Subgroups — C₉×#4893 #13142260

Order 1728.

336 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×C₉×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₉×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#822	C₂	1	2	yes	maximal, complemented
5	C₉×#819	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4893	C₃	1	1	yes	maximal, characteristic
7	C₉×#430	—	3	1	no	maximal
8	C₂×C₄×D₃×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₉×#208	C₂²	1	2	yes	—
12	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₉×ΓL(1,𝔽₂₅)	—	2	2	no	—
14	C₉×ΓL(1,𝔽₂₅)	C₄	1	4	yes	—
15	D₃×C₈×C₉	—	2	2	no	—
16	C₃×D₃×M₁₆	C₂×C₃	1	2	yes	—
17	C₂×C₃×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
18	C₃²×#108	C₂×C₃	1	1	yes	characteristic
19	C₃×#822	C₂×C₃	1	2	yes	—
20	C₃×#819	C₂×C₃	1	1	yes	characteristic
21	C₂×C₉×M₁₆	—	3	1	no	—
22	C₂×C₉×M₁₆	—	3	2	no	—
23	C₉×#108	—	2	1	no	—
24	C₉×#108	—	3	2	no	—
25	C₉×#108	—	3	1	no	—
26	C₉×#108	D₃	1	1	yes	characteristic, complemented
27	C₂²×D₃×C₉	Q₈	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₉×Dic₃	Q₈	1	1	yes	characteristic
30	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
31	C₄×D₃×C₉	D₄	1	2	yes	—
32	C₃×C₈×C₉	—	2	2	no	—
33	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
34	C₉×C₃⋊C₈	—	2	2	no	—
35	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₃×#430	—	3	1	no	—
37	#4893	C₉	1	1	yes	characteristic, direct product factor
38	C₂×C₃×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
39	C₃²×M₁₆	C₂²×C₃	1	2	yes	—
40	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₃×#208	C₂²×C₃	1	2	yes	—
42	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
43	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
44	C₃×ΓL(1,𝔽₂₅)	C₃×C₄	1	4	yes	—
45	C₃×D₃×C₈	—	2	2	no	—
46	C₂²×C₄×C₉	—	3	1	no	—
47	C₉×M₁₆	—	2	2	no	—
48	C₉×M₁₆	—	3	4	no	—
49	C₉×M₁₆	—	3	2	no	—
50	C₉×M₁₆	—	6	2	no	—
51	C₉×M₁₆	C₂×D₃	1	2	yes	—
52	C₂×C₈×C₉	—	2	1	no	—
53	C₂×C₈×C₉	—	3	1	no	—
54	C₂×C₈×C₉	—	6	2	no	—
55	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
56	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×D₃×C₉	—	4	1	no	—
58	C₂×D₃×C₉	C₄⋊C₄	1	2	yes	—
59	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
61	C₉×Dic₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₃×M₁₆	—	3	1	no	—
63	C₂×C₃×M₁₆	—	3	2	no	—
64	D₃×M₁₆	C₂×C₉	1	2	yes	—
65	C₂×ΓL(1,𝔽₂₅)	C₂×C₉	1	1	yes	characteristic
66	C₃×#108	—	2	1	no	—
67	C₃×#108	—	3	2	no	—
68	C₃×#108	—	3	1	no	—
69	C₃×#108	C₂×C₉	1	1	yes	characteristic, complemented
70	C₃×#108	C₃×D₃	1	1	yes	characteristic
71	#822	C₂×C₉	1	2	yes	complemented
72	#819	C₂×C₉	1	1	yes	characteristic, complemented
73	C₂²×C₃×D₃	C₃×Q₈	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₃×Dic₃	C₃×Q₈	1	1	yes	characteristic
76	C₃×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
77	C₃×C₄×D₃	C₃×D₄	1	2	yes	—
78	C₂³×C₉	—	3	1	no	—
79	C₃²×C₈	—	2	2	no	—
80	C₃²×C₈	C₂×C₃×C₄	1	2	yes	—
81	C₃×C₃⋊C₈	—	2	2	no	—
82	C₃×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	—	3	1	no	—
85	C₂×C₄×C₉	—	3	2	no	—
86	C₂×C₄×C₉	—	3	2	no	—
87	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
88	C₈×C₉	—	2	2	no	—
89	C₈×C₉	—	2	2	no	—
90	C₈×C₉	—	3	2	no	—
91	C₈×C₉	—	4	2	no	—
92	C₈×C₉	—	6	2	no	—
93	C₈×C₉	C₄×D₃	1	2	yes	—
94	#430	—	3	1	no	Sylow
95	C₂×C₃×C₉	—	2	1	no	—
96	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	D₃×C₉	—	2	2	no	—
98	C₂²×C₃×C₄	—	3	1	no	—
99	C₂×C₄×D₃	C₂²×C₉	1	1	yes	characteristic
100	C₃×M₁₆	—	2	2	no	—
101	C₃×M₁₆	—	3	4	no	—
102	C₃×M₁₆	—	3	2	no	—
103	C₃×M₁₆	—	6	2	no	—
104	C₃×M₁₆	C₂²×C₉	1	2	yes	—
105	C₃×M₁₆	C₂×C₃×D₃	1	2	yes	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	3	1	no	—
108	C₂×C₃×C₈	—	6	2	no	—
109	C₂×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
110	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
111	#208	C₂²×C₉	1	2	yes	—
112	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
113	ΓL(1,𝔽₂₅)	—	2	2	no	—
114	ΓL(1,𝔽₂₅)	C₄×C₉	1	4	yes	—
115	D₃×C₈	—	2	2	no	—
116	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
117	C₂×C₃×D₃	—	4	1	no	—
118	C₂×C₃×D₃	C₃×C₄⋊C₄	1	2	yes	—
119	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
121	C₃×Dic₃	C₃×C₄⋊C₄	1	2	yes	—
122	C₂²×C₉	—	2	1	no	—
123	C₂²×C₉	—	3	2	no	—
124	C₂²×C₉	—	12	1	no	—
125	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
126	C₄×C₉	—	2	1	no	—
127	C₄×C₉	—	2	1	no	—
128	C₄×C₉	—	3	2	no	—
129	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
130	C₄×C₉	D₃×D₄	1	1	yes	characteristic
131	C₂×M₁₆	—	3	1	no	—
132	C₂×M₁₆	—	3	2	no	—
133	#108	—	3	2	no	—
134	#108	—	3	1	no	—
135	#108	D₃×C₉	1	1	yes	characteristic, complemented
136	C₃×C₉	#430	1	1	yes	characteristic, complemented, Sylow
137	C₂³×C₃	—	3	1	no	—
138	C₂²×D₃	Q₈×C₉	1	1	yes	characteristic
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	3	1	no	—
141	C₂×C₃×C₄	—	3	2	no	—
142	C₂×C₃×C₄	—	3	2	no	—
143	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
145	C₂×Dic₃	Q₈×C₉	1	1	yes	characteristic
146	C₄×D₃	C₂×C₄×C₉	1	2	yes	—
147	C₄×D₃	D₄×C₉	1	2	yes	—
148	C₃×C₈	—	2	2	no	—
149	C₃×C₈	—	2	2	no	—
150	C₃×C₈	—	2	2	no	—
151	C₃×C₈	—	3	2	no	—
152	C₃×C₈	—	4	2	no	—
153	C₃×C₈	—	6	2	no	—
154	C₃×C₈	C₂×C₄×C₉	1	2	yes	—
155	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
156	C₃⋊C₈	—	2	2	no	—
157	C₃⋊C₈	C₂×C₄×C₉	1	2	yes	—
158	C₂×C₃²	—	2	1	no	—
159	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
160	C₃×D₃	—	2	2	no	—
161	C₂×C₉	—	2	1	no	—
162	C₂×C₉	—	2	1	no	—
163	C₂×C₉	—	4	1	no	—
164	C₂×C₉	—	6	2	no	—
165	C₂×C₉	D₃×C₄⋊C₄	1	1	yes	characteristic
166	C₂²×C₄	—	3	1	no	—
167	M₁₆	—	3	4	no	—
168	M₁₆	—	3	2	no	—
169	M₁₆	—	6	2	no	—
170	M₁₆	C₂×D₃×C₉	1	2	yes	—
171	C₂×C₈	—	3	1	no	—
172	C₂×C₈	—	6	2	no	—
173	C₂×C₈	C₂×D₃×C₉	1	1	yes	characteristic
174	C₂²×C₃	—	2	1	no	—
175	C₂²×C₃	—	3	2	no	—
176	C₂²×C₃	—	12	1	no	—
177	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
178	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
179	C₂×D₃	—	4	1	no	—
180	C₂×D₃	C₉×C₄⋊C₄	1	2	yes	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	3	2	no	—
184	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
185	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
186	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
187	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
188	Dic₃	C₉×C₄⋊C₄	1	2	yes	—
189	C₃²	C₃×#430	1	1	yes	characteristic
190	C₉	—	2	1	no	—
191	C₉	#4893	1	1	yes	characteristic, direct product factor
192	C₂³	—	3	1	no	—
193	C₂×C₄	—	3	1	no	—
194	C₂×C₄	—	3	2	no	—
195	C₂×C₄	—	3	2	no	—
196	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
197	C₈	—	2	2	no	—
198	C₈	—	3	2	no	—
199	C₈	—	6	2	no	—
200	C₈	C₄×D₃×C₉	1	2	yes	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	4	1	no	—
205	C₂×C₃	—	6	2	no	—
206	C₂×C₃	C₂×C₉×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₃	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
208	D₃	—	2	2	no	—
209	C₂²	—	3	2	no	—
210	C₂²	—	12	1	no	—
211	C₂²	D₃×Q₈×C₉	1	1	yes	characteristic
212	C₄	—	3	2	no	—
213	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
214	C₄	D₃×D₄×C₉	1	1	yes	characteristic
215	C₃	—	2	1	no	—
216	C₃	C₉×#430	1	1	yes	characteristic, complemented
217	C₃	C₃×#4893	1	1	yes	characteristic
218	C₂	—	2	1	no	—
219	C₂	—	6	2	no	—
220	C₂	D₃×C₉×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#108	C₂
socle	C₂×C₃²	C₂×C₃×C₄⋊C₄
prime radical	C₂²×C₃×D₃	C₃×Q₈
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#108	D₃
hypocenter	C₃	C₉×#430
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#108

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#108	C₂	semidirect
4. C₉×#822	C₂	semidirect
5. C₉×#819	C₂	semidirect
26. C₉×#108	D₃	semidirect
37. #4893	C₉	direct (N × H)
69. C₃×#108	C₂×C₉	semidirect
71. #822	C₂×C₉	semidirect
72. #819	C₂×C₉	semidirect
135. #108	D₃×C₉	semidirect
136. C₃×C₉	#430	semidirect
191. C₉	#4893	direct (N × H)
216. C₃	C₉×#430	semidirect

Maximal subgroups

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

subgroup	index
D₃×C₉×M₁₆	2
C₂×C₉×ΓL(1,𝔽₂₅)	2
C₃×C₉×#108	2
C₉×#822	2
C₉×#819	2
C₃×#4893	3
C₉×#430	3

Subgroups — C9×#4893 #13142260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4893 #13142260