Subgroups — #13144374

Order 1728.

650 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	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11720516	C₂	1	2	yes	maximal, complemented
5	C₄×C₃⋊(C₉⋊C₈)	C₂	1	2	yes	maximal, complemented
6	#10592696	—	3	1	no	maximal
7	#10588675	—	3	3	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
11	C₃×C₄²×C₉	C₂²	1	2	yes	—
12	#88532	C₄	1	4	yes	complemented
13	C₂×C₃⋊(C₉⋊C₈)	C₄	1	4	yes	complemented
14	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₃²×#117	D₃	1	1	yes	characteristic
18	C₂×#3748	—	3	1	no	—
19	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
20	#63630	—	3	2	no	—
21	C₄×C₃⋊(C₃⋊C₈)	—	3	2	no	—
22	C₉×#117	D₃	1	3	yes	—
23	C₂×#3667	—	3	3	no	—
24	C₂²×C₉⋊C₈	—	3	3	no	—
25	#63362	—	3	6	no	—
26	C₄×C₉⋊C₈	—	3	6	no	—
27	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
34	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
35	C₃⋊(C₉⋊C₈)	C₂×C₄	1	4	yes	—
36	#4997	—	9	3	no	—
37	#4997	—	9	1	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	C₃²×C₄⋊C₄	Dic₃	1	2	yes	—
40	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
41	C₃²×C₄²	C₂×D₃	1	2	yes	—
42	#3748	—	3	4	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
44	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
45	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
46	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
47	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
48	C₉×C₄⋊C₄	Dic₃	1	6	yes	—
49	C₉×C₂²⋊C₄	Dic₃	1	6	yes	—
50	C₄²×C₉	C₂×D₃	1	6	yes	—
51	#3667	—	3	12	no	—
52	C₂×C₉⋊C₈	—	3	12	no	—
53	C₂×C₉⋊C₈	—	3	3	no	—
54	C₂×C₉⋊C₈	—	3	6	no	—
55	C₂×C₉⋊C₈	—	3	3	no	—
56	C₂²×C₃×C₉	—	2	2	no	—
57	C₂²×C₃×C₉	C₄²	1	2	yes	—
58	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	—	2	4	no	—
60	C₃×C₄×C₉	C₄²	1	2	yes	—
61	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
63	C₃×#117	D₉	1	3	yes	—
64	C₃×#117	C₃⋊D₃	1	1	yes	characteristic
65	C₂×#208	—	9	3	no	—
66	C₂×#208	—	9	1	no	—
67	C₂²×C₃⋊C₈	—	9	3	no	—
68	C₂²×C₃⋊C₈	—	9	1	no	—
69	#803	—	9	6	no	—
70	#803	—	9	2	no	—
71	C₄×C₃⋊C₈	—	9	6	no	—
72	C₄×C₃⋊C₈	—	9	2	no	—
73	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
75	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
76	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
77	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
79	C₂³×C₉	C₂×Dic₃	1	3	yes	—
80	C₃⋊(C₃⋊C₈)	—	3	2	no	—
81	C₃⋊(C₃⋊C₈)	—	3	2	no	—
82	C₃⋊(C₃⋊C₈)	—	3	4	no	—
83	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
84	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
85	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
87	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
88	C₉⋊C₈	—	3	6	no	—
89	C₉⋊C₈	—	3	6	no	—
90	C₉⋊C₈	—	3	12	no	—
91	#405	—	27	1	no	Sylow
92	C₂×C₃×C₉	—	2	2	no	—
93	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₃×C₉	#131	1	2	yes	—
95	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
96	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
97	C₃×C₄⋊C₄	Dic₉	1	6	yes	—
98	C₃×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
99	C₃×C₂²⋊C₄	Dic₉	1	6	yes	—
100	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	2	yes	—
101	C₃×C₄²	C₂×D₉	1	6	yes	—
102	C₃×C₄²	C₂×C₃⋊D₃	1	2	yes	—
103	#208	—	9	12	no	—
104	#208	—	9	4	no	—
105	C₂×C₃⋊C₈	—	9	12	no	—
106	C₂×C₃⋊C₈	—	9	3	no	—
107	C₂×C₃⋊C₈	—	9	6	no	—
108	C₂×C₃⋊C₈	—	9	3	no	—
109	C₂×C₃⋊C₈	—	9	4	no	—
110	C₂×C₃⋊C₈	—	9	1	no	—
111	C₂×C₃⋊C₈	—	9	2	no	—
112	C₂×C₃⋊C₈	—	9	1	no	—
113	C₂²×C₃²	—	2	2	no	—
114	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
115	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
116	C₃²×C₄	—	2	4	no	—
117	C₃²×C₄	C₄×Dic₃	1	2	yes	—
118	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
119	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
120	C₂²×C₉	—	2	6	no	—
121	C₂²×C₉	C₄×Dic₃	1	6	yes	—
122	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
123	C₄×C₉	—	2	12	no	—
124	C₄×C₉	C₄×Dic₃	1	6	yes	—
125	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
126	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
127	C₂×M₁₆	—	27	1	no	—
128	C₂²×C₈	—	27	1	no	—
129	#117	C₃⋊D₉	1	1	yes	characteristic
130	#97	—	27	2	no	—
131	C₄×C₈	—	27	2	no	—
132	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₃	C₂×Dic₉	1	3	yes	—
134	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
135	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
136	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
137	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
138	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
139	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
140	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
141	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
143	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
145	C₃⋊C₈	—	9	6	no	—
146	C₃⋊C₈	—	9	6	no	—
147	C₃⋊C₈	—	9	12	no	—
148	C₃⋊C₈	—	9	2	no	—
149	C₃⋊C₈	—	9	2	no	—
150	C₃⋊C₈	—	9	4	no	—
151	C₂×C₃²	—	2	2	no	—
152	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
153	C₂×C₃²	#948	1	2	yes	—
154	C₂×C₉	—	2	6	no	—
155	C₂×C₉	C₂×C₄×Dic₃	1	3	yes	—
156	C₂×C₉	#948	1	6	yes	—
157	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
158	M₁₆	—	27	4	no	—
159	C₂×C₈	—	27	4	no	—
160	C₂×C₈	—	27	1	no	—
161	C₂×C₈	—	27	2	no	—
162	C₂×C₈	—	27	1	no	—
163	C₄⋊C₄	C₃⋊Dic₉	1	2	yes	—
164	C₂²⋊C₄	C₃⋊Dic₉	1	2	yes	—
165	C₄²	C₂×C₃⋊D₉	1	2	yes	—
166	C₂²×C₃	—	2	6	no	—
167	C₂²×C₃	—	2	2	no	—
168	C₂²×C₃	C₄×Dic₉	1	6	yes	—
169	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
170	C₂²×C₃	C₄×C₃⋊Dic₃	1	2	yes	—
171	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
172	C₃×C₄	—	2	12	no	—
173	C₃×C₄	—	2	4	no	—
174	C₃×C₄	C₄×Dic₉	1	6	yes	—
175	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
176	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
177	C₃×C₄	C₄×C₃⋊Dic₃	1	2	yes	—
178	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
179	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
180	C₃²	#4997	1	1	yes	characteristic
181	C₉	#4997	1	3	yes	complemented
182	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
183	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
184	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
185	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
186	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
187	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
188	C₈	—	27	2	no	—
189	C₈	—	27	2	no	—
190	C₈	—	27	4	no	—
191	C₂×C₃	—	2	6	no	—
192	C₂×C₃	—	2	2	no	—
193	C₂×C₃	C₂×C₄×Dic₉	1	3	yes	—
194	C₂×C₃	#63510	1	6	yes	—
195	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
196	C₂×C₃	#64157	1	2	yes	—
197	C₂²	—	2	2	no	—
198	C₂²	C₄×C₃⋊Dic₉	1	2	yes	—
199	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	C₄×C₃⋊Dic₉	1	2	yes	—
202	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
203	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
204	C₃	#10588675	1	3	yes	complemented
205	C₃	#10592696	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×C₄×C₃⋊Dic₉	1	1	yes	characteristic
208	C₂	#11721865	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#117	C₃⋊D₉
hypocenter	C₃×C₉	#405
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #11720516	C₂	semidirect
5. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
12. #88532	C₄	semidirect
13. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
132. C₃×C₉	#405	semidirect
181. C₉	#4997	semidirect
204. C₃	#10588675	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
C₂×#88532	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720516	2
C₄×C₃⋊(C₉⋊C₈)	2
#10592696	3
#10588675	3