Subgroups — #106×Dic₉ #11784436

Order 1152.

420 conjugacy classes of subgroups.

The subgroup lattice has 216 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	1	yes	maximal, characteristic
2	C₂×#63377	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₄×Dic₉	C₂	1	2	yes	maximal
5	#10588206	C₂	1	2	yes	maximal
6	Dic₃×#106	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₉	C₂²	1	2	yes	—
10	C₂×C₄×Dic₉	—	2	2	no	—
11	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
12	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
13	C₉×#106	C₄	1	4	yes	complemented
14	C₉⋊#95	—	2	2	no	—
15	#63377	C₄	1	4	yes	complemented
16	C₈×Dic₉	C₄	1	4	yes	complemented
17	C₂×C₃×#106	D₃	1	1	yes	characteristic
18	C₂×#817	—	3	1	no	—
19	C₂×C₈×Dic₃	—	3	1	no	—
20	Dic₃×C₄⋊C₄	—	3	2	no	—
21	#4529	—	3	2	no	—
22	C₂²×C₄×C₉	—	2	2	no	—
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₉	—	2	2	no	—
25	C₂²×Dic₉	D₄	1	1	yes	characteristic
26	C₂²×Dic₉	D₄	1	1	yes	characteristic
27	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₉	C₂×C₄	1	2	yes	—
30	C₉×C₄⋊C₄	—	2	2	no	—
31	C₉×C₄⋊C₄	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₉	—	2	2	no	—
33	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
34	C₄×Dic₉	—	4	4	no	—
35	C₄×Dic₉	C₂×C₄	1	2	yes	—
36	C₄×Dic₉	Q₈	1	2	yes	—
37	C₂×C₉⋊C₈	—	2	1	no	—
38	C₂×C₉⋊C₈	—	2	1	no	—
39	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
40	C₄×#106	—	9	1	no	Sylow
41	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
43	C₂×C₄⋊Dic₃	—	3	2	no	—
44	C₂×C₄×Dic₃	—	3	1	no	—
45	C₂×C₄×Dic₃	—	6	2	no	—
46	C₂²×C₃⋊C₈	—	3	1	no	—
47	C₃×#106	Dic₃	1	4	yes	—
48	C₃⋊#95	—	6	2	no	—
49	#817	—	3	4	no	—
50	C₈×Dic₃	—	3	4	no	—
51	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉	—	2	4	no	—
53	C₂×C₄×C₉	—	4	2	no	—
54	C₂×C₄×C₉	—	4	2	no	—
55	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₉	#41	1	1	yes	characteristic
61	C₂×Dic₉	—	2	2	no	—
62	C₂×Dic₉	—	2	4	no	—
63	C₂×Dic₉	—	4	2	no	—
64	C₂×Dic₉	—	4	2	no	—
65	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
66	C₂×Dic₉	C₄⋊C₄	1	2	yes	—
67	C₂×Dic₉	SD₁₆	1	4	yes	—
68	C₈×C₉	C₄²	1	4	yes	complemented
69	C₉⋊C₈	—	2	2	no	—
70	C₂×#106	—	9	1	no	—
71	C₂×#106	D₉	1	1	yes	characteristic
72	C₂×C₄×C₈	—	9	1	no	—
73	C₄×C₄⋊C₄	—	9	2	no	—
74	#340	—	9	2	no	—
75	C₂²×C₃×C₄	—	2	2	no	—
76	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂²×Dic₃	—	3	1	no	—
78	C₂²×Dic₃	—	3	1	no	—
79	C₂²×Dic₃	—	6	2	no	—
80	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
81	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
82	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
83	C₃×C₄⋊C₄	—	2	2	no	—
84	C₃×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
85	C₄⋊Dic₃	—	3	4	no	—
86	C₄⋊Dic₃	—	6	2	no	—
87	C₄×Dic₃	—	3	2	no	—
88	C₄×Dic₃	—	3	2	no	—
89	C₄×Dic₃	—	12	4	no	—
90	C₂×C₃⋊C₈	—	3	2	no	—
91	C₂×C₃⋊C₈	—	6	1	no	—
92	C₂×C₃⋊C₈	—	6	1	no	—
93	C₂²×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
95	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
96	C₂²×C₉	C₂×SD₁₆	1	2	yes	—
97	C₂²×C₉	#135	1	2	yes	—
98	C₄×C₉	—	4	4	no	—
99	C₄×C₉	C₂×C₄²	1	1	yes	characteristic
100	C₄×C₉	#117	1	1	yes	characteristic
101	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
102	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
103	Dic₉	—	2	2	no	—
104	Dic₉	—	4	4	no	—
105	Dic₉	#106	1	4	yes	direct product factor
106	C₂²×C₈	—	9	1	no	—
107	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	9	2	no	—
109	C₂×C₄⋊C₄	C₂×D₉	1	2	yes	—
110	C₂×C₄²	—	9	1	no	—
111	C₂×C₄²	—	18	2	no	—
112	#106	—	9	4	no	—
113	#106	Dic₉	1	4	yes	direct product factor
114	C₄×C₈	—	9	4	no	—
115	#95	—	18	2	no	—
116	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
117	C₂×C₃×C₄	—	2	4	no	—
118	C₂×C₃×C₄	—	4	2	no	—
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
122	C₂×C₃×C₄	#237	1	1	yes	characteristic
123	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
124	C₂×C₃×C₄	#239	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
126	C₂×Dic₃	—	3	2	no	—
127	C₂×Dic₃	—	3	4	no	—
128	C₂×Dic₃	—	3	2	no	—
129	C₂×Dic₃	—	6	2	no	—
130	C₂×Dic₃	—	6	4	no	—
131	C₂×Dic₃	—	12	2	no	—
132	C₂×Dic₃	—	12	2	no	—
133	C₃×C₈	C₄×Dic₃	1	4	yes	—
134	C₃⋊C₈	—	6	2	no	—
135	C₂×C₉	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₉	C₂×#106	1	1	yes	characteristic
137	C₂×C₉	#427	1	1	yes	characteristic
138	C₂×C₉	C₄×SD₁₆	1	2	yes	—
139	C₂×C₉	C₄×SD₁₆	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	9	1	no	—
142	C₂²×C₄	—	9	1	no	—
143	C₂²×C₄	—	18	2	no	—
144	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
145	C₂×C₈	—	9	2	no	—
146	C₂×C₈	—	18	1	no	—
147	C₂×C₈	—	18	1	no	—
148	C₂×C₈	C₄×D₉	1	2	yes	—
149	C₂×C₈	C₄×D₉	1	2	yes	—
150	C₂×C₈	C₂×Dic₉	1	2	yes	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	9	4	no	—
153	C₄⋊C₄	—	18	2	no	—
154	C₄⋊C₄	C₂×Dic₉	1	4	yes	—
155	C₄²	—	9	2	no	—
156	C₄²	—	9	2	no	—
157	C₄²	—	36	4	no	—
158	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
159	C₂²×C₃	#893	1	1	yes	characteristic
160	C₂²×C₃	D₃×SD₁₆	1	2	yes	—
161	C₂²×C₃	#916	1	2	yes	—
162	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
163	C₃×C₄	—	4	4	no	—
164	C₃×C₄	#887	1	1	yes	characteristic
165	C₃×C₄	#892	1	1	yes	characteristic
166	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
167	C₃×C₄	Q₈×Dic₃	1	1	yes	characteristic
168	Dic₃	—	3	4	no	—
169	Dic₃	—	6	2	no	—
170	Dic₃	—	12	4	no	—
171	C₉	C₄×#106	1	1	yes	characteristic, complemented, Sylow
172	C₂³	D₄×D₉	1	1	yes	characteristic
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	—	9	2	no	—
177	C₂×C₄	—	9	4	no	—
178	C₂×C₄	—	9	2	no	—
179	C₂×C₄	—	18	2	no	—
180	C₂×C₄	—	18	4	no	—
181	C₂×C₄	—	36	2	no	—
182	C₂×C₄	—	36	2	no	—
183	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
184	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
185	C₂×C₄	#3699	1	1	yes	characteristic
186	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
187	C₂×C₄	#3701	1	1	yes	characteristic
188	C₂×C₄	C₂²×Dic₉	1	1	yes	characteristic
189	C₈	—	18	2	no	—
190	C₈	C₄×Dic₉	1	4	yes	complemented
191	C₂×C₃	#4852	1	2	yes	—
192	C₂×C₃	D₃×#106	1	1	yes	characteristic
193	C₂×C₃	#4860	1	1	yes	characteristic
194	C₂×C₃	Dic₃×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₃	Dic₃×SD₁₆	1	2	yes	—
196	C₂²	C₄⋊C₄×D₉	1	1	yes	characteristic
197	C₂²	#63455	1	1	yes	characteristic
198	C₂²	SD₁₆×D₉	1	2	yes	—
199	C₂²	#63478	1	2	yes	—
200	C₂²	D₄×Dic₉	1	1	yes	characteristic
201	C₄	—	4	4	no	—
202	C₄	—	9	4	no	—
203	C₄	—	18	2	no	—
204	C₄	—	36	4	no	—
205	C₄	#63449	1	1	yes	characteristic
206	C₄	#63454	1	1	yes	characteristic
207	C₄	C₂×C₄×Dic₉	1	1	yes	characteristic
208	C₄	Q₈×Dic₉	1	1	yes	characteristic
209	C₃	Dic₃×#106	1	1	yes	characteristic
210	C₂	#10588532	1	2	yes	—
211	C₂	D₉×#106	1	1	yes	characteristic
212	C₂	#10588540	1	1	yes	characteristic
213	C₂	C₄⋊C₄×Dic₉	1	1	yes	characteristic
214	C₂	SD₁₆×Dic₉	1	2	yes	—

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₄×C₉	C₂×C₄²
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#106	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#106	D₉
hypocenter	C₉	C₄×#106
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
13. C₉×#106	C₄	semidirect
15. #63377	C₄	semidirect
16. C₈×Dic₉	C₄	semidirect
68. C₈×C₉	C₄²	semidirect
105. Dic₉	#106	direct (N × H)
113. #106	Dic₉	direct (N × H)
171. C₉	C₄×#106	semidirect
190. C₈	C₄×Dic₉	semidirect

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×#63377	2
C₂×C₈×Dic₉	2
C₄⋊C₄×Dic₉	2
#10588206	2
Dic₃×#106	3

Subgroups — #106×Dic9 #11784436

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #106×Dic₉ #11784436