Subgroups — C₂×C₃×C₉×#106 #13145033

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 174 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×C₃×C₉×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₃×C₉×#106	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×#106	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×#106	C₃	1	3	yes	maximal, direct product factor
6	C₂²×C₃×C₄×C₉	—	2	2	no	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
9	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
10	C₃×C₉×C₄⋊C₄	—	2	2	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
13	C₂×C₃²×C₄⋊C₄	C₂×C₃	1	2	yes	—
14	C₃²×#106	C₂×C₃	1	4	yes	—
15	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
16	C₂×C₉×C₄⋊C₄	C₂×C₃	1	6	yes	—
17	C₉×#106	C₂×C₃	1	12	yes	direct product factor
18	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₉	—	2	4	no	—
20	C₂×C₃×C₄×C₉	—	4	4	no	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
24	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₃×C₈×C₉	C₂×C₄	1	4	yes	complemented
26	C₂×C₃×#106	C₉	1	3	yes	direct product factor
27	C₂×C₃×#106	C₃²	1	1	yes	characteristic
28	C₂²×C₃²×C₄	—	2	2	no	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₃×C₄	1	4	yes	—
31	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
32	C₃²×C₄⋊C₄	—	2	2	no	—
33	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
34	C₂²×C₄×C₉	—	2	6	no	—
35	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
36	C₂×C₈×C₉	C₃×C₄	1	12	yes	complemented
37	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
38	C₉×C₄⋊C₄	—	2	6	no	—
39	C₉×C₄⋊C₄	C₂²×C₃	1	12	yes	—
40	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
41	C₂²×C₃×C₉	SD₁₆	1	4	yes	—
42	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
43	C₃×C₄×C₉	—	4	4	no	—
44	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
45	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
46	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₃×C₈	C₂×C₉	1	3	yes	—
48	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	C₂×C₉	1	6	yes	—
50	C₂×C₃×C₄⋊C₄	C₂×C₃²	1	2	yes	—
51	C₃×#106	C₂×C₉	1	12	yes	direct product factor
52	C₃×#106	C₂×C₃²	1	4	yes	—
53	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
54	C₂×C₃²×C₄	—	2	4	no	—
55	C₂×C₃²×C₄	—	4	4	no	—
56	C₂×C₃²×C₄	C₂×C₃×C₄	1	2	yes	—
57	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₃×Q₈	1	2	yes	—
59	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
60	C₂³×C₉	C₃×D₄	1	3	yes	—
61	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
62	C₂×C₄×C₉	—	2	12	no	—
63	C₂×C₄×C₉	—	4	12	no	—
64	C₂×C₄×C₉	C₂×C₃×C₄	1	6	yes	—
65	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
66	C₂×C₄×C₉	C₃×Q₈	1	6	yes	—
67	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
68	C₈×C₉	C₂×C₃×C₄	1	12	yes	complemented
69	C₂×#106	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
70	C₂×C₃×C₉	#106	1	4	yes	direct product factor
71	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₉	C₂×SD₁₆	1	2	yes	—
73	C₂²×C₃×C₄	—	2	6	no	—
74	C₂²×C₃×C₄	—	2	2	no	—
75	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
76	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
77	C₂×C₃×C₈	C₄×C₉	1	12	yes	complemented
78	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
79	C₂×C₃×C₈	C₃²×C₄	1	4	yes	—
80	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
81	C₃×C₄⋊C₄	—	2	6	no	—
82	C₃×C₄⋊C₄	—	2	2	no	—
83	C₃×C₄⋊C₄	C₂²×C₉	1	12	yes	—
84	C₃×C₄⋊C₄	C₂²×C₃²	1	4	yes	—
85	C₂²×C₃²	C₃×C₄⋊C₄	1	2	yes	—
86	C₂²×C₃²	C₃×SD₁₆	1	4	yes	—
87	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
88	C₃²×C₄	—	4	4	no	—
89	C₃²×C₄	C₃×C₄⋊C₄	1	2	yes	—
90	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
91	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
92	C₂²×C₉	C₃×C₄⋊C₄	1	6	yes	—
93	C₂²×C₉	C₃×SD₁₆	1	12	yes	—
94	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
95	C₄×C₉	—	4	12	no	—
96	C₄×C₉	C₃×C₄⋊C₄	1	6	yes	—
97	C₄×C₉	C₂²×C₃×C₄	1	3	yes	—
98	C₄×C₉	C₂×C₃×Q₈	1	3	yes	—
99	C₂²×C₈	C₂×C₃×C₉	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂×C₃×C₉	1	2	yes	—
101	#106	C₂×C₃×C₉	1	4	yes	direct product factor
102	C₃×C₉	C₂×#106	1	1	yes	characteristic, direct product factor, Sylow
103	C₂³×C₃	D₄×C₉	1	3	yes	—
104	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	12	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	4	12	no	—
108	C₂×C₃×C₄	—	4	4	no	—
109	C₂×C₃×C₄	C₂×C₄×C₉	1	6	yes	—
110	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
111	C₂×C₃×C₄	Q₈×C₉	1	6	yes	—
112	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
113	C₂×C₃×C₄	C₂×C₃²×C₄	1	2	yes	—
114	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
115	C₂×C₃×C₄	C₃²×Q₈	1	2	yes	—
116	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
117	C₃×C₈	C₂×C₄×C₉	1	12	yes	complemented
118	C₃×C₈	C₂×C₃²×C₄	1	4	yes	—
119	C₂×C₃²	C₃×#106	1	4	yes	—
120	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₃²	C₂×C₃×SD₁₆	1	2	yes	—
122	C₂×C₉	C₃×#106	1	12	yes	direct product factor
123	C₂×C₉	C₂×C₃×C₄⋊C₄	1	3	yes	—
124	C₂×C₉	C₂×C₃×SD₁₆	1	6	yes	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
127	C₂×C₈	C₃×C₄×C₉	1	4	yes	complemented
128	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	C₂²×C₃×C₉	1	4	yes	—
131	C₂²×C₃	C₉×C₄⋊C₄	1	6	yes	—
132	C₂²×C₃	C₉×SD₁₆	1	12	yes	—
133	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
134	C₂²×C₃	C₃²×C₄⋊C₄	1	2	yes	—
135	C₂²×C₃	C₃²×SD₁₆	1	4	yes	—
136	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
137	C₃×C₄	—	4	12	no	—
138	C₃×C₄	—	4	4	no	—
139	C₃×C₄	C₉×C₄⋊C₄	1	6	yes	—
140	C₃×C₄	C₂²×C₄×C₉	1	3	yes	—
141	C₃×C₄	C₂×Q₈×C₉	1	3	yes	—
142	C₃×C₄	C₃²×C₄⋊C₄	1	2	yes	—
143	C₃×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
144	C₃×C₄	C₂×C₃²×Q₈	1	1	yes	characteristic
145	C₃²	C₂×C₃×#106	1	1	yes	characteristic
146	C₉	C₂×C₃×#106	1	3	yes	direct product factor
147	C₂³	C₃×D₄×C₉	1	1	yes	characteristic
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	C₂×C₃×C₄×C₉	1	2	yes	—
151	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
152	C₂×C₄	C₃×Q₈×C₉	1	2	yes	—
153	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
154	C₈	C₂×C₃×C₄×C₉	1	4	yes	complemented
155	C₂×C₃	C₉×#106	1	12	yes	direct product factor
156	C₂×C₃	C₂×C₉×C₄⋊C₄	1	3	yes	—
157	C₂×C₃	C₂×C₉×SD₁₆	1	6	yes	—
158	C₂×C₃	C₃²×#106	1	4	yes	—
159	C₂×C₃	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₃	C₂×C₃²×SD₁₆	1	2	yes	—
161	C₂²	C₃×C₉×C₄⋊C₄	1	2	yes	—
162	C₂²	C₃×C₉×SD₁₆	1	4	yes	—
163	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
164	C₄	—	4	4	no	—
165	C₄	C₃×C₉×C₄⋊C₄	1	2	yes	—
166	C₄	C₂²×C₃×C₄×C₉	1	1	yes	characteristic
167	C₄	C₂×C₃×Q₈×C₉	1	1	yes	characteristic
168	C₃	C₂×C₉×#106	1	3	yes	direct product factor
169	C₃	C₂×C₃²×#106	1	1	yes	characteristic
170	C₂	C₃×C₉×#106	1	4	yes	direct product factor
171	C₂	C₂×C₃×C₉×C₄⋊C₄	1	1	yes	characteristic
172	C₂	C₂×C₃×C₉×SD₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃×C₉	D₄
commutator	C₄	C₂²×C₃×C₄×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle, prime radical	C₂³×C₃²	C₃×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ C₂²×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₉×#106	C₂	direct (N × H)
5. C₂×C₉×#106	C₃	direct (N × H)
8. C₂×C₃×C₈×C₉	C₄	semidirect
17. C₉×#106	C₂×C₃	direct (N × H)
25. C₃×C₈×C₉	C₂×C₄	semidirect
26. C₂×C₃×#106	C₉	direct (N × H)
36. C₂×C₈×C₉	C₃×C₄	semidirect
51. C₃×#106	C₂×C₉	direct (N × H)
68. C₈×C₉	C₂×C₃×C₄	semidirect
69. C₂×#106	C₃×C₉	direct (N × H)
70. C₂×C₃×C₉	#106	direct (N × H)
77. C₂×C₃×C₈	C₄×C₉	semidirect
101. #106	C₂×C₃×C₉	direct (N × H)
102. C₃×C₉	C₂×#106	direct (N × H)
117. C₃×C₈	C₂×C₄×C₉	semidirect
122. C₂×C₉	C₃×#106	direct (N × H)
127. C₂×C₈	C₃×C₄×C₉	semidirect
146. C₉	C₂×C₃×#106	direct (N × H)
154. C₈	C₂×C₃×C₄×C₉	semidirect
155. C₂×C₃	C₉×#106	direct (N × H)
168. C₃	C₂×C₉×#106	direct (N × H)
170. C₂	C₃×C₉×#106	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×C₃×C₉×C₄⋊C₄	2
C₃×C₉×#106	2
C₂×C₃²×#106	3
C₂×C₉×#106	3

Subgroups — C2×C3×C9×#106 #13145033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₉×#106 #13145033