Subgroups — C₉×#3189 #11767910

Order 1152.

624 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₉×#484	C₂	1	4	yes	maximal, complemented
3	C₉×#467	C₂	1	4	yes	maximal, complemented
4	C₉×#451	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₃×#3189	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	2	yes	—
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Q₁₆	C₂²	1	4	yes	complemented
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₉×C₂²⋊Q₈	—	2	2	no	—
12	C₉×C₂²⋊Q₈	—	2	2	no	—
13	C₉×C₂²⋊Q₈	C₂²	1	4	yes	—
14	C₉×C₂²⋊Q₈	C₂²	1	4	yes	—
15	C₂×C₉×C₄⋊C₄	—	2	2	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
18	C₉×#107	C₂²	1	4	yes	complemented
19	C₉×#103	C₂²	1	8	yes	complemented
20	C₉×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
22	C₃×#484	C₂×C₃	1	4	yes	—
23	C₃×#467	C₂×C₃	1	4	yes	—
24	C₃×#451	C₂×C₃	1	4	yes	—
25	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
26	C₂⁴×C₉	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₉	—	2	2	no	—
28	C₂×Q₈×C₉	—	4	4	no	—
29	C₂×Q₈×C₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₉	—	2	2	no	—
31	C₂²×C₄×C₉	—	2	2	no	—
32	C₂²×C₄×C₉	—	4	2	no	—
33	C₂²×C₄×C₉	D₄	1	2	yes	—
34	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₉×Q₁₆	—	4	4	no	—
38	C₂×C₈×C₉	—	4	2	no	—
39	C₂×C₈×C₉	C₂³	1	4	yes	—
40	C₉×C₄⋊C₄	—	2	4	no	—
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	4	no	—
45	C₉×C₂²⋊C₄	—	2	4	no	—
46	#3189	C₉	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
48	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
50	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
51	C₃×C₂²⋊Q₈	—	2	2	no	—
52	C₃×C₂²⋊Q₈	—	2	2	no	—
53	C₃×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
54	C₃×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
55	C₂×C₃×C₄⋊C₄	—	2	2	no	—
56	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
57	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
58	C₃×#107	C₂²×C₃	1	4	yes	—
59	C₃×#103	C₂²×C₃	1	8	yes	—
60	C₃×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
61	C₂³×C₉	—	4	1	no	—
62	C₂³×C₉	—	4	1	no	—
63	C₂³×C₉	Q₁₆	1	4	yes	complemented
64	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂³×C₉	C₂×D₄	1	2	yes	—
66	Q₈×C₉	—	2	8	no	—
67	Q₈×C₉	—	4	4	no	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	4	no	—
72	C₂×C₄×C₉	—	4	2	no	—
73	C₂×C₄×C₉	—	4	1	no	—
74	C₂×C₄×C₉	—	4	4	no	—
75	C₂×C₄×C₉	—	4	4	no	—
76	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
79	C₈×C₉	—	2	4	no	—
80	C₂×C₂²⋊Q₈	C₂×C₉	1	2	yes	—
81	#484	C₂×C₉	1	4	yes	complemented
82	#467	C₂×C₉	1	4	yes	complemented
83	#451	C₂×C₉	1	4	yes	complemented
84	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
85	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
86	C₂×C₃×Q₈	—	2	2	no	—
87	C₂×C₃×Q₈	—	4	4	no	—
88	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
89	C₂²×C₃×C₄	—	2	2	no	—
90	C₂²×C₃×C₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	4	2	no	—
92	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
93	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
95	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
96	C₃×Q₁₆	—	4	4	no	—
97	C₂×C₃×C₈	—	4	2	no	—
98	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
99	C₃×C₄⋊C₄	—	2	4	no	—
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	4	no	—
104	C₃×C₂²⋊C₄	—	2	4	no	—
105	C₂²×C₉	—	2	2	no	—
106	C₂²×C₉	—	2	1	no	—
107	C₂²×C₉	—	2	2	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	—	4	4	no	—
110	C₂²×C₉	C₂×Q₁₆	1	2	yes	—
111	C₂²×C₉	C₂×Q₁₆	1	4	yes	—
112	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₉	—	2	1	no	—
114	C₄×C₉	—	2	2	no	—
115	C₄×C₉	—	4	4	no	—
116	C₄×C₉	—	4	4	no	—
117	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
118	C₂²×Q₈	C₂²×C₉	1	2	yes	—
119	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
120	C₂×Q₁₆	C₂²×C₉	1	4	yes	complemented
121	C₂²×C₈	C₂²×C₉	1	2	yes	—
122	C₂²⋊Q₈	—	2	2	no	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₂²⋊Q₈	C₂²×C₉	1	4	yes	—
125	C₂²⋊Q₈	C₂²×C₉	1	4	yes	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
128	C₂×C₂²⋊C₄	—	2	2	no	—
129	#107	C₂²×C₉	1	4	yes	complemented
130	#103	C₂²×C₉	1	8	yes	complemented
131	C₂²⋊C₈	C₂²×C₉	1	4	yes	—
132	C₂³×C₃	—	4	1	no	—
133	C₂³×C₃	—	4	1	no	—
134	C₂³×C₃	C₃×Q₁₆	1	4	yes	—
135	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
137	C₃×Q₈	—	2	8	no	—
138	C₃×Q₈	—	4	4	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	—	2	4	no	—
141	C₂×C₃×C₄	—	2	4	no	—
142	C₂×C₃×C₄	—	2	4	no	—
143	C₂×C₃×C₄	—	4	1	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	4	4	no	—
146	C₂×C₃×C₄	—	4	4	no	—
147	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
149	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
150	C₃×C₈	—	2	4	no	—
151	C₂×C₉	—	2	4	no	—
152	C₂×C₉	—	2	2	no	—
153	C₂×C₉	#534	1	1	yes	characteristic
154	C₂×C₉	C₂²×Q₁₆	1	1	yes	characteristic
155	C₂×C₉	#575	1	1	yes	characteristic
156	C₂⁴	D₄×C₉	1	1	yes	characteristic
157	C₂×Q₈	—	2	2	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	C₂³×C₉	1	4	yes	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	—	2	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	D₄×C₉	1	2	yes	—
164	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
165	C₂²×C₄	C₂³×C₉	1	2	yes	—
166	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
167	Q₁₆	—	4	4	no	—
168	C₂×C₈	—	4	2	no	—
169	C₂×C₈	C₂³×C₉	1	4	yes	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₂³×C₉	1	4	yes	—
174	C₂²⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	4	4	no	—
181	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
182	C₂²×C₃	C₂×C₃×Q₁₆	1	4	yes	—
183	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
184	C₃×C₄	—	2	2	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	4	4	no	—
187	C₃×C₄	—	4	4	no	—
188	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
189	C₉	#3189	1	1	yes	characteristic, direct product factor, Sylow
190	C₂³	—	4	1	no	—
191	C₂³	—	4	1	no	—
192	C₂³	C₉×Q₁₆	1	4	yes	complemented
193	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
194	C₂³	C₂×D₄×C₉	1	2	yes	—
195	Q₈	—	2	8	no	—
196	Q₈	—	4	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
207	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
208	C₈	—	2	4	no	—
209	C₂×C₃	—	2	4	no	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	C₃×#534	1	1	yes	characteristic
212	C₂×C₃	C₂²×C₃×Q₁₆	1	1	yes	characteristic
213	C₂×C₃	C₃×#575	1	1	yes	characteristic
214	C₂²	—	2	2	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	4	4	no	—
219	C₂²	C₂×C₉×Q₁₆	1	2	yes	—
220	C₂²	C₂×C₉×Q₁₆	1	4	yes	—
221	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
222	C₄	—	2	2	no	—
223	C₄	—	2	1	no	—
224	C₄	—	4	4	no	—
225	C₄	—	4	4	no	—
226	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
227	C₃	C₃×#3189	1	1	yes	characteristic
228	C₂	—	2	4	no	—
229	C₂	—	2	2	no	—
230	C₂	C₉×#534	1	1	yes	characteristic
231	C₂	C₂²×C₉×Q₁₆	1	1	yes	characteristic
232	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	C₂⁴×C₃	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#484	C₂	semidirect
3. C₉×#467	C₂	semidirect
4. C₉×#451	C₂	semidirect
9. C₂×C₉×Q₁₆	C₂²	semidirect
18. C₉×#107	C₂²	semidirect
19. C₉×#103	C₂²	semidirect
46. #3189	C₉	direct (N × H)
63. C₂³×C₉	Q₁₆	semidirect
81. #484	C₂×C₉	semidirect
82. #467	C₂×C₉	semidirect
83. #451	C₂×C₉	semidirect
120. C₂×Q₁₆	C₂²×C₉	semidirect
129. #107	C₂²×C₉	semidirect
130. #103	C₂²×C₉	semidirect
189. C₉	#3189	direct (N × H)
192. C₂³	C₉×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₂²⋊Q₈	2
C₉×#484	2
C₉×#467	2
C₉×#451	2
C₂×C₉×C₂²⋊C₈	2
C₃×#3189	3

Subgroups — C9×#3189 #11767910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3189 #11767910