Subgroups — C₉×#5024 #13142773

Order 1728.

352 conjugacy classes of subgroups.

The subgroup lattice has 250 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	C₉×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
3	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₉×#803	C₂	1	2	yes	maximal
5	C₄×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃×#5024	C₃	1	1	yes	maximal, characteristic
7	C₉×#446	—	3	1	no	maximal
8	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
11	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	2	yes	—
13	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	—
18	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
19	C₃×#803	C₂×C₃	1	2	yes	—
20	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
21	C₄×Q₈×C₉	—	2	1	no	—
22	C₄×Q₈×C₉	D₃	1	1	yes	characteristic
23	C₉×C₄⋊C₈	—	3	1	no	—
24	C₉×C₄⋊C₈	—	3	2	no	—
25	C₉×#97	—	3	2	no	—
26	C₄×C₈×C₉	—	3	1	no	—
27	C₃×Q₈×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₉×C₃⋊C₈	—	2	1	no	—
34	C₉×C₃⋊C₈	—	2	2	no	—
35	C₉×C₃⋊C₈	Q₈	1	2	yes	complemented
36	C₃×#446	—	3	1	no	—
37	#5024	C₉	1	1	yes	characteristic, direct product factor
38	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
39	C₃²×C₄⋊C₄	C₃×C₄	1	2	yes	—
40	C₃²×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
41	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
42	C₃²×C₄²	C₂²×C₃	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
45	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	Dic₃	1	1	yes	characteristic
48	C₂×C₈×C₉	—	3	1	no	—
49	C₂×C₈×C₉	—	3	1	no	—
50	C₂×C₈×C₉	—	3	2	no	—
51	C₉×C₄⋊C₄	—	2	2	no	—
52	C₉×C₄⋊C₄	—	2	1	no	—
53	C₉×C₄⋊C₄	Dic₃	1	2	yes	—
54	C₉×C₄⋊C₄	Dic₃	1	1	yes	characteristic
55	C₄²×C₉	—	2	1	no	—
56	C₄²×C₉	—	2	2	no	—
57	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
58	C₄²×C₉	C₂×D₃	1	2	yes	—
59	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₄×C₉	—	2	2	no	—
61	C₃×C₄×C₉	—	2	1	no	—
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	M₁₆	1	2	yes	—
64	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
65	C₃×C₄×C₉	#41	1	1	yes	characteristic
66	C₃×C₄×Q₈	—	2	1	no	—
67	C₃×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
68	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
69	C₃×C₄⋊C₈	—	3	1	no	—
70	C₃×C₄⋊C₈	—	3	2	no	—
71	C₃×#97	—	3	2	no	—
72	C₃×C₄×C₈	—	3	1	no	—
73	C₄⋊(C₃⋊C₈)	C₂×C₉	1	2	yes	—
74	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic
75	#803	C₂×C₉	1	2	yes	—
76	C₄×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic
77	C₃²×Q₈	—	2	2	no	—
78	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
80	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
81	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
83	C₃×C₃⋊C₈	—	2	1	no	—
84	C₃×C₃⋊C₈	—	2	2	no	—
85	C₃×C₃⋊C₈	C₃×Q₈	1	2	yes	—
86	Q₈×C₉	—	2	4	no	—
87	Q₈×C₉	—	2	2	no	—
88	C₂×C₄×C₉	—	2	2	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
95	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
96	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
98	C₈×C₉	—	3	2	no	—
99	C₈×C₉	—	6	1	no	—
100	C₈×C₉	—	6	2	no	—
101	#446	—	3	1	no	Sylow
102	C₂×C₃×C₉	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₃×C₉	#131	1	1	yes	characteristic
105	C₂×C₃×Q₈	—	2	1	no	—
106	C₂×C₃×Q₈	C₄×C₉	1	1	yes	characteristic
107	C₂×C₃×Q₈	C₃×Dic₃	1	1	yes	characteristic
108	C₂×C₃×C₈	—	3	1	no	—
109	C₂×C₃×C₈	—	3	1	no	—
110	C₂×C₃×C₈	—	3	2	no	—
111	C₃×C₄⋊C₄	—	2	2	no	—
112	C₃×C₄⋊C₄	—	2	1	no	—
113	C₃×C₄⋊C₄	C₄×C₉	1	2	yes	—
114	C₃×C₄⋊C₄	C₄×C₉	1	1	yes	characteristic
115	C₃×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
116	C₃×C₄⋊C₄	C₃×Dic₃	1	1	yes	characteristic
117	C₃×C₄²	—	2	1	no	—
118	C₃×C₄²	—	2	2	no	—
119	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
120	C₃×C₄²	C₂²×C₉	1	2	yes	—
121	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
122	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
123	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
124	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
125	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
126	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₃²×C₄	—	2	2	no	—
128	C₃²×C₄	—	2	1	no	—
129	C₃²×C₄	—	2	2	no	—
130	C₃²×C₄	C₃×M₁₆	1	2	yes	—
131	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
132	C₃²×C₄	C₃×#41	1	1	yes	characteristic
133	C₂²×C₉	—	2	1	no	—
134	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
135	C₄×C₉	—	2	4	no	—
136	C₄×C₉	—	2	2	no	—
137	C₄×C₉	—	2	1	no	—
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	2	1	no	—
141	C₄×C₉	—	2	1	no	—
142	C₄×C₉	—	4	1	no	—
143	C₄×C₉	—	4	2	no	—
144	C₄×C₉	#208	1	2	yes	—
145	C₄×C₉	D₃×Q₈	1	1	yes	characteristic
146	C₄×C₉	#239	1	1	yes	characteristic
147	C₄×Q₈	D₃×C₉	1	1	yes	characteristic
148	C₄⋊C₈	—	3	1	no	—
149	C₄⋊C₈	—	3	2	no	—
150	#97	—	3	2	no	—
151	C₄×C₈	—	3	1	no	—
152	C₃×C₉	#446	1	1	yes	characteristic, complemented, Sylow
153	C₃×Q₈	—	2	4	no	—
154	C₃×Q₈	—	2	2	no	—
155	C₃×Q₈	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
163	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
164	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
169	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
171	C₃×C₈	—	3	2	no	—
172	C₃×C₈	—	6	1	no	—
173	C₃×C₈	—	6	2	no	—
174	C₃⋊C₈	—	2	1	no	—
175	C₃⋊C₈	—	2	2	no	—
176	C₃⋊C₈	Q₈×C₉	1	2	yes	complemented
177	C₂×C₃²	C₃×C₄×Q₈	1	1	yes	characteristic
178	C₂×C₃²	C₂×C₃×M₁₆	1	1	yes	characteristic
179	C₂×C₃²	C₃×#131	1	1	yes	characteristic
180	C₂×C₉	—	2	1	no	—
181	C₂×C₉	—	2	1	no	—
182	C₂×C₉	—	2	1	no	—
183	C₂×C₉	C₂×#208	1	1	yes	characteristic
184	C₂×C₉	Q₈×Dic₃	1	1	yes	characteristic
185	C₂×C₉	#948	1	1	yes	characteristic
186	C₂×Q₈	C₉×Dic₃	1	1	yes	characteristic
187	C₂×C₈	—	3	1	no	—
188	C₂×C₈	—	3	1	no	—
189	C₂×C₈	—	3	2	no	—
190	C₄⋊C₄	C₉×Dic₃	1	2	yes	—
191	C₄⋊C₄	C₉×Dic₃	1	1	yes	characteristic
192	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
193	C₄²	C₂×D₃×C₉	1	2	yes	—
194	C₂²×C₃	—	2	1	no	—
195	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
196	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
197	C₃×C₄	—	2	4	no	—
198	C₃×C₄	—	2	2	no	—
199	C₃×C₄	—	2	2	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	2	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	—	4	2	no	—
209	C₃×C₄	C₉×M₁₆	1	2	yes	—
210	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
211	C₃×C₄	C₉×#41	1	1	yes	characteristic
212	C₃×C₄	C₃×#208	1	2	yes	—
213	C₃×C₄	C₃×D₃×Q₈	1	1	yes	characteristic
214	C₃×C₄	C₃×#239	1	1	yes	characteristic
215	C₃²	C₃×#446	1	1	yes	characteristic
216	C₉	—	2	1	no	—
217	C₉	#5024	1	1	yes	characteristic, direct product factor
218	Q₈	—	2	2	no	—
219	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
220	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
221	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
222	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
223	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
224	C₈	—	3	2	no	—
225	C₈	—	6	1	no	—
226	C₈	—	6	2	no	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	C₄×Q₈×C₉	1	1	yes	characteristic
231	C₂×C₃	C₂×C₉×M₁₆	1	1	yes	characteristic
232	C₂×C₃	C₉×#131	1	1	yes	characteristic
233	C₂×C₃	C₂×C₃×#208	1	1	yes	characteristic
234	C₂×C₃	C₃×Q₈×Dic₃	1	1	yes	characteristic
235	C₂×C₃	C₃×#948	1	1	yes	characteristic
236	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	C₉×#208	1	2	yes	—
241	C₄	D₃×Q₈×C₉	1	1	yes	characteristic
242	C₄	C₉×#239	1	1	yes	characteristic
243	C₃	—	2	1	no	—
244	C₃	C₉×#446	1	1	yes	characteristic, complemented
245	C₃	C₃×#5024	1	1	yes	characteristic
246	C₂	C₂×C₉×#208	1	1	yes	characteristic
247	C₂	Q₈×C₉×Dic₃	1	1	yes	characteristic
248	C₂	C₉×#948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂²×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×Q₈×C₉	D₃
hypocenter	C₃	C₉×#446
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₄×Q₈×C₉

Complements

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

N	H ≅ G/N	type
35. C₉×C₃⋊C₈	Q₈	semidirect
37. #5024	C₉	direct (N × H)
152. C₃×C₉	#446	semidirect
176. C₃⋊C₈	Q₈×C₉	semidirect
217. C₉	#5024	direct (N × H)
244. C₃	C₉×#446	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
C₉×C₄⋊(C₃⋊C₈)	2
C₉×#803	2
C₄×C₉×C₃⋊C₈	2
C₃×#5024	3
C₉×#446	3

Subgroups — C9×#5024 #13142773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5024 #13142773