Subgroups — C₉×#3261 #11767982

Order 1152.

483 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#541	C₂	1	2	yes	maximal
2	C₉×#479	C₂	1	4	yes	maximal, complemented
3	C₉×#470	C₂	1	2	yes	maximal
4	C₉×#467	C₂	1	2	yes	maximal
5	C₉×#462	C₂	1	4	yes	maximal, complemented
6	C₉×#424	C₂	1	1	yes	maximal, characteristic
7	C₃×#3261	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₉×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₉×#126	—	2	4	no	—
14	C₉×#125	—	2	4	no	—
15	C₉×#125	C₂²	1	2	yes	—
16	C₉×C₂²⋊Q₈	C₂²	1	4	yes	—
17	C₄×Q₈×C₉	C₂²	1	4	yes	—
18	C₉×#117	C₂²	1	1	yes	characteristic
19	C₉×#107	C₂²	1	2	yes	—
20	C₉×#106	C₂²	1	2	yes	—
21	C₉×C₄⋊C₈	C₂²	1	4	yes	—
22	C₉×#103	C₂²	1	4	yes	—
23	C₉×#103	C₂²	1	4	yes	—
24	C₃×#541	C₂×C₃	1	2	yes	—
25	C₃×#479	C₂×C₃	1	4	yes	—
26	C₃×#470	C₂×C₃	1	2	yes	—
27	C₃×#467	C₂×C₃	1	2	yes	—
28	C₃×#462	C₂×C₃	1	4	yes	—
29	C₃×#424	C₂×C₃	1	1	yes	characteristic
30	C₂×Q₈×C₉	C₂³	1	4	yes	—
31	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₉×Q₁₆	—	4	2	no	—
33	C₉×Q₁₆	—	4	2	no	—
34	C₉×M₁₆	—	4	1	no	—
35	C₂×C₈×C₉	—	4	1	no	—
36	C₂×C₈×C₉	C₂³	1	2	yes	—
37	C₂×C₈×C₉	C₂³	1	2	yes	—
38	C₉×C₄⋊C₄	—	2	4	no	—
39	C₉×C₄⋊C₄	—	2	4	no	—
40	C₉×C₄⋊C₄	—	2	4	no	—
41	C₉×C₄⋊C₄	—	2	4	no	—
42	C₉×C₄⋊C₄	D₄	1	2	yes	—
43	C₉×C₄⋊C₄	C₂³	1	4	yes	—
44	C₉×C₂²⋊C₄	—	2	4	no	—
45	C₉×C₂²⋊C₄	D₄	1	2	yes	—
46	C₄²×C₉	—	2	4	no	—
47	C₄²×C₉	C₂³	1	2	yes	—
48	#3261	C₉	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
50	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
51	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
54	C₃×#126	—	2	4	no	—
55	C₃×#125	—	2	4	no	—
56	C₃×#125	C₂²×C₃	1	2	yes	—
57	C₃×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
58	C₃×C₄×Q₈	C₂²×C₃	1	4	yes	—
59	C₃×#117	C₂²×C₃	1	1	yes	characteristic
60	C₃×#107	C₂²×C₃	1	2	yes	—
61	C₃×#106	C₂²×C₃	1	2	yes	—
62	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
63	C₃×#103	C₂²×C₃	1	4	yes	—
64	C₃×#103	C₂²×C₃	1	4	yes	—
65	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₉	—	2	4	no	—
67	Q₈×C₉	—	2	4	no	—
68	C₂×C₄×C₉	—	2	1	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	4	no	—
72	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
75	C₈×C₉	—	2	2	no	—
76	C₈×C₉	—	2	2	no	—
77	#541	C₂×C₉	1	2	yes	—
78	#479	C₂×C₉	1	4	yes	complemented
79	#470	C₂×C₉	1	2	yes	—
80	#467	C₂×C₉	1	2	yes	—
81	#462	C₂×C₉	1	4	yes	complemented
82	#424	C₂×C₉	1	1	yes	characteristic
83	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
84	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₃×Q₁₆	—	4	2	no	—
86	C₃×Q₁₆	—	4	2	no	—
87	C₃×M₁₆	—	4	1	no	—
88	C₂×C₃×C₈	—	4	1	no	—
89	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
90	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
91	C₃×C₄⋊C₄	—	2	4	no	—
92	C₃×C₄⋊C₄	—	2	4	no	—
93	C₃×C₄⋊C₄	—	2	4	no	—
94	C₃×C₄⋊C₄	—	2	4	no	—
95	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
96	C₃×C₄⋊C₄	C₂³×C₃	1	4	yes	—
97	C₃×C₂²⋊C₄	—	2	4	no	—
98	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
99	C₃×C₄²	—	2	4	no	—
100	C₃×C₄²	C₂³×C₃	1	2	yes	—
101	C₂²×C₉	—	2	1	no	—
102	C₂²×C₉	—	2	2	no	—
103	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
104	C₄×C₉	—	2	1	no	—
105	C₄×C₉	—	2	4	no	—
106	C₄×C₉	—	4	4	no	—
107	C₄×C₉	—	4	4	no	—
108	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
109	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
111	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
112	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
113	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
114	C₄⋊Q₈	C₂²×C₉	1	2	yes	—
115	#126	—	2	4	no	—
116	#125	—	2	4	no	—
117	#125	C₂²×C₉	1	2	yes	—
118	C₂²⋊Q₈	C₂²×C₉	1	4	yes	—
119	C₄×Q₈	C₂²×C₉	1	4	yes	—
120	#117	C₂²×C₉	1	1	yes	characteristic
121	#107	C₂²×C₉	1	2	yes	—
122	#106	C₂²×C₉	1	2	yes	—
123	C₄⋊C₈	C₂²×C₉	1	4	yes	—
124	#103	C₂²×C₉	1	4	yes	—
125	#103	C₂²×C₉	1	4	yes	—
126	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
127	C₃×Q₈	—	2	4	no	—
128	C₃×Q₈	—	2	4	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	4	no	—
132	C₂×C₃×C₄	—	2	4	no	—
133	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
134	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
136	C₃×C₈	—	2	2	no	—
137	C₃×C₈	—	2	2	no	—
138	C₂×C₉	—	4	1	no	—
139	C₂×C₉	#537	1	1	yes	characteristic
140	C₂×C₉	#578	1	2	yes	—
141	C₂×Q₈	C₂³×C₉	1	4	yes	—
142	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
143	Q₁₆	—	4	2	no	—
144	Q₁₆	—	4	2	no	—
145	M₁₆	—	4	1	no	—
146	C₂×C₈	—	4	1	no	—
147	C₂×C₈	C₂³×C₉	1	2	yes	—
148	C₂×C₈	C₂³×C₉	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	D₄×C₉	1	2	yes	—
154	C₄⋊C₄	C₂³×C₉	1	4	yes	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	D₄×C₉	1	2	yes	—
157	C₄²	—	2	4	no	—
158	C₄²	C₂³×C₉	1	2	yes	—
159	C₂²×C₃	—	2	2	no	—
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
162	C₃×C₄	—	2	4	no	—
163	C₃×C₄	—	2	1	no	—
164	C₃×C₄	—	4	4	no	—
165	C₃×C₄	—	4	4	no	—
166	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
167	C₃×C₄	C₃×2₋¹⁺⁴	1	1	yes	characteristic
168	C₉	#3261	1	1	yes	characteristic, direct product factor, Sylow
169	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
170	Q₈	—	2	4	no	—
171	Q₈	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
177	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
178	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	2	no	—
181	C₂×C₃	—	4	1	no	—
182	C₂×C₃	C₃×#537	1	1	yes	characteristic
183	C₂×C₃	C₃×#578	1	2	yes	—
184	C₂²	—	2	1	no	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
187	C₄	—	2	1	no	—
188	C₄	—	2	4	no	—
189	C₄	—	4	4	no	—
190	C₄	—	4	4	no	—
191	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
192	C₄	C₉×2₋¹⁺⁴	1	1	yes	characteristic
193	C₃	C₃×#3261	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	C₉×#537	1	1	yes	characteristic
196	C₂	C₉×#578	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#479	C₂	semidirect
5. C₉×#462	C₂	semidirect
48. #3261	C₉	direct (N × H)
78. #479	C₂×C₉	semidirect
81. #462	C₂×C₉	semidirect
168. C₉	#3261	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×#541	2
C₉×#479	2
C₉×#470	2
C₉×#467	2
C₉×#462	2
C₉×#424	2
C₃×#3261	3

Subgroups — C9×#3261 #11767982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3261 #11767982