Subgroups — C₉×#5731 #13153312

Order 1728.

644 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#946	C₂	1	2	yes	maximal
3	Q₈×C₉×Dic₃	C₂	1	2	yes	maximal, complemented
4	C₉×#898	C₂	1	4	yes	maximal
5	C₉×#889	C₂	1	4	yes	maximal, complemented
6	C₉×#872	C₂	1	1	yes	maximal, characteristic
7	C₉×#870	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5731	C₃	1	1	yes	maximal, characteristic
9	C₉×#541	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
11	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
15	C₉×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
16	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
18	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
19	C₉×Dic₃⋊C₄	C₂²	1	2	yes	—
20	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₄×C₉×Dic₃	C₂²	1	1	yes	characteristic
22	C₄×C₉×Dic₃	C₂²	1	4	yes	—
23	C₃²×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#946	C₂×C₃	1	2	yes	—
25	C₃×Q₈×Dic₃	C₂×C₃	1	2	yes	—
26	C₃×#898	C₂×C₃	1	4	yes	—
27	C₃×#889	C₂×C₃	1	4	yes	—
28	C₃×#872	C₂×C₃	1	1	yes	characteristic
29	C₃×#870	C₂×C₃	1	1	yes	characteristic
30	C₉×C₄⋊Q₈	—	2	1	no	—
31	C₉×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
32	C₉×#126	—	3	4	no	—
33	C₉×#125	—	3	4	no	—
34	C₉×#125	—	3	1	no	—
35	C₉×C₂²⋊Q₈	—	3	2	no	—
36	C₄×Q₈×C₉	—	3	2	no	—
37	C₉×#117	—	3	1	no	—
38	C₂²×D₃×C₉	C₂³	1	1	yes	characteristic
39	C₃×Q₈×C₉	—	2	4	no	—
40	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
41	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
42	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
44	C₂×C₉×Dic₃	C₂³	1	2	yes	—
45	C₂×C₉×Dic₃	C₂³	1	4	yes	—
46	C₄×D₃×C₉	—	2	2	no	—
47	C₃×#541	—	3	1	no	—
48	#5731	C₉	1	1	yes	characteristic, direct product factor
49	C₂×C₃²×Q₈	C₂²×C₃	1	2	yes	—
50	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
51	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
52	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
53	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
54	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
55	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
56	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
57	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
58	C₃×Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
59	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
60	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
61	C₃×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₂×Q₈×C₉	—	2	2	no	—
63	C₂×Q₈×C₉	C₂×D₃	1	2	yes	—
64	C₂²×C₄×C₉	—	3	1	no	—
65	C₉×C₄⋊C₄	—	2	4	no	—
66	C₉×C₄⋊C₄	—	3	2	no	—
67	C₉×C₄⋊C₄	—	3	2	no	—
68	C₉×C₄⋊C₄	—	3	4	no	—
69	C₉×C₄⋊C₄	—	3	4	no	—
70	C₉×C₄⋊C₄	—	3	4	no	—
71	C₉×C₄⋊C₄	C₂×D₃	1	4	yes	—
72	C₉×C₂²⋊C₄	—	3	2	no	—
73	C₉×C₂²⋊C₄	—	3	4	no	—
74	C₄²×C₉	—	2	1	no	—
75	C₄²×C₉	—	3	1	no	—
76	C₄²×C₉	—	3	4	no	—
77	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
78	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
79	C₂×D₃×C₉	—	2	1	no	—
80	C₂×D₃×C₉	—	2	2	no	—
81	C₃×C₄×C₉	—	2	4	no	—
82	C₃×C₄×C₉	—	2	2	no	—
83	C₃×C₄×C₉	#41	1	2	yes	—
84	C₉×Dic₃	—	2	1	no	—
85	C₉×Dic₃	—	2	2	no	—
86	C₉×Dic₃	—	2	4	no	—
87	C₃×C₄⋊Q₈	—	2	1	no	—
88	C₃×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
89	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
90	C₃×#126	—	3	4	no	—
91	C₃×#125	—	3	4	no	—
92	C₃×#125	—	3	1	no	—
93	C₃×C₂²⋊Q₈	—	3	2	no	—
94	C₃×C₄×Q₈	—	3	2	no	—
95	C₃×#117	—	3	1	no	—
96	#946	C₂×C₉	1	2	yes	—
97	Q₈×Dic₃	C₂×C₉	1	2	yes	complemented
98	#898	C₂×C₉	1	4	yes	—
99	#889	C₂×C₉	1	4	yes	complemented
100	#872	C₂×C₉	1	1	yes	characteristic
101	#870	C₂×C₉	1	1	yes	characteristic, complemented
102	C₂²×C₃×D₃	C₂³×C₃	1	1	yes	characteristic
103	C₃²×Q₈	—	2	4	no	—
104	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
105	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
106	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
107	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
108	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
109	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
110	C₃×C₄×D₃	—	2	2	no	—
111	C₂³×C₉	—	3	1	no	—
112	Q₈×C₉	—	2	4	no	—
113	Q₈×C₉	—	4	4	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	1	no	—
116	C₂×C₄×C₉	—	2	4	no	—
117	C₂×C₄×C₉	—	3	2	no	—
118	C₂×C₄×C₉	—	3	4	no	—
119	C₂×C₄×C₉	—	3	1	no	—
120	C₂×C₄×C₉	—	6	2	no	—
121	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
122	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
123	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
124	#541	—	3	1	no	Sylow
125	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
126	C₂×C₃×C₉	2₋¹⁺⁴	1	2	yes	—
127	D₃×C₉	—	4	1	no	—
128	C₂×C₃×Q₈	—	2	2	no	—
129	C₂×C₃×Q₈	C₂²×C₉	1	2	yes	—
130	C₂×C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
131	C₂²×C₃×C₄	—	3	1	no	—
132	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	3	2	no	—
135	C₃×C₄⋊C₄	—	3	4	no	—
136	C₃×C₄⋊C₄	—	3	4	no	—
137	C₃×C₄⋊C₄	—	3	2	no	—
138	C₃×C₄⋊C₄	—	3	4	no	—
139	C₃×C₄⋊C₄	C₂²×C₉	1	4	yes	—
140	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
141	C₃×C₂²⋊C₄	—	3	2	no	—
142	C₃×C₂²⋊C₄	—	3	4	no	—
143	C₃×C₄²	—	2	1	no	—
144	C₃×C₄²	—	3	1	no	—
145	C₃×C₄²	—	3	4	no	—
146	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
147	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
148	(C₂×D₃)⋊C₄	C₂²×C₉	1	2	yes	—
149	(C₂×D₃)⋊C₄	C₂²×C₉	1	4	yes	—
150	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
151	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
152	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
153	Dic₃⋊C₄	C₂²×C₉	1	2	yes	—
154	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
155	C₄×Dic₃	C₂²×C₉	1	1	yes	characteristic
156	C₄×Dic₃	C₂²×C₉	1	4	yes	—
157	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
158	C₂×C₃×D₃	—	2	1	no	—
159	C₂×C₃×D₃	—	2	2	no	—
160	C₃²×C₄	—	2	2	no	—
161	C₃²×C₄	—	2	4	no	—
162	C₃²×C₄	C₃×#41	1	2	yes	—
163	C₃×Dic₃	—	2	1	no	—
164	C₃×Dic₃	—	2	2	no	—
165	C₃×Dic₃	—	2	4	no	—
166	C₂²×C₉	—	2	1	no	—
167	C₂²×C₉	—	6	1	no	—
168	C₂²×C₉	—	6	2	no	—
169	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
170	C₄×C₉	—	2	4	no	—
171	C₄×C₉	—	2	4	no	—
172	C₄×C₉	—	2	2	no	—
173	C₄×C₉	—	2	2	no	—
174	C₄×C₉	—	4	4	no	—
175	C₄×C₉	—	6	1	no	—
176	C₄×C₉	—	6	2	no	—
177	C₄×C₉	—	6	4	no	—
178	C₄×C₉	#237	1	2	yes	—
179	C₄⋊Q₈	D₃×C₉	1	1	yes	characteristic, complemented
180	#126	—	3	4	no	—
181	#125	—	3	4	no	—
182	#125	—	3	1	no	—
183	C₂²⋊Q₈	—	3	2	no	—
184	C₄×Q₈	—	3	2	no	—
185	#117	—	3	1	no	—
186	C₃×C₉	#541	1	1	yes	characteristic, complemented, Sylow
187	C₂³×C₃	—	3	1	no	—
188	C₂²×D₃	C₂³×C₉	1	1	yes	characteristic
189	C₃×Q₈	—	2	4	no	—
190	C₃×Q₈	—	2	4	no	—
191	C₃×Q₈	—	4	4	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	3	2	no	—
196	C₂×C₃×C₄	—	3	4	no	—
197	C₂×C₃×C₄	—	3	1	no	—
198	C₂×C₃×C₄	—	6	2	no	—
199	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
200	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
201	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
203	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
204	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
205	C₂×Dic₃	C₂³×C₉	1	2	yes	—
206	C₂×Dic₃	C₂³×C₉	1	4	yes	—
207	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
208	C₄×D₃	—	2	2	no	—
209	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
210	C₂×C₃²	C₃×2₋¹⁺⁴	1	2	yes	—
211	C₃×D₃	—	4	1	no	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	2	no	—
214	C₂×C₉	—	12	1	no	—
215	C₂×C₉	C₂×#237	1	1	yes	characteristic
216	C₂×C₉	#1007	1	2	yes	—
217	C₂×Q₈	C₂×D₃×C₉	1	2	yes	—
218	C₂²×C₄	—	3	1	no	—
219	C₄⋊C₄	—	3	2	no	—
220	C₄⋊C₄	—	3	2	no	—
221	C₄⋊C₄	—	3	4	no	—
222	C₄⋊C₄	—	3	4	no	—
223	C₄⋊C₄	—	3	4	no	—
224	C₄⋊C₄	C₂×D₃×C₉	1	4	yes	—
225	C₂²⋊C₄	—	3	2	no	—
226	C₂²⋊C₄	—	3	4	no	—
227	C₄²	—	3	1	no	—
228	C₄²	—	3	4	no	—
229	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	6	1	no	—
232	C₂²×C₃	—	6	2	no	—
233	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
234	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
235	C₂×D₃	—	2	1	no	—
236	C₂×D₃	—	2	2	no	—
237	C₃×C₄	—	2	4	no	—
238	C₃×C₄	—	2	4	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	2	4	no	—
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	4	4	no	—
244	C₃×C₄	—	6	1	no	—
245	C₃×C₄	—	6	2	no	—
246	C₃×C₄	—	6	4	no	—
247	C₃×C₄	C₉×#41	1	2	yes	—
248	C₃×C₄	C₃×#237	1	2	yes	—
249	Dic₃	—	2	1	no	—
250	Dic₃	—	2	2	no	—
251	Dic₃	—	2	4	no	—
252	C₃²	C₃×#541	1	1	yes	characteristic
253	C₉	—	2	1	no	—
254	C₉	#5731	1	1	yes	characteristic, direct product factor
255	C₂³	—	3	1	no	—
256	Q₈	—	2	4	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	3	4	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
262	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
263	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	12	1	no	—
267	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
268	C₂×C₃	C₉×2₋¹⁺⁴	1	2	yes	—
269	C₂×C₃	C₂×C₃×#237	1	1	yes	characteristic
270	C₂×C₃	C₃×#1007	1	2	yes	—
271	D₃	—	4	1	no	—
272	C₂²	—	6	1	no	—
273	C₂²	—	6	2	no	—
274	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	6	1	no	—
278	C₄	—	6	2	no	—
279	C₄	—	6	4	no	—
280	C₄	C₉×#237	1	2	yes	—
281	C₃	—	2	1	no	—
282	C₃	C₉×#541	1	1	yes	characteristic, complemented
283	C₃	C₃×#5731	1	1	yes	characteristic
284	C₂	—	12	1	no	—
285	C₂	C₂×C₉×#237	1	1	yes	characteristic
286	C₂	C₉×#1007	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle, cosocle	C₂²×C₃²	C₂⁴×C₃
prime radical	C₂²×C₃×D₃	C₂³×C₃
hypercenter	C₉×C₄⋊Q₈	D₃
hypocenter	C₃	C₉×#541
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×C₉×C₄⋊Q₈	C₂	semidirect
3. Q₈×C₉×Dic₃	C₂	semidirect
5. C₉×#889	C₂	semidirect
7. C₉×#870	C₂	semidirect
31. C₉×C₄⋊Q₈	D₃	semidirect
48. #5731	C₉	direct (N × H)
88. C₃×C₄⋊Q₈	C₂×C₉	semidirect
97. Q₈×Dic₃	C₂×C₉	semidirect
99. #889	C₂×C₉	semidirect
101. #870	C₂×C₉	semidirect
179. C₄⋊Q₈	D₃×C₉	semidirect
186. C₃×C₉	#541	semidirect
254. C₉	#5731	direct (N × H)
282. C₃	C₉×#541	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
C₉×#946	2
Q₈×C₉×Dic₃	2
C₉×#898	2
C₉×#889	2
C₉×#872	2
C₉×#870	2
C₃×#5731	3
C₉×#541	3

Subgroups — C9×#5731 #13153312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5731 #13153312