Subgroups — C₂×C₄×C₉×C₃⋊C₈ #13142371

Order 1728.

594 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	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₃⋊C₈	C₂	1	2	yes	maximal
3	C₄×C₉×C₃⋊C₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×C₄×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic
5	C₂×C₄×C₈×C₉	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₉	C₄	1	2	yes	—
9	C₃×C₄²×C₉	C₂²	1	2	yes	—
10	C₂×C₉×C₃⋊C₈	C₄	1	8	yes	direct product factor
11	C₂×C₉×C₃⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	C₂×C₃	1	2	yes	—
14	C₃×C₄×C₃⋊C₈	C₂×C₃	1	4	yes	—
15	C₂²×C₈×C₉	—	3	2	no	—
16	C₂×C₄²×C₉	—	2	1	no	—
17	C₂×C₄²×C₉	D₃	1	1	yes	characteristic
18	C₄×C₈×C₉	—	3	4	no	—
19	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	C₈	1	8	yes	complemented
21	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
23	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₉×C₃⋊C₈	C₂×C₄	1	8	yes	direct product factor
26	C₂×C₃×C₄×C₈	—	3	1	no	—
27	C₂×C₄×C₃⋊C₈	C₉	1	1	yes	characteristic, direct product factor
28	C₂²×C₃²×C₄	C₃×C₄	1	2	yes	—
29	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₃²×C₄²	C₃×C₄	1	2	yes	—
31	C₃²×C₄²	C₂²×C₃	1	2	yes	—
32	C₂×C₃×C₃⋊C₈	C₃×C₄	1	8	yes	—
33	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	4	yes	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	Dic₃	1	2	yes	—
37	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
38	C₂×C₈×C₉	—	3	8	no	—
39	C₂×C₈×C₉	—	3	4	no	—
40	C₄²×C₉	—	2	2	no	—
41	C₄²×C₉	—	2	2	no	—
42	C₄²×C₉	Dic₃	1	2	yes	—
43	C₄²×C₉	C₂×D₃	1	2	yes	—
44	C₂²×C₃×C₉	C₄²	1	2	yes	—
45	C₂²×C₃×C₉	C₂×C₈	1	4	yes	—
46	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₄×C₉	C₄²	1	2	yes	—
48	C₃×C₄×C₉	C₂×C₈	1	8	yes	complemented
49	C₃×C₄×C₉	C₂²×C₄	1	2	yes	—
50	C₂²×C₃×C₈	—	3	2	no	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
54	C₂²×C₃⋊C₈	C₂×C₉	1	2	yes	—
55	C₃×C₄×C₈	—	3	4	no	—
56	C₄×C₃⋊C₈	C₂×C₉	1	4	yes	direct product factor
57	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₃×C₈	1	8	yes	—
59	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
60	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
62	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
63	C₂³×C₉	—	2	1	no	—
64	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
65	C₃×C₃⋊C₈	C₂×C₃×C₄	1	8	yes	—
66	C₂×C₄×C₉	—	2	8	no	—
67	C₂×C₄×C₉	—	2	4	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	C₃⋊C₈	1	8	yes	direct product factor
72	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
73	C₂×C₄×C₉	C₂×Dic₃	1	4	yes	—
74	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
76	C₈×C₉	—	3	8	no	—
77	C₂×C₄×C₈	—	3	1	no	Sylow
78	C₂×C₃×C₉	C₄×C₈	1	4	yes	complemented
79	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₃×C₉	C₂²×C₈	1	2	yes	—
81	C₂²×C₃×C₄	—	2	2	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	C₄×C₉	1	2	yes	—
84	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
86	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
87	C₂×C₃×C₈	—	3	8	no	—
88	C₂×C₃×C₈	—	3	4	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	C₄×C₉	1	2	yes	—
92	C₃×C₄²	C₂²×C₉	1	2	yes	—
93	C₃×C₄²	C₃×Dic₃	1	2	yes	—
94	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
95	C₂×C₃⋊C₈	C₄×C₉	1	8	yes	direct product factor
96	C₂×C₃⋊C₈	C₂²×C₉	1	4	yes	—
97	C₂²×C₃²	C₃×C₄²	1	2	yes	—
98	C₂²×C₃²	C₂×C₃×C₈	1	4	yes	—
99	C₂²×C₃²	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₃²×C₄	C₃×C₄²	1	2	yes	—
101	C₃²×C₄	C₂×C₃×C₈	1	8	yes	—
102	C₃²×C₄	C₂²×C₃×C₄	1	2	yes	—
103	C₂²×C₉	—	2	4	no	—
104	C₂²×C₉	—	2	2	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	C₂×C₃⋊C₈	1	4	yes	—
107	C₂²×C₉	C₄×Dic₃	1	2	yes	—
108	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
109	C₄×C₉	—	2	8	no	—
110	C₄×C₉	—	2	2	no	—
111	C₄×C₉	—	2	2	no	—
112	C₄×C₉	C₂×C₃⋊C₈	1	8	yes	direct product factor
113	C₄×C₉	C₄×Dic₃	1	2	yes	—
114	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
115	C₂²×C₈	—	3	2	no	—
116	C₂×C₄²	D₃×C₉	1	1	yes	characteristic
117	C₄×C₈	—	3	4	no	—
118	C₃×C₉	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	C₂×C₄×C₉	1	1	yes	characteristic
121	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	C₈×C₉	1	8	yes	complemented
128	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
129	C₂×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
130	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
131	C₂×C₃×C₄	C₃×C₃⋊C₈	1	8	yes	—
132	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
133	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
134	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
135	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
137	C₃×C₈	—	3	8	no	—
138	C₃⋊C₈	C₂×C₄×C₉	1	8	yes	direct product factor
139	C₂×C₃²	C₃×C₄×C₈	1	4	yes	—
140	C₂×C₃²	C₂×C₃×C₄²	1	1	yes	characteristic
141	C₂×C₃²	C₂²×C₃×C₈	1	2	yes	—
142	C₂×C₉	—	2	2	no	—
143	C₂×C₉	—	2	1	no	—
144	C₂×C₉	—	2	4	no	—
145	C₂×C₉	C₄×C₃⋊C₈	1	4	yes	direct product factor
146	C₂×C₉	C₂²×C₃⋊C₈	1	2	yes	—
147	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
148	C₂²×C₄	C₉×Dic₃	1	2	yes	—
149	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
150	C₂×C₈	—	3	8	no	—
151	C₂×C₈	—	3	4	no	—
152	C₄²	C₉×Dic₃	1	2	yes	—
153	C₄²	C₂×D₃×C₉	1	2	yes	—
154	C₂²×C₃	—	2	1	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	C₄²×C₉	1	2	yes	—
158	C₂²×C₃	C₂×C₈×C₉	1	4	yes	—
159	C₂²×C₃	C₂²×C₄×C₉	1	1	yes	characteristic
160	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	4	yes	—
161	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
162	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
163	C₃×C₄	—	2	2	no	—
164	C₃×C₄	—	2	8	no	—
165	C₃×C₄	—	2	2	no	—
166	C₃×C₄	C₄²×C₉	1	2	yes	—
167	C₃×C₄	C₂×C₈×C₉	1	8	yes	complemented
168	C₃×C₄	C₂²×C₄×C₉	1	2	yes	—
169	C₃×C₄	C₂×C₃×C₃⋊C₈	1	8	yes	—
170	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
171	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
172	C₃²	C₂×C₃×C₄×C₈	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic, direct product factor
175	C₂³	C₂×C₉×Dic₃	1	1	yes	characteristic
176	C₂×C₄	C₉×C₃⋊C₈	1	8	yes	direct product factor
177	C₂×C₄	C₄×D₃×C₉	1	4	yes	—
178	C₂×C₄	C₂×C₉×Dic₃	1	4	yes	—
179	C₂×C₄	C₂×C₉×Dic₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
181	C₈	—	3	8	no	—
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	C₄×C₈×C₉	1	4	yes	complemented
186	C₂×C₃	C₂×C₄²×C₉	1	1	yes	characteristic
187	C₂×C₃	C₂²×C₈×C₉	1	2	yes	—
188	C₂×C₃	C₃×C₄×C₃⋊C₈	1	4	yes	—
189	C₂×C₃	C₂²×C₃×C₃⋊C₈	1	2	yes	—
190	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
191	C₂²	C₂×C₉×C₃⋊C₈	1	4	yes	—
192	C₂²	C₄×C₉×Dic₃	1	2	yes	—
193	C₂²	C₂²×C₉×Dic₃	1	1	yes	characteristic
194	C₄	C₂×C₉×C₃⋊C₈	1	8	yes	direct product factor
195	C₄	C₄×C₉×Dic₃	1	2	yes	—
196	C₄	C₂×C₄×D₃×C₉	1	2	yes	—
197	C₃	—	2	1	no	—
198	C₃	C₂×C₄×C₈×C₉	1	1	yes	characteristic, complemented
199	C₃	C₂×C₃×C₄×C₃⋊C₈	1	1	yes	characteristic
200	C₂	C₄×C₉×C₃⋊C₈	1	4	yes	direct product factor
201	C₂	C₂²×C₉×C₃⋊C₈	1	2	yes	—
202	C₂	C₂×C₄×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄²×C₉	D₃
commutator, hypocenter	C₃	C₂×C₄×C₈×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂×C₃×C₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
3. C₄×C₉×C₃⋊C₈	C₂	direct (N × H)
10. C₂×C₉×C₃⋊C₈	C₄	direct (N × H)
20. C₂×C₃×C₄×C₉	C₈	semidirect
25. C₉×C₃⋊C₈	C₂×C₄	direct (N × H)
27. C₂×C₄×C₃⋊C₈	C₉	direct (N × H)
48. C₃×C₄×C₉	C₂×C₈	semidirect
56. C₄×C₃⋊C₈	C₂×C₉	direct (N × H)
71. C₂×C₄×C₉	C₃⋊C₈	direct (N × H)
78. C₂×C₃×C₉	C₄×C₈	semidirect
95. C₂×C₃⋊C₈	C₄×C₉	direct (N × H)
112. C₄×C₉	C₂×C₃⋊C₈	direct (N × H)
118. C₃×C₉	C₂×C₄×C₈	semidirect
127. C₂×C₃×C₄	C₈×C₉	semidirect
138. C₃⋊C₈	C₂×C₄×C₉	direct (N × H)
145. C₂×C₉	C₄×C₃⋊C₈	direct (N × H)
167. C₃×C₄	C₂×C₈×C₉	semidirect
174. C₉	C₂×C₄×C₃⋊C₈	direct (N × H)
176. C₂×C₄	C₉×C₃⋊C₈	direct (N × H)
185. C₂×C₃	C₄×C₈×C₉	semidirect
194. C₄	C₂×C₉×C₃⋊C₈	direct (N × H)
198. C₃	C₂×C₄×C₈×C₉	semidirect
200. C₂	C₄×C₉×C₃⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂²×C₉×C₃⋊C₈	2
C₄×C₉×C₃⋊C₈	2
C₂×C₃×C₄×C₃⋊C₈	3
C₂×C₄×C₈×C₉	3

Subgroups — C2×C4×C9×C3⋊C8 #13142371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₉×C₃⋊C₈ #13142371