Subgroups — #13135937

Order 1728.

450 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#11721815	C₂	1	1	yes	maximal, characteristic
3	C₂×#88532	C₂	1	1	yes	maximal, characteristic
4	#10589425	—	3	1	no	maximal
5	#10588232	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₉×M₁₆	—	2	1	no	—
8	C₃×C₉×M₁₆	C₄	1	2	yes	complemented
9	C₂×C₃×C₈×C₉	—	2	1	no	—
10	#88551	C₄	1	2	yes	—
11	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
12	C₄×C₃⋊Dic₉	—	2	1	no	—
13	#88532	—	2	1	no	—
14	#88532	—	2	1	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₄	1	2	yes	complemented
16	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
17	#64135	—	3	1	no	—
18	C₂×#3748	—	3	1	no	—
19	C₂×C₉×M₁₆	D₃	1	3	yes	—
20	#63488	—	3	3	no	—
21	C₂×#3667	—	3	3	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
26	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
29	C₂×C₃⋊Dic₉	—	2	2	no	—
30	C₃×C₈×C₉	—	2	2	no	—
31	C₃⋊(C₉⋊C₈)	—	2	2	no	—
32	#4559	—	9	3	no	—
33	#4559	—	9	1	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₃²×M₁₆	—	2	1	no	—
36	C₃²×M₁₆	Dic₃	1	2	yes	—
37	C₂×C₃²×C₈	—	2	1	no	—
38	#3757	—	3	2	no	—
39	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
40	C₄×C₃⋊Dic₃	—	6	1	no	—
41	#3748	—	6	1	no	—
42	#3748	—	6	1	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
45	C₉×M₁₆	—	2	3	no	—
46	C₉×M₁₆	Dic₃	1	6	yes	complemented
47	C₂×C₈×C₉	—	2	3	no	—
48	#3676	—	3	6	no	—
49	C₄⋊Dic₉	—	6	3	no	—
50	C₄×Dic₉	—	6	3	no	—
51	#3667	—	6	3	no	—
52	#3667	—	6	3	no	—
53	C₂×C₉⋊C₈	—	3	6	no	—
54	C₂²×C₃×C₉	—	4	1	no	—
55	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
56	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
58	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
61	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
62	C₃⋊Dic₉	—	4	2	no	—
63	C₂×C₃×M₁₆	D₉	1	3	yes	—
64	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
65	#926	—	9	3	no	—
66	#926	—	9	1	no	—
67	C₂×#208	—	9	3	no	—
68	C₂×#208	—	9	1	no	—
69	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
71	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	#67	1	1	yes	characteristic
75	C₂×C₃²×C₄	#67	1	1	yes	characteristic
76	C₂×C₃⋊Dic₃	—	6	2	no	—
77	C₂³×C₉	C₄×D₃	1	3	yes	—
78	C₃²×C₈	—	2	2	no	—
79	C₃⋊(C₃⋊C₈)	—	6	2	no	—
80	C₂×C₄×C₉	Dic₆	1	3	yes	—
81	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
82	C₂×C₄×C₉	D₁₂	1	3	yes	—
83	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
84	C₂×C₄×C₉	#67	1	3	yes	—
85	C₂×C₄×C₉	#67	1	3	yes	—
86	C₂×Dic₉	—	6	6	no	—
87	C₈×C₉	—	2	6	no	—
88	C₉⋊C₈	—	6	6	no	—
89	#344	—	27	1	no	Sylow
90	C₂×C₃×C₉	—	2	1	no	—
91	C₂×C₃×C₉	—	2	1	no	—
92	C₂×C₃×C₉	—	2	1	no	—
93	C₂×C₃×C₉	#95	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
95	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
96	C₃×M₁₆	—	2	3	no	—
97	C₃×M₁₆	—	2	1	no	—
98	C₃×M₁₆	Dic₉	1	6	yes	complemented
99	C₃×M₁₆	C₃⋊Dic₃	1	2	yes	—
100	C₂×C₃×C₈	—	2	3	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	#217	—	9	6	no	—
103	#217	—	9	2	no	—
104	C₄⋊Dic₃	—	18	3	no	—
105	C₄⋊Dic₃	—	18	1	no	—
106	C₄×Dic₃	—	18	3	no	—
107	C₄×Dic₃	—	18	1	no	—
108	#208	—	18	3	no	—
109	#208	—	18	3	no	—
110	#208	—	18	1	no	—
111	#208	—	18	1	no	—
112	C₂×C₃⋊C₈	—	9	6	no	—
113	C₂×C₃⋊C₈	—	9	2	no	—
114	C₂²×C₃²	—	4	1	no	—
115	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
116	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
117	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
118	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
119	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
120	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
121	C₃²×C₄	#217	1	1	yes	characteristic
122	C₃⋊Dic₃	—	12	2	no	—
123	C₂²×C₉	—	4	3	no	—
124	C₂²×C₉	C₄×Dic₃	1	3	yes	—
125	C₂²×C₉	Dic₃⋊C₄	1	3	yes	—
126	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
127	C₄×C₉	Dic₃⋊C₄	1	3	yes	—
128	C₄×C₉	C₄⋊Dic₃	1	3	yes	—
129	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
130	C₄×C₉	#217	1	3	yes	—
131	Dic₉	—	12	6	no	—
132	C₂×M₁₆	—	27	1	no	—
133	C₂×M₁₆	C₃⋊D₉	1	1	yes	characteristic
134	#117	—	27	1	no	—
135	C₃×C₉	#344	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₃	C₄×D₉	1	3	yes	—
137	C₂³×C₃	C₄×C₃⋊D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	Dic₁₈	1	3	yes	—
139	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
140	C₂×C₃×C₄	D₃₆	1	3	yes	—
141	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
142	C₂×C₃×C₄	#611	1	3	yes	—
143	C₂×C₃×C₄	#611	1	3	yes	—
144	C₂×C₃×C₄	#634	1	1	yes	characteristic
145	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
146	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
147	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
148	C₂×C₃×C₄	#638	1	1	yes	characteristic
149	C₂×C₃×C₄	#638	1	1	yes	characteristic
150	C₂×Dic₃	—	18	6	no	—
151	C₂×Dic₃	—	18	2	no	—
152	C₃×C₈	—	2	6	no	—
153	C₃×C₈	—	2	2	no	—
154	C₃⋊C₈	—	18	6	no	—
155	C₃⋊C₈	—	18	2	no	—
156	C₂×C₃²	—	2	1	no	—
157	C₂×C₃²	—	2	1	no	—
158	C₂×C₃²	—	2	1	no	—
159	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
160	C₂×C₉	—	2	3	no	—
161	C₂×C₉	—	2	3	no	—
162	C₂×C₉	—	2	3	no	—
163	C₂×C₉	C₃⋊#95	1	3	yes	—
164	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
165	M₁₆	—	2	1	no	—
166	M₁₆	—	54	1	no	—
167	M₁₆	—	54	1	no	—
168	M₁₆	C₃⋊Dic₉	1	2	yes	complemented
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	27	2	no	—
171	C₄⋊C₄	—	54	1	no	—
172	C₂²⋊C₄	—	27	2	no	—
173	C₄²	—	54	1	no	—
174	C₂²×C₃	—	4	3	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	C₄×Dic₉	1	3	yes	—
177	C₂²×C₃	Dic₉⋊C₄	1	3	yes	—
178	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
179	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
180	C₂²×C₃	#3750	1	1	yes	characteristic
181	C₂²×C₃	#3752	1	1	yes	characteristic
182	C₃×C₄	Dic₉⋊C₄	1	3	yes	—
183	C₃×C₄	C₄⋊Dic₉	1	3	yes	—
184	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
185	C₃×C₄	#3676	1	3	yes	—
186	C₃×C₄	#3750	1	1	yes	characteristic
187	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
188	C₃×C₄	#3752	1	1	yes	characteristic
189	C₃×C₄	#3757	1	1	yes	characteristic
190	Dic₃	—	36	2	no	—
191	Dic₃	—	36	6	no	—
192	C₃²	#4559	1	1	yes	characteristic
193	C₉	#4559	1	3	yes	complemented
194	C₂³	C₄×C₃⋊D₉	1	1	yes	characteristic
195	C₂×C₄	—	54	2	no	—
196	C₂×C₄	#6226	1	1	yes	characteristic
197	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
198	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
200	C₂×C₄	#6233	1	1	yes	characteristic
201	C₂×C₄	#6233	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	54	2	no	—
204	C₂×C₃	—	2	3	no	—
205	C₂×C₃	—	2	3	no	—
206	C₂×C₃	—	2	3	no	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	C₉⋊#95	1	3	yes	—
211	C₂×C₃	#63658	1	1	yes	characteristic
212	C₂²	—	4	1	no	—
213	C₂²	C₄×C₃⋊Dic₉	1	1	yes	characteristic
214	C₂²	#88534	1	1	yes	characteristic
215	C₂²	#88536	1	1	yes	characteristic
216	C₄	—	108	2	no	—
217	C₄	#88534	1	1	yes	characteristic
218	C₄	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
219	C₄	#88536	1	1	yes	characteristic
220	C₄	#88551	1	1	yes	characteristic
221	C₃	#10588232	1	3	yes	complemented
222	C₃	#10589425	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	#11720572	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#88536
commutator	C₂²×C₃×C₉	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₉×M₁₆	C₂
socle	C₂×C₃²	C₃⋊#95
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂²×C₃×C₄×C₉	C₂²
hypercenter	C₂×M₁₆	C₃⋊D₉
hypocenter	C₃×C₉	#344
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
8. C₃×C₉×M₁₆	C₄	semidirect
15. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
46. C₉×M₁₆	Dic₃	semidirect
98. C₃×M₁₆	Dic₉	semidirect
135. C₃×C₉	#344	semidirect
168. M₁₆	C₃⋊Dic₉	semidirect
193. C₉	#4559	semidirect
221. C₃	#10588232	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×M₁₆	2
#11721815	2
C₂×#88532	2
#10589425	3
#10588232	3