Subgroups — C₉×#2961 #11767682

Order 1152.

522 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic
2	C₉×#446	C₂	1	8	yes	maximal, complemented
3	C₉×#424	C₂	1	2	yes	maximal
4	C₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₉×#405	C₂	1	2	yes	maximal
6	C₂×C₉×#97	C₂	1	1	yes	maximal, characteristic
7	C₃×#2961	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×C₄⋊Q₈	C₄	1	2	yes	—
12	C₉×#125	C₄	1	2	yes	—
13	C₉×C₂²⋊Q₈	C₄	1	4	yes	—
14	C₄×Q₈×C₉	C₂²	1	4	yes	—
15	C₉×#117	C₂²	1	2	yes	—
16	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
17	C₉×C₄⋊C₈	C₂²	1	4	yes	—
18	C₉×C₄⋊C₈	C₂²	1	8	yes	—
19	C₉×#97	C₂²	1	4	yes	—
20	C₉×#97	C₂²	1	4	yes	—
21	C₄×C₈×C₉	C₂²	1	4	yes	—
22	C₃×#533	C₂×C₃	1	1	yes	characteristic
23	C₃×#446	C₂×C₃	1	8	yes	—
24	C₃×#424	C₂×C₃	1	2	yes	—
25	C₃×#423	C₂×C₃	1	1	yes	characteristic
26	C₃×#405	C₂×C₃	1	2	yes	—
27	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
28	C₂×Q₈×C₉	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₉×M₁₆	—	2	4	no	—
32	C₂×C₈×C₉	—	2	2	no	—
33	C₂×C₈×C₉	Q₈	1	4	yes	complemented
34	C₂×C₈×C₉	C₂³	1	2	yes	—
35	C₂×C₈×C₉	C₂³	1	4	yes	—
36	C₂×C₈×C₉	C₂³	1	2	yes	—
37	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₉×C₄⋊C₄	C₂×C₄	1	8	yes	—
39	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
40	C₉×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₉	C₂³	1	2	yes	—
44	C₄²×C₉	C₂³	1	4	yes	—
45	#2961	C₉	1	1	yes	characteristic, direct product factor, Sylow
46	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
47	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₃×C₄⋊Q₈	C₃×C₄	1	2	yes	—
50	C₃×#125	C₃×C₄	1	2	yes	—
51	C₃×C₂²⋊Q₈	C₃×C₄	1	4	yes	—
52	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
53	C₃×#117	C₂²×C₃	1	2	yes	—
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
56	C₃×C₄⋊C₈	C₂²×C₃	1	8	yes	—
57	C₃×#97	C₂²×C₃	1	4	yes	—
58	C₃×#97	C₂²×C₃	1	4	yes	—
59	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
60	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
61	Q₈×C₉	—	4	4	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	C₂²×C₄	1	8	yes	—
66	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₉	#41	1	2	yes	—
69	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
70	C₈×C₉	—	2	2	no	—
71	C₈×C₉	—	2	4	no	—
72	C₈×C₉	C₂×Q₈	1	4	yes	—
73	#533	C₂×C₉	1	1	yes	characteristic
74	#446	C₂×C₉	1	8	yes	complemented
75	#424	C₂×C₉	1	2	yes	—
76	#423	C₂×C₉	1	1	yes	characteristic
77	#405	C₂×C₉	1	2	yes	—
78	C₂×#97	C₂×C₉	1	1	yes	characteristic
79	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
80	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
81	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
82	C₃×M₁₆	—	2	4	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
85	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
86	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
87	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
88	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
89	C₃×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
90	C₃×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
91	C₃×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
92	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
93	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
94	C₃×C₄²	C₂³×C₃	1	4	yes	—
95	C₃×C₄²	C₂³×C₃	1	2	yes	—
96	C₂²×C₉	—	2	2	no	—
97	C₂²×C₉	C₄×Q₈	1	2	yes	—
98	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	2	2	no	—
101	C₄×C₉	—	2	8	no	—
102	C₄×C₉	C₄×Q₈	1	2	yes	—
103	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
104	C₄×C₉	C₂×#41	1	1	yes	characteristic
105	C₂×M₁₆	C₂²×C₉	1	2	yes	—
106	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
107	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
108	C₄⋊Q₈	C₄×C₉	1	2	yes	—
109	#125	C₄×C₉	1	2	yes	—
110	C₂²⋊Q₈	C₄×C₉	1	4	yes	—
111	C₄×Q₈	C₂²×C₉	1	4	yes	—
112	#117	C₂²×C₉	1	2	yes	—
113	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
114	C₄⋊C₈	C₂²×C₉	1	4	yes	—
115	C₄⋊C₈	C₂²×C₉	1	8	yes	—
116	#97	C₂²×C₉	1	4	yes	—
117	#97	C₂²×C₉	1	4	yes	—
118	C₄×C₈	C₂²×C₉	1	4	yes	—
119	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₃×Q₈	—	4	4	no	—
121	C₂×C₃×C₄	—	4	2	no	—
122	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
123	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
124	C₂×C₃×C₄	C₂²×C₃×C₄	1	8	yes	—
125	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
127	C₂×C₃×C₄	C₃×#41	1	2	yes	—
128	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
129	C₃×C₈	—	2	2	no	—
130	C₃×C₈	—	2	4	no	—
131	C₃×C₈	C₂×C₃×Q₈	1	4	yes	—
132	C₂×C₉	—	2	2	no	—
133	C₂×C₉	C₂×C₄×Q₈	1	1	yes	characteristic
134	C₂×C₉	#568	1	2	yes	—
135	C₂×Q₈	C₂×C₄×C₉	1	4	yes	—
136	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
137	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
138	M₁₆	—	2	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	Q₈×C₉	1	4	yes	complemented
141	C₂×C₈	C₂³×C₉	1	4	yes	—
142	C₂×C₈	C₂³×C₉	1	2	yes	—
143	C₂×C₈	C₂³×C₉	1	2	yes	—
144	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
145	C₄⋊C₄	C₂×C₄×C₉	1	8	yes	—
146	C₄⋊C₄	C₂×C₄×C₉	1	4	yes	—
147	C₂²⋊C₄	C₂×C₄×C₉	1	4	yes	—
148	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
149	C₄²	C₂×C₄×C₉	1	1	yes	characteristic
150	C₄²	C₂³×C₉	1	4	yes	—
151	C₄²	C₂³×C₉	1	2	yes	—
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	C₃×C₄×Q₈	1	2	yes	—
154	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
155	C₃×C₄	—	2	2	no	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	8	no	—
158	C₃×C₄	C₃×C₄×Q₈	1	2	yes	—
159	C₃×C₄	C₂²×C₃×Q₈	1	1	yes	characteristic
160	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
161	C₉	#2961	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³	C₂²×C₄×C₉	1	1	yes	characteristic
163	Q₈	—	4	4	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
166	C₂×C₄	C₂²×C₄×C₉	1	2	yes	—
167	C₂×C₄	C₂²×C₄×C₉	1	8	yes	—
168	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
169	C₂×C₄	C₂×Q₈×C₉	1	2	yes	—
170	C₂×C₄	C₉×#41	1	2	yes	—
171	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
172	C₈	—	2	2	no	—
173	C₈	—	2	4	no	—
174	C₈	C₂×Q₈×C₉	1	4	yes	—
175	C₂×C₃	—	2	2	no	—
176	C₂×C₃	C₂×C₃×C₄×Q₈	1	1	yes	characteristic
177	C₂×C₃	C₃×#568	1	2	yes	—
178	C₂²	—	2	2	no	—
179	C₂²	C₄×Q₈×C₉	1	2	yes	—
180	C₂²	C₂³×C₄×C₉	1	1	yes	characteristic
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	8	no	—
184	C₄	C₄×Q₈×C₉	1	2	yes	—
185	C₄	C₂²×Q₈×C₉	1	1	yes	characteristic
186	C₄	C₂×C₉×#41	1	1	yes	characteristic
187	C₃	C₃×#2961	1	1	yes	characteristic
188	C₂	—	2	2	no	—
189	C₂	C₂×C₄×Q₈×C₉	1	1	yes	characteristic
190	C₂	C₉×#568	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₉×#446	C₂	semidirect
33. C₂×C₈×C₉	Q₈	semidirect
45. #2961	C₉	direct (N × H)
74. #446	C₂×C₉	semidirect
140. C₂×C₈	Q₈×C₉	semidirect
161. C₉	#2961	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₉×#446	2
C₉×#424	2
C₉×#423	2
C₉×#405	2
C₂×C₉×#97	2
C₃×#2961	3

Subgroups — C9×#2961 #11767682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2961 #11767682