Subgroups — C₉×#1985 #11766706

Order 1152.

420 conjugacy classes of subgroups.

The subgroup lattice has 201 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×C₉×#97	C₂	1	1	yes	maximal, characteristic
4	C₉×#336	C₂	1	4	yes	maximal
5	C₃×#1985	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	C₉×#97	—	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×#97	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	4	no	—
23	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₉	—	2	2	no	—
29	C₂×C₈×C₉	—	2	2	no	—
30	C₂×C₈×C₉	—	2	2	no	—
31	C₂×C₈×C₉	—	2	4	no	—
32	C₂×C₈×C₉	Q₈	1	4	yes	complemented
33	C₉×C₄⋊C₄	—	2	4	no	—
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₄²×C₉	—	2	2	no	—
36	C₄²×C₉	—	2	2	no	—
37	C₄²×C₉	D₄	1	2	yes	—
38	C₄²×C₉	D₄	1	2	yes	—
39	#1985	C₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
41	C₂²×C₃×C₈	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	—	2	2	no	—
45	C₂×C₃×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄⋊C₄	C₃×C₄	1	2	yes	—
47	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
48	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₄⋊C₈	—	2	2	no	—
50	C₃×#97	—	2	2	no	—
51	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
52	Q₈×C₉	—	2	8	no	—
53	C₂×C₄×C₉	—	2	2	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	M₁₆	1	4	yes	—
62	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
65	C₂×C₄×C₉	#41	1	2	yes	—
66	C₈×C₉	—	2	4	no	—
67	C₈×C₉	—	4	2	no	—
68	C₂×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
69	C₂×C₄⋊C₈	C₂×C₉	1	1	yes	characteristic
70	C₂×#97	C₂×C₉	1	1	yes	characteristic
71	#336	C₂×C₉	1	4	yes	—
72	C₂×C₃×Q₈	—	2	2	no	—
73	C₂×C₃×Q₈	—	2	4	no	—
74	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
75	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
78	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
79	C₂×C₃×C₈	—	2	2	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	4	no	—
83	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
84	C₃×C₄⋊C₄	—	2	4	no	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄²	—	2	2	no	—
87	C₃×C₄²	—	2	2	no	—
88	C₃×C₄²	C₃×D₄	1	2	yes	—
89	C₃×C₄²	C₃×D₄	1	2	yes	—
90	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₉	C₄×Q₈	1	2	yes	—
92	C₂²×C₉	C₂×M₁₆	1	2	yes	—
93	C₂²×C₉	#131	1	2	yes	—
94	C₄×C₉	—	2	2	no	—
95	C₄×C₉	—	2	4	no	—
96	C₄×C₉	—	2	2	no	—
97	C₄×C₉	—	4	2	no	—
98	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
99	C₄×C₉	#124	1	1	yes	characteristic
100	C₄×C₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂²×Q₈	C₄×C₉	1	1	yes	characteristic
102	C₂²×C₈	C₂²×C₉	1	2	yes	—
103	C₂²×C₈	C₂²×C₉	1	2	yes	—
104	C₄×Q₈	—	2	2	no	—
105	C₄×Q₈	—	2	2	no	—
106	C₂×C₄⋊C₄	C₄×C₉	1	1	yes	characteristic
107	C₂×C₄⋊C₄	C₄×C₉	1	2	yes	—
108	C₂×C₄²	C₂²×C₉	1	2	yes	—
109	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
110	C₄⋊C₈	—	2	2	no	—
111	#97	—	2	2	no	—
112	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
113	C₃×Q₈	—	2	8	no	—
114	C₂×C₃×C₄	—	2	2	no	—
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	2	no	—
119	C₂×C₃×C₄	—	2	4	no	—
120	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
121	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
122	C₂×C₃×C₄	C₃×M₁₆	1	4	yes	—
123	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
126	C₂×C₃×C₄	C₃×#41	1	2	yes	—
127	C₃×C₈	—	2	4	no	—
128	C₃×C₈	—	4	2	no	—
129	C₂×C₉	#391	1	1	yes	characteristic
130	C₂×C₉	#407	1	1	yes	characteristic
131	C₂×C₉	#408	1	1	yes	characteristic
132	C₂×C₉	#446	1	4	yes	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	2	4	no	—
135	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
136	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₄×C₉	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
139	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	Q₈×C₉	1	4	yes	complemented
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	2	no	—
149	C₄²	D₄×C₉	1	2	yes	—
150	C₄²	D₄×C₉	1	2	yes	—
151	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
153	C₂²×C₃	C₂×C₃×M₁₆	1	2	yes	—
154	C₂²×C₃	C₃×#131	1	2	yes	—
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	2	4	no	—
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	—	4	2	no	—
159	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
160	C₃×C₄	C₃×#124	1	1	yes	characteristic
161	C₃×C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
162	C₉	#1985	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
164	Q₈	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
172	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
173	C₂×C₄	C₉×M₁₆	1	4	yes	—
174	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
175	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
176	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
177	C₂×C₄	C₉×#41	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	4	2	no	—
180	C₂×C₃	C₃×#391	1	1	yes	characteristic
181	C₂×C₃	C₃×#407	1	1	yes	characteristic
182	C₂×C₃	C₃×#408	1	1	yes	characteristic
183	C₂×C₃	C₃×#446	1	4	yes	—
184	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²	C₄×Q₈×C₉	1	2	yes	—
186	C₂²	C₂×C₉×M₁₆	1	2	yes	—
187	C₂²	C₉×#131	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	4	2	no	—
192	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
193	C₄	C₉×#124	1	1	yes	characteristic
194	C₄	C₉×C₄⋊Q₈	1	1	yes	characteristic
195	C₃	C₃×#1985	1	1	yes	characteristic
196	C₂	C₉×#391	1	1	yes	characteristic
197	C₂	C₉×#407	1	1	yes	characteristic
198	C₂	C₉×#408	1	1	yes	characteristic
199	C₂	C₉×#446	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
32. C₂×C₈×C₉	Q₈	semidirect
39. #1985	C₉	direct (N × H)
144. C₂×C₈	Q₈×C₉	semidirect
162. C₉	#1985	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×C₉×#97	2
C₉×#336	2
C₃×#1985	3

Subgroups — C9×#1985 #11766706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1985 #11766706