Subgroups — #13144375

Order 1728.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 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	#11721815	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#11720526	C₂	1	2	yes	maximal, complemented
5	#11720525	C₂	1	2	yes	maximal, complemented
6	#10592697	—	3	1	no	maximal
7	#10588676	—	3	3	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
11	C₃×C₄²×C₉	—	2	1	no	—
12	#88551	—	2	1	no	—
13	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
14	C₄×C₃⋊Dic₉	—	2	1	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₄	1	4	yes	complemented
16	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃²×#117	D₃	1	1	yes	characteristic
19	#64135	—	3	1	no	—
20	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
21	#63635	—	3	2	no	—
22	#63634	—	3	2	no	—
23	C₉×#117	D₃	1	3	yes	—
24	#63488	—	3	3	no	—
25	C₂²×C₉⋊C₈	—	3	3	no	—
26	#63367	—	3	6	no	—
27	#63366	—	3	6	no	—
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
33	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
34	C₂×C₃⋊Dic₉	—	2	2	no	—
35	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
36	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
37	#4998	—	9	3	no	—
38	#4998	—	9	1	no	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₃²×C₄⋊C₄	C₂×D₃	1	2	yes	—
41	C₃²×C₂²⋊C₄	—	2	1	no	—
42	C₃²×C₄²	—	2	1	no	—
43	#3757	—	6	1	no	—
44	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
45	C₄×C₃⋊Dic₃	—	6	1	no	—
46	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
47	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
48	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
49	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
50	C₉×C₄⋊C₄	C₂×D₃	1	6	yes	—
51	C₉×C₂²⋊C₄	—	2	3	no	—
52	C₄²×C₉	—	2	3	no	—
53	#3676	—	6	3	no	—
54	C₄⋊Dic₉	—	3	6	no	—
55	C₄×Dic₉	—	6	3	no	—
56	C₂×C₉⋊C₈	—	3	12	no	—
57	C₂×C₉⋊C₈	—	3	3	no	—
58	C₂×C₉⋊C₈	—	3	3	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
62	C₃×C₄×C₉	—	4	2	no	—
63	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
64	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₃⋊Dic₉	—	4	2	no	—
67	C₃×#117	D₉	1	3	yes	—
68	C₃×#117	C₃⋊D₃	1	1	yes	characteristic
69	#926	—	9	3	no	—
70	#926	—	9	1	no	—
71	C₂²×C₃⋊C₈	—	9	3	no	—
72	C₂²×C₃⋊C₈	—	9	1	no	—
73	#808	—	9	6	no	—
74	#808	—	9	2	no	—
75	#807	—	9	6	no	—
76	#807	—	9	2	no	—
77	C₂³×C₃²	#67	1	1	yes	characteristic
78	C₂×C₃²×C₄	—	2	2	no	—
79	C₂×C₃²×C₄	Dic₆	1	2	yes	—
80	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
81	C₂×C₃²×C₄	#67	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂×C₃⋊Dic₃	—	6	2	no	—
84	C₂³×C₉	#67	1	3	yes	—
85	C₃⋊(C₃⋊C₈)	—	3	2	no	—
86	C₃⋊(C₃⋊C₈)	—	3	2	no	—
87	C₂×C₄×C₉	—	2	6	no	—
88	C₂×C₄×C₉	Dic₆	1	6	yes	—
89	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
90	C₂×C₄×C₉	#67	1	3	yes	—
91	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
92	C₂×Dic₉	—	6	6	no	—
93	C₉⋊C₈	—	3	6	no	—
94	C₉⋊C₈	—	3	6	no	—
95	#427	—	27	1	no	Sylow
96	C₂×C₃×C₉	—	2	2	no	—
97	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₃×C₉	#135	1	2	yes	—
99	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
100	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
101	C₃×C₄⋊C₄	C₂×D₉	1	6	yes	—
102	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	2	yes	—
103	C₃×C₂²⋊C₄	—	2	3	no	—
104	C₃×C₂²⋊C₄	—	2	1	no	—
105	C₃×C₄²	—	2	3	no	—
106	C₃×C₄²	—	2	1	no	—
107	#217	—	18	3	no	—
108	#217	—	18	1	no	—
109	C₄⋊Dic₃	—	9	6	no	—
110	C₄⋊Dic₃	—	9	2	no	—
111	C₄×Dic₃	—	18	3	no	—
112	C₄×Dic₃	—	18	1	no	—
113	C₂×C₃⋊C₈	—	9	12	no	—
114	C₂×C₃⋊C₈	—	9	3	no	—
115	C₂×C₃⋊C₈	—	9	3	no	—
116	C₂×C₃⋊C₈	—	9	4	no	—
117	C₂×C₃⋊C₈	—	9	1	no	—
118	C₂×C₃⋊C₈	—	9	1	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
121	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
122	C₃²×C₄	—	4	2	no	—
123	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
124	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
125	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
126	C₃⋊Dic₃	—	12	2	no	—
127	C₂²×C₉	—	2	6	no	—
128	C₂²×C₉	Dic₃⋊C₄	1	6	yes	—
129	C₂²×C₉	C₂×#67	1	3	yes	—
130	C₄×C₉	—	4	6	no	—
131	C₄×C₉	Dic₃⋊C₄	1	6	yes	—
132	C₄×C₉	C₂×Dic₆	1	3	yes	—
133	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
134	Dic₉	—	12	6	no	—
135	C₂²×C₈	—	27	1	no	—
136	#117	—	27	1	no	—
137	#117	C₃⋊D₉	1	1	yes	characteristic
138	#107	—	27	2	no	—
139	#106	—	27	2	no	—
140	C₃×C₉	#427	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₃	#611	1	3	yes	—
142	C₂³×C₃	#638	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	6	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
146	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
147	C₂×C₃×C₄	#611	1	3	yes	—
148	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
149	C₂×C₃×C₄	#634	1	2	yes	—
150	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
151	C₂×C₃×C₄	#638	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
153	C₂×Dic₃	—	18	6	no	—
154	C₂×Dic₃	—	18	2	no	—
155	C₃⋊C₈	—	9	6	no	—
156	C₃⋊C₈	—	9	6	no	—
157	C₃⋊C₈	—	9	2	no	—
158	C₃⋊C₈	—	9	2	no	—
159	C₂×C₃²	—	2	2	no	—
160	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
161	C₂×C₃²	#950	1	2	yes	—
162	C₂×C₉	—	2	6	no	—
163	C₂×C₉	C₂×Dic₃⋊C₄	1	3	yes	—
164	C₂×C₉	#950	1	6	yes	—
165	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
166	C₂×C₈	—	27	4	no	—
167	C₂×C₈	—	27	1	no	—
168	C₂×C₈	—	27	1	no	—
169	C₄⋊C₄	—	27	2	no	—
170	C₄⋊C₄	C₂×C₃⋊D₉	1	2	yes	—
171	C₂²⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	—	54	1	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	54	1	no	—
175	C₂²×C₃	—	2	6	no	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	Dic₉⋊C₄	1	6	yes	—
178	C₂²×C₃	C₂×#611	1	3	yes	—
179	C₂²×C₃	#3750	1	2	yes	—
180	C₂²×C₃	C₂×#638	1	1	yes	characteristic
181	C₃×C₄	—	4	6	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	Dic₉⋊C₄	1	6	yes	—
184	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
185	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
186	C₃×C₄	#3750	1	2	yes	—
187	C₃×C₄	C₂×#634	1	1	yes	characteristic
188	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
189	Dic₃	—	36	6	no	—
190	Dic₃	—	36	2	no	—
191	C₃²	#4998	1	1	yes	characteristic
192	C₉	#4998	1	3	yes	complemented
193	C₂³	#6233	1	1	yes	characteristic
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	54	2	no	—
196	C₂×C₄	#6226	1	2	yes	—
197	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
198	C₂×C₄	#6233	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
200	C₈	—	27	2	no	—
201	C₈	—	27	2	no	—
202	C₂×C₃	—	2	6	no	—
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	C₂×Dic₉⋊C₄	1	3	yes	—
205	C₂×C₃	#63513	1	6	yes	—
206	C₂×C₃	C₂×#3750	1	1	yes	characteristic
207	C₂×C₃	#64159	1	2	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	#88534	1	2	yes	—
210	C₂²	C₂×#6233	1	1	yes	characteristic
211	C₄	—	4	2	no	—
212	C₄	—	108	2	no	—
213	C₄	#88534	1	2	yes	—
214	C₄	C₂×#6226	1	1	yes	characteristic
215	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
216	C₃	#10588676	1	3	yes	complemented
217	C₃	#10592697	1	1	yes	characteristic
218	C₂	—	2	2	no	—
219	C₂	C₂×#88534	1	1	yes	characteristic
220	C₂	#11721868	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#6233
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂²×C₃²	C₂×#67
prime radical	C₂³×C₃²	#67
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#117	C₃⋊D₉
hypocenter	C₃×C₉	#427
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
4. #11720526	C₂	semidirect
5. #11720525	C₂	semidirect
15. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
140. C₃×C₉	#427	semidirect
192. C₉	#4998	semidirect
216. C₃	#10588676	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
#11721815	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720526	2
#11720525	2
#10592697	3
#10588676	3