Subgroups — C₂×C₄⋊(C₃⋊(C₉⋊C₈)) #13144227

Order 1728.

690 conjugacy classes of subgroups.

The subgroup lattice has 210 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×C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal
3	C₄⋊(C₃⋊(C₉⋊C₈))	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	—	3	1	no	maximal
5	C₂×C₄⋊(C₉⋊C₈)	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₄²×C₉	C₄	1	2	yes	—
10	C₃×C₄²×C₉	C₂²	1	2	yes	—
11	C₂×C₃⋊(C₉⋊C₈)	—	2	4	no	—
12	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
14	C₂²×C₃⋊(C₃⋊C₈)	—	3	2	no	—
15	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	4	no	—
16	C₂×C₄²×C₉	D₃	1	3	yes	—
17	C₂²×C₉⋊C₈	—	3	6	no	—
18	C₄⋊(C₉⋊C₈)	—	3	12	no	—
19	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	—	2	2	no	—
21	C₂×C₃×C₄×C₉	C₈	1	4	yes	complemented
22	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
25	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
26	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
27	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
28	C₃⋊(C₉⋊C₈)	—	2	4	no	—
29	C₂×C₄⋊(C₃⋊C₈)	—	9	3	no	—
30	C₂×C₄⋊(C₃⋊C₈)	—	9	1	no	—
31	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
32	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
34	C₃²×C₄²	Dic₃	1	2	yes	—
35	C₃²×C₄²	C₂×D₃	1	2	yes	—
36	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	6	4	no	—
38	C₂²×C₄×C₉	Dic₃	1	3	yes	—
39	C₂²×C₄×C₉	Dic₃	1	3	yes	—
40	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
41	C₄²×C₉	Dic₃	1	6	yes	—
42	C₄²×C₉	C₂×D₃	1	6	yes	—
43	C₂×C₉⋊C₈	—	3	12	no	—
44	C₂×C₉⋊C₈	—	6	12	no	—
45	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
46	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
47	C₂²×C₃×C₉	M₁₆	1	2	yes	—
48	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄×C₉	—	2	2	no	—
50	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
51	C₃×C₄×C₉	C₂×C₈	1	4	yes	complemented
52	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
53	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₃×C₄²	D₉	1	3	yes	—
55	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
56	C₂²×C₃⋊C₈	—	9	6	no	—
57	C₂²×C₃⋊C₈	—	9	2	no	—
58	C₄⋊(C₃⋊C₈)	—	9	12	no	—
59	C₄⋊(C₃⋊C₈)	—	9	4	no	—
60	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
61	C₂×C₃²×C₄	—	2	2	no	—
62	C₂×C₃²×C₄	C₃⋊C₈	1	4	yes	—
63	C₂×C₃²×C₄	Dic₆	1	2	yes	—
64	C₂×C₃²×C₄	D₁₂	1	2	yes	—
65	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
66	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
67	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
68	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
69	C₂³×C₉	C₂×Dic₃	1	3	yes	—
70	C₃⋊(C₃⋊C₈)	—	6	4	no	—
71	C₂×C₄×C₉	—	2	6	no	—
72	C₂×C₄×C₉	C₃⋊C₈	1	12	yes	complemented
73	C₂×C₄×C₉	Dic₆	1	6	yes	—
74	C₂×C₄×C₉	D₁₂	1	6	yes	—
75	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
76	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
77	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
78	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
79	C₉⋊C₈	—	6	12	no	—
80	C₂×C₄⋊C₈	—	27	1	no	Sylow
81	C₂×C₃×C₉	C₄⋊C₈	1	4	yes	complemented
82	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₃×C₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂²×C₃×C₄	Dic₉	1	3	yes	—
86	C₂²×C₃×C₄	Dic₉	1	3	yes	—
87	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
88	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
91	C₃×C₄²	Dic₉	1	6	yes	—
92	C₃×C₄²	C₂×D₉	1	6	yes	—
93	C₃×C₄²	C₃⋊Dic₃	1	2	yes	—
94	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
95	C₂×C₃⋊C₈	—	9	12	no	—
96	C₂×C₃⋊C₈	—	9	4	no	—
97	C₂×C₃⋊C₈	—	18	12	no	—
98	C₂×C₃⋊C₈	—	18	4	no	—
99	C₂²×C₃²	C₂×C₃⋊C₈	1	2	yes	—
100	C₂²×C₃²	#208	1	2	yes	—
101	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
102	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
103	C₃²×C₄	—	2	2	no	—
104	C₃²×C₄	C₂×C₃⋊C₈	1	4	yes	—
105	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
106	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
107	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
108	C₂²×C₉	C₂×C₃⋊C₈	1	6	yes	—
109	C₂²×C₉	#208	1	6	yes	—
110	C₂²×C₉	C₄⋊Dic₃	1	6	yes	—
111	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
112	C₄×C₉	—	2	6	no	—
113	C₄×C₉	C₂×C₃⋊C₈	1	12	yes	complemented
114	C₄×C₉	C₄⋊Dic₃	1	6	yes	—
115	C₄×C₉	C₂×Dic₆	1	3	yes	—
116	C₄×C₉	C₂×D₁₂	1	3	yes	—
117	C₂²×C₈	—	27	2	no	—
118	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
119	C₄⋊C₈	—	27	4	no	—
120	C₃×C₉	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
121	C₂³×C₃	C₂×Dic₉	1	3	yes	—
122	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
123	C₂×C₃×C₄	—	2	6	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	C₉⋊C₈	1	12	yes	complemented
126	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
127	C₂×C₃×C₄	D₃₆	1	6	yes	—
128	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
129	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
130	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
131	C₂×C₃×C₄	C₃⋊(C₃⋊C₈)	1	4	yes	—
132	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
133	C₂×C₃×C₄	#634	1	2	yes	—
134	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	2	yes	—
135	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
136	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
137	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
139	C₃⋊C₈	—	18	12	no	—
140	C₃⋊C₈	—	18	4	no	—
141	C₂×C₃²	C₄⋊(C₃⋊C₈)	1	4	yes	—
142	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
143	C₂×C₃²	C₂×#208	1	1	yes	characteristic
144	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
145	C₂×C₉	C₄⋊(C₃⋊C₈)	1	12	yes	complemented
146	C₂×C₉	C₂²×C₃⋊C₈	1	3	yes	—
147	C₂×C₉	C₂×#208	1	3	yes	—
148	C₂×C₉	C₂×C₄⋊Dic₃	1	3	yes	—
149	C₂²×C₄	C₃⋊Dic₉	1	1	yes	characteristic
150	C₂²×C₄	C₃⋊Dic₉	1	1	yes	characteristic
151	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
152	C₂×C₈	—	27	4	no	—
153	C₂×C₈	—	54	4	no	—
154	C₄²	C₃⋊Dic₉	1	2	yes	—
155	C₄²	C₂×C₃⋊D₉	1	2	yes	—
156	C₂²×C₃	C₂×C₉⋊C₈	1	6	yes	—
157	C₂²×C₃	#3667	1	6	yes	—
158	C₂²×C₃	C₄⋊Dic₉	1	6	yes	—
159	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
160	C₂²×C₃	C₂×C₃⋊(C₃⋊C₈)	1	2	yes	—
161	C₂²×C₃	#3748	1	2	yes	—
162	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
163	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	2	6	no	—
166	C₃×C₄	C₂×C₉⋊C₈	1	12	yes	complemented
167	C₃×C₄	C₄⋊Dic₉	1	6	yes	—
168	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
169	C₃×C₄	C₂×D₃₆	1	3	yes	—
170	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	4	yes	—
171	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	2	yes	—
172	C₃×C₄	C₂×#634	1	1	yes	characteristic
173	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
174	C₃²	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
175	C₉	C₂×C₄⋊(C₃⋊C₈)	1	3	yes	complemented
176	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₃⋊(C₉⋊C₈)	1	4	yes	complemented
179	C₂×C₄	#6226	1	2	yes	—
180	C₂×C₄	C₄⋊(C₃⋊D₉)	1	2	yes	—
181	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
182	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
183	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
184	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
185	C₈	—	54	4	no	—
186	C₂×C₃	C₄⋊(C₉⋊C₈)	1	12	yes	complemented
187	C₂×C₃	C₂²×C₉⋊C₈	1	3	yes	—
188	C₂×C₃	C₂×#3667	1	3	yes	—
189	C₂×C₃	C₂×C₄⋊Dic₉	1	3	yes	—
190	C₂×C₃	C₄⋊(C₃⋊(C₃⋊C₈))	1	4	yes	—
191	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
192	C₂×C₃	C₂×#3748	1	1	yes	characteristic
193	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
194	C₂²	C₂×C₃⋊(C₉⋊C₈)	1	2	yes	—
195	C₂²	#88532	1	2	yes	—
196	C₂²	C₄⋊(C₃⋊Dic₉)	1	2	yes	—
197	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	C₂×C₃⋊(C₉⋊C₈)	1	4	yes	complemented
200	C₄	C₄⋊(C₃⋊Dic₉)	1	2	yes	—
201	C₄	C₂×#6226	1	1	yes	characteristic
202	C₄	C₂×C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
203	C₃	C₂×C₄⋊(C₉⋊C₈)	1	3	yes	complemented
204	C₃	C₂×C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
205	C₂	C₄⋊(C₃⋊(C₉⋊C₈))	1	4	yes	direct product factor
206	C₂	C₂²×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
207	C₂	C₂×#88532	1	1	yes	characteristic
208	C₂	C₂×C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×C₄²	C₃⋊D₉
hypocenter	C₃×C₉	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₃⋊(C₉⋊C₈))	C₂	direct (N × H)
21. C₂×C₃×C₄×C₉	C₈	semidirect
51. C₃×C₄×C₉	C₂×C₈	semidirect
72. C₂×C₄×C₉	C₃⋊C₈	semidirect
81. C₂×C₃×C₉	C₄⋊C₈	semidirect
113. C₄×C₉	C₂×C₃⋊C₈	semidirect
120. C₃×C₉	C₂×C₄⋊C₈	semidirect
125. C₂×C₃×C₄	C₉⋊C₈	semidirect
145. C₂×C₉	C₄⋊(C₃⋊C₈)	semidirect
166. C₃×C₄	C₂×C₉⋊C₈	semidirect
175. C₉	C₂×C₄⋊(C₃⋊C₈)	semidirect
178. C₂×C₄	C₃⋊(C₉⋊C₈)	semidirect
186. C₂×C₃	C₄⋊(C₉⋊C₈)	semidirect
199. C₄	C₂×C₃⋊(C₉⋊C₈)	semidirect
203. C₃	C₂×C₄⋊(C₉⋊C₈)	semidirect
205. C₂	C₄⋊(C₃⋊(C₉⋊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×C₃⋊(C₉⋊C₈)	2
C₄⋊(C₃⋊(C₉⋊C₈))	2
C₂×C₄⋊(C₃⋊(C₃⋊C₈))	3
C₂×C₄⋊(C₉⋊C₈)	3

Subgroups — C2×C4⋊(C3⋊(C9⋊C8)) #13144227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₃⋊(C₉⋊C₈)) #13144227