Subgroups — C₉×#3131 #11767852

Order 1152.

600 conjugacy classes of subgroups.

The subgroup lattice has 237 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#532	C₂	1	1	yes	maximal, characteristic
3	C₉×#488	C₂	1	4	yes	maximal, complemented
4	C₉×#487	C₂	1	2	yes	maximal, complemented
5	C₉×#486	C₂	1	2	yes	maximal, complemented
6	C₉×#416	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3131	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	2	yes	—
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₉×#127	C₂²	1	1	yes	characteristic
14	C₉×#125	C₂²	1	2	yes	—
15	C₉×#124	C₂²	1	2	yes	—
16	C₉×C₂²⋊Q₈	—	2	2	no	—
17	C₉×#121	—	2	2	no	—
18	C₄×Q₈×C₉	—	2	2	no	—
19	C₄×D₄×C₉	—	2	2	no	—
20	C₉×#117	C₂²	1	2	yes	—
21	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
22	C₉×#103	C₂²	1	8	yes	—
23	C₉×#102	C₂²	1	8	yes	—
24	C₄×C₈×C₉	C₂²	1	2	yes	—
25	C₄×C₈×C₉	C₂²	1	2	yes	—
26	C₃×#533	C₂×C₃	1	1	yes	characteristic
27	C₃×#532	C₂×C₃	1	1	yes	characteristic
28	C₃×#488	C₂×C₃	1	4	yes	—
29	C₃×#487	C₂×C₃	1	2	yes	—
30	C₃×#486	C₂×C₃	1	2	yes	—
31	C₃×#416	C₂×C₃	1	4	yes	—
32	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
33	C₉×#41	—	2	8	no	—
34	C₂×Q₈×C₉	—	2	2	no	—
35	C₂×Q₈×C₉	C₂³	1	2	yes	—
36	C₂×D₄×C₉	—	2	2	no	—
37	C₂×D₄×C₉	—	2	2	no	—
38	C₂×D₄×C₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₉	—	2	2	no	—
40	C₂²×C₄×C₉	D₄	1	2	yes	—
41	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₉	—	2	4	no	—
43	C₂×C₈×C₉	C₂³	1	4	yes	—
44	C₉×C₄⋊C₄	—	2	2	no	—
45	C₉×C₄⋊C₄	—	2	4	no	—
46	C₉×C₄⋊C₄	—	2	2	no	—
47	C₉×C₄⋊C₄	C₂³	1	4	yes	—
48	C₉×C₂²⋊C₄	—	2	2	no	—
49	C₉×C₂²⋊C₄	—	2	4	no	—
50	C₄²×C₉	—	2	2	no	—
51	C₄²×C₉	D₄	1	2	yes	—
52	C₄²×C₉	C₂³	1	1	yes	characteristic
53	C₄²×C₉	C₂³	1	1	yes	characteristic
54	#3131	C₉	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
56	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
57	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
59	C₃×#127	C₂²×C₃	1	1	yes	characteristic
60	C₃×#125	C₂²×C₃	1	2	yes	—
61	C₃×#124	C₂²×C₃	1	2	yes	—
62	C₃×C₂²⋊Q₈	—	2	2	no	—
63	C₃×#121	—	2	2	no	—
64	C₃×C₄×Q₈	—	2	2	no	—
65	C₃×C₄×D₄	—	2	2	no	—
66	C₃×#117	C₂²×C₃	1	2	yes	—
67	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
68	C₃×#103	C₂²×C₃	1	8	yes	—
69	C₃×#102	C₂²×C₃	1	8	yes	—
70	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
71	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
72	C₂³×C₉	—	2	2	no	—
73	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₉	—	2	4	no	—
75	Q₈×C₉	—	4	4	no	—
76	D₄×C₉	—	2	4	no	—
77	D₄×C₉	—	4	4	no	—
78	D₄×C₉	—	4	4	no	—
79	C₂×C₄×C₉	—	2	4	no	—
80	C₂×C₄×C₉	—	2	4	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	4	4	no	—
83	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉	#41	1	4	yes	—
87	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
88	C₈×C₉	—	2	4	no	—
89	#533	C₂×C₉	1	1	yes	characteristic, complemented
90	#532	C₂×C₉	1	1	yes	characteristic
91	#488	C₂×C₉	1	4	yes	complemented
92	#487	C₂×C₉	1	2	yes	complemented
93	#486	C₂×C₉	1	2	yes	complemented
94	#416	C₂×C₉	1	4	yes	complemented
95	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic, complemented
96	C₃×#41	—	2	8	no	—
97	C₂×C₃×Q₈	—	2	2	no	—
98	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
99	C₂×C₃×D₄	—	2	2	no	—
100	C₂×C₃×D₄	—	2	2	no	—
101	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
104	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
105	C₂×C₃×C₈	—	2	4	no	—
106	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
107	C₃×C₄⋊C₄	—	2	2	no	—
108	C₃×C₄⋊C₄	—	2	4	no	—
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
111	C₃×C₂²⋊C₄	—	2	2	no	—
112	C₃×C₂²⋊C₄	—	2	4	no	—
113	C₃×C₄²	—	2	2	no	—
114	C₃×C₄²	C₃×D₄	1	2	yes	—
115	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
116	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
117	C₂²×C₉	—	2	2	no	—
118	C₂²×C₉	—	4	2	no	—
119	C₂²×C₉	—	4	4	no	—
120	C₂²×C₉	#124	1	2	yes	—
121	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₉	—	2	2	no	—
123	C₄×C₉	—	4	2	no	—
124	C₄×C₉	—	4	4	no	—
125	C₄×C₉	#124	1	2	yes	—
126	C₄×C₉	#135	1	4	yes	—
127	C₄×C₉	C₂×#41	1	2	yes	—
128	C₂×#41	C₂²×C₉	1	2	yes	—
129	C₂²×C₈	C₂²×C₉	1	2	yes	—
130	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
131	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
132	#127	C₂²×C₉	1	1	yes	characteristic
133	#125	C₂²×C₉	1	2	yes	—
134	#124	C₂²×C₉	1	2	yes	—
135	C₂²⋊Q₈	—	2	2	no	—
136	#121	—	2	2	no	—
137	C₄×Q₈	—	2	2	no	—
138	C₄×D₄	—	2	2	no	—
139	#117	C₂²×C₉	1	2	yes	—
140	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
141	#103	C₂²×C₉	1	8	yes	—
142	#102	C₂²×C₉	1	8	yes	—
143	C₄×C₈	C₂²×C₉	1	2	yes	—
144	C₄×C₈	C₂²×C₉	1	2	yes	—
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
147	C₃×Q₈	—	2	4	no	—
148	C₃×Q₈	—	4	4	no	—
149	C₃×D₄	—	2	4	no	—
150	C₃×D₄	—	4	4	no	—
151	C₃×D₄	—	4	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	4	4	no	—
156	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
157	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
159	C₂×C₃×C₄	C₃×#41	1	4	yes	—
160	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
161	C₃×C₈	—	2	4	no	—
162	C₂×C₉	—	2	2	no	—
163	C₂×C₉	—	8	2	no	—
164	C₂×C₉	C₂×#124	1	1	yes	characteristic
165	C₂×C₉	C₂×#135	1	2	yes	—
166	#41	—	2	8	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	C₂³×C₉	1	2	yes	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	2	2	no	—
171	C₂×D₄	C₂³×C₉	1	2	yes	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	D₄×C₉	1	2	yes	—
174	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	C₂³×C₉	1	4	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	C₂³×C₉	1	4	yes	—
181	C₂²⋊C₄	—	2	2	no	—
182	C₂²⋊C₄	—	2	4	no	—
183	C₄²	—	2	2	no	—
184	C₄²	D₄×C₉	1	2	yes	—
185	C₄²	C₂³×C₉	1	1	yes	characteristic
186	C₄²	C₂³×C₉	1	1	yes	characteristic
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	4	4	no	—
189	C₂²×C₃	—	4	2	no	—
190	C₂²×C₃	C₃×#124	1	2	yes	—
191	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	4	4	no	—
194	C₃×C₄	—	4	2	no	—
195	C₃×C₄	C₃×#124	1	2	yes	—
196	C₃×C₄	C₃×#135	1	4	yes	—
197	C₃×C₄	C₂×C₃×#41	1	2	yes	—
198	C₉	#3131	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	2	2	no	—
200	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	4	4	no	—
203	D₄	—	2	4	no	—
204	D₄	—	4	4	no	—
205	D₄	—	4	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
211	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
212	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
213	C₂×C₄	C₉×#41	1	4	yes	—
214	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
215	C₈	—	2	4	no	—
216	C₂×C₃	—	2	2	no	—
217	C₂×C₃	—	8	2	no	—
218	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
219	C₂×C₃	C₂×C₃×#135	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	4	no	—
223	C₂²	C₉×#124	1	2	yes	—
224	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	4	no	—
228	C₄	C₉×#124	1	2	yes	—
229	C₄	C₉×#135	1	4	yes	—
230	C₄	C₂×C₉×#41	1	2	yes	—
231	C₃	C₃×#3131	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	—	8	2	no	—
234	C₂	C₂×C₉×#124	1	1	yes	characteristic
235	C₂	C₂×C₉×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#532	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#533	C₂	semidirect
3. C₉×#488	C₂	semidirect
4. C₉×#487	C₂	semidirect
5. C₉×#486	C₂	semidirect
6. C₉×#416	C₂	semidirect
7. C₂×C₄×C₈×C₉	C₂	semidirect
54. #3131	C₉	direct (N × H)
89. #533	C₂×C₉	semidirect
91. #488	C₂×C₉	semidirect
92. #487	C₂×C₉	semidirect
93. #486	C₂×C₉	semidirect
94. #416	C₂×C₉	semidirect
95. C₂×C₄×C₈	C₂×C₉	semidirect
198. C₉	#3131	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×#533	2
C₉×#532	2
C₉×#488	2
C₉×#487	2
C₉×#486	2
C₉×#416	2
C₂×C₄×C₈×C₉	2
C₃×#3131	3

Subgroups — C9×#3131 #11767852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3131 #11767852