Subgroups — C₉×#1984 #11766705

Order 1152.

438 conjugacy classes of subgroups.

The subgroup lattice has 213 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
4	C₉×#336	C₂	1	4	yes	maximal
5	C₃×#1984	C₃	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₄×Q₈×C₉	—	2	2	no	—
10	C₄×Q₈×C₉	—	2	2	no	—
11	C₂×C₉×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₄²×C₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊C₈	—	2	2	no	—
16	C₄×C₈×C₉	—	2	2	no	—
17	C₂×C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
20	C₃×#336	C₂×C₃	1	4	yes	—
21	C₂×Q₈×C₉	—	2	2	no	—
22	C₂×Q₈×C₉	—	2	2	no	—
23	C₂×Q₈×C₉	C₈	1	4	yes	complemented
24	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₉	—	2	2	no	—
30	C₂×C₈×C₉	—	2	2	no	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₂×C₈×C₉	—	2	4	no	—
33	C₂×C₈×C₉	Q₈	1	4	yes	complemented
34	C₉×C₄⋊C₄	—	2	4	no	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₄²×C₉	—	2	2	no	—
37	C₄²×C₉	—	2	2	no	—
38	C₄²×C₉	D₄	1	2	yes	—
39	C₄²×C₉	D₄	1	2	yes	—
40	#1984	C₉	1	1	yes	characteristic, direct product factor, Sylow
41	C₂²×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
42	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
43	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
44	C₃×C₄×Q₈	—	2	2	no	—
45	C₃×C₄×Q₈	—	2	2	no	—
46	C₂×C₃×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
48	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
49	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
50	C₃×C₄⋊C₈	—	2	2	no	—
51	C₃×C₄×C₈	—	2	2	no	—
52	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
53	Q₈×C₉	—	2	8	no	—
54	C₂×C₄×C₉	—	2	2	no	—
55	C₂×C₄×C₉	—	2	2	no	—
56	C₂×C₄×C₉	—	2	2	no	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	4	no	—
59	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₉	C₂×C₈	1	4	yes	—
62	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
63	C₂×C₄×C₉	M₁₆	1	2	yes	—
64	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₉	#41	1	2	yes	—
68	C₈×C₉	—	2	4	no	—
69	C₈×C₉	—	4	2	no	—
70	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
71	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
72	C₂×C₄×C₈	C₂×C₉	1	1	yes	characteristic
73	#336	C₂×C₉	1	4	yes	—
74	C₂×C₃×Q₈	—	2	2	no	—
75	C₂×C₃×Q₈	—	2	2	no	—
76	C₂×C₃×Q₈	C₃×C₈	1	4	yes	—
77	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
78	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
80	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	—	2	4	no	—
86	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
87	C₃×C₄⋊C₄	—	2	4	no	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	C₃×D₄	1	2	yes	—
92	C₃×C₄²	C₃×D₄	1	2	yes	—
93	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₉	C₄×Q₈	1	2	yes	—
95	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
96	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
97	C₂²×C₉	#131	1	2	yes	—
98	C₄×C₉	—	2	4	no	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	4	2	no	—
101	C₄×C₉	C₂²⋊C₈	1	4	yes	—
102	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
103	C₄×C₉	#124	1	1	yes	characteristic
104	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
105	C₂²×Q₈	C₄×C₉	1	1	yes	characteristic
106	C₂²×C₈	C₂²×C₉	1	2	yes	—
107	C₂²×C₈	C₂²×C₉	1	2	yes	—
108	C₄×Q₈	—	2	2	no	—
109	C₄×Q₈	—	2	2	no	—
110	C₂×C₄⋊C₄	C₄×C₉	1	1	yes	characteristic
111	C₂×C₄⋊C₄	C₄×C₉	1	2	yes	—
112	C₂×C₄²	C₂²×C₉	1	2	yes	—
113	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₃×Q₈	—	2	8	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	2	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
124	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₃×C₈	1	4	yes	—
126	C₂×C₃×C₄	C₂×C₃×C₈	1	2	yes	—
127	C₂×C₃×C₄	C₃×M₁₆	1	2	yes	—
128	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
131	C₂×C₃×C₄	C₃×#41	1	2	yes	—
132	C₃×C₈	—	2	4	no	—
133	C₃×C₈	—	4	2	no	—
134	C₂×C₉	#391	1	1	yes	characteristic
135	C₂×C₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂×C₉	#408	1	1	yes	characteristic
137	C₂×C₉	C₈×Q₈	1	2	yes	—
138	C₂×C₉	#446	1	2	yes	—
139	C₂×Q₈	—	2	2	no	—
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	C₈×C₉	1	4	yes	complemented
142	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
143	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
145	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	Q₈×C₉	1	4	yes	complemented
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	D₄×C₉	1	2	yes	—
157	C₄²	D₄×C₉	1	2	yes	—
158	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
160	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
161	C₂²×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
162	C₂²×C₃	C₃×#131	1	2	yes	—
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	4	2	no	—
166	C₃×C₄	C₃×C₂²⋊C₈	1	4	yes	—
167	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
168	C₃×C₄	C₃×#124	1	1	yes	characteristic
169	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
170	C₉	#1984	1	1	yes	characteristic, direct product factor, Sylow
171	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
172	Q₈	—	2	8	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
179	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
180	C₂×C₄	C₂×C₈×C₉	1	4	yes	—
181	C₂×C₄	C₂×C₈×C₉	1	2	yes	—
182	C₂×C₄	C₉×M₁₆	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
185	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
186	C₂×C₄	C₉×#41	1	2	yes	—
187	C₈	—	2	4	no	—
188	C₈	—	4	2	no	—
189	C₂×C₃	C₃×#391	1	1	yes	characteristic
190	C₂×C₃	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
191	C₂×C₃	C₃×#408	1	1	yes	characteristic
192	C₂×C₃	C₃×C₈×Q₈	1	2	yes	—
193	C₂×C₃	C₃×#446	1	2	yes	—
194	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
195	C₂²	C₄×Q₈×C₉	1	2	yes	—
196	C₂²	C₂²×C₈×C₉	1	1	yes	characteristic
197	C₂²	C₂×C₉×M₁₆	1	1	yes	characteristic
198	C₂²	C₉×#131	1	2	yes	—
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	C₉×C₂²⋊C₈	1	4	yes	—
203	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
204	C₄	C₉×#124	1	1	yes	characteristic
205	C₄	C₉×C₄⋊Q₈	1	1	yes	characteristic
206	C₃	C₃×#1984	1	1	yes	characteristic
207	C₂	C₉×#391	1	1	yes	characteristic
208	C₂	C₂×C₉×C₂²⋊C₈	1	1	yes	characteristic
209	C₂	C₉×#408	1	1	yes	characteristic
210	C₂	C₈×Q₈×C₉	1	2	yes	—
211	C₂	C₉×#446	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂³
commutator	C₂²	C₂²×C₈×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂²×C₃×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
23. C₂×Q₈×C₉	C₈	semidirect
33. C₂×C₈×C₉	Q₈	semidirect
40. #1984	C₉	direct (N × H)
141. C₂×Q₈	C₈×C₉	semidirect
151. C₂×C₈	Q₈×C₉	semidirect
170. C₉	#1984	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₉	2
C₂×C₉×C₄⋊C₈	2
C₂×C₄×C₈×C₉	2
C₉×#336	2
C₃×#1984	3

Subgroups — C9×#1984 #11766705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1984 #11766705