Subgroups — C₃×C₉×#381 #13144901

Order 1728.

1090 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₃×C₉×#95	C₂	1	4	yes	maximal, complemented
4	C₃²×#381	C₃	1	1	yes	maximal, characteristic
5	C₉×#381	C₃	1	3	yes	maximal, direct product factor
6	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	4	yes	complemented
9	C₂²×C₃×C₄×C₉	C₂²	1	4	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₃×C₉×C₂²⋊C₄	—	2	4	no	—
12	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	2	yes	—
14	C₂³×C₄×C₉	C₂×C₃	1	3	yes	—
15	C₃²×#95	C₂×C₃	1	4	yes	—
16	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	6	yes	—
17	C₉×#95	C₂×C₃	1	12	yes	complemented
18	C₂³×C₃×C₉	—	2	4	no	—
19	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
20	C₂³×C₃×C₉	D₄	1	4	yes	—
21	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	—	2	8	no	—
23	C₂×C₃×C₄×C₉	—	2	4	no	—
24	C₂×C₃×C₄×C₉	—	2	4	no	—
25	C₂×C₃×C₄×C₉	—	2	8	no	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
27	C₃×#381	C₉	1	3	yes	direct product factor
28	C₃×#381	C₃²	1	1	yes	characteristic
29	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
30	C₂²×C₃²×C₄	—	2	4	no	—
31	C₂²×C₃²×C₄	C₃×C₄	1	4	yes	—
32	C₂²×C₃²×C₄	C₂²×C₃	1	4	yes	—
33	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
34	C₂⁴×C₉	C₂²×C₃	1	3	yes	—
35	C₃²×C₂²⋊C₄	—	2	4	no	—
36	C₂²×C₄×C₉	—	2	12	no	—
37	C₂²×C₄×C₉	C₃×C₄	1	12	yes	complemented
38	C₂²×C₄×C₉	C₂²×C₃	1	12	yes	—
39	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
40	C₉×C₂²⋊C₄	—	2	12	no	—
41	C₂²×C₃×C₉	—	2	4	no	—
42	C₂²×C₃×C₉	—	2	8	no	—
43	C₂²×C₃×C₉	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₃×C₉	C₂×D₄	1	2	yes	—
46	C₂²×C₃×C₉	#41	1	4	yes	—
47	C₃×C₄×C₉	—	2	4	no	—
48	C₃×C₄×C₉	—	4	4	no	—
49	C₂³×C₃×C₄	C₂×C₉	1	3	yes	—
50	C₂³×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
51	C₂×C₃×C₂²⋊C₄	C₂×C₉	1	6	yes	—
52	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	2	yes	—
53	C₃×#95	C₂×C₉	1	12	yes	complemented
54	C₃×#95	C₂×C₃²	1	4	yes	—
55	C₂³×C₃²	—	2	4	no	—
56	C₂³×C₃²	C₂×C₃×C₄	1	2	yes	—
57	C₂³×C₃²	C₃×D₄	1	4	yes	—
58	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
59	C₂×C₃²×C₄	—	2	8	no	—
60	C₂×C₃²×C₄	—	2	4	no	—
61	C₂×C₃²×C₄	—	2	4	no	—
62	C₂×C₃²×C₄	—	2	8	no	—
63	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
64	C₂³×C₉	—	2	12	no	—
65	C₂³×C₉	C₂×C₃×C₄	1	6	yes	—
66	C₂³×C₉	C₃×D₄	1	12	yes	—
67	C₂³×C₉	C₂³×C₃	1	3	yes	—
68	C₂×C₄×C₉	—	2	24	no	—
69	C₂×C₄×C₉	—	2	12	no	—
70	C₂×C₄×C₉	—	2	12	no	—
71	C₂×C₄×C₉	—	2	24	no	—
72	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
73	#381	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₃×C₉	—	2	4	no	—
75	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₉	#117	1	2	yes	—
77	C₂×C₃×C₉	#123	1	4	yes	—
78	C₂⁴×C₃	C₂²×C₉	1	3	yes	—
79	C₂⁴×C₃	C₂²×C₃²	1	1	yes	characteristic
80	C₂²×C₃×C₄	—	2	12	no	—
81	C₂²×C₃×C₄	—	2	4	no	—
82	C₂²×C₃×C₄	C₄×C₉	1	12	yes	complemented
83	C₂²×C₃×C₄	C₂²×C₉	1	12	yes	—
84	C₂²×C₃×C₄	C₂²×C₉	1	6	yes	—
85	C₂²×C₃×C₄	C₃²×C₄	1	4	yes	—
86	C₂²×C₃×C₄	C₂²×C₃²	1	4	yes	—
87	C₂²×C₃×C₄	C₂²×C₃²	1	2	yes	—
88	C₃×C₂²⋊C₄	—	2	12	no	—
89	C₃×C₂²⋊C₄	—	2	4	no	—
90	C₂²×C₃²	—	2	4	no	—
91	C₂²×C₃²	—	2	8	no	—
92	C₂²×C₃²	C₃×C₂²⋊C₄	1	4	yes	—
93	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
94	C₂²×C₃²	C₂×C₃×D₄	1	2	yes	—
95	C₂²×C₃²	C₃×#41	1	4	yes	—
96	C₃²×C₄	—	2	4	no	—
97	C₃²×C₄	—	4	4	no	—
98	C₂²×C₉	—	2	12	no	—
99	C₂²×C₉	—	2	24	no	—
100	C₂²×C₉	C₃×C₂²⋊C₄	1	12	yes	—
101	C₂²×C₉	C₂²×C₃×C₄	1	3	yes	—
102	C₂²×C₉	C₂×C₃×D₄	1	6	yes	—
103	C₂²×C₉	C₃×#41	1	12	yes	—
104	C₄×C₉	—	2	12	no	—
105	C₄×C₉	—	4	12	no	—
106	C₂³×C₄	C₂×C₃×C₉	1	1	yes	characteristic
107	C₂×C₂²⋊C₄	C₂×C₃×C₉	1	2	yes	—
108	#95	C₂×C₃×C₉	1	4	yes	complemented
109	C₃×C₉	#381	1	1	yes	characteristic, direct product factor, Sylow
110	C₂³×C₃	—	2	12	no	—
111	C₂³×C₃	—	2	4	no	—
112	C₂³×C₃	C₂×C₄×C₉	1	6	yes	—
113	C₂³×C₃	D₄×C₉	1	12	yes	—
114	C₂³×C₃	C₂³×C₉	1	3	yes	—
115	C₂³×C₃	C₂×C₃²×C₄	1	2	yes	—
116	C₂³×C₃	C₃²×D₄	1	4	yes	—
117	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	24	no	—
119	C₂×C₃×C₄	—	2	12	no	—
120	C₂×C₃×C₄	—	2	8	no	—
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	12	no	—
123	C₂×C₃×C₄	—	2	24	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	8	no	—
126	C₂×C₃×C₄	C₂×C₄×C₉	1	12	yes	—
127	C₂×C₃×C₄	C₂×C₃²×C₄	1	4	yes	—
128	C₂×C₃²	—	2	4	no	—
129	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₃²	C₃×#117	1	2	yes	—
131	C₂×C₃²	C₃×#123	1	4	yes	—
132	C₂×C₉	—	2	12	no	—
133	C₂×C₉	C₂×C₃×C₂²⋊C₄	1	3	yes	—
134	C₂×C₉	C₃×#117	1	6	yes	—
135	C₂×C₉	C₃×#123	1	12	yes	—
136	C₂⁴	C₂²×C₃×C₉	1	1	yes	characteristic
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₃×C₄×C₉	1	4	yes	complemented
139	C₂²×C₄	C₂²×C₃×C₉	1	4	yes	—
140	C₂²×C₄	C₂²×C₃×C₉	1	2	yes	—
141	C₂²⋊C₄	—	2	4	no	—
142	C₂²×C₃	—	2	12	no	—
143	C₂²×C₃	—	2	24	no	—
144	C₂²×C₃	—	2	4	no	—
145	C₂²×C₃	—	2	8	no	—
146	C₂²×C₃	C₉×C₂²⋊C₄	1	12	yes	—
147	C₂²×C₃	C₂²×C₄×C₉	1	3	yes	—
148	C₂²×C₃	C₂×D₄×C₉	1	6	yes	—
149	C₂²×C₃	C₉×#41	1	12	yes	—
150	C₂²×C₃	C₃²×C₂²⋊C₄	1	4	yes	—
151	C₂²×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
152	C₂²×C₃	C₂×C₃²×D₄	1	2	yes	—
153	C₂²×C₃	C₃²×#41	1	4	yes	—
154	C₃×C₄	—	2	12	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	—	4	12	no	—
157	C₃×C₄	—	4	4	no	—
158	C₃²	C₃×#381	1	1	yes	characteristic
159	C₉	C₃×#381	1	3	yes	direct product factor
160	C₂³	—	2	4	no	—
161	C₂³	C₂×C₃×C₄×C₉	1	2	yes	—
162	C₂³	C₃×D₄×C₉	1	4	yes	—
163	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	8	no	—
168	C₂×C₄	C₂×C₃×C₄×C₉	1	4	yes	—
169	C₂×C₃	—	2	12	no	—
170	C₂×C₃	—	2	4	no	—
171	C₂×C₃	C₂×C₉×C₂²⋊C₄	1	3	yes	—
172	C₂×C₃	C₉×#117	1	6	yes	—
173	C₂×C₃	C₉×#123	1	12	yes	—
174	C₂×C₃	C₂×C₃²×C₂²⋊C₄	1	1	yes	characteristic
175	C₂×C₃	C₃²×#117	1	2	yes	—
176	C₂×C₃	C₃²×#123	1	4	yes	—
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	C₃×C₉×C₂²⋊C₄	1	4	yes	—
180	C₂²	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
181	C₂²	C₂×C₃×D₄×C₉	1	2	yes	—
182	C₂²	C₃×C₉×#41	1	4	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	4	4	no	—
185	C₃	C₉×#381	1	3	yes	direct product factor
186	C₃	C₃²×#381	1	1	yes	characteristic
187	C₂	—	2	4	no	—
188	C₂	C₂×C₃×C₉×C₂²⋊C₄	1	1	yes	characteristic
189	C₂	C₃×C₉×#117	1	2	yes	—
190	C₂	C₃×C₉×#123	1	4	yes	—

	subgroup	G/H
center	C₂³×C₃×C₉	C₂³
commutator	C₂²	C₂²×C₃×C₄×C₉
Frattini, cosocle	C₂³×C₃	C₂³×C₃²
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂⁴×C₃²	C₂²×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#95	C₂	semidirect
5. C₉×#381	C₃	direct (N × H)
8. C₂²×C₃×C₄×C₉	C₄	semidirect
17. C₉×#95	C₂×C₃	semidirect
27. C₃×#381	C₉	direct (N × H)
37. C₂²×C₄×C₉	C₃×C₄	semidirect
53. C₃×#95	C₂×C₉	semidirect
73. #381	C₃×C₉	direct (N × H)
82. C₂²×C₃×C₄	C₄×C₉	semidirect
108. #95	C₂×C₃×C₉	semidirect
109. C₃×C₉	#381	direct (N × H)
138. C₂²×C₄	C₃×C₄×C₉	semidirect
159. C₉	C₃×#381	direct (N × H)
185. C₃	C₉×#381	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×C₃×C₉×C₂²⋊C₄	2
C₃×C₉×#95	2
C₃²×#381	3
C₉×#381	3

Subgroups — C3×C9×#381 #13144901

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#381 #13144901