—
25	C₃⋊#338	C₂×C₃	1	2	yes	complemented
26	C₃²×#41	—	2	2	no	—
27	C₃²×#41	—	2	2	no	—
28	C₂×C₃²×Q₈	D₄	1	1	yes	characteristic
29	C₂×C₃²×D₄	D₄	1	1	yes	characteristic
30	C₂×C₃²×D₄	Q₈	1	2	yes	—
31	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₃²×M₁₆	—	2	2	no	—
34	C₃²×M₁₆	—	2	4	no	—
35	C₂×C₃²×C₈	—	2	4	no	—
36	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₈	D₄	1	4	yes	—
40	C₃×#208	—	2	4	no	—
41	C₃×#208	—	2	4	no	—
42	C₂×C₃×C₃⋊C₈	—	2	4	no	—
43	#1813	—	3	1	no	Sylow
44	C₂×C₃×#41	—	2	1	no	—
45	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
46	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
47	C₃×#131	—	2	8	no	—
48	C₃×#131	—	2	4	no	—
49	C₃×#131	—	2	4	no	—
50	C₂×C₃×M₁₆	—	2	2	no	—
51	C₂×C₃×M₁₆	—	2	1	no	—
52	C₂×C₃×M₁₆	—	3	4	no	—
53	C₂×C₃×M₁₆	Dic₃	1	2	yes	—
54	C₂×C₃×M₁₆	C₃×C₄	1	2	yes	—
55	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₃×C₈	—	2	2	no	—
58	C₂²×C₃×C₈	—	2	1	no	—
59	C₂²×C₃×C₈	Dic₃	1	2	yes	—
60	C₂²×C₃×C₈	C₃×C₄	1	2	yes	—
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×#208	C₂²×C₃	1	4	yes	—
64	C₃×#108	—	6	4	no	—
65	C₃×#101	—	6	2	no	—
66	C₃×#100	—	6	2	no	—
67	#836	—	2	2	no	—
68	#833	—	2	2	no	—
69	#819	—	2	4	no	—
70	C₂³×C₃²	C₄⋊C₄	1	2	yes	—
71	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
72	C₃²×Q₈	—	2	2	no	—
73	C₃²×D₄	—	2	4	no	—
74	C₃²×D₄	—	2	2	no	—
75	C₂×C₃²×C₄	—	2	2	no	—
76	C₂×C₃²×C₄	—	2	2	no	—
77	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
78	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
81	C₂×C₃²×C₄	#41	1	2	yes	—
82	C₃²×C₈	—	2	2	no	—
83	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
84	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
85	C₃×C₃⋊C₈	—	4	4	no	—
86	C₂×#131	C₃×D₃	1	1	yes	characteristic
87	C₂×#108	—	3	2	no	—
88	#413	—	3	2	no	—
89	#338	—	3	2	no	—
90	C₃×#41	—	2	4	no	—
91	C₃×#41	—	2	4	no	—
92	C₃×#41	—	2	2	no	—
93	C₃×#41	—	2	2	no	—
94	C₃×#41	—	2	2	no	—
95	C₃×#41	—	2	2	no	—
96	C₂×C₃×Q₈	—	2	1	no	—
97	C₂×C₃×Q₈	D₁₂	1	1	yes	characteristic
98	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
99	C₂×C₃×D₄	—	2	2	no	—
100	C₂×C₃×D₄	—	2	1	no	—
101	C₂×C₃×D₄	Dic₆	1	2	yes	—
102	C₂×C₃×D₄	D₁₂	1	1	yes	characteristic
103	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×D₄	C₃×Q₈	1	2	yes	—
105	C₂²×C₃×C₄	—	2	2	no	—
106	C₂²×C₃×C₄	—	2	1	no	—
107	C₂²×C₃×C₄	C₂×Dic₃	1	2	yes	—
108	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
111	C₃×M₁₆	—	2	4	no	—
112	C₃×M₁₆	—	2	8	no	—
113	C₃×M₁₆	—	2	2	no	—
114	C₃×M₁₆	—	2	2	no	—
115	C₃×M₁₆	—	2	4	no	—
116	C₃×M₁₆	—	2	4	no	—
117	C₃×M₁₆	—	6	4	no	—
118	C₃×M₁₆	—	6	4	no	—
119	C₂×C₃×C₈	—	2	8	no	—
120	C₂×C₃×C₈	—	2	4	no	—
121	C₂×C₃×C₈	—	2	4	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	4	no	—
126	C₂×C₃×C₈	—	6	4	no	—
127	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
128	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
129	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
130	C₂×C₃×C₈	#67	1	4	yes	—
131	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
132	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
135	#208	—	2	4	no	—
136	#208	—	2	4	no	—
137	C₂×C₃⋊C₈	—	2	4	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	—	4	1	no	—
140	C₂²×C₃²	—	4	2	no	—
141	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂²×C₃²	C₂²⋊Q₈	1	2	yes	—
143	C₃²×C₄	—	2	2	no	—
144	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	#117	1	1	yes	characteristic
146	C₃²×C₄	#121	1	2	yes	—
147	C₂×#41	C₂×C₃×D₃	1	1	yes	characteristic
148	#131	—	2	4	no	—
149	C₂×M₁₆	—	3	4	no	—
150	C₂×M₁₆	C₃×Dic₃	1	2	yes	—
151	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
152	C₂²×C₈	C₃×Dic₃	1	2	yes	—
153	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
154	#108	—	6	4	no	—
155	#101	—	6	2	no	—
156	#100	—	6	2	no	—
157	C₂³×C₃	—	2	2	no	—
158	C₂³×C₃	—	2	1	no	—
159	C₂³×C₃	C₄⋊Dic₃	1	2	yes	—
160	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
161	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
162	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
163	C₃×Q₈	—	2	4	no	—
164	C₃×Q₈	—	2	2	no	—
165	C₃×Q₈	—	2	2	no	—
166	C₃×D₄	—	2	8	no	—
167	C₃×D₄	—	2	4	no	—
168	C₃×D₄	—	2	4	no	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	2	2	no	—
171	C₃×D₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	4	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	2	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
184	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
185	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
186	C₂×C₃×C₄	#235	1	2	yes	—
187	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×#67	1	2	yes	—
189	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
192	C₂×C₃×C₄	C₃×#41	1	2	yes	—
193	C₃×C₈	—	2	4	no	—
194	C₃×C₈	—	2	2	no	—
195	C₃×C₈	—	2	2	no	—
196	C₃×C₈	—	2	2	no	—
197	C₃×C₈	—	2	2	no	—
198	C₃×C₈	—	12	4	no	—
199	C₃×C₈	#217	1	2	yes	—
200	C₃×C₈	#217	1	2	yes	—
201	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
202	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
203	C₃⋊C₈	—	4	4	no	—
204	C₂×C₃²	—	2	1	no	—
205	C₂×C₃²	—	2	2	no	—
206	C₂×C₃²	—	4	2	no	—
207	C₂×C₃²	#380	1	1	yes	characteristic
208	#41	—	2	2	no	—
209	#41	—	2	2	no	—
210	C₂×Q₈	C₃×D₁₂	1	1	yes	characteristic
211	C₂×D₄	C₃×Dic₆	1	2	yes	—
212	C₂×D₄	C₃×D₁₂	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₃×Dic₃	1	2	yes	—
214	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
215	M₁₆	—	2	2	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	6	4	no	—
218	M₁₆	—	6	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	6	4	no	—
221	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
222	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
223	C₂×C₈	C₂×C₃×Dic₃	1	1	yes	characteristic
224	C₂×C₈	C₃×#67	1	4	yes	—
225	C₂²×C₃	—	2	4	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	1	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	#924	1	2	yes	—
237	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
238	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
239	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
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	2	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	#926	1	1	yes	characteristic
247	C₃×C₄	#930	1	2	yes	—
248	C₃×C₄	C₂×#217	1	1	yes	characteristic
249	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
250	C₃×C₄	C₃×#117	1	1	yes	characteristic
251	C₃×C₄	C₃×#121	1	2	yes	—
252	C₃²	#1813	1	1	yes	characteristic, complemented, Sylow
253	C₂³	C₃×C₄⋊Dic₃	1	2	yes	—
254	C₂³	C₂×C₃×Dic₆	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	C₃×C₄⋊Dic₃	1	2	yes	—
261	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
262	C₂×C₄	C₃×#235	1	2	yes	—
263	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₃×#67	1	2	yes	—
265	C₈	—	2	2	no	—
266	C₈	—	12	4	no	—
267	C₈	C₃×#217	1	2	yes	—
268	C₈	C₃×#217	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	#5212	1	1	yes	characteristic
280	C₂×C₃	C₃×#380	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	C₃×#924	1	2	yes	—
285	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	C₃×#926	1	1	yes	characteristic
288	C₄	C₃×#930	1	2	yes	—
289	C₄	C₂×C₃×#217	1	1	yes	characteristic
290	C₃	—	2	1	no	—
291	C₃	#72014	1	1	yes	characteristic, direct product factor
292	C₃	C₃×#1813	1	1	yes	characteristic, complemented
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	—	4	2	no	—
296	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×#1813
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. C₂×C₃×#819	C₂	semidirect
4. C₃×C₃⋊#338	C₂	semidirect
6. #72014	C₃	direct (N × H)
23. C₂×#819	C₂×C₃	semidirect
25. C₃⋊#338	C₂×C₃	semidirect
252. C₃²	#1813	semidirect
291. C₃	#72014	direct (N × H)
292. C₃	C₃×#1813	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×#5237	2
C₂×C₃×#819	2
C₃×C₃⋊#338	2
C₃×#1813	3
#72014	3

Subgroups — C3×#72014 #11785515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72014 #11785515