Subgroups — C₂²×C₈×C₃⋊D₉ #13154306

Order 1728.

1690 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×C₃⋊D₉	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₈×C₃⋊D₃	—	3	1	no	maximal
6	C₂²×C₈×D₉	—	3	3	no	maximal
7	C₂³×C₃⋊D₉	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₄	1	1	yes	characteristic
10	C₂×C₄×C₃⋊D₉	C₄	1	6	yes	—
11	C₂×C₄×C₃⋊D₉	C₂²	1	6	yes	—
12	C₂×C₃×C₈×C₉	C₂²	1	6	yes	complemented
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	6	yes	complemented
14	C₈×C₃⋊D₉	C₂²	1	16	yes	direct product factor
15	C₂²×C₄×C₃⋊D₃	—	3	1	no	—
16	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
17	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
18	C₂×C₈×C₃⋊D₃	—	3	12	no	—
19	C₂²×C₄×D₉	—	3	3	no	—
20	C₂²×C₈×C₉	D₃	1	3	yes	complemented
21	C₂²×C₉⋊C₈	—	3	3	no	—
22	C₂×C₈×D₉	—	3	36	no	—
23	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃⋊D₉	C₈	1	8	yes	direct product factor
25	C₂²×C₃⋊D₉	C₂×C₄	1	6	yes	—
26	C₂×C₃×C₄×C₉	C₂×C₄	1	3	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	3	yes	—
28	C₂×C₃⋊Dic₉	C₂×C₄	1	6	yes	—
29	C₄×C₃⋊D₉	C₂×C₄	1	12	yes	—
30	C₄×C₃⋊D₉	C₂³	1	4	yes	—
31	C₃×C₈×C₉	C₂³	1	4	yes	complemented
32	C₃⋊(C₉⋊C₈)	C₂³	1	4	yes	complemented
33	C₂²×D₃×C₈	—	9	3	no	—
34	C₂²×D₃×C₈	—	9	1	no	—
35	C₂³×C₃⋊D₃	—	3	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	—	3	1	no	—
38	C₂×C₄×C₃⋊D₃	—	3	6	no	—
39	C₂×C₄×C₃⋊D₃	—	3	6	no	—
40	C₂³×D₉	—	3	3	no	—
41	C₂×C₃²×C₈	C₂×D₃	1	6	yes	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	6	no	—
43	C₈×C₃⋊D₃	—	3	16	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
45	C₂²×Dic₉	—	3	3	no	—
46	C₂×C₄×D₉	—	3	18	no	—
47	C₂×C₄×D₉	—	3	18	no	—
48	C₂×C₈×C₉	C₂×D₃	1	18	yes	complemented
49	C₂×C₉⋊C₈	—	3	18	no	—
50	C₈×D₉	—	3	48	no	—
51	C₂²×C₃×C₉	C₂×C₈	1	4	yes	complemented
52	C₂²×C₃×C₉	C₂²×C₄	1	3	yes	—
53	C₂×C₃⋊D₉	C₂×C₈	1	24	yes	direct product factor
54	C₂×C₃⋊D₉	C₂²×C₄	1	4	yes	—
55	C₃×C₄×C₉	C₂²×C₄	1	3	yes	—
56	C₃×C₄×C₉	C₂⁴	1	1	yes	characteristic
57	C₃⋊Dic₉	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×D₃	—	9	3	no	—
59	C₂²×C₄×D₃	—	9	1	no	—
60	C₂²×C₃×C₈	D₉	1	3	yes	complemented
61	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
62	C₂²×C₃⋊C₈	—	9	3	no	—
63	C₂²×C₃⋊C₈	—	9	1	no	—
64	C₂×D₃×C₈	—	9	36	no	—
65	C₂×D₃×C₈	—	9	12	no	—
66	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
67	C₂²×C₃⋊D₃	—	3	8	no	—
68	C₂²×C₃⋊D₃	—	3	6	no	—
69	C₂×C₃²×C₄	C₄×D₃	1	3	yes	—
70	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
71	C₂×C₃⋊Dic₃	—	3	6	no	—
72	C₄×C₃⋊D₃	—	3	12	no	—
73	C₄×C₃⋊D₃	—	3	4	no	—
74	C₂³×C₉	C₄×D₃	1	3	yes	—
75	C₂²×D₉	—	3	24	no	—
76	C₂²×D₉	—	3	18	no	—
77	C₃²×C₈	C₂²×D₃	1	4	yes	—
78	C₃⋊(C₃⋊C₈)	—	3	4	no	—
79	C₂×C₄×C₉	C₄×D₃	1	9	yes	—
80	C₂×C₄×C₉	C₂²×D₃	1	9	yes	—
81	C₂×Dic₉	—	3	18	no	—
82	C₄×D₉	—	3	36	no	—
83	C₄×D₉	—	3	12	no	—
84	C₈×C₉	C₂²×D₃	1	12	yes	complemented
85	C₉⋊C₈	—	3	12	no	—
86	C₂³×C₈	—	27	1	no	Sylow
87	C₂×C₃×C₉	C₂²×C₈	1	6	yes	complemented
88	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
89	C₃⋊D₉	C₂²×C₈	1	8	yes	direct product factor
90	C₂³×D₃	—	9	3	no	—
91	C₂³×D₃	—	9	1	no	—
92	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
93	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
94	C₂²×Dic₃	—	9	3	no	—
95	C₂²×Dic₃	—	9	1	no	—
96	C₂×C₄×D₃	—	9	18	no	—
97	C₂×C₄×D₃	—	9	18	no	—
98	C₂×C₄×D₃	—	9	6	no	—
99	C₂×C₄×D₃	—	9	6	no	—
100	C₂×C₃×C₈	C₂×D₉	1	18	yes	complemented
101	C₂×C₃×C₈	C₂×C₃⋊D₃	1	6	yes	—
102	C₂×C₃⋊C₈	—	9	18	no	—
103	C₂×C₃⋊C₈	—	9	6	no	—
104	D₃×C₈	—	9	48	no	—
105	D₃×C₈	—	9	16	no	—
106	C₂²×C₃²	D₃×C₈	1	4	yes	—
107	C₂²×C₃²	C₂×C₄×D₃	1	3	yes	—
108	C₂×C₃⋊D₃	—	3	24	no	—
109	C₂×C₃⋊D₃	—	3	4	no	—
110	C₃²×C₄	C₂×C₄×D₃	1	3	yes	—
111	C₃²×C₄	C₂³×D₃	1	1	yes	characteristic
112	C₃⋊Dic₃	—	3	4	no	—
113	C₂²×C₉	D₃×C₈	1	12	yes	complemented
114	C₂²×C₉	C₂×C₄×D₃	1	9	yes	—
115	C₂×D₉	—	3	72	no	—
116	C₂×D₉	—	3	12	no	—
117	C₄×C₉	C₂×C₄×D₃	1	9	yes	—
118	C₄×C₉	C₂³×D₃	1	3	yes	—
119	Dic₉	—	3	12	no	—
120	C₂³×C₄	—	27	1	no	—
121	C₂²×C₈	—	27	1	no	—
122	C₂²×C₈	—	27	12	no	—
123	C₂²×C₈	C₃⋊D₉	1	1	yes	characteristic, direct product factor
124	C₃×C₉	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₃	C₄×D₉	1	3	yes	—
126	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
127	C₂²×D₃	—	9	24	no	—
128	C₂²×D₃	—	9	18	no	—
129	C₂²×D₃	—	9	8	no	—
130	C₂²×D₃	—	9	6	no	—
131	C₂×C₃×C₄	C₄×D₉	1	9	yes	—
132	C₂×C₃×C₄	C₂²×D₉	1	9	yes	—
133	C₂×C₃×C₄	C₄×C₃⋊D₃	1	3	yes	—
134	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
135	C₂×Dic₃	—	9	18	no	—
136	C₂×Dic₃	—	9	6	no	—
137	C₄×D₃	—	9	4	no	—
138	C₄×D₃	—	9	36	no	—
139	C₄×D₃	—	9	12	no	—
140	C₄×D₃	—	9	12	no	—
141	C₃×C₈	C₂²×D₉	1	12	yes	complemented
142	C₃×C₈	C₂²×C₃⋊D₃	1	4	yes	—
143	C₃⋊C₈	—	9	4	no	—
144	C₃⋊C₈	—	9	12	no	—
145	C₂×C₃²	C₂×D₃×C₈	1	6	yes	—
146	C₂×C₃²	C₂²×C₄×D₃	1	1	yes	characteristic
147	C₃⋊D₃	—	3	8	no	—
148	C₂×C₉	C₂×D₃×C₈	1	18	yes	complemented
149	C₂×C₉	C₂²×C₄×D₃	1	3	yes	—
150	D₉	—	3	24	no	—
151	C₂⁴	—	27	1	no	—
152	C₂²×C₄	—	27	1	no	—
153	C₂²×C₄	—	27	6	no	—
154	C₂²×C₄	—	27	6	no	—
155	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
156	C₂×C₈	—	27	6	no	—
157	C₂×C₈	—	27	16	no	—
158	C₂×C₈	C₂×C₃⋊D₉	1	6	yes	direct product factor
159	C₂²×C₃	C₈×D₉	1	12	yes	complemented
160	C₂²×C₃	C₂×C₄×D₉	1	9	yes	—
161	C₂²×C₃	C₈×C₃⋊D₃	1	4	yes	—
162	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	3	yes	—
163	C₂×D₃	—	9	4	no	—
164	C₂×D₃	—	9	24	no	—
165	C₂×D₃	—	9	12	no	—
166	C₂×D₃	—	9	72	no	—
167	C₃×C₄	C₂×C₄×D₉	1	9	yes	—
168	C₃×C₄	C₂³×D₉	1	3	yes	—
169	C₃×C₄	C₂×C₄×C₃⋊D₃	1	3	yes	—
170	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
171	Dic₃	—	9	4	no	—
172	Dic₃	—	9	12	no	—
173	C₃²	C₂²×D₃×C₈	1	1	yes	characteristic
174	C₉	C₂²×D₃×C₈	1	3	yes	complemented
175	C₂³	—	27	6	no	—
176	C₂³	—	27	8	no	—
177	C₂³	C₄×C₃⋊D₉	1	1	yes	characteristic
178	C₂×C₄	—	27	6	no	—
179	C₂×C₄	—	27	4	no	—
180	C₂×C₄	—	27	12	no	—
181	C₂×C₄	C₄×C₃⋊D₉	1	3	yes	—
182	C₂×C₄	C₂²×C₃⋊D₉	1	3	yes	—
183	C₈	—	27	4	no	—
184	C₈	C₂²×C₃⋊D₉	1	4	yes	direct product factor
185	C₂×C₃	C₂×C₈×D₉	1	18	yes	complemented
186	C₂×C₃	C₂²×C₄×D₉	1	3	yes	—
187	C₂×C₃	C₂×C₈×C₃⋊D₃	1	6	yes	—
188	C₂×C₃	C₂²×C₄×C₃⋊D₃	1	1	yes	characteristic
189	D₃	—	9	8	no	—
190	D₃	—	9	24	no	—
191	C₂²	—	27	24	no	—
192	C₂²	—	27	4	no	—
193	C₂²	C₈×C₃⋊D₉	1	4	yes	direct product factor
194	C₂²	C₂×C₄×C₃⋊D₉	1	3	yes	—
195	C₄	—	27	4	no	—
196	C₄	C₂×C₄×C₃⋊D₉	1	3	yes	—
197	C₄	C₂³×C₃⋊D₉	1	1	yes	characteristic
198	C₃	C₂²×C₈×D₉	1	3	yes	complemented
199	C₃	C₂²×C₈×C₃⋊D₃	1	1	yes	characteristic
200	C₂	—	27	8	no	—
201	C₂	C₂×C₈×C₃⋊D₉	1	6	yes	direct product factor
202	C₂	C₂²×C₄×C₃⋊D₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₈	C₃⋊D₉
commutator, hypocenter	C₃×C₉	C₂³×C₈
Frattini	C₃×C₄	C₂³×C₃⋊D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂³×C₃²	C₄×D₃
prime radical	C₂³×C₃⋊D₉	C₄
cosocle	C₃×C₄×C₉	C₂⁴
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₈×C₉	C₂	semidirect
3. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
4. C₂×C₈×C₃⋊D₉	C₂	direct (N × H)
12. C₂×C₃×C₈×C₉	C₂²	semidirect
13. C₂×C₃⋊(C₉⋊C₈)	C₂²	semidirect
14. C₈×C₃⋊D₉	C₂²	direct (N × H)
20. C₂²×C₈×C₉	D₃	semidirect
24. C₂²×C₃⋊D₉	C₈	direct (N × H)
31. C₃×C₈×C₉	C₂³	semidirect
32. C₃⋊(C₉⋊C₈)	C₂³	semidirect
48. C₂×C₈×C₉	C₂×D₃	semidirect
51. C₂²×C₃×C₉	C₂×C₈	semidirect
53. C₂×C₃⋊D₉	C₂×C₈	direct (N × H)
60. C₂²×C₃×C₈	D₉	semidirect
84. C₈×C₉	C₂²×D₃	semidirect
87. C₂×C₃×C₉	C₂²×C₈	semidirect
89. C₃⋊D₉	C₂²×C₈	direct (N × H)
100. C₂×C₃×C₈	C₂×D₉	semidirect
113. C₂²×C₉	D₃×C₈	semidirect
123. C₂²×C₈	C₃⋊D₉	direct (N × H)
124. C₃×C₉	C₂³×C₈	semidirect
141. C₃×C₈	C₂²×D₉	semidirect
148. C₂×C₉	C₂×D₃×C₈	semidirect
158. C₂×C₈	C₂×C₃⋊D₉	direct (N × H)
159. C₂²×C₃	C₈×D₉	semidirect
174. C₉	C₂²×D₃×C₈	semidirect
184. C₈	C₂²×C₃⋊D₉	direct (N × H)
185. C₂×C₃	C₂×C₈×D₉	semidirect
193. C₂²	C₈×C₃⋊D₉	direct (N × H)
198. C₃	C₂²×C₈×D₉	semidirect
201. C₂	C₂×C₈×C₃⋊D₉	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×C₄×C₃⋊D₉	2
C₂²×C₃×C₈×C₉	2
C₂²×C₃⋊(C₉⋊C₈)	2
C₂×C₈×C₃⋊D₉	2
C₂²×C₈×C₃⋊D₃	3
C₂²×C₈×D₉	3

Subgroups — C22×C8×C3⋊D9 #13154306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×C₃⋊D₉ #13154306