Subgroups — C₂×C₉×#818 #13143087

Order 1728.

466 conjugacy classes of subgroups.

The subgroup lattice has 230 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×#818	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#818	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×#107	—	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×#818	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×#107	—	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×#107	—	3	1	no	—
28	C₂×#818	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	D₈	1	2	yes	—
46	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
47	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
49	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
50	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
51	C₉×Dic₃	—	4	4	no	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	C₂×C₉	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄⋊C₄	—	3	2	no	—
56	C₂×C₄⋊Dic₃	C₂×C₉	1	2	yes	—
57	C₃×#107	—	3	4	no	—
58	#818	C₂×C₉	1	4	yes	direct product factor
59	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
61	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
63	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
64	C₂×C₃×Dic₃	—	2	4	no	—
65	C₂×C₃×Dic₃	—	4	4	no	—
66	C₂³×C₉	—	2	1	no	—
67	C₂³×C₉	D₁₂	1	1	yes	characteristic
68	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	—	6	4	no	—
74	C₂×C₄×C₉	—	12	4	no	—
75	C₂×C₄×C₉	Dic₆	1	2	yes	—
76	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
77	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
78	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
79	C₈×C₉	—	2	4	no	—
80	C₈×C₉	C₂×Dic₃	1	4	yes	complemented
81	C₂×#107	—	3	1	no	Sylow
82	C₂×C₃×C₉	#107	1	4	yes	complemented
83	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	6	2	no	—
88	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	2	2	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	C₄×C₉	1	4	yes	complemented
94	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
95	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
96	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
97	C₃×C₄⋊C₄	—	3	4	no	—
98	C₃×C₄⋊C₄	—	6	2	no	—
99	C₄⋊Dic₃	—	2	2	no	—
100	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
101	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
102	C₂²×C₃²	C₃×D₈	1	2	yes	—
103	C₂²×C₃²	C₃×Q₁₆	1	2	yes	—
104	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
105	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
106	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
107	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
108	C₃×Dic₃	—	4	4	no	—
109	C₂²×C₉	—	2	2	no	—
110	C₂²×C₉	—	2	2	no	—
111	C₂²×C₉	—	2	2	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	D₂₄	1	2	yes	—
114	C₂²×C₉	Dic₁₂	1	2	yes	—
115	C₂²×C₉	C₄⋊Dic₃	1	2	yes	—
116	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
117	C₄×C₉	—	2	2	no	—
118	C₄×C₉	—	2	1	no	—
119	C₄×C₉	—	2	1	no	—
120	C₄×C₉	—	12	4	no	—
121	C₄×C₉	C₄⋊Dic₃	1	2	yes	—
122	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
123	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
124	C₂²×C₈	D₃×C₉	1	1	yes	characteristic
125	C₂×C₄⋊C₄	—	3	2	no	—
126	#107	—	3	4	no	—
127	C₃×C₉	C₂×#107	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₃	—	2	1	no	—
129	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
130	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	6	4	no	—
136	C₂×C₃×C₄	—	12	4	no	—
137	C₂×C₃×C₄	C₂×C₄×C₉	1	2	yes	—
138	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
139	C₂×C₃×C₄	Q₈×C₉	1	2	yes	—
140	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
141	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
142	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
144	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
145	C₂×Dic₃	—	2	4	no	—
146	C₂×Dic₃	—	4	4	no	—
147	C₃×C₈	—	2	4	no	—
148	C₃×C₈	C₂×C₄×C₉	1	4	yes	complemented
149	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
150	C₂×C₃²	C₃×#107	1	4	yes	—
151	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
153	C₂×C₃²	C₂×C₃×Q₁₆	1	1	yes	characteristic
154	C₂×C₉	—	2	1	no	—
155	C₂×C₉	—	2	1	no	—
156	C₂×C₉	—	2	1	no	—
157	C₂×C₉	—	2	4	no	—
158	C₂×C₉	#818	1	4	yes	direct product factor
159	C₂×C₉	C₂×D₂₄	1	1	yes	characteristic
160	C₂×C₉	C₂×Dic₁₂	1	1	yes	characteristic
161	C₂×C₉	C₂×C₄⋊Dic₃	1	1	yes	characteristic
162	C₂²×C₄	—	6	2	no	—
163	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
164	C₂×C₈	C₉×Dic₃	1	4	yes	complemented
165	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
166	C₄⋊C₄	—	3	4	no	—
167	C₄⋊C₄	—	6	2	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	C₉×C₄⋊C₄	1	2	yes	—
173	C₂²×C₃	C₉×D₈	1	2	yes	—
174	C₂²×C₃	C₉×Q₁₆	1	2	yes	—
175	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
176	C₂²×C₃	C₃×D₂₄	1	2	yes	—
177	C₂²×C₃	C₃×Dic₁₂	1	2	yes	—
178	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
179	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	12	4	no	—
184	C₃×C₄	C₉×C₄⋊C₄	1	2	yes	—
185	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
186	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
187	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
188	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
189	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
190	Dic₃	—	4	4	no	—
191	C₃²	C₂×C₃×#107	1	1	yes	characteristic
192	C₉	—	2	1	no	—
193	C₉	C₂×#818	1	1	yes	characteristic, direct product factor
194	C₂³	C₉×D₁₂	1	1	yes	characteristic
195	C₂×C₄	—	6	4	no	—
196	C₂×C₄	—	12	4	no	—
197	C₂×C₄	C₉×Dic₆	1	2	yes	—
198	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
199	C₂×C₄	C₂×C₉×Dic₃	1	2	yes	—
200	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
201	C₈	C₂×C₉×Dic₃	1	4	yes	complemented
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	2	4	no	—
206	C₂×C₃	C₉×#107	1	4	yes	complemented
207	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
208	C₂×C₃	C₂×C₉×D₈	1	1	yes	characteristic
209	C₂×C₃	C₂×C₉×Q₁₆	1	1	yes	characteristic
210	C₂×C₃	C₃×#818	1	4	yes	—
211	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
212	C₂×C₃	C₂×C₃×Dic₁₂	1	1	yes	characteristic
213	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
214	C₂²	C₉×D₂₄	1	2	yes	—
215	C₂²	C₉×Dic₁₂	1	2	yes	—
216	C₂²	C₉×C₄⋊Dic₃	1	2	yes	—
217	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
218	C₄	—	12	4	no	—
219	C₄	C₉×C₄⋊Dic₃	1	2	yes	—
220	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
221	C₄	C₂²×C₉×Dic₃	1	1	yes	characteristic
222	C₃	—	2	1	no	—
223	C₃	C₂×C₉×#107	1	1	yes	characteristic, complemented
224	C₃	C₂×C₃×#818	1	1	yes	characteristic
225	C₂	C₉×#818	1	4	yes	direct product factor
226	C₂	C₂×C₉×D₂₄	1	1	yes	characteristic
227	C₂	C₂×C₉×Dic₁₂	1	1	yes	characteristic
228	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×#107
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×#818	C₂	direct (N × H)
8. C₂×C₃×C₈×C₉	C₄	semidirect
26. C₃×C₈×C₉	C₂×C₄	semidirect
28. C₂×#818	C₉	direct (N × H)
40. C₂×C₈×C₉	Dic₃	semidirect
58. #818	C₂×C₉	direct (N × H)
80. C₈×C₉	C₂×Dic₃	semidirect
82. C₂×C₃×C₉	#107	semidirect
93. C₂×C₃×C₈	C₄×C₉	semidirect
127. C₃×C₉	C₂×#107	semidirect
148. C₃×C₈	C₂×C₄×C₉	semidirect
158. C₂×C₉	#818	direct (N × H)
164. C₂×C₈	C₉×Dic₃	semidirect
193. C₉	C₂×#818	direct (N × H)
201. C₈	C₂×C₉×Dic₃	semidirect
206. C₂×C₃	C₉×#107	semidirect
223. C₃	C₂×C₉×#107	semidirect
225. C₂	C₉×#818	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×#818	2
C₂×C₃×#818	3
C₂×C₉×#107	3

Subgroups — C2×C9×#818 #13143087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#818 #13143087