Subgroups — C₉×#2094 #11766815

Order 1152.

435 conjugacy classes of subgroups.

The subgroup lattice has 225 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#106	C₂	1	2	yes	maximal, complemented
2	C₂×C₉×#102	C₂	1	2	yes	maximal
3	C₉×#390	C₂	1	1	yes	maximal, characteristic
4	C₉×#384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2094	C₃	1	1	yes	maximal, characteristic
7	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₂×C₉×C₄⋊C₄	—	2	2	no	—
10	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×#106	—	2	4	no	—
15	C₉×#102	—	2	4	no	—
16	C₂×C₃×#106	C₂×C₃	1	2	yes	—
17	C₂×C₃×#102	C₂×C₃	1	2	yes	—
18	C₃×#390	C₂×C₃	1	1	yes	characteristic
19	C₃×#384	C₂×C₃	1	1	yes	characteristic
20	C₃×#336	C₂×C₃	1	1	yes	characteristic
21	C₂⁴×C₉	—	2	1	no	—
22	C₂×D₄×C₉	—	2	2	no	—
23	C₂×D₄×C₉	—	2	2	no	—
24	C₂×D₄×C₉	—	2	1	no	—
25	C₂×D₄×C₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₉	—	2	1	no	—
27	C₂²×C₄×C₉	—	2	2	no	—
28	C₂²×C₄×C₉	D₄	1	2	yes	—
29	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₉	—	2	2	no	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₂×C₈×C₉	D₄	1	4	yes	—
33	C₉×C₄⋊C₄	—	2	2	no	—
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₉×C₄⋊C₄	—	4	4	no	—
37	C₉×C₂²⋊C₄	—	4	4	no	—
38	C₄²×C₉	—	2	2	no	—
39	#2094	C₉	1	1	yes	characteristic, direct product factor, Sylow
40	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
42	C₂×C₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
44	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
46	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
47	C₃×#106	—	2	4	no	—
48	C₃×#102	—	2	4	no	—
49	C₂³×C₉	—	2	2	no	—
50	C₂³×C₉	—	4	1	no	—
51	C₂³×C₉	—	4	1	no	—
52	C₂³×C₉	—	4	2	no	—
53	C₂³×C₉	—	4	2	no	—
54	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
55	D₄×C₉	—	2	4	no	—
56	D₄×C₉	—	4	2	no	—
57	C₂×C₄×C₉	—	2	2	no	—
58	C₂×C₄×C₉	—	2	2	no	—
59	C₂×C₄×C₉	—	4	1	no	—
60	C₂×C₄×C₉	—	4	2	no	—
61	C₂×C₄×C₉	—	4	2	no	—
62	C₂×C₄×C₉	—	4	4	no	—
63	C₂×C₄×C₉	SD₁₆	1	4	yes	—
64	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
65	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₄×C₉	#41	1	2	yes	—
67	C₂×C₄×C₉	#41	1	1	yes	characteristic
68	C₈×C₉	—	2	4	no	—
69	C₂×#106	C₂×C₉	1	2	yes	complemented
70	C₂×#102	C₂×C₉	1	2	yes	—
71	#390	C₂×C₉	1	1	yes	characteristic
72	#384	C₂×C₉	1	1	yes	characteristic, complemented
73	#336	C₂×C₉	1	1	yes	characteristic, complemented
74	C₂⁴×C₃	—	2	1	no	—
75	C₂×C₃×D₄	—	2	2	no	—
76	C₂×C₃×D₄	—	2	2	no	—
77	C₂×C₃×D₄	—	2	1	no	—
78	C₂×C₃×D₄	C₃×Q₈	1	2	yes	—
79	C₂²×C₃×C₄	—	2	1	no	—
80	C₂²×C₃×C₄	—	2	2	no	—
81	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
82	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
86	C₃×C₄⋊C₄	—	2	2	no	—
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	4	4	no	—
90	C₃×C₂²⋊C₄	—	4	4	no	—
91	C₃×C₄²	—	2	2	no	—
92	C₂²×C₉	—	4	2	no	—
93	C₂²×C₉	—	4	4	no	—
94	C₂²×C₉	—	8	2	no	—
95	C₂²×C₉	—	8	1	no	—
96	C₂²×C₉	—	8	1	no	—
97	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂²×C₉	#123	1	1	yes	characteristic
99	C₂²×C₉	#127	1	1	yes	characteristic
100	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
101	C₂²×C₉	AGL(1,ℤ/8)	1	2	yes	—
102	C₄×C₉	—	2	4	no	—
103	C₄×C₉	—	8	1	no	—
104	C₄×C₉	—	8	2	no	—
105	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
106	C₄×C₉	#123	1	2	yes	—
107	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
108	C₂²×C₈	C₂²×C₉	1	2	yes	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
112	C₂×C₂²⋊C₄	—	2	2	no	—
113	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
114	#106	—	2	4	no	—
115	#102	—	2	4	no	—
116	C₂³×C₃	—	2	2	no	—
117	C₂³×C₃	—	4	1	no	—
118	C₂³×C₃	—	4	1	no	—
119	C₂³×C₃	—	4	2	no	—
120	C₂³×C₃	—	4	2	no	—
121	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
122	C₃×D₄	—	2	4	no	—
123	C₃×D₄	—	4	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	4	1	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	—	4	2	no	—
129	C₂×C₃×C₄	—	4	4	no	—
130	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
131	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
132	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₃×#41	1	2	yes	—
134	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
135	C₃×C₈	—	2	4	no	—
136	C₂×C₉	—	8	2	no	—
137	C₂×C₉	#399	1	1	yes	characteristic
138	C₂×C₉	#476	1	2	yes	—
139	C₂×C₉	#481	1	2	yes	—
140	C₂×C₉	#492	1	1	yes	characteristic
141	C₂×C₉	#496	1	1	yes	characteristic
142	C₂⁴	—	2	1	no	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	—	2	2	no	—
145	C₂×D₄	—	2	1	no	—
146	C₂×D₄	Q₈×C₉	1	2	yes	—
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	2	no	—
149	C₂²×C₄	D₄×C₉	1	2	yes	—
150	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	D₄×C₉	1	4	yes	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₂²⋊C₄	—	4	4	no	—
159	C₄²	—	2	2	no	—
160	C₂²×C₃	—	4	2	no	—
161	C₂²×C₃	—	4	4	no	—
162	C₂²×C₃	—	8	2	no	—
163	C₂²×C₃	—	8	1	no	—
164	C₂²×C₃	—	8	1	no	—
165	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
166	C₂²×C₃	C₃×#123	1	1	yes	characteristic
167	C₂²×C₃	C₃×#127	1	1	yes	characteristic
168	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
169	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	2	yes	—
170	C₃×C₄	—	2	4	no	—
171	C₃×C₄	—	8	1	no	—
172	C₃×C₄	—	8	2	no	—
173	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
174	C₃×C₄	C₃×#123	1	2	yes	—
175	C₉	#2094	1	1	yes	characteristic, direct product factor, Sylow
176	C₂³	—	2	2	no	—
177	C₂³	—	4	1	no	—
178	C₂³	—	4	1	no	—
179	C₂³	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
182	D₄	—	2	4	no	—
183	D₄	—	4	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	—	4	4	no	—
190	C₂×C₄	C₉×SD₁₆	1	4	yes	—
191	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
192	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
193	C₂×C₄	C₉×#41	1	2	yes	—
194	C₂×C₄	C₉×#41	1	1	yes	characteristic
195	C₈	—	2	4	no	—
196	C₂×C₃	—	8	2	no	—
197	C₂×C₃	C₃×#399	1	1	yes	characteristic
198	C₂×C₃	C₃×#476	1	2	yes	—
199	C₂×C₃	C₃×#481	1	2	yes	—
200	C₂×C₃	C₃×#492	1	1	yes	characteristic
201	C₂×C₃	C₃×#496	1	1	yes	characteristic
202	C₂²	—	4	2	no	—
203	C₂²	—	4	4	no	—
204	C₂²	—	8	2	no	—
205	C₂²	—	8	1	no	—
206	C₂²	—	8	1	no	—
207	C₂²	C₉×C₂²⋊Q₈	1	1	yes	characteristic
208	C₂²	C₉×#123	1	1	yes	characteristic
209	C₂²	C₉×#127	1	1	yes	characteristic
210	C₂²	C₂×C₉×SD₁₆	1	2	yes	—
211	C₂²	C₉×AGL(1,ℤ/8)	1	2	yes	—
212	C₄	—	2	4	no	—
213	C₄	—	8	1	no	—
214	C₄	—	8	2	no	—
215	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
216	C₄	C₉×#123	1	2	yes	—
217	C₃	C₃×#2094	1	1	yes	characteristic
218	C₂	—	8	2	no	—
219	C₂	C₉×#399	1	1	yes	characteristic
220	C₂	C₉×#476	1	2	yes	—
221	C₂	C₉×#481	1	2	yes	—
222	C₂	C₉×#492	1	1	yes	characteristic
223	C₂	C₉×#496	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂²×C₃×D₄	C₂²×C₃
LCS	G ▷ C₂²×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
1. C₂×C₉×#106	C₂	semidirect
4. C₉×#384	C₂	semidirect
5. C₉×#336	C₂	semidirect
39. #2094	C₉	direct (N × H)
69. C₂×#106	C₂×C₉	semidirect
72. #384	C₂×C₉	semidirect
73. #336	C₂×C₉	semidirect
175. C₉	#2094	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×C₉×#106	2
C₂×C₉×#102	2
C₉×#390	2
C₉×#384	2
C₉×#336	2
C₃×#2094	3

Subgroups — C9×#2094 #11766815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2094 #11766815