Subgroups — C₉×#2979 #11767700

Order 1152.

615 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#532	C₂	1	1	yes	maximal, characteristic
2	C₉×#436	C₂	1	4	yes	maximal, complemented
3	C₈×D₄×C₉	C₂	1	4	yes	maximal, complemented
4	C₉×#431	C₂	1	2	yes	maximal, complemented
5	C₉×#408	C₂	1	4	yes	maximal, complemented
6	C₃×#2979	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	C₂²	1	2	yes	—
8	C₂×C₉×M₁₆	C₂²	1	4	yes	—
9	C₂²×C₈×C₉	C₂²	1	4	yes	—
10	C₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₉×#127	C₄	1	1	yes	characteristic
12	C₉×#124	C₄	1	2	yes	—
13	C₉×#121	C₄	1	4	yes	—
14	C₄×D₄×C₉	C₂²	1	4	yes	—
15	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
16	C₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₉×C₄⋊C₈	C₂²	1	2	yes	—
18	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₉×C₂²⋊C₈	C₂²	1	8	yes	—
20	C₄×C₈×C₉	C₂²	1	4	yes	—
21	C₃×#532	C₂×C₃	1	1	yes	characteristic
22	C₃×#436	C₂×C₃	1	4	yes	—
23	C₃×C₈×D₄	C₂×C₃	1	4	yes	—
24	C₃×#431	C₂×C₃	1	2	yes	—
25	C₃×#408	C₂×C₃	1	4	yes	—
26	C₉×#41	—	4	2	no	—
27	C₉×#41	—	4	2	no	—
28	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
30	C₂×D₄×C₉	C₂×C₄	1	4	yes	—
31	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂³	1	4	yes	—
35	C₉×M₁₆	—	4	4	no	—
36	C₂×C₈×C₉	—	4	4	no	—
37	C₂×C₈×C₉	C₂³	1	4	yes	—
38	C₂×C₈×C₉	C₂³	1	4	yes	—
39	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₉×C₂²⋊C₄	C₂×C₄	1	8	yes	—
41	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₉	C₂³	1	2	yes	—
44	#2979	C₉	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
46	C₂×C₃×M₁₆	C₂²×C₃	1	4	yes	—
47	C₂²×C₃×C₈	C₂²×C₃	1	4	yes	—
48	C₃×C₄⋊Q₈	C₃×C₄	1	1	yes	characteristic
49	C₃×#127	C₃×C₄	1	1	yes	characteristic
50	C₃×#124	C₃×C₄	1	2	yes	—
51	C₃×#121	C₃×C₄	1	4	yes	—
52	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
53	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
55	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
56	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
57	C₃×C₂²⋊C₈	C₂²×C₃	1	8	yes	—
58	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
59	C₂³×C₉	C₂²×C₄	1	4	yes	—
60	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
61	Q₈×C₉	—	4	2	no	—
62	D₄×C₉	—	2	8	no	—
63	D₄×C₉	—	4	2	no	—
64	D₄×C₉	—	4	4	no	—
65	C₂×C₄×C₉	—	2	4	no	—
66	C₂×C₄×C₉	—	2	4	no	—
67	C₂×C₄×C₉	—	2	4	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₉	C₂²×C₄	1	4	yes	—
73	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
75	C₈×C₉	—	2	4	no	—
76	C₈×C₉	—	2	4	no	—
77	#532	C₂×C₉	1	1	yes	characteristic
78	#436	C₂×C₉	1	4	yes	complemented
79	C₈×D₄	C₂×C₉	1	4	yes	complemented
80	#431	C₂×C₉	1	2	yes	complemented
81	#408	C₂×C₉	1	4	yes	complemented
82	C₃×#41	—	4	2	no	—
83	C₃×#41	—	4	2	no	—
84	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
85	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
86	C₂×C₃×D₄	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
88	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
89	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
91	C₃×M₁₆	—	4	4	no	—
92	C₂×C₃×C₈	—	4	4	no	—
93	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
94	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
95	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
96	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	8	yes	—
97	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
98	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
99	C₃×C₄²	C₂³×C₃	1	2	yes	—
100	C₂²×C₉	—	2	1	no	—
101	C₂²×C₉	—	2	4	no	—
102	C₂²×C₉	—	2	2	no	—
103	C₂²×C₉	—	2	8	no	—
104	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
105	C₄×C₉	—	2	1	no	—
106	C₄×C₉	—	2	4	no	—
107	C₄×C₉	—	2	2	no	—
108	C₄×C₉	#131	1	4	yes	—
109	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
110	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×#41	C₂²×C₉	1	2	yes	—
112	C₂×M₁₆	C₂²×C₉	1	4	yes	—
113	C₂²×C₈	C₂²×C₉	1	4	yes	—
114	C₄⋊Q₈	C₄×C₉	1	1	yes	characteristic
115	#127	C₄×C₉	1	1	yes	characteristic
116	#124	C₄×C₉	1	2	yes	—
117	#121	C₄×C₉	1	4	yes	—
118	C₄×D₄	C₂²×C₉	1	4	yes	—
119	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
120	C₄⋊C₈	C₂²×C₉	1	2	yes	—
121	C₄⋊C₈	C₂²×C₉	1	2	yes	—
122	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
123	C₂²⋊C₈	C₂²×C₉	1	8	yes	—
124	C₄×C₈	C₂²×C₉	1	4	yes	—
125	C₂³×C₃	C₂²×C₃×C₄	1	4	yes	—
126	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
127	C₃×Q₈	—	4	2	no	—
128	C₃×D₄	—	2	8	no	—
129	C₃×D₄	—	4	2	no	—
130	C₃×D₄	—	4	4	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	—	2	4	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
138	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
139	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
140	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
141	C₃×C₈	—	2	4	no	—
142	C₃×C₈	—	2	4	no	—
143	C₂×C₉	—	4	1	no	—
144	C₂×C₉	—	4	4	no	—
145	C₂×C₉	#518	1	1	yes	characteristic
146	C₂×C₉	C₂×#131	1	2	yes	—
147	#41	—	4	2	no	—
148	#41	—	4	2	no	—
149	C₂×Q₈	C₂×C₄×C₉	1	2	yes	—
150	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
151	C₂×D₄	C₂×C₄×C₉	1	4	yes	—
152	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
153	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
154	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₉	1	4	yes	—
156	M₁₆	—	4	4	no	—
157	C₂×C₈	—	4	4	no	—
158	C₂×C₈	C₂³×C₉	1	4	yes	—
159	C₂×C₈	C₂³×C₉	1	4	yes	—
160	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
161	C₂²⋊C₄	C₂×C₄×C₉	1	8	yes	—
162	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
163	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
164	C₄²	C₂³×C₉	1	2	yes	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	—	2	8	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	4	no	—
169	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	4	no	—
173	C₃×C₄	C₃×#131	1	4	yes	—
174	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
175	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
176	C₉	#2979	1	1	yes	characteristic, direct product factor, Sylow
177	C₂³	C₂²×C₄×C₉	1	4	yes	—
178	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
179	Q₈	—	4	2	no	—
180	D₄	—	2	8	no	—
181	D₄	—	4	2	no	—
182	D₄	—	4	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
189	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
190	C₂×C₄	C₂²×C₄×C₉	1	4	yes	—
191	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
192	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₈	—	2	4	no	—
195	C₂×C₃	—	4	1	no	—
196	C₂×C₃	—	4	4	no	—
197	C₂×C₃	C₃×#518	1	1	yes	characteristic
198	C₂×C₃	C₂×C₃×#131	1	2	yes	—
199	C₂²	—	2	2	no	—
200	C₂²	—	2	8	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	4	no	—
203	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	4	no	—
207	C₄	C₉×#131	1	4	yes	—
208	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
209	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
210	C₃	C₃×#2979	1	1	yes	characteristic
211	C₂	—	4	1	no	—
212	C₂	—	4	4	no	—
213	C₂	C₉×#518	1	1	yes	characteristic
214	C₂	C₂×C₉×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₉	C₂⁴
commutator	C₂²	C₂³×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂³×C₃×C₄
prime radical	C₃×#532	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#436	C₂	semidirect
3. C₈×D₄×C₉	C₂	semidirect
4. C₉×#431	C₂	semidirect
5. C₉×#408	C₂	semidirect
44. #2979	C₉	direct (N × H)
78. #436	C₂×C₉	semidirect
79. C₈×D₄	C₂×C₉	semidirect
80. #431	C₂×C₉	semidirect
81. #408	C₂×C₉	semidirect
176. C₉	#2979	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×#532	2
C₉×#436	2
C₈×D₄×C₉	2
C₉×#431	2
C₉×#408	2
C₃×#2979	3

Subgroups — C9×#2979 #11767700

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2979 #11767700