Subgroups — #13144376

Order 1728.

590 conjugacy classes of subgroups.

The subgroup lattice has 246 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×C₄⋊(C₃⋊Dic₉)	C₂	1	1	yes	maximal, characteristic
3	C₂×#88532	C₂	1	1	yes	maximal, characteristic
4	#11720526	C₂	1	2	yes	maximal, complemented
5	#11720525	C₂	1	2	yes	maximal, complemented
6	#10592694	—	3	1	no	maximal
7	#10588679	—	3	3	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	—	2	1	no	—
10	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
13	C₃×C₄²×C₉	—	2	1	no	—
14	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
15	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
16	#88532	C₄	1	4	yes	complemented
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
18	C₃²×#117	D₃	1	1	yes	characteristic
19	C₂×C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
20	C₂×#3748	—	3	1	no	—
21	#63635	—	3	2	no	—
22	#63634	—	3	2	no	—
23	C₉×#117	D₃	1	3	yes	—
24	C₂×C₄⋊Dic₉	—	3	3	no	—
25	C₂×#3667	—	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	1	no	—
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊Dic₉	—	2	2	no	—
36	C₂×C₃⋊Dic₉	—	4	2	no	—
37	C₃⋊(C₉⋊C₈)	C₂×C₄	1	4	yes	complemented
38	#4995	—	9	3	no	—
39	#4995	—	9	1	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃⋊Dic₃	—	6	1	no	—
42	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
43	C₃²×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
44	C₃²×C₂²⋊C₄	—	2	1	no	—
45	C₃²×C₄²	—	2	1	no	—
46	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
47	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
48	#3748	—	3	4	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
50	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
51	C₂²×Dic₉	—	6	3	no	—
52	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
53	C₉×C₄⋊C₄	C₂×D₃	1	3	yes	—
54	C₉×C₂²⋊C₄	—	2	3	no	—
55	C₄²×C₉	—	2	3	no	—
56	C₄⋊Dic₉	—	3	6	no	—
57	C₄⋊Dic₉	—	6	3	no	—
58	#3667	—	3	12	no	—
59	C₂×C₉⋊C₈	—	3	6	no	—
60	C₂²×C₃×C₉	—	2	1	no	—
61	C₂²×C₃×C₉	—	2	1	no	—
62	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₃×C₄×C₉	—	4	1	no	—
65	C₃×C₄×C₉	—	4	1	no	—
66	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
67	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
69	C₃⋊Dic₉	—	4	2	no	—
70	C₃×#117	D₉	1	3	yes	—
71	C₃×#117	C₃⋊D₃	1	1	yes	characteristic
72	C₂×C₄⋊Dic₃	—	9	3	no	—
73	C₂×C₄⋊Dic₃	—	9	1	no	—
74	C₂×#208	—	9	3	no	—
75	C₂×#208	—	9	1	no	—
76	#808	—	9	6	no	—
77	#808	—	9	2	no	—
78	#807	—	9	6	no	—
79	#807	—	9	2	no	—
80	C₂³×C₃²	#67	1	1	yes	characteristic
81	C₂×C₃²×C₄	—	2	1	no	—
82	C₂×C₃²×C₄	—	2	1	no	—
83	C₂×C₃²×C₄	Dic₆	1	2	yes	—
84	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
85	C₂×C₃²×C₄	#67	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	6	2	no	—
88	C₂×C₃⋊Dic₃	—	12	2	no	—
89	C₂³×C₉	#67	1	3	yes	—
90	C₃⋊(C₃⋊C₈)	—	3	4	no	—
91	C₂×C₄×C₉	—	2	3	no	—
92	C₂×C₄×C₉	—	2	3	no	—
93	C₂×C₄×C₉	Dic₆	1	6	yes	—
94	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
95	C₂×C₄×C₉	#67	1	3	yes	—
96	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
97	C₂×Dic₉	—	6	6	no	—
98	C₂×Dic₉	—	12	6	no	—
99	C₉⋊C₈	—	3	12	no	—
100	#428	—	27	1	no	Sylow
101	C₂×C₃×C₉	—	2	2	no	—
102	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₃×C₉	#137	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
106	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
107	C₂²×Dic₃	—	18	3	no	—
108	C₂²×Dic₃	—	18	1	no	—
109	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
110	C₃×C₄⋊C₄	C₂×D₉	1	3	yes	—
111	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
112	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
113	C₃×C₂²⋊C₄	—	2	3	no	—
114	C₃×C₂²⋊C₄	—	2	1	no	—
115	C₃×C₄²	—	2	3	no	—
116	C₃×C₄²	—	2	1	no	—
117	C₄⋊Dic₃	—	9	6	no	—
118	C₄⋊Dic₃	—	9	2	no	—
119	C₄⋊Dic₃	—	18	3	no	—
120	C₄⋊Dic₃	—	18	1	no	—
121	#208	—	9	12	no	—
122	#208	—	9	4	no	—
123	C₂×C₃⋊C₈	—	9	6	no	—
124	C₂×C₃⋊C₈	—	9	2	no	—
125	C₂²×C₃²	—	2	1	no	—
126	C₂²×C₃²	—	2	1	no	—
127	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
128	C₂²×C₃²	C₂×#67	1	1	yes	characteristic
129	C₃²×C₄	—	4	1	no	—
130	C₃²×C₄	—	4	1	no	—
131	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
132	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
133	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
134	C₃⋊Dic₃	—	12	2	no	—
135	C₂²×C₉	—	2	3	no	—
136	C₂²×C₉	—	2	3	no	—
137	C₂²×C₉	Dic₃⋊C₄	1	6	yes	—
138	C₂²×C₉	C₂×#67	1	3	yes	—
139	C₄×C₉	—	4	3	no	—
140	C₄×C₉	—	4	3	no	—
141	C₄×C₉	Dic₃⋊C₄	1	6	yes	—
142	C₄×C₉	C₂×Dic₆	1	3	yes	—
143	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
144	Dic₉	—	12	6	no	—
145	C₂×M₁₆	—	27	1	no	—
146	#117	C₃⋊D₉	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	27	1	no	—
148	#107	—	27	2	no	—
149	#106	—	27	2	no	—
150	C₃×C₉	#428	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₃	#611	1	3	yes	—
152	C₂³×C₃	#638	1	1	yes	characteristic
153	C₂×C₃×C₄	—	2	3	no	—
154	C₂×C₃×C₄	—	2	3	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	Dic₁₈	1	6	yes	—
158	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
159	C₂×C₃×C₄	#611	1	3	yes	—
160	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
161	C₂×C₃×C₄	#634	1	2	yes	—
162	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
163	C₂×C₃×C₄	#638	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
165	C₂×Dic₃	—	18	6	no	—
166	C₂×Dic₃	—	18	2	no	—
167	C₂×Dic₃	—	36	2	no	—
168	C₂×Dic₃	—	36	6	no	—
169	C₃⋊C₈	—	9	12	no	—
170	C₃⋊C₈	—	9	4	no	—
171	C₂×C₃²	—	2	2	no	—
172	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
173	C₂×C₃²	#949	1	1	yes	characteristic
174	C₂×C₃²	#951	1	1	yes	characteristic
175	C₂×C₉	—	2	6	no	—
176	C₂×C₉	C₂×Dic₃⋊C₄	1	3	yes	—
177	C₂×C₉	#949	1	3	yes	—
178	C₂×C₉	#951	1	3	yes	—
179	C₂²×C₄	—	54	1	no	—
180	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
181	M₁₆	—	27	4	no	—
182	C₂×C₈	—	27	2	no	—
183	C₄⋊C₄	—	27	2	no	—
184	C₄⋊C₄	—	54	1	no	—
185	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
186	C₄⋊C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
187	C₂²⋊C₄	—	2	1	no	—
188	C₄²	—	2	1	no	—
189	C₂²×C₃	—	2	3	no	—
190	C₂²×C₃	—	2	3	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	Dic₉⋊C₄	1	6	yes	—
194	C₂²×C₃	C₂×#611	1	3	yes	—
195	C₂²×C₃	#3750	1	2	yes	—
196	C₂²×C₃	C₂×#638	1	1	yes	characteristic
197	C₃×C₄	—	4	3	no	—
198	C₃×C₄	—	4	3	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	4	1	no	—
201	C₃×C₄	Dic₉⋊C₄	1	6	yes	—
202	C₃×C₄	C₂×Dic₁₈	1	3	yes	—
203	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
204	C₃×C₄	#3750	1	2	yes	—
205	C₃×C₄	C₂×#634	1	1	yes	characteristic
206	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
207	Dic₃	—	36	6	no	—
208	Dic₃	—	36	2	no	—
209	C₃²	#4995	1	1	yes	characteristic
210	C₉	#4995	1	3	yes	complemented
211	C₂³	#6233	1	1	yes	characteristic
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	54	2	no	—
215	C₂×C₄	—	108	2	no	—
216	C₂×C₄	#6226	1	2	yes	—
217	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
218	C₂×C₄	#6233	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
220	C₈	—	27	4	no	—
221	C₂×C₃	—	2	6	no	—
222	C₂×C₃	—	2	2	no	—
223	C₂×C₃	C₂×Dic₉⋊C₄	1	3	yes	—
224	C₂×C₃	#63511	1	3	yes	—
225	C₂×C₃	#63512	1	3	yes	—
226	C₂×C₃	C₂×#3750	1	1	yes	characteristic
227	C₂×C₃	#64158	1	1	yes	characteristic
228	C₂×C₃	#64160	1	1	yes	characteristic
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#88534	1	2	yes	—
232	C₂²	C₂×#6233	1	1	yes	characteristic
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	108	2	no	—
236	C₄	#88534	1	2	yes	—
237	C₄	C₂×#6226	1	1	yes	characteristic
238	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
239	C₃	#10588679	1	3	yes	complemented
240	C₃	#10592694	1	1	yes	characteristic
241	C₂	—	2	2	no	—
242	C₂	C₂×#88534	1	1	yes	characteristic
243	C₂	#11721866	1	1	yes	characteristic
244	C₂	#11721867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#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	#428
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ 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
16. #88532	C₄	semidirect
37. C₃⋊(C₉⋊C₈)	C₂×C₄	semidirect
150. C₃×C₉	#428	semidirect
210. C₉	#4995	semidirect
239. C₃	#10588679	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×C₄⋊(C₃⋊Dic₉)	2
C₂×#88532	2
#11720526	2
#11720525	2
#10592694	3
#10588679	3