Subgroups — C₉×#1981 #11766702

Order 1152.

486 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	C₂×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#103	C₂	1	4	yes	maximal
4	C₄×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₃×#1981	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	C₂²	1	2	yes	—
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₉×C₄⋊Q₈	—	2	2	no	—
9	C₉×C₄⋊Q₈	C₄	1	4	yes	complemented
10	C₂×C₉×C₄⋊C₄	—	2	1	no	—
11	C₂×C₉×C₄⋊C₄	—	2	1	no	—
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₉	—	2	1	no	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊C₈	—	2	2	no	—
16	C₉×#103	—	2	8	no	—
17	C₉×#95	—	2	1	no	—
18	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×#103	C₂×C₃	1	4	yes	—
21	C₃×C₄×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
22	C₂×Q₈×C₉	—	2	2	no	—
23	C₂×Q₈×C₉	—	2	4	no	—
24	C₂×Q₈×C₉	—	2	4	no	—
25	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₉	—	2	2	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₉	—	2	4	no	—
32	C₂×C₈×C₉	—	2	2	no	—
33	C₉×C₄⋊C₄	—	2	4	no	—
34	C₉×C₄⋊C₄	—	2	4	no	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₉×C₄⋊C₄	D₄	1	4	yes	—
37	C₄²×C₉	—	4	2	no	—
38	C₄²×C₉	C₂×C₄	1	2	yes	—
39	C₄²×C₉	D₄	1	2	yes	—
40	#1981	C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
42	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
43	C₃×C₄⋊Q₈	—	2	2	no	—
44	C₃×C₄⋊Q₈	C₃×C₄	1	4	yes	—
45	C₂×C₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₃×C₄⋊C₄	—	2	1	no	—
47	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₂×C₃×C₄²	—	2	1	no	—
49	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₃×C₄⋊C₈	—	2	2	no	—
51	C₃×#103	—	2	8	no	—
52	C₃×#95	—	2	1	no	—
53	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
54	Q₈×C₉	—	2	8	no	—
55	Q₈×C₉	—	4	2	no	—
56	Q₈×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	4	no	—
59	C₂×C₄×C₉	—	2	4	no	—
60	C₂×C₄×C₉	—	4	4	no	—
61	C₂×C₄×C₉	—	4	2	no	—
62	C₂×C₄×C₉	—	4	2	no	—
63	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₉	SD₁₆	1	2	yes	—
66	C₂×C₄×C₉	Q₁₆	1	2	yes	—
67	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₄×C₉	#41	1	2	yes	—
71	C₈×C₉	—	4	2	no	—
72	C₂×C₄⋊Q₈	C₂×C₉	1	1	yes	characteristic
73	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
74	C₂×#103	C₂×C₉	1	4	yes	—
75	C₄×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
76	C₂×C₃×Q₈	—	2	2	no	—
77	C₂×C₃×Q₈	—	2	4	no	—
78	C₂×C₃×Q₈	—	2	4	no	—
79	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
80	C₂²×C₃×C₄	—	2	2	no	—
81	C₂²×C₃×C₄	—	2	2	no	—
82	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₂×C₃×C₈	—	2	4	no	—
86	C₂×C₃×C₈	—	2	2	no	—
87	C₃×C₄⋊C₄	—	2	4	no	—
88	C₃×C₄⋊C₄	—	2	4	no	—
89	C₃×C₄⋊C₄	—	2	2	no	—
90	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
91	C₃×C₄²	—	4	2	no	—
92	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
93	C₃×C₄²	C₃×D₄	1	2	yes	—
94	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₉	#121	1	2	yes	—
96	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
97	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
98	C₂²×C₉	#137	1	2	yes	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	4	4	no	—
101	C₄×C₉	—	4	4	no	—
102	C₄×C₉	#103	1	4	yes	complemented
103	C₄×C₉	C₄×D₄	1	2	yes	—
104	C₄×C₉	#124	1	1	yes	characteristic
105	C₄×C₉	#127	1	1	yes	characteristic
106	C₂²×Q₈	C₂²×C₉	1	2	yes	—
107	C₂²×C₈	C₂²×C₉	1	2	yes	—
108	C₄⋊Q₈	—	2	2	no	—
109	C₄⋊Q₈	C₄×C₉	1	4	yes	complemented
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
113	C₂×C₄²	—	2	1	no	—
114	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
115	C₄⋊C₈	—	2	2	no	—
116	#103	—	2	8	no	—
117	#95	—	2	1	no	—
118	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
119	C₃×Q₈	—	2	8	no	—
120	C₃×Q₈	—	4	2	no	—
121	C₃×Q₈	—	4	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	4	4	no	—
126	C₂×C₃×C₄	—	4	2	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
129	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
130	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
131	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
132	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₃×#41	1	2	yes	—
136	C₃×C₈	—	4	2	no	—
137	C₂×C₉	#390	1	1	yes	characteristic
138	C₂×C₉	C₂×#103	1	1	yes	characteristic
139	C₂×C₉	#419	1	1	yes	characteristic
140	C₂×C₉	#461	1	2	yes	—
141	C₂×C₉	#462	1	2	yes	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	4	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	C₂×C₄×C₉	1	4	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
149	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	2	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	D₄×C₉	1	4	yes	—
157	C₄²	—	4	2	no	—
158	C₄²	C₂×C₄×C₉	1	2	yes	—
159	C₄²	D₄×C₉	1	2	yes	—
160	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
161	C₂²×C₃	C₃×#121	1	2	yes	—
162	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
163	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
164	C₂²×C₃	C₃×#137	1	2	yes	—
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	—	4	4	no	—
167	C₃×C₄	—	4	4	no	—
168	C₃×C₄	C₃×#103	1	4	yes	—
169	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
170	C₃×C₄	C₃×#124	1	1	yes	characteristic
171	C₃×C₄	C₃×#127	1	1	yes	characteristic
172	C₉	#1981	1	1	yes	characteristic, direct product factor, Sylow
173	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
174	Q₈	—	2	8	no	—
175	Q₈	—	4	2	no	—
176	Q₈	—	4	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
184	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
185	C₂×C₄	C₉×SD₁₆	1	2	yes	—
186	C₂×C₄	C₉×Q₁₆	1	2	yes	—
187	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
188	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
189	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
190	C₂×C₄	C₉×#41	1	2	yes	—
191	C₈	—	4	2	no	—
192	C₂×C₃	C₃×#390	1	1	yes	characteristic
193	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
194	C₂×C₃	C₃×#419	1	1	yes	characteristic
195	C₂×C₃	C₃×#461	1	2	yes	—
196	C₂×C₃	C₃×#462	1	2	yes	—
197	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
198	C₂²	C₉×#121	1	2	yes	—
199	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
200	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
201	C₂²	C₉×#137	1	2	yes	—
202	C₄	—	2	2	no	—
203	C₄	—	4	4	no	—
204	C₄	—	4	4	no	—
205	C₄	C₉×#103	1	4	yes	complemented
206	C₄	C₄×D₄×C₉	1	2	yes	—
207	C₄	C₉×#124	1	1	yes	characteristic
208	C₄	C₉×#127	1	1	yes	characteristic
209	C₃	C₃×#1981	1	1	yes	characteristic
210	C₂	C₉×#390	1	1	yes	characteristic
211	C₂	C₂×C₉×#103	1	1	yes	characteristic
212	C₂	C₉×#419	1	1	yes	characteristic
213	C₂	C₉×#461	1	2	yes	—
214	C₂	C₉×#462	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, prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₉×C₄⋊Q₈	C₄	semidirect
40. #1981	C₉	direct (N × H)
102. C₄×C₉	#103	semidirect
109. C₄⋊Q₈	C₄×C₉	semidirect
172. C₉	#1981	direct (N × H)
205. C₄	C₉×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×C₄⋊C₈	2
C₂×C₉×#103	2
C₄×C₉×C₄⋊C₄	2
C₃×#1981	3

Subgroups — C9×#1981 #11766702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1981 #11766702