Subgroups — C₂×C₉×#817 #13143086

Order 1728.

466 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₉×#817	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#817	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×#106	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₉×Dic₃	—	2	2	no	—
8	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
9	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
10	C₉×C₄⋊Dic₃	—	2	2	no	—
11	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
12	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	2	yes	—
14	C₃×#817	C₂×C₃	1	4	yes	—
15	C₂²×C₈×C₉	—	2	1	no	—
16	C₂²×C₈×C₉	D₃	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	—	3	2	no	—
18	C₉×#106	—	3	4	no	—
19	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₉×Dic₃	—	2	4	no	—
25	C₂×C₉×Dic₃	—	4	4	no	—
26	C₃×C₈×C₉	C₂×C₄	1	4	yes	complemented
27	C₂×C₃×#106	—	3	1	no	—
28	C₂×#817	C₉	1	1	yes	characteristic, direct product factor
29	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	—	2	2	no	—
31	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
32	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
33	C₃×C₄⋊Dic₃	—	2	2	no	—
34	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	—	6	2	no	—
37	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
38	C₂×C₈×C₉	—	2	4	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₂×C₈×C₉	Dic₃	1	4	yes	complemented
41	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
42	C₉×C₄⋊C₄	—	3	4	no	—
43	C₉×C₄⋊C₄	—	6	2	no	—
44	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
45	C₂²×C₃×C₉	SD₁₆	1	4	yes	—
46	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
47	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
48	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
50	C₉×Dic₃	—	4	4	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	3	2	no	—
55	C₂×C₄⋊Dic₃	C₂×C₉	1	2	yes	—
56	C₃×#106	—	3	4	no	—
57	#817	C₂×C₉	1	4	yes	direct product factor
58	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
60	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
62	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
63	C₂×C₃×Dic₃	—	2	4	no	—
64	C₂×C₃×Dic₃	—	4	4	no	—
65	C₂³×C₉	—	2	1	no	—
66	C₂³×C₉	D₁₂	1	1	yes	characteristic
67	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	—	6	4	no	—
73	C₂×C₄×C₉	—	12	4	no	—
74	C₂×C₄×C₉	Dic₆	1	2	yes	—
75	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
76	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
77	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
78	C₈×C₉	—	2	4	no	—
79	C₈×C₉	C₂×Dic₃	1	4	yes	complemented
80	C₂×#106	—	3	1	no	Sylow
81	C₂×C₃×C₉	#106	1	4	yes	complemented
82	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₉	C₂×SD₁₆	1	2	yes	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	6	2	no	—
86	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
88	C₂²×Dic₃	—	2	2	no	—
89	C₂×C₃×C₈	—	2	4	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	C₄×C₉	1	4	yes	complemented
92	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
93	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
94	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
95	C₃×C₄⋊C₄	—	3	4	no	—
96	C₃×C₄⋊C₄	—	6	2	no	—
97	C₄⋊Dic₃	—	2	2	no	—
98	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
99	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
100	C₂²×C₃²	C₃×SD₁₆	1	4	yes	—
101	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
102	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
103	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
104	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
105	C₃×Dic₃	—	4	4	no	—
106	C₂²×C₉	—	2	4	no	—
107	C₂²×C₉	—	2	2	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	#204	1	4	yes	—
110	C₂²×C₉	C₄⋊Dic₃	1	2	yes	—
111	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
112	C₄×C₉	—	2	2	no	—
113	C₄×C₉	—	2	1	no	—
114	C₄×C₉	—	2	1	no	—
115	C₄×C₉	—	12	4	no	—
116	C₄×C₉	C₄⋊Dic₃	1	2	yes	—
117	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
118	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
119	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	3	2	no	—
121	#106	—	3	4	no	—
122	C₃×C₉	C₂×#106	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	2	1	no	—
124	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
125	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	6	4	no	—
131	C₂×C₃×C₄	—	12	4	no	—
132	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
133	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
134	C₂×C₃×C₄	Q₈×C₉	1	2	yes	—
135	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
136	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
137	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
139	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
140	C₂×Dic₃	—	2	4	no	—
141	C₂×Dic₃	—	4	4	no	—
142	C₃×C₈	—	2	4	no	—
143	C₃×C₈	C₂×C₄×C₉	1	4	yes	complemented
144	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
145	C₂×C₃²	C₃×#106	1	4	yes	—
146	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₃²	C₂×C₃×SD₁₆	1	2	yes	—
148	C₂×C₉	—	2	2	no	—
149	C₂×C₉	—	2	1	no	—
150	C₂×C₉	—	2	4	no	—
151	C₂×C₉	#817	1	4	yes	direct product factor
152	C₂×C₉	C₂×#204	1	2	yes	—
153	C₂×C₉	C₂×C₄⋊Dic₃	1	1	yes	characteristic
154	C₂²×C₄	—	6	2	no	—
155	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
156	C₂×C₈	C₉×Dic₃	1	4	yes	complemented
157	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
158	C₄⋊C₄	—	3	4	no	—
159	C₄⋊C₄	—	6	2	no	—
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	4	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	C₉×C₄⋊C₄	1	2	yes	—
164	C₂²×C₃	C₉×SD₁₆	1	4	yes	—
165	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
166	C₂²×C₃	C₃×#204	1	4	yes	—
167	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
168	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	12	4	no	—
173	C₃×C₄	C₉×C₄⋊C₄	1	2	yes	—
174	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
175	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
176	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
177	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
178	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
179	Dic₃	—	4	4	no	—
180	C₃²	C₂×C₃×#106	1	1	yes	characteristic
181	C₉	—	2	1	no	—
182	C₉	C₂×#817	1	1	yes	characteristic, direct product factor
183	C₂³	C₉×D₁₂	1	1	yes	characteristic
184	C₂×C₄	—	6	4	no	—
185	C₂×C₄	—	12	4	no	—
186	C₂×C₄	C₉×Dic₆	1	2	yes	—
187	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
188	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
189	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
190	C₈	C₂×C₉×Dic₃	1	4	yes	complemented
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	4	no	—
194	C₂×C₃	C₉×#106	1	4	yes	complemented
195	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₃	C₂×C₉×SD₁₆	1	2	yes	—
197	C₂×C₃	C₃×#817	1	4	yes	—
198	C₂×C₃	C₂×C₃×#204	1	2	yes	—
199	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
200	C₂²	C₉×#204	1	4	yes	—
201	C₂²	C₉×C₄⋊Dic₃	1	2	yes	—
202	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
203	C₄	—	12	4	no	—
204	C₄	C₉×C₄⋊Dic₃	1	2	yes	—
205	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
206	C₄	C₂²×C₉×Dic₃	1	1	yes	characteristic
207	C₃	—	2	1	no	—
208	C₃	C₂×C₉×#106	1	1	yes	characteristic, complemented
209	C₃	C₂×C₃×#817	1	1	yes	characteristic
210	C₂	C₉×#817	1	4	yes	direct product factor
211	C₂	C₂×C₉×#204	1	2	yes	—
212	C₂	C₂×C₉×C₄⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	D₁₂
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle, prime radical	C₂³×C₃²	C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×C₈×C₉	D₃
hypocenter	C₃	C₂×C₉×#106
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂²×C₄×C₉ ◁ C₂²×C₈×C₉

Complements

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

N	H ≅ G/N	type
3. C₉×#817	C₂	direct (N × H)
8. C₂×C₃×C₈×C₉	C₄	semidirect
26. C₃×C₈×C₉	C₂×C₄	semidirect
28. C₂×#817	C₉	direct (N × H)
40. C₂×C₈×C₉	Dic₃	semidirect
57. #817	C₂×C₉	direct (N × H)
79. C₈×C₉	C₂×Dic₃	semidirect
81. C₂×C₃×C₉	#106	semidirect
91. C₂×C₃×C₈	C₄×C₉	semidirect
122. C₃×C₉	C₂×#106	semidirect
143. C₃×C₈	C₂×C₄×C₉	semidirect
151. C₂×C₉	#817	direct (N × H)
156. C₂×C₈	C₉×Dic₃	semidirect
182. C₉	C₂×#817	direct (N × H)
190. C₈	C₂×C₉×Dic₃	semidirect
194. C₂×C₃	C₉×#106	semidirect
208. C₃	C₂×C₉×#106	semidirect
210. C₂	C₉×#817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₂×C₉×C₄⋊Dic₃	2
C₉×#817	2
C₂×C₃×#817	3
C₂×C₉×#106	3

Subgroups — C2×C9×#817 #13143086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#817 #13143086