Subgroups — C₃×#68847 #11786025

Order 1152.

553 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×#4975	C₂	1	2	yes	maximal
3	C₃²×#340	C₂	1	1	yes	maximal, characteristic
4	C₃×#4551	C₂	1	1	yes	maximal, characteristic
5	C₃×#4529	C₂	1	2	yes	maximal
6	C₃×#1834	—	3	1	no	maximal
7	#68847	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₃×Dic₃⋊C₄	—	2	2	no	—
12	C₂×C₃×Dic₃⋊C₄	—	2	2	no	—
13	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₈×Dic₃	C₄	1	4	yes	complemented
16	C₂×C₃×C₄×C₈	—	3	1	no	—
17	C₃×#384	—	3	2	no	—
18	C₂×C₈×Dic₃	C₂×C₃	1	1	yes	characteristic
19	#4975	C₂×C₃	1	2	yes	—
20	C₃×#340	—	2	1	no	—
21	C₃×#340	—	3	1	no	—
22	C₃×#340	—	3	2	no	—
23	C₃×#340	D₃	1	1	yes	characteristic
24	C₃×#340	C₂×C₃	1	1	yes	characteristic
25	#4551	C₂×C₃	1	1	yes	characteristic
26	#4529	C₂×C₃	1	2	yes	—
27	C₂²×C₃²×C₄	—	2	2	no	—
28	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₂²×C₃×Dic₃	—	2	2	no	—
30	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	D₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	—	2	2	no	—
33	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
34	C₃²×C₄⋊C₄	—	2	2	no	—
35	C₃²×C₄⋊C₄	—	2	2	no	—
36	C₃×C₄⋊Dic₃	—	2	2	no	—
37	C₃×C₄⋊Dic₃	—	2	2	no	—
38	C₃×Dic₃⋊C₄	—	4	4	no	—
39	C₃×Dic₃⋊C₄	—	4	4	no	—
40	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
41	C₃×C₄×Dic₃	Q₈	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	—	2	2	no	—
43	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
44	#1834	—	3	1	no	Sylow
45	C₂²×C₃×C₈	—	2	1	no	—
46	C₂²×C₃×C₈	—	3	1	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	—	2	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×C₄⋊C₄	C₂×D₃	1	2	yes	—
54	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
55	C₂×C₃×C₄²	—	3	1	no	—
56	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
57	C₂×Dic₃⋊C₄	—	2	2	no	—
58	C₂×Dic₃⋊C₄	—	2	2	no	—
59	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
61	C₃×C₄×C₈	—	3	4	no	—
62	C₈×Dic₃	C₃×C₄	1	4	yes	complemented
63	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	—	4	2	no	—
65	C₂×C₃²×C₄	—	4	2	no	—
66	C₂×C₃²×C₄	—	4	2	no	—
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃²×C₄	#41	1	2	yes	—
70	C₂×C₃²×C₄	#41	1	2	yes	—
71	C₂×C₃×Dic₃	—	4	2	no	—
72	C₂×C₃×Dic₃	—	4	2	no	—
73	C₂×C₃×Dic₃	—	4	2	no	—
74	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₃×Dic₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₃×Dic₃	D₈	1	2	yes	—
77	C₂×C₃×Dic₃	SD₁₆	1	4	yes	—
78	C₂×C₃×Dic₃	Q₁₆	1	2	yes	—
79	C₃²×C₈	—	2	2	no	—
80	C₃×C₃⋊C₈	—	2	2	no	—
81	C₂×C₄×C₈	—	3	1	no	—
82	#384	—	3	2	no	—
83	#340	—	3	1	no	—
84	#340	—	3	2	no	—
85	#340	C₃×D₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	2	no	—
87	C₂²×C₃×C₄	—	2	2	no	—
88	C₂²×C₃×C₄	—	2	1	no	—
89	C₂²×C₃×C₄	—	3	1	no	—
90	C₂²×C₃×C₄	—	3	1	no	—
91	C₂²×C₃×C₄	—	4	2	no	—
92	C₂²×C₃×C₄	—	6	2	no	—
93	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
95	C₂²×Dic₃	—	2	2	no	—
96	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
97	C₂²×Dic₃	C₃×D₄	1	1	yes	characteristic
98	C₂×C₃×C₈	—	2	2	no	—
99	C₂×C₃×C₈	—	2	2	no	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	3	2	no	—
102	C₂×C₃×C₈	—	4	2	no	—
103	C₂×C₃×C₈	—	6	2	no	—
104	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
105	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
106	C₃×C₄⋊C₄	—	2	4	no	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	4	2	no	—
112	C₃×C₄⋊C₄	—	6	2	no	—
113	C₃×C₄⋊C₄	—	6	2	no	—
114	C₃×C₄⋊C₄	—	12	4	no	—
115	C₃×C₄⋊C₄	—	12	4	no	—
116	C₃×C₄²	—	3	2	no	—
117	C₃×C₄²	—	3	2	no	—
118	C₄⋊Dic₃	—	2	2	no	—
119	C₄⋊Dic₃	—	2	2	no	—
120	Dic₃⋊C₄	—	4	4	no	—
121	Dic₃⋊C₄	—	4	4	no	—
122	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
123	C₄×Dic₃	C₃×Q₈	1	2	yes	—
124	C₂×C₃⋊C₈	—	2	2	no	—
125	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
126	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₃²	#124	1	2	yes	—
128	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
129	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
130	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
131	C₃²×C₄	—	8	2	no	—
132	C₃²×C₄	#117	1	1	yes	characteristic
133	C₃²×C₄	#117	1	1	yes	characteristic
134	C₃²×C₄	#125	1	2	yes	—
135	C₃×Dic₃	—	8	2	no	—
136	C₃×Dic₃	#106	1	4	yes	—
137	C₃×Dic₃	#107	1	4	yes	—
138	C₂²×C₈	—	3	1	no	—
139	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
140	C₂×C₄⋊C₄	—	3	2	no	—
141	C₂×C₄⋊C₄	—	6	2	no	—
142	C₂×C₄⋊C₄	—	6	2	no	—
143	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
144	C₂×C₄²	—	3	1	no	—
145	C₄×C₈	—	3	4	no	—
146	C₂³×C₃	—	2	1	no	—
147	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
148	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	3	2	no	—
154	C₂×C₃×C₄	—	3	2	no	—
155	C₂×C₃×C₄	—	3	2	no	—
156	C₂×C₃×C₄	—	3	4	no	—
157	C₂×C₃×C₄	—	3	2	no	—
158	C₂×C₃×C₄	—	4	4	no	—
159	C₂×C₃×C₄	—	4	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	—	8	2	no	—
166	C₂×C₃×C₄	—	8	2	no	—
167	C₂×C₃×C₄	—	12	2	no	—
168	C₂×C₃×C₄	—	12	2	no	—
169	C₂×C₃×C₄	—	12	2	no	—
170	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	#235	1	2	yes	—
172	C₂×C₃×C₄	#237	1	2	yes	—
173	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
176	C₂×C₃×C₄	C₃×#41	1	2	yes	—
177	C₂×C₃×C₄	C₃×#41	1	2	yes	—
178	C₂×Dic₃	—	4	2	no	—
179	C₂×Dic₃	—	4	2	no	—
180	C₂×Dic₃	—	4	2	no	—
181	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
182	C₂×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
183	C₂×Dic₃	C₃×D₈	1	2	yes	—
184	C₂×Dic₃	C₃×SD₁₆	1	4	yes	—
185	C₂×Dic₃	C₃×Q₁₆	1	2	yes	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	4	2	no	—
189	C₃×C₈	—	6	2	no	—
190	C₃⋊C₈	—	2	2	no	—
191	C₂×C₃²	#382	1	1	yes	characteristic
192	C₂×C₃²	C₂×#106	1	1	yes	characteristic
193	C₂×C₃²	C₂×#107	1	1	yes	characteristic
194	C₂×C₃²	#486	1	2	yes	—
195	C₂×C₃²	#487	1	2	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	3	1	no	—
199	C₂²×C₄	—	6	2	no	—
200	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	6	2	no	—
208	C₄⋊C₄	—	6	2	no	—
209	C₄⋊C₄	—	12	4	no	—
210	C₄⋊C₄	—	12	4	no	—
211	C₄²	—	3	2	no	—
212	C₄²	—	3	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	#885	1	2	yes	—
219	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
220	C₂²×C₃	D₃×D₈	1	1	yes	characteristic
221	C₂²×C₃	D₃×SD₁₆	1	2	yes	—
222	C₂²×C₃	D₃×Q₁₆	1	1	yes	characteristic
223	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
224	C₂²×C₃	C₃×#124	1	2	yes	—
225	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
226	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
227	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	3	4	no	—
232	C₃×C₄	—	3	4	no	—
233	C₃×C₄	—	8	4	no	—
234	C₃×C₄	—	8	2	no	—
235	C₃×C₄	—	8	2	no	—
236	C₃×C₄	—	24	2	no	—
237	C₃×C₄	#872	1	1	yes	characteristic
238	C₃×C₄	#877	1	1	yes	characteristic
239	C₃×C₄	#889	1	2	yes	—
240	C₃×C₄	C₃×#117	1	1	yes	characteristic
241	C₃×C₄	C₃×#117	1	1	yes	characteristic
242	C₃×C₄	C₃×#125	1	2	yes	—
243	Dic₃	—	8	2	no	—
244	Dic₃	C₃×#106	1	4	yes	—
245	Dic₃	C₃×#107	1	4	yes	—
246	C₃²	#1834	1	1	yes	characteristic, complemented, Sylow
247	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	3	2	no	—
250	C₂×C₄	—	3	2	no	—
251	C₂×C₄	—	3	4	no	—
252	C₂×C₄	—	3	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	12	2	no	—
257	C₂×C₄	—	12	2	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
260	C₂×C₄	C₃×#235	1	2	yes	—
261	C₂×C₄	C₃×#237	1	2	yes	—
262	C₂×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
263	C₈	—	2	2	no	—
264	C₈	—	6	2	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	—	2	2	no	—
268	C₂×C₃	—	2	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	#4650	1	1	yes	characteristic
271	C₂×C₃	#4760	1	2	yes	—
272	C₂×C₃	#4792	1	2	yes	—
273	C₂×C₃	D₃×#106	1	1	yes	characteristic
274	C₂×C₃	D₃×#107	1	1	yes	characteristic
275	C₂×C₃	C₃×#382	1	1	yes	characteristic
276	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
277	C₂×C₃	C₂×C₃×#107	1	1	yes	characteristic
278	C₂×C₃	C₃×#486	1	2	yes	—
279	C₂×C₃	C₃×#487	1	2	yes	—
280	C₂²	C₃×#885	1	2	yes	—
281	C₂²	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
282	C₂²	C₃×D₃×D₈	1	1	yes	characteristic
283	C₂²	C₃×D₃×SD₁₆	1	2	yes	—
284	C₂²	C₃×D₃×Q₁₆	1	1	yes	characteristic
285	C₄	—	3	4	no	—
286	C₄	—	3	4	no	—
287	C₄	—	8	2	no	—
288	C₄	—	24	2	no	—
289	C₄	C₃×#872	1	1	yes	characteristic
290	C₄	C₃×#877	1	1	yes	characteristic
291	C₄	C₃×#889	1	2	yes	—
292	C₃	—	2	1	no	—
293	C₃	#68847	1	1	yes	characteristic, direct product factor
294	C₃	C₃×#1834	1	1	yes	characteristic, complemented
295	C₂	C₃×#4650	1	1	yes	characteristic
296	C₂	C₃×#4760	1	2	yes	—
297	C₂	C₃×#4792	1	2	yes	—
298	C₂	C₃×D₃×#106	1	1	yes	characteristic
299	C₂	C₃×D₃×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#340	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#340	D₃
hypocenter	C₃	C₃×#1834
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#340

Complements

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

N	H ≅ G/N	type
7. #68847	C₃	direct (N × H)
15. C₃×C₈×Dic₃	C₄	semidirect
62. C₈×Dic₃	C₃×C₄	semidirect
246. C₃²	#1834	semidirect
293. C₃	#68847	direct (N × H)
294. C₃	C₃×#1834	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×Dic₃	2
C₃×#4975	2
C₃²×#340	2
C₃×#4551	2
C₃×#4529	2
C₃×#1834	3
#68847	3

Subgroups — C3×#68847 #11786025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68847 #11786025