Subgroups — C₉×#3039 #11767760

Order 1152.

720 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#137	C₂	1	2	yes	maximal
2	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₉×#41	C₂	1	1	yes	maximal, characteristic
4	C₉×#462	C₂	1	4	yes	maximal, complemented
5	C₉×#461	C₂	1	4	yes	maximal, complemented
6	C₉×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#419	C₂	1	2	yes	maximal, complemented
8	C₃×#3039	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₉	C₂²	1	2	yes	—
11	C₉×#137	—	2	8	no	—
12	C₂×C₉×Q₁₆	C₂²	1	4	yes	—
13	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
14	C₂×C₉×M₁₆	C₂²	1	2	yes	—
15	C₉×C₄⋊Q₈	—	2	1	no	—
16	C₉×C₄⋊Q₈	—	2	1	no	—
17	C₉×C₄⋊Q₈	C₂²	1	4	yes	—
18	C₄×Q₈×C₉	C₂²	1	2	yes	—
19	C₄×D₄×C₉	—	2	2	no	—
20	C₄×D₄×C₉	C₂²	1	2	yes	—
21	C₉×#117	—	2	1	no	—
22	C₉×#117	C₂²	1	1	yes	characteristic
23	C₂×C₉×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄²×C₉	—	2	1	no	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	4	yes	—
27	C₉×#103	C₂²	1	8	yes	—
28	C₂×C₃×#137	C₂×C₃	1	2	yes	—
29	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
30	C₃×C₄×#41	C₂×C₃	1	1	yes	characteristic
31	C₃×#462	C₂×C₃	1	4	yes	—
32	C₃×#461	C₂×C₃	1	4	yes	—
33	C₃×#423	C₂×C₃	1	1	yes	characteristic
34	C₃×#419	C₂×C₃	1	2	yes	—
35	C₉×#41	—	2	2	no	—
36	C₉×#41	D₄	1	4	yes	—
37	C₂×Q₈×C₉	—	2	2	no	—
38	C₂×Q₈×C₉	—	2	4	no	—
39	C₂×Q₈×C₉	—	2	4	no	—
40	C₂×Q₈×C₉	C₂³	1	4	yes	—
41	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
42	C₂×D₄×C₉	—	2	1	no	—
43	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₉	—	2	1	no	—
45	C₂²×C₄×C₉	—	2	2	no	—
46	C₂²×C₄×C₉	—	2	2	no	—
47	C₂²×C₄×C₉	D₄	1	2	yes	—
48	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
49	C₉×Q₁₆	—	2	8	no	—
50	C₉×SD₁₆	—	2	8	no	—
51	C₉×M₁₆	—	2	4	no	—
52	C₂×C₈×C₉	C₂³	1	4	yes	—
53	C₉×C₄⋊C₄	—	2	4	no	—
54	C₉×C₄⋊C₄	—	2	4	no	—
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	C₂³	1	2	yes	—
57	C₉×C₂²⋊C₄	—	2	1	no	—
58	C₉×C₂²⋊C₄	—	2	2	no	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	—	2	2	no	—
61	C₄²×C₉	D₄	1	1	yes	characteristic
62	C₄²×C₉	D₄	1	1	yes	characteristic
63	C₄²×C₉	C₂³	1	2	yes	—
64	#3039	C₉	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
66	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
67	C₃×#137	—	2	8	no	—
68	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
69	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
70	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
71	C₃×C₄⋊Q₈	—	2	1	no	—
72	C₃×C₄⋊Q₈	—	2	1	no	—
73	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
74	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
75	C₃×C₄×D₄	—	2	2	no	—
76	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
77	C₃×#117	—	2	1	no	—
78	C₃×#117	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×C₄⋊C₄	—	2	2	no	—
80	C₂×C₃×C₄²	—	2	1	no	—
81	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
82	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
83	C₃×#103	C₂²×C₃	1	8	yes	—
84	C₂³×C₉	—	2	1	no	—
85	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₉	—	2	1	no	—
87	Q₈×C₉	—	2	4	no	—
88	Q₈×C₉	—	2	4	no	—
89	Q₈×C₉	—	4	4	no	—
90	Q₈×C₉	C₂×D₄	1	2	yes	—
91	D₄×C₉	—	2	4	no	—
92	D₄×C₉	—	2	1	no	—
93	D₄×C₉	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	4	no	—
96	C₂×C₄×C₉	—	2	4	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	4	4	no	—
100	C₂×C₄×C₉	—	4	4	no	—
101	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₉	#41	1	2	yes	—
106	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
107	C₈×C₉	—	2	4	no	—
108	C₂×#137	C₂×C₉	1	2	yes	—
109	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic, complemented
110	C₄×#41	C₂×C₉	1	1	yes	characteristic
111	#462	C₂×C₉	1	4	yes	complemented
112	#461	C₂×C₉	1	4	yes	complemented
113	#423	C₂×C₉	1	1	yes	characteristic, complemented
114	#419	C₂×C₉	1	2	yes	complemented
115	C₃×#41	—	2	2	no	—
116	C₃×#41	C₃×D₄	1	4	yes	—
117	C₂×C₃×Q₈	—	2	2	no	—
118	C₂×C₃×Q₈	—	2	4	no	—
119	C₂×C₃×Q₈	—	2	4	no	—
120	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
121	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
122	C₂×C₃×D₄	—	2	1	no	—
123	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	2	1	no	—
125	C₂²×C₃×C₄	—	2	2	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
128	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
129	C₃×Q₁₆	—	2	8	no	—
130	C₃×SD₁₆	—	2	8	no	—
131	C₃×M₁₆	—	2	4	no	—
132	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₄⋊C₄	—	2	4	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
137	C₃×C₂²⋊C₄	—	2	1	no	—
138	C₃×C₂²⋊C₄	—	2	2	no	—
139	C₃×C₄²	—	2	1	no	—
140	C₃×C₄²	—	2	2	no	—
141	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
142	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
143	C₃×C₄²	C₂³×C₃	1	2	yes	—
144	C₂²×C₉	—	2	2	no	—
145	C₂²×C₉	—	2	2	no	—
146	C₂²×C₉	—	4	2	no	—
147	C₂²×C₉	#121	1	2	yes	—
148	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₉	—	2	2	no	—
150	C₄×C₉	—	2	2	no	—
151	C₄×C₉	—	4	4	no	—
152	C₄×C₉	—	4	2	no	—
153	C₄×C₉	#121	1	2	yes	—
154	C₄×C₉	#137	1	4	yes	complemented
155	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
156	C₄×C₉	C₂×#41	1	1	yes	characteristic
157	C₂×#41	C₂²×C₉	1	1	yes	characteristic
158	C₂²×Q₈	C₂²×C₉	1	2	yes	—
159	#137	—	2	8	no	—
160	C₂×Q₁₆	C₂²×C₉	1	4	yes	—
161	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
162	C₂×M₁₆	C₂²×C₉	1	2	yes	—
163	C₄⋊Q₈	—	2	1	no	—
164	C₄⋊Q₈	—	2	1	no	—
165	C₄⋊Q₈	C₂²×C₉	1	4	yes	—
166	C₄×Q₈	C₂²×C₉	1	2	yes	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	C₂²×C₉	1	2	yes	—
169	#117	—	2	1	no	—
170	#117	C₂²×C₉	1	1	yes	characteristic
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	1	no	—
173	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
174	C₄⋊C₈	C₂²×C₉	1	4	yes	—
175	#103	C₂²×C₉	1	8	yes	—
176	C₂³×C₃	—	2	1	no	—
177	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
178	C₃×Q₈	—	2	1	no	—
179	C₃×Q₈	—	2	4	no	—
180	C₃×Q₈	—	2	4	no	—
181	C₃×Q₈	—	4	4	no	—
182	C₃×Q₈	C₂×C₃×D₄	1	2	yes	—
183	C₃×D₄	—	2	4	no	—
184	C₃×D₄	—	2	1	no	—
185	C₃×D₄	C₂×C₃×D₄	1	2	yes	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	4	4	no	—
192	C₂×C₃×C₄	—	4	4	no	—
193	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
196	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
197	C₂×C₃×C₄	C₃×#41	1	2	yes	—
198	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
199	C₃×C₈	—	2	4	no	—
200	C₂×C₉	—	2	2	no	—
201	C₂×C₉	—	4	2	no	—
202	C₂×C₉	C₂×#121	1	1	yes	characteristic
203	C₂×C₉	C₂×#137	1	2	yes	—
204	#41	—	2	2	no	—
205	#41	D₄×C₉	1	4	yes	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	C₂³×C₉	1	4	yes	—
210	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	D₄×C₉	1	2	yes	—
217	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
218	Q₁₆	—	2	8	no	—
219	SD₁₆	—	2	8	no	—
220	M₁₆	—	2	4	no	—
221	C₂×C₈	C₂³×C₉	1	4	yes	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	C₂³×C₉	1	2	yes	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	—	2	2	no	—
230	C₄²	D₄×C₉	1	1	yes	characteristic
231	C₄²	D₄×C₉	1	1	yes	characteristic
232	C₄²	C₂³×C₉	1	2	yes	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	—	4	2	no	—
236	C₂²×C₃	C₃×#121	1	2	yes	—
237	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	4	2	no	—
241	C₃×C₄	—	4	4	no	—
242	C₃×C₄	C₃×#121	1	2	yes	—
243	C₃×C₄	C₃×#137	1	4	yes	—
244	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
245	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
246	C₉	#3039	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	2	1	no	—
248	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
249	Q₈	—	2	1	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	2	4	no	—
252	Q₈	—	4	4	no	—
253	Q₈	C₂×D₄×C₉	1	2	yes	—
254	D₄	—	2	4	no	—
255	D₄	—	2	1	no	—
256	D₄	C₂×D₄×C₉	1	2	yes	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
265	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
266	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
267	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
268	C₂×C₄	C₉×#41	1	2	yes	—
269	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₂×C₃	—	2	2	no	—
272	C₂×C₃	—	4	2	no	—
273	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
274	C₂×C₃	C₂×C₃×#137	1	2	yes	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	C₉×#121	1	2	yes	—
279	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	4	2	no	—
284	C₄	C₉×#121	1	2	yes	—
285	C₄	C₉×#137	1	4	yes	complemented
286	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
287	C₄	C₂×C₉×#41	1	1	yes	characteristic
288	C₃	C₃×#3039	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	4	2	no	—
291	C₂	C₂×C₉×#121	1	1	yes	characteristic
292	C₂	C₂×C₉×#137	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
2. C₂×C₉×C₄⋊Q₈	C₂	semidirect
4. C₉×#462	C₂	semidirect
5. C₉×#461	C₂	semidirect
6. C₉×#423	C₂	semidirect
7. C₉×#419	C₂	semidirect
64. #3039	C₉	direct (N × H)
109. C₂×C₄⋊Q₈	C₂×C₉	semidirect
111. #462	C₂×C₉	semidirect
112. #461	C₂×C₉	semidirect
113. #423	C₂×C₉	semidirect
114. #419	C₂×C₉	semidirect
154. C₄×C₉	#137	semidirect
246. C₉	#3039	direct (N × H)
285. C₄	C₉×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#137	2
C₂×C₉×C₄⋊Q₈	2
C₄×C₉×#41	2
C₉×#462	2
C₉×#461	2
C₉×#423	2
C₉×#419	2
C₃×#3039	3

Subgroups — C9×#3039 #11767760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3039 #11767760