Subgroups — D₃×C₉×#95 #13141321

Order 1728.

1066 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃×C₉	C₂	1	3	yes	maximal
2	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×C₃⋊#95	C₂	1	3	yes	maximal, complemented
4	C₃×D₃×#95	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×#95	—	3	1	no	maximal
6	C₂³×D₃×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
8	C₂²×C₉×Dic₃	C₂²	1	3	yes	—
9	C₂×C₄×D₃×C₉	—	2	12	no	—
10	C₂×C₄×D₃×C₉	C₄	1	12	yes	complemented
11	C₂²×C₃×C₄×D₃	C₂×C₃	1	3	yes	—
12	C₂³×C₄×C₉	—	3	3	no	—
13	C₃²×#95	C₂×C₃	1	1	yes	characteristic
14	C₃×C₃⋊#95	C₂×C₃	1	3	yes	—
15	C₉×#95	—	2	1	no	—
16	C₉×#95	—	3	3	no	—
17	C₉×#95	D₃	1	1	yes	characteristic, direct product factor
18	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
19	C₂²×D₃×C₉	C₂×C₄	1	6	yes	—
20	C₂²×D₃×C₉	D₄	1	6	yes	—
21	C₂²×D₃×C₉	Q₈	1	2	yes	—
22	C₂×C₃×C₄×C₉	—	2	6	no	—
23	C₂×C₃×C₄×C₉	C₂×C₄	1	6	yes	complemented
24	C₂×C₉×Dic₃	—	2	6	no	—
25	C₂×C₉×Dic₃	C₂×C₄	1	6	yes	complemented
26	C₄×D₃×C₉	—	2	24	no	—
27	C₂×C₃×#95	—	3	1	no	—
28	D₃×#95	C₉	1	1	yes	characteristic, direct product factor
29	C₂³×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
31	C₂²×C₃×Dic₃	C₂²×C₃	1	3	yes	—
32	C₂×C₃×C₄×D₃	—	2	12	no	—
33	C₂×C₃×C₄×D₃	C₃×C₄	1	12	yes	—
34	C₂⁴×C₉	—	3	1	no	—
35	C₂²×C₄×C₉	—	2	3	no	—
36	C₂²×C₄×C₉	—	3	3	no	—
37	C₂²×C₄×C₉	—	3	12	no	—
38	C₂²×C₄×C₉	—	6	12	no	—
39	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
40	C₂²×C₃×C₉	C₂²×C₄	1	3	yes	—
41	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
42	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×D₃×C₉	C₄²	1	4	yes	—
44	C₂×D₃×C₉	C₂²⋊C₄	1	12	yes	—
45	C₂×D₃×C₉	C₄⋊C₄	1	12	yes	—
46	C₃×C₄×C₉	—	2	6	no	—
47	C₉×Dic₃	—	2	6	no	—
48	C₂³×C₃×C₄	—	3	3	no	—
49	C₂²×C₄×D₃	C₂×C₉	1	3	yes	—
50	C₃×#95	—	2	1	no	—
51	C₃×#95	—	3	3	no	—
52	C₃×#95	C₂×C₉	1	1	yes	characteristic, complemented
53	C₃×#95	C₃×D₃	1	1	yes	characteristic
54	C₃⋊#95	C₂×C₉	1	3	yes	complemented
55	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
56	C₂²×C₃×D₃	C₂×C₃×C₄	1	6	yes	—
57	C₂²×C₃×D₃	C₃×D₄	1	6	yes	—
58	C₂²×C₃×D₃	C₃×Q₈	1	2	yes	—
59	C₂×C₃²×C₄	—	2	6	no	—
60	C₂×C₃²×C₄	C₂×C₃×C₄	1	6	yes	—
61	C₂×C₃×Dic₃	—	2	6	no	—
62	C₂×C₃×Dic₃	C₂×C₃×C₄	1	6	yes	—
63	C₃×C₄×D₃	—	2	24	no	—
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	—	3	6	no	—
66	C₂³×C₉	—	3	6	no	—
67	C₂³×C₉	—	3	2	no	—
68	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₄×C₉	—	2	6	no	—
70	C₂×C₄×C₉	—	2	6	no	—
71	C₂×C₄×C₉	—	3	6	no	—
72	C₂×C₄×C₉	—	4	6	no	—
73	C₂×C₄×C₉	—	6	24	no	—
74	C₂×C₄×C₉	—	6	6	no	—
75	C₂×C₄×C₉	C₄×D₃	1	6	yes	complemented
76	C₂×#95	—	3	1	no	Sylow
77	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	3	yes	—
79	C₂×C₃×C₉	C₂×C₄⋊C₄	1	3	yes	—
80	D₃×C₉	#95	1	8	yes	direct product factor
81	C₂⁴×C₃	—	3	1	no	—
82	C₂³×D₃	C₂²×C₉	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	3	no	—
84	C₂²×C₃×C₄	—	3	3	no	—
85	C₂²×C₃×C₄	—	3	12	no	—
86	C₂²×C₃×C₄	—	6	12	no	—
87	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
88	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
89	C₂²×Dic₃	C₂²×C₉	1	3	yes	—
90	C₂×C₄×D₃	—	2	12	no	—
91	C₂×C₄×D₃	C₄×C₉	1	12	yes	complemented
92	C₂²×C₃²	C₂²×C₃×C₄	1	3	yes	—
93	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
94	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
95	C₂×C₃×D₃	C₃×C₄²	1	4	yes	—
96	C₂×C₃×D₃	C₃×C₂²⋊C₄	1	12	yes	—
97	C₂×C₃×D₃	C₃×C₄⋊C₄	1	12	yes	—
98	C₃²×C₄	—	2	6	no	—
99	C₃×Dic₃	—	2	6	no	—
100	C₂²×C₉	—	2	3	no	—
101	C₂²×C₉	—	2	3	no	—
102	C₂²×C₉	—	2	1	no	—
103	C₂²×C₉	—	3	4	no	—
104	C₂²×C₉	—	3	12	no	—
105	C₂²×C₉	—	3	12	no	—
106	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
107	C₂²×C₉	D₃×D₄	1	3	yes	—
108	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
109	C₄×C₉	—	2	6	no	—
110	C₄×C₉	—	4	6	no	—
111	C₄×C₉	—	6	6	no	—
112	C₂³×C₄	—	3	3	no	—
113	#95	—	3	3	no	—
114	#95	D₃×C₉	1	1	yes	characteristic, direct product factor
115	C₃×C₉	C₂×#95	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₃	—	2	1	no	—
117	C₂³×C₃	—	3	6	no	—
118	C₂³×C₃	—	3	6	no	—
119	C₂³×C₃	—	3	2	no	—
120	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
121	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
122	C₂²×D₃	C₂×C₄×C₉	1	6	yes	—
123	C₂²×D₃	D₄×C₉	1	6	yes	—
124	C₂²×D₃	Q₈×C₉	1	2	yes	—
125	C₂×C₃×C₄	—	2	6	no	—
126	C₂×C₃×C₄	—	2	6	no	—
127	C₂×C₃×C₄	—	2	6	no	—
128	C₂×C₃×C₄	—	3	6	no	—
129	C₂×C₃×C₄	—	4	6	no	—
130	C₂×C₃×C₄	—	6	24	no	—
131	C₂×C₃×C₄	—	6	6	no	—
132	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	complemented
133	C₂×C₃×C₄	C₃×C₄×D₃	1	6	yes	—
134	C₂×Dic₃	—	2	6	no	—
135	C₂×Dic₃	C₂×C₄×C₉	1	6	yes	complemented
136	C₄×D₃	—	2	24	no	—
137	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
138	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	3	yes	—
139	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	3	yes	—
140	C₃×D₃	C₃×#95	1	8	yes	—
141	C₂×C₉	—	2	3	no	—
142	C₂×C₉	—	2	3	no	—
143	C₂×C₉	—	2	1	no	—
144	C₂×C₉	—	3	8	no	—
145	C₂×C₉	C₄²×D₃	1	1	yes	characteristic
146	C₂×C₉	D₃×C₂²⋊C₄	1	3	yes	—
147	C₂×C₉	D₃×C₄⋊C₄	1	3	yes	—
148	C₂⁴	—	3	1	no	—
149	C₂²×C₄	—	3	3	no	—
150	C₂²×C₄	—	3	12	no	—
151	C₂²×C₄	—	6	12	no	—
152	C₂²×C₄	C₂×D₃×C₉	1	3	yes	—
153	C₂²×C₃	—	2	3	no	—
154	C₂²×C₃	—	2	3	no	—
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	3	12	no	—
157	C₂²×C₃	—	3	12	no	—
158	C₂²×C₃	—	3	4	no	—
159	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
160	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
161	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
162	C₂²×C₃	C₂×C₃×C₄×D₃	1	3	yes	—
163	C₂²×C₃	C₃×D₃×D₄	1	3	yes	—
164	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
165	C₂×D₃	C₄²×C₉	1	4	yes	—
166	C₂×D₃	C₉×C₂²⋊C₄	1	12	yes	—
167	C₂×D₃	C₉×C₄⋊C₄	1	12	yes	—
168	C₃×C₄	—	2	6	no	—
169	C₃×C₄	—	2	6	no	—
170	C₃×C₄	—	4	6	no	—
171	C₃×C₄	—	6	6	no	—
172	Dic₃	—	2	6	no	—
173	C₃²	C₂×C₃×#95	1	1	yes	characteristic
174	C₉	—	2	1	no	—
175	C₉	D₃×#95	1	1	yes	characteristic, direct product factor
176	C₂³	—	3	6	no	—
177	C₂³	—	3	6	no	—
178	C₂³	—	3	2	no	—
179	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
180	C₂×C₄	—	2	6	no	—
181	C₂×C₄	—	3	6	no	—
182	C₂×C₄	—	6	24	no	—
183	C₂×C₄	—	6	6	no	—
184	C₂×C₄	C₄×D₃×C₉	1	6	yes	complemented
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	2	3	no	—
187	C₂×C₃	—	2	3	no	—
188	C₂×C₃	—	3	8	no	—
189	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
190	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	3	yes	—
191	C₂×C₃	C₂×C₉×C₄⋊C₄	1	3	yes	—
192	C₂×C₃	C₃×C₄²×D₃	1	1	yes	characteristic
193	C₂×C₃	C₃×D₃×C₂²⋊C₄	1	3	yes	—
194	C₂×C₃	C₃×D₃×C₄⋊C₄	1	3	yes	—
195	D₃	C₉×#95	1	8	yes	direct product factor
196	C₂²	—	3	4	no	—
197	C₂²	—	3	12	no	—
198	C₂²	—	3	12	no	—
199	C₂²	C₂×C₄×D₃×C₉	1	3	yes	—
200	C₂²	D₃×D₄×C₉	1	3	yes	—
201	C₂²	D₃×Q₈×C₉	1	1	yes	characteristic
202	C₄	—	2	6	no	—
203	C₄	—	6	6	no	—
204	C₃	—	2	1	no	—
205	C₃	C₂×C₉×#95	1	1	yes	characteristic, complemented
206	C₃	C₃×D₃×#95	1	1	yes	characteristic
207	C₂	—	3	8	no	—
208	C₂	C₄²×D₃×C₉	1	1	yes	characteristic
209	C₂	D₃×C₉×C₂²⋊C₄	1	3	yes	—
210	C₂	D₃×C₉×C₄⋊C₄	1	3	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₃×C₉×#95	C₂
socle, cosocle	C₂³×C₃²	C₂³×C₃
prime radical	C₂³×C₃×D₃	C₂²×C₃
hypercenter	C₉×#95	D₃
hypocenter	C₃	C₂×C₉×#95
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₉×#95

Complements

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

N	H ≅ G/N	type
2. C₃×C₉×#95	C₂	semidirect
3. C₉×C₃⋊#95	C₂	semidirect
10. C₂×C₄×D₃×C₉	C₄	semidirect
17. C₉×#95	D₃	direct (N × H)
23. C₂×C₃×C₄×C₉	C₂×C₄	semidirect
25. C₂×C₉×Dic₃	C₂×C₄	semidirect
28. D₃×#95	C₉	direct (N × H)
52. C₃×#95	C₂×C₉	semidirect
54. C₃⋊#95	C₂×C₉	semidirect
75. C₂×C₄×C₉	C₄×D₃	semidirect
80. D₃×C₉	#95	direct (N × H)
91. C₂×C₄×D₃	C₄×C₉	semidirect
114. #95	D₃×C₉	direct (N × H)
115. C₃×C₉	C₂×#95	semidirect
132. C₂×C₃×C₄	C₂×C₄×C₉	semidirect
135. C₂×Dic₃	C₂×C₄×C₉	semidirect
175. C₉	D₃×#95	direct (N × H)
184. C₂×C₄	C₄×D₃×C₉	semidirect
195. D₃	C₉×#95	direct (N × H)
205. C₃	C₂×C₉×#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₃×C₉	2
C₃×C₉×#95	2
C₉×C₃⋊#95	2
C₃×D₃×#95	3
C₂×C₉×#95	3

Subgroups — D3×C9×#95 #13141321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×C₉×#95 #13141321