Subgroups — C₉×#3245 #11767966

Order 1152.

654 conjugacy classes of subgroups.

The subgroup lattice has 231 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	2	yes	maximal, complemented
2	C₉×#474	C₂	1	4	yes	maximal, complemented
3	C₉×#466	C₂	1	4	yes	maximal, complemented
4	C₉×#459	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#3245	C₃	1	1	yes	maximal, characteristic
7	C₂×C₉×#41	—	2	2	no	—
8	C₂×C₉×D₈	C₂²	1	4	yes	—
9	C₂²×C₈×C₉	C₂²	1	2	yes	—
10	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₉×#127	C₂²	1	2	yes	—
12	C₉×#124	—	2	2	no	—
13	C₉×#121	—	2	2	no	—
14	C₉×#121	C₂²	1	4	yes	—
15	C₄×D₄×C₉	—	2	2	no	—
16	C₄×D₄×C₉	C₂²	1	4	yes	—
17	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₉×#107	C₂²	1	4	yes	—
19	C₉×C₄⋊C₈	C₂²	1	4	yes	—
20	C₉×#102	C₂²	1	8	yes	—
21	C₃×#532	C₂×C₃	1	2	yes	—
22	C₃×#474	C₂×C₃	1	4	yes	—
23	C₃×#466	C₂×C₃	1	4	yes	—
24	C₃×#459	C₂×C₃	1	4	yes	—
25	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
26	C₉×#41	—	4	8	no	—
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×D₄×C₉	—	2	4	no	—
29	C₂×D₄×C₉	—	2	2	no	—
30	C₂×D₄×C₉	—	2	2	no	—
31	C₂×D₄×C₉	C₂³	1	4	yes	—
32	C₂²×C₄×C₉	—	2	4	no	—
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₉×D₈	—	4	4	no	—
37	C₂×C₈×C₉	—	4	2	no	—
38	C₂×C₈×C₉	C₂³	1	4	yes	—
39	C₉×C₄⋊C₄	—	2	2	no	—
40	C₉×C₄⋊C₄	C₂³	1	4	yes	—
41	C₉×C₂²⋊C₄	—	2	4	no	—
42	C₉×C₂²⋊C₄	—	2	4	no	—
43	C₄²×C₉	D₄	1	2	yes	—
44	C₄²×C₉	C₂³	1	2	yes	—
45	#3245	C₉	1	1	yes	characteristic, direct product factor, Sylow
46	C₂×C₃×#41	—	2	2	no	—
47	C₂×C₃×D₈	C₂²×C₃	1	4	yes	—
48	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
49	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
50	C₃×#127	C₂²×C₃	1	2	yes	—
51	C₃×#124	—	2	2	no	—
52	C₃×#121	—	2	2	no	—
53	C₃×#121	C₂²×C₃	1	4	yes	—
54	C₃×C₄×D₄	—	2	2	no	—
55	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
56	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
57	C₃×#107	C₂²×C₃	1	4	yes	—
58	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
59	C₃×#102	C₂²×C₃	1	8	yes	—
60	C₂³×C₉	—	2	4	no	—
61	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
62	Q₈×C₉	—	4	4	no	—
63	D₄×C₉	—	2	4	no	—
64	D₄×C₉	—	2	4	no	—
65	D₄×C₉	—	4	4	no	—
66	D₄×C₉	—	4	8	no	—
67	D₄×C₉	—	4	4	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	4	1	no	—
72	C₂×C₄×C₉	—	4	8	no	—
73	C₂×C₄×C₉	D₈	1	4	yes	complemented
74	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
78	C₈×C₉	—	2	4	no	—
79	#532	C₂×C₉	1	2	yes	complemented
80	#474	C₂×C₉	1	4	yes	complemented
81	#466	C₂×C₉	1	4	yes	complemented
82	#459	C₂×C₉	1	4	yes	complemented
83	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
84	C₃×#41	—	4	8	no	—
85	C₂×C₃×Q₈	—	2	2	no	—
86	C₂×C₃×D₄	—	2	4	no	—
87	C₂×C₃×D₄	—	2	2	no	—
88	C₂×C₃×D₄	—	2	2	no	—
89	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
90	C₂²×C₃×C₄	—	2	4	no	—
91	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
93	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
94	C₃×D₈	—	4	4	no	—
95	C₂×C₃×C₈	—	4	2	no	—
96	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
99	C₃×C₂²⋊C₄	—	2	4	no	—
100	C₃×C₂²⋊C₄	—	2	4	no	—
101	C₃×C₄²	C₃×D₄	1	2	yes	—
102	C₃×C₄²	C₂³×C₃	1	2	yes	—
103	C₂²×C₉	—	2	1	no	—
104	C₂²×C₉	—	2	1	no	—
105	C₂²×C₉	—	4	4	no	—
106	C₂²×C₉	—	4	4	no	—
107	C₂²×C₉	—	4	4	no	—
108	C₂²×C₉	C₂×D₈	1	2	yes	—
109	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
110	C₄×C₉	—	2	1	no	—
111	C₄×C₉	—	2	2	no	—
112	C₄×C₉	—	4	4	no	—
113	C₄×C₉	C₂×D₈	1	4	yes	—
114	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×#41	—	2	2	no	—
117	C₂×D₈	C₂²×C₉	1	4	yes	—
118	C₂²×C₈	C₂²×C₉	1	2	yes	—
119	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
120	#127	C₂²×C₉	1	2	yes	—
121	#124	—	2	2	no	—
122	#121	—	2	2	no	—
123	#121	C₂²×C₉	1	4	yes	—
124	C₄×D₄	—	2	2	no	—
125	C₄×D₄	C₂²×C₉	1	4	yes	—
126	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
127	#107	C₂²×C₉	1	4	yes	—
128	C₄⋊C₈	C₂²×C₉	1	4	yes	—
129	#102	C₂²×C₉	1	8	yes	—
130	C₂³×C₃	—	2	4	no	—
131	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
132	C₃×Q₈	—	4	4	no	—
133	C₃×D₄	—	2	4	no	—
134	C₃×D₄	—	2	4	no	—
135	C₃×D₄	—	4	4	no	—
136	C₃×D₄	—	4	8	no	—
137	C₃×D₄	—	4	4	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	4	1	no	—
142	C₂×C₃×C₄	—	4	8	no	—
143	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
144	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
146	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
147	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
148	C₃×C₈	—	2	4	no	—
149	C₂×C₉	—	2	2	no	—
150	C₂×C₉	—	8	4	no	—
151	C₂×C₉	#537	1	1	yes	characteristic
152	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
153	C₂×C₉	#575	1	1	yes	characteristic
154	#41	—	4	8	no	—
155	C₂×Q₈	—	2	2	no	—
156	C₂×D₄	—	2	4	no	—
157	C₂×D₄	—	2	2	no	—
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	C₂³×C₉	1	4	yes	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
162	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
163	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
164	D₈	—	4	4	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	C₂³×C₉	1	4	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	C₂³×C₉	1	4	yes	—
169	C₂²⋊C₄	—	2	4	no	—
170	C₂²⋊C₄	—	2	4	no	—
171	C₄²	D₄×C₉	1	2	yes	—
172	C₄²	C₂³×C₉	1	2	yes	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	4	4	no	—
176	C₂²×C₃	—	4	4	no	—
177	C₂²×C₃	—	4	4	no	—
178	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
179	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	4	4	no	—
183	C₃×C₄	C₂×C₃×D₈	1	4	yes	—
184	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
185	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
186	C₉	#3245	1	1	yes	characteristic, direct product factor, Sylow
187	C₂³	—	2	4	no	—
188	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
189	Q₈	—	4	4	no	—
190	D₄	—	2	4	no	—
191	D₄	—	2	4	no	—
192	D₄	—	4	4	no	—
193	D₄	—	4	8	no	—
194	D₄	—	4	4	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	8	no	—
200	C₂×C₄	C₉×D₈	1	4	yes	complemented
201	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
202	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
203	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
204	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
205	C₈	—	2	4	no	—
206	C₂×C₃	—	2	2	no	—
207	C₂×C₃	—	8	4	no	—
208	C₂×C₃	C₃×#537	1	1	yes	characteristic
209	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
210	C₂×C₃	C₃×#575	1	1	yes	characteristic
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	4	no	—
215	C₂²	—	4	4	no	—
216	C₂²	C₂×C₉×D₈	1	2	yes	—
217	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	2	no	—
220	C₄	—	4	4	no	—
221	C₄	C₂×C₉×D₈	1	4	yes	—
222	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
223	C₄	C₉×2₋¹⁺⁴	1	1	yes	characteristic
224	C₃	C₃×#3245	1	1	yes	characteristic
225	C₂	—	2	2	no	—
226	C₂	—	8	4	no	—
227	C₂	C₉×#537	1	1	yes	characteristic
228	C₂	C₂²×C₉×D₈	1	1	yes	characteristic
229	C₂	C₉×#575	1	1	yes	characteristic

	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	C₃×#3245	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×#532	C₂	semidirect
2. C₉×#474	C₂	semidirect
3. C₉×#466	C₂	semidirect
4. C₉×#459	C₂	semidirect
5. C₂×C₉×C₄⋊C₈	C₂	semidirect
45. #3245	C₉	direct (N × H)
73. C₂×C₄×C₉	D₈	semidirect
79. #532	C₂×C₉	semidirect
80. #474	C₂×C₉	semidirect
81. #466	C₂×C₉	semidirect
82. #459	C₂×C₉	semidirect
83. C₂×C₄⋊C₈	C₂×C₉	semidirect
186. C₉	#3245	direct (N × H)
200. C₂×C₄	C₉×D₈	semidirect

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×#474	2
C₉×#466	2
C₉×#459	2
C₂×C₉×C₄⋊C₈	2
C₃×#3245	3

Subgroups — C9×#3245 #11767966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3245 #11767966