Subgroups — C₂×C₃×#63509 #13153644

Order 1728.

1146 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₃₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#63509	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×(C₂×D₉)⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#947	—	3	1	no	maximal
7	C₂×#63509	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃×D₉	C₂²	1	2	yes	—
9	C₂×C₃×Q₈×C₉	—	2	4	no	—
10	C₂×C₃×Q₈×C₉	C₂²	1	4	yes	complemented
11	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
12	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
13	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
14	C₂×C₃×D₃₆	—	2	4	no	—
15	C₂×C₃×D₃₆	C₂²	1	4	yes	—
16	C₃×(C₂×D₉)⋊C₄	C₂²	1	16	yes	complemented
17	C₃×C₄×Dic₉	C₂²	1	4	yes	complemented
18	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
19	C₂²×C₃×D₁₂	—	3	1	no	—
20	C₃×#947	—	3	8	no	—
21	C₂×C₃×(C₂×D₃)⋊C₄	—	3	4	no	—
22	C₂×C₃×C₄×Dic₃	—	3	1	no	—
23	C₂²×Q₈×C₉	—	2	1	no	—
24	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×D₃₆	C₂×C₃	1	1	yes	characteristic
26	#63509	C₂×C₃	1	8	yes	direct product factor
27	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	4	yes	complemented
28	C₂×C₄×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
30	C₂²×C₃×D₉	—	2	4	no	—
31	C₂²×C₃×D₉	—	2	8	no	—
32	C₂²×C₃×D₉	C₂³	1	4	yes	—
33	C₃×Q₈×C₉	—	2	8	no	—
34	C₂×C₃×C₄×C₉	—	2	4	no	—
35	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
36	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
37	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
38	C₂×C₃×Dic₉	—	2	4	no	—
39	C₂×C₃×Dic₉	C₂³	1	4	yes	—
40	C₃×D₃₆	—	2	8	no	—
41	C₂×C₃×#124	—	9	1	no	—
42	C₂×#947	—	3	1	no	—
43	C₂³×C₃×D₃	—	3	2	no	—
44	C₂×C₃²×Q₈	—	2	4	no	—
45	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
46	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
47	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×Dic₃	—	3	2	no	—
49	C₂×C₃×D₁₂	—	3	4	no	—
50	C₂×C₃×D₁₂	—	6	4	no	—
51	C₂³×D₉	C₂²×C₃	1	2	yes	—
52	C₃×(C₂×D₃)⋊C₄	—	3	16	no	—
53	C₃×C₄×Dic₃	—	3	4	no	—
54	C₂×Q₈×C₉	—	2	8	no	—
55	C₂×Q₈×C₉	—	2	4	no	—
56	C₂×Q₈×C₉	—	2	4	no	—
57	C₂×Q₈×C₉	C₂²×C₃	1	4	yes	complemented
58	C₂²×C₄×C₉	—	2	2	no	—
59	C₂²×C₄×C₉	—	2	1	no	—
60	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
61	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
62	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
63	C₂×D₃₆	—	2	4	no	—
64	C₂×D₃₆	C₂²×C₃	1	4	yes	—
65	(C₂×D₉)⋊C₄	C₂²×C₃	1	16	yes	complemented
66	C₄×Dic₉	C₂²×C₃	1	4	yes	complemented
67	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
68	C₂²×C₃×C₉	#41	1	4	yes	—
69	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
70	C₂×C₃×D₉	—	2	4	no	—
71	C₂×C₃×D₉	—	2	8	no	—
72	C₂×C₃×D₉	—	4	8	no	—
73	C₃×C₄×C₉	—	2	4	no	—
74	C₃×C₄×C₉	C₂×D₄	1	4	yes	—
75	C₃×Dic₉	—	2	4	no	—
76	C₂²×C₃×Q₈	—	2	1	no	—
77	C₂²×C₃×Q₈	D₉	1	1	yes	characteristic, complemented
78	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
79	C₂²×C₃×D₄	—	9	1	no	—
80	C₂²×D₁₂	—	3	1	no	—
81	C₃×#124	—	9	8	no	—
82	C₂×C₃×C₂²⋊C₄	—	9	4	no	—
83	C₂×C₃×C₄²	—	9	1	no	—
84	#947	—	3	8	no	—
85	C₂×(C₂×D₃)⋊C₄	—	3	4	no	—
86	C₂×C₄×Dic₃	—	3	1	no	—
87	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
88	C₂²×C₃×D₃	—	3	4	no	—
89	C₂²×C₃×D₃	—	6	8	no	—
90	C₂²×C₃×D₃	—	6	4	no	—
91	C₃²×Q₈	—	2	8	no	—
92	C₂×C₃²×C₄	—	2	4	no	—
93	C₂×C₃²×C₄	#67	1	4	yes	—
94	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
95	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
96	C₂×C₃×Dic₃	—	3	4	no	—
97	C₂×C₃×Dic₃	—	6	4	no	—
98	C₃×D₁₂	—	6	8	no	—
99	C₂³×C₉	—	2	1	no	—
100	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
101	C₂²×D₉	—	2	4	no	—
102	C₂²×D₉	—	2	8	no	—
103	C₂²×D₉	C₂³×C₃	1	4	yes	—
104	Q₈×C₉	—	2	16	no	—
105	Q₈×C₉	—	2	8	no	—
106	C₂×C₄×C₉	—	2	8	no	—
107	C₂×C₄×C₉	—	2	4	no	—
108	C₂×C₄×C₉	—	2	4	no	—
109	C₂×C₄×C₉	—	2	4	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	C₃×D₄	1	4	yes	—
112	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
113	C₂×C₄×C₉	C₂³×C₃	1	2	yes	—
114	C₂×Dic₉	—	2	4	no	—
115	C₂×Dic₉	C₂³×C₃	1	4	yes	—
116	D₃₆	—	2	8	no	—
117	C₂×#124	—	9	1	no	Sylow
118	C₂×C₃×C₉	#124	1	4	yes	complemented
119	C₂×C₃×C₉	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₃×C₉	C₂×#41	1	2	yes	—
121	C₃×D₉	—	4	4	no	—
122	C₂⁴×C₃	—	9	2	no	—
123	C₂³×D₃	—	3	2	no	—
124	C₂×C₃×Q₈	—	2	8	no	—
125	C₂×C₃×Q₈	—	2	4	no	—
126	C₂×C₃×Q₈	—	2	4	no	—
127	C₂×C₃×Q₈	—	2	4	no	—
128	C₂×C₃×Q₈	C₂×D₉	1	4	yes	complemented
129	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
130	C₂×C₃×D₄	—	9	4	no	—
131	C₂×C₃×D₄	—	18	4	no	—
132	C₂²×C₃×C₄	—	2	2	no	—
133	C₂²×C₃×C₄	—	2	1	no	—
134	C₂²×C₃×C₄	—	9	2	no	—
135	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
136	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
138	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
139	C₂²×Dic₃	—	3	2	no	—
140	C₂×D₁₂	—	3	4	no	—
141	C₂×D₁₂	—	6	4	no	—
142	C₃×C₂²⋊C₄	—	9	16	no	—
143	C₃×C₄²	—	9	4	no	—
144	(C₂×D₃)⋊C₄	—	3	16	no	—
145	C₄×Dic₃	—	3	4	no	—
146	C₂²×C₃²	#239	1	4	yes	—
147	C₂²×C₃²	C₂×#67	1	2	yes	—
148	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
149	C₂×C₃×D₃	—	6	8	no	—
150	C₂×C₃×D₃	—	6	4	no	—
151	C₂×C₃×D₃	—	12	8	no	—
152	C₃²×C₄	—	2	4	no	—
153	C₃²×C₄	C₂×#67	1	4	yes	—
154	C₃×Dic₃	—	6	4	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	4	no	—
157	C₂²×C₉	—	2	2	no	—
158	C₂²×C₉	C₂×C₃×D₄	1	2	yes	—
159	C₂²×C₉	C₃×#41	1	4	yes	—
160	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
161	C₂×D₉	—	2	4	no	—
162	C₂×D₉	—	2	8	no	—
163	C₂×D₉	—	4	8	no	—
164	C₄×C₉	—	2	8	no	—
165	C₄×C₉	—	2	4	no	—
166	C₄×C₉	—	2	4	no	—
167	C₄×C₉	C₂×C₃×D₄	1	4	yes	—
168	Dic₉	—	2	4	no	—
169	C₂²×Q₈	C₃×D₉	1	1	yes	characteristic, complemented
170	C₂²×D₄	—	9	1	no	—
171	#124	—	9	8	no	—
172	C₂×C₂²⋊C₄	—	9	4	no	—
173	C₂×C₄²	—	9	1	no	—
174	C₃×C₉	C₂×#124	1	1	yes	characteristic, complemented, Sylow
175	C₂³×C₃	—	2	1	no	—
176	C₂³×C₃	—	9	4	no	—
177	C₂³×C₃	—	18	8	no	—
178	C₂³×C₃	—	18	4	no	—
179	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
180	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
181	C₂²×D₃	—	3	4	no	—
182	C₂²×D₃	—	6	8	no	—
183	C₂²×D₃	—	6	4	no	—
184	C₃×Q₈	—	2	16	no	—
185	C₃×Q₈	—	2	8	no	—
186	C₃×Q₈	—	2	8	no	—
187	C₃×D₄	—	18	8	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	9	4	no	—
195	C₂×C₃×C₄	—	18	4	no	—
196	C₂×C₃×C₄	#611	1	4	yes	—
197	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
198	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
199	C₂×C₃×C₄	C₃×#67	1	4	yes	—
200	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
201	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
202	C₂×Dic₃	—	3	4	no	—
203	C₂×Dic₃	—	6	4	no	—
204	D₁₂	—	6	8	no	—
205	C₂×C₃²	#947	1	4	yes	—
206	C₂×C₃²	C₂×#239	1	2	yes	—
207	C₂×C₃²	C₂²×#67	1	1	yes	characteristic
208	C₃×D₃	—	12	4	no	—
209	C₂×C₉	—	2	4	no	—
210	C₂×C₉	—	2	2	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	C₃×#124	1	4	yes	complemented
213	C₂×C₉	C₂²×C₃×D₄	1	1	yes	characteristic
214	C₂×C₉	C₂×C₃×#41	1	2	yes	—
215	D₉	—	4	4	no	—
216	C₂⁴	—	9	2	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	C₂×C₃×D₉	1	4	yes	complemented
219	C₂×D₄	—	9	4	no	—
220	C₂×D₄	—	18	4	no	—
221	C₂²×C₄	—	9	2	no	—
222	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
223	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
224	C₂²⋊C₄	—	9	16	no	—
225	C₄²	—	9	4	no	—
226	C₂²×C₃	—	2	1	no	—
227	C₂²×C₃	—	2	4	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	18	8	no	—
230	C₂²×C₃	—	18	4	no	—
231	C₂²×C₃	—	36	8	no	—
232	C₂²×C₃	#3701	1	4	yes	—
233	C₂²×C₃	C₂×#611	1	2	yes	—
234	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
235	C₂²×C₃	C₃×#239	1	4	yes	—
236	C₂²×C₃	C₂×C₃×#67	1	2	yes	—
237	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
238	C₂×D₃	—	6	8	no	—
239	C₂×D₃	—	6	4	no	—
240	C₂×D₃	—	12	8	no	—
241	C₃×C₄	—	2	8	no	—
242	C₃×C₄	—	2	4	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	18	4	no	—
246	C₃×C₄	C₂×#611	1	4	yes	—
247	C₃×C₄	C₂×C₃×#67	1	4	yes	—
248	Dic₃	—	6	4	no	—
249	C₃²	C₂×#947	1	1	yes	characteristic
250	C₉	—	2	1	no	—
251	C₉	C₂×C₃×#124	1	1	yes	characteristic, complemented
252	C₂³	—	9	4	no	—
253	C₂³	—	18	8	no	—
254	C₂³	—	18	4	no	—
255	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
256	Q₈	—	2	8	no	—
257	D₄	—	18	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	9	4	no	—
260	C₂×C₄	—	18	4	no	—
261	C₂×C₄	C₃×#611	1	4	yes	—
262	C₂×C₄	C₂²×C₃×D₉	1	4	yes	—
263	C₂×C₄	C₂²×C₃×D₉	1	2	yes	—
264	C₂×C₃	—	2	2	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	4	no	—
267	C₂×C₃	—	36	4	no	—
268	C₂×C₃	#63509	1	4	yes	direct product factor
269	C₂×C₃	C₂×#3701	1	2	yes	—
270	C₂×C₃	C₂²×#611	1	1	yes	characteristic
271	C₂×C₃	C₃×#947	1	4	yes	—
272	C₂×C₃	C₂×C₃×#239	1	2	yes	—
273	C₂×C₃	C₂²×C₃×#67	1	1	yes	characteristic
274	D₃	—	12	4	no	—
275	C₂²	—	18	8	no	—
276	C₂²	—	18	4	no	—
277	C₂²	—	36	8	no	—
278	C₂²	C₃×#3701	1	4	yes	—
279	C₂²	C₂×C₃×#611	1	2	yes	—
280	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	18	4	no	—
283	C₄	C₂×C₃×#611	1	4	yes	—
284	C₃	—	2	1	no	—
285	C₃	C₂×#63509	1	1	yes	characteristic, direct product factor
286	C₃	C₂×C₃×#947	1	1	yes	characteristic
287	C₂	—	36	4	no	—
288	C₂	C₃×#63509	1	4	yes	direct product factor
289	C₂	C₂×C₃×#3701	1	2	yes	—
290	C₂	C₂²×C₃×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂⁴×C₃
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂²×C₃×Q₈×C₉	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂²×C₃×D₃₆	C₂
hypercenter	C₂²×C₃×Q₈	D₉
hypocenter	C₉	C₂×C₃×#124
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Q₈×C₉	C₂	semidirect
3. C₃×#63509	C₂	direct (N × H)
4. C₂×C₃×(C₂×D₉)⋊C₄	C₂	semidirect
5. C₂×C₃×C₄×Dic₉	C₂	semidirect
7. C₂×#63509	C₃	direct (N × H)
10. C₂×C₃×Q₈×C₉	C₂²	semidirect
16. C₃×(C₂×D₉)⋊C₄	C₂²	semidirect
17. C₃×C₄×Dic₉	C₂²	semidirect
24. C₂²×Q₈×C₉	C₂×C₃	semidirect
26. #63509	C₂×C₃	direct (N × H)
27. C₂×(C₂×D₉)⋊C₄	C₂×C₃	semidirect
28. C₂×C₄×Dic₉	C₂×C₃	semidirect
57. C₂×Q₈×C₉	C₂²×C₃	semidirect
65. (C₂×D₉)⋊C₄	C₂²×C₃	semidirect
66. C₄×Dic₉	C₂²×C₃	semidirect
77. C₂²×C₃×Q₈	D₉	semidirect
118. C₂×C₃×C₉	#124	semidirect
128. C₂×C₃×Q₈	C₂×D₉	semidirect
169. C₂²×Q₈	C₃×D₉	semidirect
174. C₃×C₉	C₂×#124	semidirect
212. C₂×C₉	C₃×#124	semidirect
218. C₂×Q₈	C₂×C₃×D₉	semidirect
251. C₉	C₂×C₃×#124	semidirect
268. C₂×C₃	#63509	direct (N × H)
285. C₃	C₂×#63509	direct (N × H)
288. C₂	C₃×#63509	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×Q₈×C₉	2
C₂²×C₃×D₃₆	2
C₃×#63509	2
C₂×C₃×(C₂×D₉)⋊C₄	2
C₂×C₃×C₄×Dic₉	2
C₂×C₃×#947	3
C₂×#63509	3

Subgroups — C2×C3×#63509 #13153644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63509 #13153644