Subgroups — C₉×#5569 #13152675

Order 1728.

670 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₉×#890	C₂	1	3	yes	maximal
5	C₉×#888	C₂	1	3	yes	maximal
6	C₉×#870	C₂	1	3	yes	maximal
7	C₄×C₉×Dic₆	C₂	1	3	yes	maximal
8	C₃×#5569	C₃	1	1	yes	maximal, characteristic
9	C₉×#557	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×Dic₆	C₂²	1	3	yes	—
12	C₃×C₉×C₄⋊C₄	C₂²	1	3	yes	—
13	C₃×C₄²×C₉	C₂²	1	3	yes	—
14	C₉×C₄⋊Dic₃	C₂²	1	3	yes	—
15	C₉×C₄⋊Dic₃	C₂²	1	3	yes	—
16	C₉×C₄⋊Dic₃	C₂²	1	3	yes	—
17	C₉×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
18	C₉×Dic₃⋊C₄	C₂²	1	3	yes	—
19	C₉×Dic₃⋊C₄	C₂²	1	6	yes	—
20	C₄×C₉×Dic₃	C₂²	1	3	yes	—
21	C₄×C₉×Dic₃	C₂²	1	3	yes	—
22	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
23	C₃×Q₈×Dic₃	C₂×C₃	1	1	yes	characteristic
24	C₃×Q₈×Dic₃	C₂×C₃	1	1	yes	characteristic
25	C₃×#890	C₂×C₃	1	3	yes	—
26	C₃×#888	C₂×C₃	1	3	yes	—
27	C₃×#870	C₂×C₃	1	3	yes	—
28	C₃×C₄×Dic₆	C₂×C₃	1	3	yes	—
29	C₉×C₄⋊Q₈	—	3	3	no	—
30	C₉×#125	—	3	3	no	—
31	C₉×#125	—	3	3	no	—
32	C₄×Q₈×C₉	—	2	1	no	—
33	C₄×Q₈×C₉	—	3	1	no	—
34	C₄×Q₈×C₉	—	3	1	no	—
35	C₄×Q₈×C₉	—	3	3	no	—
36	C₄×Q₈×C₉	D₃	1	1	yes	characteristic
37	C₃×Q₈×C₉	Q₈	1	4	yes	complemented
38	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
39	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
41	C₂×C₉×Dic₃	C₂³	1	3	yes	—
42	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
43	C₂×C₉×Dic₃	C₂³	1	3	yes	—
44	C₂×C₉×Dic₃	C₂³	1	1	yes	characteristic
45	C₉×Dic₆	—	2	6	no	—
46	C₃×#557	—	3	1	no	—
47	#5569	C₉	1	1	yes	characteristic, direct product factor
48	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×Dic₆	C₂²×C₃	1	3	yes	—
50	C₃²×C₄⋊C₄	C₂²×C₃	1	3	yes	—
51	C₃²×C₄²	C₂²×C₃	1	3	yes	—
52	C₃×C₄⋊Dic₃	C₂²×C₃	1	3	yes	—
53	C₃×C₄⋊Dic₃	C₂²×C₃	1	3	yes	—
54	C₃×C₄⋊Dic₃	C₂²×C₃	1	3	yes	—
55	C₃×Dic₃⋊C₄	C₂²×C₃	1	6	yes	—
56	C₃×Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
57	C₃×Dic₃⋊C₄	C₂²×C₃	1	3	yes	—
58	C₃×C₄×Dic₃	C₂²×C₃	1	3	yes	—
59	C₃×C₄×Dic₃	C₂²×C₃	1	3	yes	—
60	C₂×Q₈×C₉	—	2	1	no	—
61	C₂×Q₈×C₉	—	3	3	no	—
62	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
63	C₉×C₄⋊C₄	—	2	3	no	—
64	C₉×C₄⋊C₄	—	3	1	no	—
65	C₉×C₄⋊C₄	—	3	3	no	—
66	C₉×C₄⋊C₄	—	3	3	no	—
67	C₉×C₄⋊C₄	—	3	3	no	—
68	C₉×C₄⋊C₄	—	3	3	no	—
69	C₉×C₄⋊C₄	—	3	6	no	—
70	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
71	C₄²×C₉	—	2	3	no	—
72	C₄²×C₉	—	3	3	no	—
73	C₄²×C₉	—	3	3	no	—
74	C₄²×C₉	C₂×D₃	1	3	yes	—
75	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
76	C₃×C₄×C₉	—	2	3	no	—
77	C₃×C₄×C₉	—	2	1	no	—
78	C₃×C₄×C₉	C₂×Q₈	1	6	yes	—
79	C₉×Dic₃	—	2	3	no	—
80	C₉×Dic₃	—	2	1	no	—
81	C₉×Dic₃	—	2	3	no	—
82	C₉×Dic₃	#41	1	2	yes	—
83	C₃×C₄⋊Q₈	—	3	3	no	—
84	C₃×#125	—	3	3	no	—
85	C₃×#125	—	3	3	no	—
86	C₃×C₄×Q₈	—	2	1	no	—
87	C₃×C₄×Q₈	—	3	1	no	—
88	C₃×C₄×Q₈	—	3	1	no	—
89	C₃×C₄×Q₈	—	3	3	no	—
90	C₃×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
91	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
92	Q₈×Dic₃	C₂×C₉	1	1	yes	characteristic
93	Q₈×Dic₃	C₂×C₉	1	1	yes	characteristic
94	#890	C₂×C₉	1	3	yes	—
95	#888	C₂×C₉	1	3	yes	—
96	#870	C₂×C₉	1	3	yes	—
97	C₄×Dic₆	C₂×C₉	1	3	yes	—
98	C₃²×Q₈	C₃×Q₈	1	4	yes	—
99	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
101	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
102	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
103	C₂×C₃×Dic₃	C₂³×C₃	1	3	yes	—
104	C₂×C₃×Dic₃	C₂³×C₃	1	3	yes	—
105	C₂×C₃×Dic₃	C₂³×C₃	1	1	yes	characteristic
106	C₃×Dic₆	—	2	6	no	—
107	Q₈×C₉	—	2	4	no	—
108	Q₈×C₉	—	6	6	no	—
109	Q₈×C₉	Dic₆	1	4	yes	complemented
110	C₂×C₄×C₉	—	2	1	no	—
111	C₂×C₄×C₉	—	2	3	no	—
112	C₂×C₄×C₉	—	2	3	no	—
113	C₂×C₄×C₉	—	3	1	no	—
114	C₂×C₄×C₉	—	3	1	no	—
115	C₂×C₄×C₉	—	3	3	no	—
116	C₂×C₄×C₉	—	3	3	no	—
117	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
119	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
120	#557	—	3	1	no	Sylow
121	C₂×C₃×C₉	C₂²×Q₈	1	1	yes	characteristic
122	C₂×C₃×C₉	C₂×#41	1	1	yes	characteristic
123	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₃×Q₈	—	2	1	no	—
125	C₂×C₃×Q₈	—	3	3	no	—
126	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
127	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
128	C₂×Dic₆	C₂²×C₉	1	3	yes	—
129	C₃×C₄⋊C₄	—	2	3	no	—
130	C₃×C₄⋊C₄	—	3	1	no	—
131	C₃×C₄⋊C₄	—	3	3	no	—
132	C₃×C₄⋊C₄	—	3	6	no	—
133	C₃×C₄⋊C₄	—	3	3	no	—
134	C₃×C₄⋊C₄	—	3	3	no	—
135	C₃×C₄⋊C₄	—	3	3	no	—
136	C₃×C₄⋊C₄	C₂²×C₉	1	3	yes	—
137	C₃×C₄⋊C₄	C₂×C₃×D₃	1	3	yes	—
138	C₃×C₄²	—	2	3	no	—
139	C₃×C₄²	—	3	3	no	—
140	C₃×C₄²	—	3	3	no	—
141	C₃×C₄²	C₂²×C₉	1	3	yes	—
142	C₃×C₄²	C₂×C₃×D₃	1	3	yes	—
143	C₄⋊Dic₃	C₂²×C₉	1	3	yes	—
144	C₄⋊Dic₃	C₂²×C₉	1	3	yes	—
145	C₄⋊Dic₃	C₂²×C₉	1	3	yes	—
146	Dic₃⋊C₄	C₂²×C₉	1	1	yes	characteristic
147	Dic₃⋊C₄	C₂²×C₉	1	3	yes	—
148	Dic₃⋊C₄	C₂²×C₉	1	6	yes	—
149	C₄×Dic₃	C₂²×C₉	1	3	yes	—
150	C₄×Dic₃	C₂²×C₉	1	3	yes	—
151	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
152	C₃²×C₄	—	2	1	no	—
153	C₃²×C₄	—	2	3	no	—
154	C₃²×C₄	C₂×C₃×Q₈	1	6	yes	—
155	C₃×Dic₃	—	2	3	no	—
156	C₃×Dic₃	—	2	1	no	—
157	C₃×Dic₃	—	2	3	no	—
158	C₃×Dic₃	C₃×#41	1	2	yes	—
159	C₂²×C₉	—	2	1	no	—
160	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
161	C₄×C₉	—	2	2	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	3	no	—
164	C₄×C₉	—	2	6	no	—
165	C₄×C₉	—	3	2	no	—
166	C₄×C₉	—	4	3	no	—
167	C₄×C₉	—	6	1	no	—
168	C₄×C₉	—	6	3	no	—
169	C₄×C₉	—	6	3	no	—
170	C₄×C₉	C₂×Dic₆	1	6	yes	—
171	C₄⋊Q₈	—	3	3	no	—
172	#125	—	3	3	no	—
173	#125	—	3	3	no	—
174	C₄×Q₈	—	3	1	no	—
175	C₄×Q₈	—	3	1	no	—
176	C₄×Q₈	—	3	3	no	—
177	C₄×Q₈	D₃×C₉	1	1	yes	characteristic
178	C₃×C₉	#557	1	1	yes	characteristic, complemented, Sylow
179	C₃×Q₈	—	2	4	no	—
180	C₃×Q₈	—	6	6	no	—
181	C₃×Q₈	Q₈×C₉	1	4	yes	complemented
182	C₃×Q₈	C₃×Dic₆	1	4	yes	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	3	no	—
185	C₂×C₃×C₄	—	2	3	no	—
186	C₂×C₃×C₄	—	3	1	no	—
187	C₂×C₃×C₄	—	3	1	no	—
188	C₂×C₃×C₄	—	3	3	no	—
189	C₂×C₃×C₄	—	3	3	no	—
190	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
192	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
193	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
195	C₂×C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
196	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
197	C₂×Dic₃	C₂³×C₉	1	3	yes	—
198	C₂×Dic₃	C₂³×C₉	1	3	yes	—
199	C₂×Dic₃	C₂³×C₉	1	1	yes	characteristic
200	Dic₆	—	2	6	no	—
201	C₂×C₃²	C₂²×C₃×Q₈	1	1	yes	characteristic
202	C₂×C₃²	C₂×C₃×#41	1	1	yes	characteristic
203	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
204	C₂×C₉	—	2	1	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	C₂²×Dic₆	1	1	yes	characteristic
208	C₂×C₉	#1007	1	1	yes	characteristic
209	C₂×C₉	D₃×#41	1	1	yes	characteristic
210	C₂×Q₈	—	3	3	no	—
211	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
212	C₄⋊C₄	—	3	1	no	—
213	C₄⋊C₄	—	3	3	no	—
214	C₄⋊C₄	—	3	3	no	—
215	C₄⋊C₄	—	3	3	no	—
216	C₄⋊C₄	—	3	3	no	—
217	C₄⋊C₄	—	3	6	no	—
218	C₄⋊C₄	C₂×D₃×C₉	1	3	yes	—
219	C₄²	—	3	3	no	—
220	C₄²	—	3	3	no	—
221	C₄²	C₂×D₃×C₉	1	3	yes	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
224	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	3	no	—
228	C₃×C₄	—	2	3	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	6	no	—
231	C₃×C₄	—	3	2	no	—
232	C₃×C₄	—	4	3	no	—
233	C₃×C₄	—	6	1	no	—
234	C₃×C₄	—	6	3	no	—
235	C₃×C₄	—	6	3	no	—
236	C₃×C₄	C₂×Q₈×C₉	1	6	yes	—
237	C₃×C₄	C₂×C₃×Dic₆	1	6	yes	—
238	Dic₃	—	2	3	no	—
239	Dic₃	—	2	1	no	—
240	Dic₃	—	2	3	no	—
241	Dic₃	C₉×#41	1	2	yes	—
242	C₃²	C₃×#557	1	1	yes	characteristic
243	C₉	—	2	1	no	—
244	C₉	#5569	1	1	yes	characteristic, direct product factor
245	Q₈	—	6	6	no	—
246	Q₈	C₉×Dic₆	1	4	yes	complemented
247	C₂×C₄	—	3	1	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	3	no	—
250	C₂×C₄	—	3	3	no	—
251	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
253	C₂×C₄	C₂²×D₃×C₉	1	3	yes	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	C₂²×Q₈×C₉	1	1	yes	characteristic
258	C₂×C₃	C₂×C₉×#41	1	1	yes	characteristic
259	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
260	C₂×C₃	C₂²×C₃×Dic₆	1	1	yes	characteristic
261	C₂×C₃	C₃×#1007	1	1	yes	characteristic
262	C₂×C₃	C₃×D₃×#41	1	1	yes	characteristic
263	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	2	3	no	—
266	C₄	—	3	2	no	—
267	C₄	—	6	1	no	—
268	C₄	—	6	3	no	—
269	C₄	—	6	3	no	—
270	C₄	C₂×C₉×Dic₆	1	6	yes	—
271	C₃	—	2	1	no	—
272	C₃	C₉×#557	1	1	yes	characteristic, complemented
273	C₃	C₃×#5569	1	1	yes	characteristic
274	C₂	C₂²×C₉×Dic₆	1	1	yes	characteristic
275	C₂	C₉×#1007	1	1	yes	characteristic
276	C₂	D₃×C₉×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical, cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₄×Q₈×C₉	D₃
hypocenter	C₃	C₉×#557
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄×Q₈×C₉

Complements

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

N	H ≅ G/N	type
37. C₃×Q₈×C₉	Q₈	semidirect
47. #5569	C₉	direct (N × H)
109. Q₈×C₉	Dic₆	semidirect
178. C₃×C₉	#557	semidirect
181. C₃×Q₈	Q₈×C₉	semidirect
244. C₉	#5569	direct (N × H)
246. Q₈	C₉×Dic₆	semidirect
272. C₃	C₉×#557	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
Q₈×C₉×Dic₃	2
C₉×#890	2
C₉×#888	2
C₉×#870	2
C₄×C₉×Dic₆	2
C₃×#5569	3
C₉×#557	3

Subgroups — C9×#5569 #13152675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5569 #13152675