Subgroups — C₉×#3138 #11767859

Order 1152.

726 conjugacy classes of subgroups.

The subgroup lattice has 225 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	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic
3	C₉×#490	C₂	1	4	yes	maximal, complemented
4	C₉×#486	C₂	1	4	yes	maximal, complemented
5	C₉×#418	C₂	1	4	yes	maximal, complemented
6	C₄×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#3138	C₃	1	1	yes	maximal, characteristic
8	C₂²×D₄×C₉	—	2	2	no	—
9	C₂²×D₄×C₉	C₂²	1	2	yes	—
10	C₂×C₉×M₁₆	C₂²	1	2	yes	—
11	C₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
12	C₉×#127	—	2	2	no	—
13	C₉×#127	C₂²	1	2	yes	—
14	C₉×#127	C₂²	1	2	yes	—
15	C₉×#125	C₂²	1	2	yes	complemented
16	C₉×C₂²⋊Q₈	—	2	2	no	—
17	C₄×Q₈×C₉	—	2	2	no	—
18	C₉×#117	C₂²	1	2	yes	—
19	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
20	C₉×#102	C₂²	1	16	yes	complemented
21	C₉×#97	C₂²	1	2	yes	complemented
22	C₄×C₈×C₉	C₂²	1	2	yes	complemented
23	C₃×#533	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
25	C₃×#490	C₂×C₃	1	4	yes	—
26	C₃×#486	C₂×C₃	1	4	yes	—
27	C₃×#418	C₂×C₃	1	4	yes	—
28	C₃×C₄×M₁₆	C₂×C₃	1	1	yes	characteristic
29	C₂⁴×C₉	—	2	2	no	—
30	C₂×Q₈×C₉	—	2	2	no	—
31	C₂×D₄×C₉	—	2	2	no	—
32	C₂×D₄×C₉	—	2	4	no	—
33	C₂×D₄×C₉	—	2	4	no	—
34	C₂×D₄×C₉	—	2	8	no	—
35	C₂×D₄×C₉	—	4	8	no	—
36	C₂×D₄×C₉	C₂³	1	4	yes	—
37	C₂²×C₄×C₉	D₄	1	2	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₉×M₁₆	—	2	4	no	—
40	C₂×C₈×C₉	C₂³	1	4	yes	—
41	C₉×C₄⋊C₄	—	2	4	no	—
42	C₉×C₄⋊C₄	—	2	2	no	—
43	C₉×C₄⋊C₄	C₂³	1	4	yes	—
44	C₉×C₂²⋊C₄	—	2	2	no	—
45	C₄²×C₉	—	2	2	no	—
46	C₄²×C₉	D₄	1	2	yes	—
47	C₄²×C₉	C₂³	1	1	yes	characteristic
48	C₄²×C₉	C₂³	1	1	yes	characteristic
49	#3138	C₉	1	1	yes	characteristic, direct product factor, Sylow
50	C₂²×C₃×D₄	—	2	2	no	—
51	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
52	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
53	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
54	C₃×#127	—	2	2	no	—
55	C₃×#127	C₂²×C₃	1	2	yes	—
56	C₃×#127	C₂²×C₃	1	2	yes	—
57	C₃×#125	C₂²×C₃	1	2	yes	—
58	C₃×C₂²⋊Q₈	—	2	2	no	—
59	C₃×C₄×Q₈	—	2	2	no	—
60	C₃×#117	C₂²×C₃	1	2	yes	—
61	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
62	C₃×#102	C₂²×C₃	1	16	yes	—
63	C₃×#97	C₂²×C₃	1	2	yes	—
64	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
65	C₂³×C₉	—	2	4	no	—
66	C₂³×C₉	—	4	8	no	—
67	C₂³×C₉	—	4	4	no	—
68	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
69	Q₈×C₉	—	4	4	no	—
70	D₄×C₉	—	2	8	no	—
71	D₄×C₉	—	4	8	no	—
72	D₄×C₉	—	4	4	no	—
73	D₄×C₉	—	4	8	no	—
74	C₂×C₄×C₉	—	2	4	no	—
75	C₂×C₄×C₉	—	2	4	no	—
76	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₉	#41	1	4	yes	—
80	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
81	C₈×C₉	—	2	4	no	—
82	#533	C₂×C₉	1	1	yes	characteristic, complemented
83	C₂×#127	C₂×C₉	1	1	yes	characteristic
84	#490	C₂×C₉	1	4	yes	complemented
85	#486	C₂×C₉	1	4	yes	complemented
86	#418	C₂×C₉	1	4	yes	complemented
87	C₄×M₁₆	C₂×C₉	1	1	yes	characteristic, complemented
88	C₂⁴×C₃	—	2	2	no	—
89	C₂×C₃×Q₈	—	2	2	no	—
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂×C₃×D₄	—	2	4	no	—
92	C₂×C₃×D₄	—	2	4	no	—
93	C₂×C₃×D₄	—	2	8	no	—
94	C₂×C₃×D₄	—	4	8	no	—
95	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
96	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
97	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
98	C₃×M₁₆	—	2	4	no	—
99	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
100	C₃×C₄⋊C₄	—	2	4	no	—
101	C₃×C₄⋊C₄	—	2	2	no	—
102	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
103	C₃×C₂²⋊C₄	—	2	2	no	—
104	C₃×C₄²	—	2	2	no	—
105	C₃×C₄²	C₃×D₄	1	2	yes	—
106	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
107	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
108	C₂²×C₉	—	2	2	no	—
109	C₂²×C₉	—	4	4	no	—
110	C₂²×C₉	—	4	8	no	—
111	C₂²×C₉	—	8	8	no	—
112	C₂²×C₉	#124	1	2	yes	—
113	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₉	—	2	2	no	—
115	C₄×C₉	—	4	4	no	—
116	C₄×C₉	#124	1	2	yes	—
117	C₄×C₉	AGL(1,ℤ/8)	1	4	yes	—
118	C₄×C₉	C₂×#41	1	2	yes	—
119	C₂²×D₄	—	2	2	no	—
120	C₂²×D₄	C₂²×C₉	1	2	yes	—
121	C₂×M₁₆	C₂²×C₉	1	2	yes	—
122	C₄⋊Q₈	C₂²×C₉	1	2	yes	complemented
123	#127	—	2	2	no	—
124	#127	C₂²×C₉	1	2	yes	—
125	#127	C₂²×C₉	1	2	yes	—
126	#125	C₂²×C₉	1	2	yes	complemented
127	C₂²⋊Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	#117	C₂²×C₉	1	2	yes	—
130	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
131	#102	C₂²×C₉	1	16	yes	complemented
132	#97	C₂²×C₉	1	2	yes	complemented
133	C₄×C₈	C₂²×C₉	1	2	yes	complemented
134	C₂³×C₃	—	2	4	no	—
135	C₂³×C₃	—	4	8	no	—
136	C₂³×C₃	—	4	4	no	—
137	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
138	C₃×Q₈	—	4	4	no	—
139	C₃×D₄	—	2	8	no	—
140	C₃×D₄	—	4	8	no	—
141	C₃×D₄	—	4	4	no	—
142	C₃×D₄	—	4	8	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
148	C₂×C₃×C₄	C₃×#41	1	4	yes	—
149	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
150	C₃×C₈	—	2	4	no	—
151	C₂×C₉	—	2	2	no	—
152	C₂×C₉	—	8	4	no	—
153	C₂×C₉	C₂×#124	1	1	yes	characteristic
154	C₂×C₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
155	C₂⁴	—	2	2	no	—
156	C₂×Q₈	—	2	2	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	2	4	no	—
160	C₂×D₄	—	2	8	no	—
161	C₂×D₄	—	4	8	no	—
162	C₂×D₄	C₂³×C₉	1	4	yes	—
163	C₂²×C₄	D₄×C₉	1	2	yes	—
164	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
165	M₁₆	—	2	4	no	—
166	C₂×C₈	C₂³×C₉	1	4	yes	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	C₂³×C₉	1	4	yes	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	D₄×C₉	1	2	yes	—
173	C₄²	C₂³×C₉	1	1	yes	characteristic
174	C₄²	C₂³×C₉	1	1	yes	characteristic
175	C₂²×C₃	—	2	2	no	—
176	C₂²×C₃	—	4	8	no	—
177	C₂²×C₃	—	4	4	no	—
178	C₂²×C₃	—	8	8	no	—
179	C₂²×C₃	C₃×#124	1	2	yes	—
180	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	4	4	no	—
183	C₃×C₄	C₃×#124	1	2	yes	—
184	C₃×C₄	C₃×AGL(1,ℤ/8)	1	4	yes	—
185	C₃×C₄	C₂×C₃×#41	1	2	yes	—
186	C₉	#3138	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	—	2	4	no	—
188	C₂³	—	4	8	no	—
189	C₂³	—	4	4	no	—
190	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	D₄	—	2	8	no	—
193	D₄	—	4	8	no	—
194	D₄	—	4	4	no	—
195	D₄	—	4	8	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
199	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
201	C₂×C₄	C₉×#41	1	4	yes	—
202	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
203	C₈	—	2	4	no	—
204	C₂×C₃	—	2	2	no	—
205	C₂×C₃	—	8	4	no	—
206	C₂×C₃	C₂×C₃×#124	1	1	yes	characteristic
207	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	4	4	no	—
210	C₂²	—	4	8	no	—
211	C₂²	—	8	8	no	—
212	C₂²	C₉×#124	1	2	yes	—
213	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
214	C₄	—	2	2	no	—
215	C₄	—	4	4	no	—
216	C₄	C₉×#124	1	2	yes	—
217	C₄	C₉×AGL(1,ℤ/8)	1	4	yes	—
218	C₄	C₂×C₉×#41	1	2	yes	—
219	C₃	C₃×#3138	1	1	yes	characteristic
220	C₂	—	2	2	no	—
221	C₂	—	8	4	no	—
222	C₂	C₂×C₉×#124	1	1	yes	characteristic
223	C₂	C₂×C₉×AGL(1,ℤ/8)	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×#127	C₂×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. C₉×#533	C₂	semidirect
3. C₉×#490	C₂	semidirect
4. C₉×#486	C₂	semidirect
5. C₉×#418	C₂	semidirect
6. C₄×C₉×M₁₆	C₂	semidirect
11. C₉×C₄⋊Q₈	C₂²	semidirect
15. C₉×#125	C₂²	semidirect
20. C₉×#102	C₂²	semidirect
21. C₉×#97	C₂²	semidirect
22. C₄×C₈×C₉	C₂²	semidirect
49. #3138	C₉	direct (N × H)
82. #533	C₂×C₉	semidirect
84. #490	C₂×C₉	semidirect
85. #486	C₂×C₉	semidirect
86. #418	C₂×C₉	semidirect
87. C₄×M₁₆	C₂×C₉	semidirect
122. C₄⋊Q₈	C₂²×C₉	semidirect
126. #125	C₂²×C₉	semidirect
131. #102	C₂²×C₉	semidirect
132. #97	C₂²×C₉	semidirect
133. C₄×C₈	C₂²×C₉	semidirect
186. C₉	#3138	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
C₂×C₉×#127	2
C₉×#490	2
C₉×#486	2
C₉×#418	2
C₄×C₉×M₁₆	2
C₃×#3138	3

Subgroups — C9×#3138 #11767859

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3138 #11767859