Subgroups — C₉×#1808 #11766529

Order 1152.

504 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	C₂³×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#107	C₂	1	2	yes	maximal, complemented
3	C₉×#380	C₂	1	2	yes	maximal
4	C₉×#340	C₂	1	2	yes	maximal, complemented
5	C₃×#1808	C₃	1	1	yes	maximal, characteristic
6	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₉	—	2	4	no	—
8	C₂²×C₈×C₉	C₄	1	4	yes	complemented
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
13	C₉×#107	—	2	4	no	—
14	C₉×#95	—	2	2	no	—
15	C₂³×C₃×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×#107	C₂×C₃	1	2	yes	—
17	C₃×#380	C₂×C₃	1	2	yes	—
18	C₃×#340	C₂×C₃	1	2	yes	—
19	C₂⁴×C₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₉	—	2	2	no	—
21	C₂²×C₄×C₉	—	2	2	no	—
22	C₂²×C₄×C₉	—	2	4	no	—
23	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₉	—	2	2	no	—
28	C₂×C₈×C₉	—	2	8	no	—
29	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₉	D₄	1	4	yes	—
32	C₉×C₄⋊C₄	—	2	4	no	—
33	C₉×C₄⋊C₄	—	2	4	no	—
34	C₉×C₂²⋊C₄	—	4	4	no	—
35	#1808	C₉	1	1	yes	characteristic, direct product factor, Sylow
36	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
37	C₂²×C₃×C₈	—	2	4	no	—
38	C₂²×C₃×C₈	C₃×C₄	1	4	yes	—
39	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
42	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
43	C₃×#107	—	2	4	no	—
44	C₃×#95	—	2	2	no	—
45	C₂³×C₉	—	2	2	no	—
46	C₂³×C₉	—	2	2	no	—
47	C₂³×C₉	C₄⋊C₄	1	2	yes	—
48	C₂³×C₉	D₈	1	2	yes	—
49	C₂³×C₉	Q₁₆	1	2	yes	—
50	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉	—	2	2	no	—
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	—	2	2	no	—
54	C₂×C₄×C₉	—	2	4	no	—
55	C₂×C₄×C₉	—	4	4	no	—
56	C₂×C₄×C₉	—	4	4	no	—
57	C₂×C₄×C₉	—	4	4	no	—
58	C₂×C₄×C₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₄×C₉	#41	1	2	yes	—
63	C₈×C₉	—	2	2	no	—
64	C₈×C₉	C₂²⋊C₄	1	4	yes	complemented
65	C₂³×C₈	C₂×C₉	1	1	yes	characteristic
66	C₂×#107	C₂×C₉	1	2	yes	complemented
67	#380	C₂×C₉	1	2	yes	—
68	#340	C₂×C₉	1	2	yes	complemented
69	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	—	2	2	no	—
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	—	2	4	no	—
73	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
74	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
76	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₂×C₃×C₈	—	2	2	no	—
78	C₂×C₃×C₈	—	2	8	no	—
79	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
80	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
81	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
82	C₃×C₄⋊C₄	—	2	4	no	—
83	C₃×C₄⋊C₄	—	2	4	no	—
84	C₃×C₂²⋊C₄	—	4	4	no	—
85	C₂²×C₉	—	2	2	no	—
86	C₂²×C₉	—	2	2	no	—
87	C₂²×C₉	—	2	4	no	—
88	C₂²×C₉	—	2	4	no	—
89	C₂²×C₉	#107	1	4	yes	complemented
90	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₉	#121	1	2	yes	—
92	C₂²×C₉	C₂×D₈	1	1	yes	characteristic
93	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
94	C₂²×C₉	#135	1	2	yes	—
95	C₄×C₉	—	2	2	no	—
96	C₄×C₉	—	8	4	no	—
97	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₄×C₉	#117	1	1	yes	characteristic
99	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
101	C₂²×C₈	—	2	4	no	—
102	C₂²×C₈	C₄×C₉	1	4	yes	complemented
103	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
104	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
106	C₂×C₂²⋊C₄	—	2	2	no	—
107	#107	—	2	4	no	—
108	#95	—	2	2	no	—
109	C₂³×C₃	—	2	2	no	—
110	C₂³×C₃	—	2	2	no	—
111	C₂³×C₃	C₃×C₄⋊C₄	1	2	yes	—
112	C₂³×C₃	C₃×D₈	1	2	yes	—
113	C₂³×C₃	C₃×Q₁₆	1	2	yes	—
114	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	4	no	—
119	C₂×C₃×C₄	—	4	4	no	—
120	C₂×C₃×C₄	—	4	4	no	—
121	C₂×C₃×C₄	—	4	4	no	—
122	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
123	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
126	C₂×C₃×C₄	C₃×#41	1	2	yes	—
127	C₃×C₈	—	2	2	no	—
128	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
129	C₂×C₉	—	2	4	no	—
130	C₂×C₉	#380	1	1	yes	characteristic
131	C₂×C₉	C₂×#107	1	1	yes	characteristic
132	C₂×C₉	#427	1	1	yes	characteristic
133	C₂×C₉	#466	1	2	yes	—
134	C₂×C₉	#467	1	2	yes	—
135	C₂⁴	D₄×C₉	1	1	yes	characteristic
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
140	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
141	C₂²×C₄	Q₈×C₉	1	2	yes	—
142	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
146	C₂×C₈	C₂×C₄×C₉	1	2	yes	—
147	C₂×C₈	D₄×C₉	1	4	yes	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₂²⋊C₄	—	4	4	no	—
151	C₂²×C₃	—	2	2	no	—
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	2	4	no	—
155	C₂²×C₃	C₃×#107	1	4	yes	—
156	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₃	C₃×#121	1	2	yes	—
158	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
159	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
160	C₂²×C₃	C₃×#135	1	2	yes	—
161	C₃×C₄	—	2	2	no	—
162	C₃×C₄	—	8	4	no	—
163	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
164	C₃×C₄	C₃×#117	1	1	yes	characteristic
165	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
166	C₉	#1808	1	1	yes	characteristic, direct product factor, Sylow
167	C₂³	—	2	2	no	—
168	C₂³	—	2	2	no	—
169	C₂³	C₉×C₄⋊C₄	1	2	yes	—
170	C₂³	C₉×D₈	1	2	yes	—
171	C₂³	C₉×Q₁₆	1	2	yes	—
172	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	C₉×C₄⋊C₄	1	2	yes	—
181	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
183	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
184	C₂×C₄	C₉×#41	1	2	yes	—
185	C₈	—	2	2	no	—
186	C₈	C₉×C₂²⋊C₄	1	4	yes	complemented
187	C₂×C₃	—	2	4	no	—
188	C₂×C₃	C₃×#380	1	1	yes	characteristic
189	C₂×C₃	C₂×C₃×#107	1	1	yes	characteristic
190	C₂×C₃	C₃×#427	1	1	yes	characteristic
191	C₂×C₃	C₃×#466	1	2	yes	—
192	C₂×C₃	C₃×#467	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	4	no	—
197	C₂²	C₉×#107	1	4	yes	complemented
198	C₂²	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
199	C₂²	C₉×#121	1	2	yes	—
200	C₂²	C₂×C₉×D₈	1	1	yes	characteristic
201	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
202	C₂²	C₉×#135	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	8	4	no	—
205	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
206	C₄	C₉×#117	1	1	yes	characteristic
207	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
208	C₃	C₃×#1808	1	1	yes	characteristic
209	C₂	—	2	4	no	—
210	C₂	C₉×#380	1	1	yes	characteristic
211	C₂	C₂×C₉×#107	1	1	yes	characteristic
212	C₂	C₉×#427	1	1	yes	characteristic
213	C₂	C₉×#466	1	2	yes	—
214	C₂	C₉×#467	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂⁴×C₃	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#107	C₂	semidirect
4. C₉×#340	C₂	semidirect
8. C₂²×C₈×C₉	C₄	semidirect
35. #1808	C₉	direct (N × H)
64. C₈×C₉	C₂²⋊C₄	semidirect
66. C₂×#107	C₂×C₉	semidirect
68. #340	C₂×C₉	semidirect
89. C₂²×C₉	#107	semidirect
102. C₂²×C₈	C₄×C₉	semidirect
166. C₉	#1808	direct (N × H)
186. C₈	C₉×C₂²⋊C₄	semidirect
197. C₂²	C₉×#107	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₈×C₉	2
C₂×C₉×#107	2
C₉×#380	2
C₉×#340	2
C₃×#1808	3

Subgroups — C9×#1808 #11766529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1808 #11766529