Subgroups — C₉×#3127 #11767848

Order 1152.

537 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	C₉×#541	C₂	1	1	yes	maximal, characteristic
2	C₉×#519	C₂	1	1	yes	maximal, characteristic
3	C₉×#485	C₂	1	2	yes	maximal
4	C₉×#479	C₂	1	2	yes	maximal, complemented
5	C₉×#462	C₂	1	2	yes	maximal, complemented
6	C₉×#451	C₂	1	2	yes	maximal
7	C₉×#441	C₂	1	2	yes	maximal, complemented
8	C₄×C₉×Q₁₆	C₂	1	2	yes	maximal, complemented
9	C₉×#433	C₂	1	1	yes	maximal, characteristic
10	C₃×#3127	C₃	1	1	yes	maximal, characteristic
11	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₉×#126	—	2	2	no	—
16	C₉×#125	—	2	2	no	—
17	C₉×#125	C₂²	1	1	yes	characteristic
18	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₄×Q₈×C₉	—	2	2	no	—
20	C₄×Q₈×C₉	C₂²	1	2	yes	—
21	C₄×Q₈×C₉	C₂²	1	2	yes	—
22	C₄×Q₈×C₉	C₂²	1	2	yes	—
23	C₉×#117	—	2	2	no	—
24	C₉×#117	C₂²	1	1	yes	characteristic
25	C₉×#117	C₂²	1	1	yes	characteristic
26	C₉×#107	C₂²	1	2	yes	—
27	C₉×#106	C₂²	1	2	yes	—
28	C₉×C₄⋊C₈	C₂²	1	2	yes	—
29	C₉×#103	C₂²	1	2	yes	—
30	C₉×#103	C₂²	1	2	yes	—
31	C₉×#103	C₂²	1	2	yes	—
32	C₉×#103	C₂²	1	2	yes	—
33	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₉×#97	C₂²	1	1	yes	characteristic
35	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
36	C₃×#541	C₂×C₃	1	1	yes	characteristic
37	C₃×#519	C₂×C₃	1	1	yes	characteristic
38	C₃×#485	C₂×C₃	1	2	yes	—
39	C₃×#479	C₂×C₃	1	2	yes	—
40	C₃×#462	C₂×C₃	1	2	yes	—
41	C₃×#451	C₂×C₃	1	2	yes	—
42	C₃×#441	C₂×C₃	1	2	yes	—
43	C₃×C₄×Q₁₆	C₂×C₃	1	2	yes	—
44	C₃×#433	C₂×C₃	1	1	yes	characteristic
45	C₂×Q₈×C₉	—	2	4	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	C₂³	1	2	yes	—
48	C₂×Q₈×C₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
51	C₉×Q₁₆	—	2	4	no	—
52	C₉×Q₁₆	—	2	4	no	—
53	C₂×C₈×C₉	C₂³	1	2	yes	—
54	C₂×C₈×C₉	C₂³	1	2	yes	—
55	C₉×C₄⋊C₄	—	2	2	no	—
56	C₉×C₄⋊C₄	—	2	2	no	—
57	C₉×C₄⋊C₄	—	2	2	no	—
58	C₉×C₄⋊C₄	—	2	2	no	—
59	C₉×C₄⋊C₄	—	2	2	no	—
60	C₉×C₄⋊C₄	—	2	2	no	—
61	C₉×C₄⋊C₄	D₄	1	2	yes	—
62	C₉×C₄⋊C₄	C₂³	1	2	yes	—
63	C₉×C₄⋊C₄	C₂³	1	2	yes	—
64	C₉×C₂²⋊C₄	—	2	1	no	—
65	C₉×C₂²⋊C₄	—	2	2	no	—
66	C₉×C₂²⋊C₄	D₄	1	2	yes	—
67	C₄²×C₉	—	2	2	no	—
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	—	2	1	no	—
70	C₄²×C₉	—	2	2	no	—
71	C₄²×C₉	C₂³	1	1	yes	characteristic
72	C₄²×C₉	C₂³	1	1	yes	characteristic
73	#3127	C₉	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
76	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
77	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₃×#126	—	2	2	no	—
79	C₃×#125	—	2	2	no	—
80	C₃×#125	C₂²×C₃	1	1	yes	characteristic
81	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
82	C₃×C₄×Q₈	—	2	2	no	—
83	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
84	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
85	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
86	C₃×#117	—	2	2	no	—
87	C₃×#117	C₂²×C₃	1	1	yes	characteristic
88	C₃×#117	C₂²×C₃	1	1	yes	characteristic
89	C₃×#107	C₂²×C₃	1	2	yes	—
90	C₃×#106	C₂²×C₃	1	2	yes	—
91	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
92	C₃×#103	C₂²×C₃	1	2	yes	—
93	C₃×#103	C₂²×C₃	1	2	yes	—
94	C₃×#103	C₂²×C₃	1	2	yes	—
95	C₃×#103	C₂²×C₃	1	2	yes	—
96	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	C₃×#97	C₂²×C₃	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
99	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₉	—	2	4	no	—
101	Q₈×C₉	—	2	2	no	—
102	Q₈×C₉	—	2	2	no	—
103	Q₈×C₉	—	2	2	no	—
104	Q₈×C₉	#41	1	4	yes	—
105	C₂×C₄×C₉	—	2	4	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
112	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
113	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	2	2	no	—
117	#541	C₂×C₉	1	1	yes	characteristic
118	#519	C₂×C₉	1	1	yes	characteristic
119	#485	C₂×C₉	1	2	yes	—
120	#479	C₂×C₉	1	2	yes	complemented
121	#462	C₂×C₉	1	2	yes	complemented
122	#451	C₂×C₉	1	2	yes	—
123	#441	C₂×C₉	1	2	yes	complemented
124	C₄×Q₁₆	C₂×C₉	1	2	yes	complemented
125	#433	C₂×C₉	1	1	yes	characteristic
126	C₂×C₃×Q₈	—	2	4	no	—
127	C₂×C₃×Q₈	—	2	1	no	—
128	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
129	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
130	C₂²×C₃×C₄	—	2	1	no	—
131	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₃×Q₁₆	—	2	4	no	—
133	C₃×Q₁₆	—	2	4	no	—
134	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
135	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
143	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
144	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
145	C₃×C₂²⋊C₄	—	2	1	no	—
146	C₃×C₂²⋊C₄	—	2	2	no	—
147	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
148	C₃×C₄²	—	2	2	no	—
149	C₃×C₄²	—	2	2	no	—
150	C₃×C₄²	—	2	1	no	—
151	C₃×C₄²	—	2	2	no	—
152	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	2	no	—
156	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
157	C₄×C₉	—	2	4	no	—
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	2	2	no	—
160	C₄×C₉	—	2	2	no	—
161	C₄×C₉	—	4	2	no	—
162	C₄×C₉	—	4	2	no	—
163	C₄×C₉	—	4	2	no	—
164	C₄×C₉	C₂×#41	1	2	yes	—
165	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
166	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
167	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
168	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
169	#126	—	2	2	no	—
170	#125	—	2	2	no	—
171	#125	C₂²×C₉	1	1	yes	characteristic
172	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
173	C₄×Q₈	—	2	2	no	—
174	C₄×Q₈	C₂²×C₉	1	2	yes	—
175	C₄×Q₈	C₂²×C₉	1	2	yes	—
176	C₄×Q₈	C₂²×C₉	1	2	yes	—
177	#117	—	2	2	no	—
178	#117	C₂²×C₉	1	1	yes	characteristic
179	#117	C₂²×C₉	1	1	yes	characteristic
180	#107	C₂²×C₉	1	2	yes	—
181	#106	C₂²×C₉	1	2	yes	—
182	C₄⋊C₈	C₂²×C₉	1	2	yes	—
183	#103	C₂²×C₉	1	2	yes	—
184	#103	C₂²×C₉	1	2	yes	—
185	#103	C₂²×C₉	1	2	yes	—
186	#103	C₂²×C₉	1	2	yes	—
187	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
188	#97	C₂²×C₉	1	1	yes	characteristic
189	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
190	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
191	C₃×Q₈	—	2	4	no	—
192	C₃×Q₈	—	2	2	no	—
193	C₃×Q₈	—	2	2	no	—
194	C₃×Q₈	—	2	2	no	—
195	C₃×Q₈	C₃×#41	1	4	yes	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
203	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
204	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
206	C₃×C₈	—	2	2	no	—
207	C₃×C₈	—	2	2	no	—
208	C₂×C₉	—	4	1	no	—
209	C₂×C₉	#525	1	1	yes	characteristic
210	C₂×C₉	#578	1	2	yes	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	C₂³×C₉	1	2	yes	—
214	C₂×Q₈	C₂³×C₉	1	2	yes	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	2	4	no	—
219	C₂×C₈	C₂³×C₉	1	2	yes	—
220	C₂×C₈	C₂³×C₉	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	D₄×C₉	1	2	yes	—
228	C₄⋊C₄	C₂³×C₉	1	2	yes	—
229	C₄⋊C₄	C₂³×C₉	1	2	yes	—
230	C₂²⋊C₄	—	2	1	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	D₄×C₉	1	2	yes	—
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	2	no	—
237	C₄²	C₂³×C₉	1	1	yes	characteristic
238	C₄²	C₂³×C₉	1	1	yes	characteristic
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	1	no	—
241	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	4	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	4	2	no	—
247	C₃×C₄	—	4	2	no	—
248	C₃×C₄	—	4	2	no	—
249	C₃×C₄	C₂×C₃×#41	1	2	yes	—
250	C₉	#3127	1	1	yes	characteristic, direct product factor, Sylow
251	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
252	Q₈	—	2	4	no	—
253	Q₈	—	2	2	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	2	2	no	—
256	Q₈	C₉×#41	1	4	yes	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
264	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
265	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
266	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₂×C₃	—	4	1	no	—
270	C₂×C₃	C₃×#525	1	1	yes	characteristic
271	C₂×C₃	C₃×#578	1	2	yes	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	4	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	4	2	no	—
282	C₄	C₂×C₉×#41	1	2	yes	—
283	C₃	C₃×#3127	1	1	yes	characteristic
284	C₂	—	4	1	no	—
285	C₂	C₉×#525	1	1	yes	characteristic
286	C₂	C₉×#578	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	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×#479	C₂	semidirect
5. C₉×#462	C₂	semidirect
7. C₉×#441	C₂	semidirect
8. C₄×C₉×Q₁₆	C₂	semidirect
73. #3127	C₉	direct (N × H)
120. #479	C₂×C₉	semidirect
121. #462	C₂×C₉	semidirect
123. #441	C₂×C₉	semidirect
124. C₄×Q₁₆	C₂×C₉	semidirect
250. C₉	#3127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#541	2
C₉×#519	2
C₉×#485	2
C₉×#479	2
C₉×#462	2
C₉×#451	2
C₉×#441	2
C₄×C₉×Q₁₆	2
C₉×#433	2
C₃×#3127	3

Subgroups — C9×#3127 #11767848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3127 #11767848