Subgroups — C₉×#3080 #11767801

Order 1152.

576 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
3	C₉×#479	C₂	1	4	yes	maximal, complemented
4	C₉×#478	C₂	1	4	yes	maximal, complemented
5	C₉×#428	C₂	1	2	yes	maximal, complemented
6	C₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#417	C₂	1	2	yes	maximal
8	C₃×#3080	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	2	yes	—
11	C₉×#125	—	2	1	no	—
12	C₉×#125	—	2	1	no	—
13	C₉×#125	C₂²	1	4	yes	—
14	C₄×Q₈×C₉	C₂²	1	2	yes	—
15	C₄×D₄×C₉	—	2	2	no	—
16	C₄×D₄×C₉	C₂²	1	2	yes	—
17	C₉×#117	—	2	1	no	—
18	C₉×#117	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	—	2	2	no	—
20	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₉	—	2	1	no	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₉×#107	C₂²	1	4	yes	—
24	C₉×#106	C₂²	1	4	yes	—
25	C₉×C₄⋊C₈	C₂²	1	4	yes	—
26	C₉×#103	C₂²	1	4	yes	—
27	C₉×#102	C₂²	1	4	yes	—
28	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic
29	C₃×C₄×#41	C₂×C₃	1	1	yes	characteristic
30	C₃×#479	C₂×C₃	1	4	yes	—
31	C₃×#478	C₂×C₃	1	4	yes	—
32	C₃×#428	C₂×C₃	1	2	yes	—
33	C₃×#423	C₂×C₃	1	1	yes	characteristic
34	C₃×#417	C₂×C₃	1	2	yes	—
35	C₉×#41	—	2	2	no	—
36	C₉×#41	Q₈	1	4	yes	—
37	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₉	—	2	1	no	—
39	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	—	2	2	no	—
42	C₂²×C₄×C₉	—	2	2	no	—
43	C₂²×C₄×C₉	D₄	1	2	yes	—
44	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₉×M₁₆	—	2	4	no	—
46	C₂×C₈×C₉	C₂³	1	4	yes	—
47	C₉×C₄⋊C₄	—	2	4	no	—
48	C₉×C₄⋊C₄	—	2	4	no	—
49	C₉×C₄⋊C₄	—	2	2	no	—
50	C₉×C₄⋊C₄	—	2	2	no	—
51	C₉×C₄⋊C₄	C₂³	1	4	yes	—
52	C₉×C₄⋊C₄	C₂³	1	2	yes	—
53	C₉×C₂²⋊C₄	—	2	1	no	—
54	C₉×C₂²⋊C₄	—	2	2	no	—
55	C₄²×C₉	—	2	1	no	—
56	C₄²×C₉	—	2	2	no	—
57	C₄²×C₉	D₄	1	1	yes	characteristic
58	C₄²×C₉	D₄	1	1	yes	characteristic
59	C₄²×C₉	C₂³	1	2	yes	—
60	#3080	C₉	1	1	yes	characteristic, direct product factor, Sylow
61	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
63	C₃×#125	—	2	1	no	—
64	C₃×#125	—	2	1	no	—
65	C₃×#125	C₂²×C₃	1	4	yes	—
66	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
67	C₃×C₄×D₄	—	2	2	no	—
68	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
69	C₃×#117	—	2	1	no	—
70	C₃×#117	C₂²×C₃	1	1	yes	characteristic
71	C₂×C₃×C₄⋊C₄	—	2	2	no	—
72	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
73	C₂×C₃×C₄²	—	2	1	no	—
74	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
75	C₃×#107	C₂²×C₃	1	4	yes	—
76	C₃×#106	C₂²×C₃	1	4	yes	—
77	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
78	C₃×#103	C₂²×C₃	1	4	yes	—
79	C₃×#102	C₂²×C₃	1	4	yes	—
80	C₂³×C₉	—	2	1	no	—
81	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₉	—	2	1	no	—
83	Q₈×C₉	C₂×Q₈	1	2	yes	—
84	D₄×C₉	—	2	4	no	—
85	D₄×C₉	—	2	1	no	—
86	D₄×C₉	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	4	no	—
89	C₂×C₄×C₉	—	2	2	no	—
90	C₂×C₄×C₉	—	2	4	no	—
91	C₂×C₄×C₉	—	4	2	no	—
92	C₂×C₄×C₉	—	4	4	no	—
93	C₂×C₄×C₉	—	4	4	no	—
94	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₉	#41	1	2	yes	—
99	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
100	C₈×C₉	—	2	4	no	—
101	C₂×#125	C₂×C₉	1	1	yes	characteristic, complemented
102	C₄×#41	C₂×C₉	1	1	yes	characteristic
103	#479	C₂×C₉	1	4	yes	complemented
104	#478	C₂×C₉	1	4	yes	complemented
105	#428	C₂×C₉	1	2	yes	complemented
106	#423	C₂×C₉	1	1	yes	characteristic, complemented
107	#417	C₂×C₉	1	2	yes	—
108	C₃×#41	—	2	2	no	—
109	C₃×#41	C₃×Q₈	1	4	yes	—
110	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₃×D₄	—	2	1	no	—
112	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	2	2	no	—
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
117	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
118	C₃×M₁₆	—	2	4	no	—
119	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
120	C₃×C₄⋊C₄	—	2	4	no	—
121	C₃×C₄⋊C₄	—	2	4	no	—
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
125	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
126	C₃×C₂²⋊C₄	—	2	1	no	—
127	C₃×C₂²⋊C₄	—	2	2	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
131	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
132	C₃×C₄²	C₂³×C₃	1	2	yes	—
133	C₂²×C₉	—	2	2	no	—
134	C₂²×C₉	—	2	2	no	—
135	C₂²×C₉	—	4	2	no	—
136	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
137	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₉	—	2	2	no	—
139	C₄×C₉	—	2	2	no	—
140	C₄×C₉	—	2	2	no	—
141	C₄×C₉	—	4	2	no	—
142	C₄×C₉	—	4	4	no	—
143	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
144	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
145	C₄×C₉	C₂×#41	1	1	yes	characteristic
146	C₂×#41	C₂²×C₉	1	1	yes	characteristic
147	C₂×M₁₆	C₂²×C₉	1	2	yes	—
148	#125	—	2	1	no	—
149	#125	—	2	1	no	—
150	#125	C₂²×C₉	1	4	yes	—
151	C₄×Q₈	C₂²×C₉	1	2	yes	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	C₂²×C₉	1	2	yes	—
154	#117	—	2	1	no	—
155	#117	C₂²×C₉	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
160	#107	C₂²×C₉	1	4	yes	—
161	#106	C₂²×C₉	1	4	yes	—
162	C₄⋊C₈	C₂²×C₉	1	4	yes	—
163	#103	C₂²×C₉	1	4	yes	—
164	#102	C₂²×C₉	1	4	yes	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
167	C₃×Q₈	—	2	1	no	—
168	C₃×Q₈	C₂×C₃×Q₈	1	2	yes	—
169	C₃×D₄	—	2	4	no	—
170	C₃×D₄	—	2	1	no	—
171	C₃×D₄	C₂×C₃×Q₈	1	2	yes	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	4	no	—
176	C₂×C₃×C₄	—	4	2	no	—
177	C₂×C₃×C₄	—	4	4	no	—
178	C₂×C₃×C₄	—	4	4	no	—
179	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
183	C₂×C₃×C₄	C₃×#41	1	2	yes	—
184	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
185	C₃×C₈	—	2	4	no	—
186	C₂×C₉	—	2	2	no	—
187	C₂×C₉	—	4	2	no	—
188	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
189	C₂×C₉	#575	1	2	yes	—
190	#41	—	2	2	no	—
191	#41	Q₈×C₉	1	4	yes	—
192	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	D₄×C₉	1	2	yes	—
199	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	C₂×C₈	C₂³×C₉	1	4	yes	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	C₂³×C₉	1	4	yes	—
207	C₄⋊C₄	C₂³×C₉	1	2	yes	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	2	no	—
212	C₄²	D₄×C₉	1	1	yes	characteristic
213	C₄²	D₄×C₉	1	1	yes	characteristic
214	C₄²	C₂³×C₉	1	2	yes	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	C₃×C₂²⋊Q₈	1	2	yes	—
219	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	2	2	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	4	2	no	—
224	C₃×C₄	—	4	4	no	—
225	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
226	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
227	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
228	C₉	#3080	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	2	1	no	—
230	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	C₂×Q₈×C₉	1	2	yes	—
233	D₄	—	2	4	no	—
234	D₄	—	2	1	no	—
235	D₄	C₂×Q₈×C₉	1	2	yes	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
245	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
246	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
247	C₂×C₄	C₉×#41	1	2	yes	—
248	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₂×C₃	—	2	2	no	—
251	C₂×C₃	—	4	2	no	—
252	C₂×C₃	C₂×C₃×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₃	C₃×#575	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₉×C₂²⋊Q₈	1	2	yes	—
258	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	4	no	—
264	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
265	C₄	C₂²×Q₈×C₉	1	1	yes	characteristic
266	C₄	C₂×C₉×#41	1	1	yes	characteristic
267	C₃	C₃×#3080	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	2	no	—
270	C₂	C₂×C₉×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂	C₉×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂×C₃×#41	C₂²×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#125	C₂	semidirect
3. C₉×#479	C₂	semidirect
4. C₉×#478	C₂	semidirect
5. C₉×#428	C₂	semidirect
6. C₉×#423	C₂	semidirect
60. #3080	C₉	direct (N × H)
101. C₂×#125	C₂×C₉	semidirect
103. #479	C₂×C₉	semidirect
104. #478	C₂×C₉	semidirect
105. #428	C₂×C₉	semidirect
106. #423	C₂×C₉	semidirect
228. C₉	#3080	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#125	2
C₄×C₉×#41	2
C₉×#479	2
C₉×#478	2
C₉×#428	2
C₉×#423	2
C₉×#417	2
C₃×#3080	3

Subgroups — C9×#3080 #11767801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3080 #11767801