Subgroups — #13135939

Order 1728.

550 conjugacy classes of subgroups.

The subgroup lattice has 240 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×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	#10589421	—	3	1	no	maximal
5	#10588235	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₉	—	2	1	no	—
8	C₃×C₉×M₁₆	—	2	1	no	—
9	C₃×C₉×M₁₆	C₄	1	2	yes	complemented
10	C₂×C₃×C₈×C₉	—	2	1	no	—
11	#88551	—	2	1	no	—
12	C₄⋊(C₃⋊Dic₉)	C₄	1	2	yes	complemented
13	C₄×C₃⋊Dic₉	—	2	1	no	—
14	C₄×C₃⋊Dic₉	—	2	1	no	—
15	C₄×C₃⋊Dic₉	C₄	1	2	yes	complemented
16	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
17	#64135	—	3	1	no	—
18	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
19	C₂×C₉×M₁₆	D₃	1	3	yes	—
20	#63488	—	3	3	no	—
21	C₂×C₄×Dic₉	—	3	3	no	—
22	C₂³×C₃×C₉	D₄	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	C₂×C₄	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	4	no	—
30	C₂×C₃⋊Dic₉	—	2	2	no	—
31	C₂×C₃⋊Dic₉	—	2	2	no	—
32	C₃×C₈×C₉	—	2	2	no	—
33	#4555	—	9	3	no	—
34	#4555	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂²×C₃⋊Dic₃	—	6	1	no	—
37	C₃²×M₁₆	—	2	1	no	—
38	C₃²×M₁₆	Dic₃	1	2	yes	—
39	C₂×C₃²×C₈	—	2	1	no	—
40	#3757	—	6	1	no	—
41	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
42	C₄×C₃⋊Dic₃	—	3	2	no	—
43	C₄×C₃⋊Dic₃	—	6	1	no	—
44	C₄×C₃⋊Dic₃	—	6	1	no	—
45	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
46	C₂²×Dic₉	—	6	3	no	—
47	C₉×M₁₆	—	2	3	no	—
48	C₉×M₁₆	Dic₃	1	6	yes	complemented
49	C₂×C₈×C₉	—	2	3	no	—
50	#3676	—	6	3	no	—
51	C₄⋊Dic₉	—	3	6	no	—
52	C₄×Dic₉	—	3	6	no	—
53	C₄×Dic₉	—	6	3	no	—
54	C₄×Dic₉	—	6	3	no	—
55	C₂²×C₃×C₉	—	2	2	no	—
56	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
59	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
60	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
62	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
63	C₃⋊Dic₉	—	2	4	no	—
64	C₃⋊Dic₉	—	4	2	no	—
65	C₂×C₃×M₁₆	D₉	1	3	yes	—
66	C₂×C₃×M₁₆	C₃⋊D₃	1	1	yes	characteristic
67	#926	—	9	3	no	—
68	#926	—	9	1	no	—
69	C₂×C₄×Dic₃	—	9	3	no	—
70	C₂×C₄×Dic₃	—	9	1	no	—
71	C₂³×C₃²	D₁₂	1	1	yes	characteristic
72	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	#67	1	1	yes	characteristic
77	C₂×C₃²×C₄	#67	1	1	yes	characteristic
78	C₂×C₃⋊Dic₃	—	6	4	no	—
79	C₂×C₃⋊Dic₃	—	6	2	no	—
80	C₂×C₃⋊Dic₃	—	6	2	no	—
81	C₂³×C₉	D₁₂	1	3	yes	—
82	C₃²×C₈	—	2	2	no	—
83	C₂×C₄×C₉	Dic₆	1	3	yes	—
84	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
85	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
87	C₂×C₄×C₉	#67	1	3	yes	—
88	C₂×C₄×C₉	#67	1	3	yes	—
89	C₂×Dic₉	—	6	12	no	—
90	C₂×Dic₉	—	6	6	no	—
91	C₂×Dic₉	—	6	6	no	—
92	C₈×C₉	—	2	6	no	—
93	#339	—	27	1	no	Sylow
94	C₂×C₃×C₉	—	2	1	no	—
95	C₂×C₃×C₉	—	2	1	no	—
96	C₂×C₃×C₉	#95	1	1	yes	characteristic
97	C₂×C₃×C₉	#104	1	2	yes	—
98	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
99	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
100	C₂²×Dic₃	—	18	3	no	—
101	C₂²×Dic₃	—	18	1	no	—
102	C₃×M₁₆	—	2	3	no	—
103	C₃×M₁₆	—	2	1	no	—
104	C₃×M₁₆	Dic₉	1	6	yes	complemented
105	C₃×M₁₆	C₃⋊Dic₃	1	2	yes	—
106	C₂×C₃×C₈	—	2	3	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	#217	—	18	3	no	—
109	#217	—	18	1	no	—
110	C₄⋊Dic₃	—	9	6	no	—
111	C₄⋊Dic₃	—	9	2	no	—
112	C₄×Dic₃	—	9	6	no	—
113	C₄×Dic₃	—	9	2	no	—
114	C₄×Dic₃	—	18	3	no	—
115	C₄×Dic₃	—	18	3	no	—
116	C₄×Dic₃	—	18	1	no	—
117	C₄×Dic₃	—	18	1	no	—
118	C₂²×C₃²	—	2	2	no	—
119	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
120	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
121	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
122	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
123	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
124	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
125	C₃²×C₄	#217	1	1	yes	characteristic
126	C₃⋊Dic₃	—	6	4	no	—
127	C₃⋊Dic₃	—	12	2	no	—
128	C₂²×C₉	—	2	6	no	—
129	C₂²×C₉	C₄⋊Dic₃	1	3	yes	—
130	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
131	C₂²×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
132	C₄×C₉	C₄×Dic₃	1	3	yes	—
133	C₄×C₉	Dic₃⋊C₄	1	3	yes	—
134	C₄×C₉	Dic₃⋊C₄	1	3	yes	—
135	C₄×C₉	#217	1	3	yes	—
136	Dic₉	—	6	12	no	—
137	Dic₉	—	12	6	no	—
138	C₂×M₁₆	C₃⋊D₉	1	1	yes	characteristic
139	#117	—	27	1	no	—
140	C₂×C₄²	—	27	1	no	—
141	C₃×C₉	#339	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	D₃₆	1	3	yes	—
143	C₂³×C₃	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
144	C₂×C₃×C₄	Dic₁₈	1	3	yes	—
145	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
146	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
147	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
148	C₂×C₃×C₄	#611	1	3	yes	—
149	C₂×C₃×C₄	#611	1	3	yes	—
150	C₂×C₃×C₄	#634	1	1	yes	characteristic
151	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
152	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₄	#638	1	1	yes	characteristic
155	C₂×C₃×C₄	#638	1	1	yes	characteristic
156	C₂×Dic₃	—	18	4	no	—
157	C₂×Dic₃	—	18	12	no	—
158	C₂×Dic₃	—	18	6	no	—
159	C₂×Dic₃	—	18	2	no	—
160	C₂×Dic₃	—	18	6	no	—
161	C₂×Dic₃	—	18	2	no	—
162	C₃×C₈	—	2	6	no	—
163	C₃×C₈	—	2	2	no	—
164	C₂×C₃²	—	2	1	no	—
165	C₂×C₃²	—	2	1	no	—
166	C₂×C₃²	#825	1	2	yes	—
167	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
168	C₂×C₉	—	2	3	no	—
169	C₂×C₉	—	2	3	no	—
170	C₂×C₉	#825	1	6	yes	—
171	C₂×C₉	C₃⋊#95	1	3	yes	—
172	C₂²×C₄	—	54	1	no	—
173	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
174	M₁₆	—	2	1	no	—
175	M₁₆	C₃⋊Dic₉	1	2	yes	complemented
176	C₂×C₈	—	2	1	no	—
177	C₄⋊C₄	—	27	2	no	—
178	C₂²⋊C₄	—	54	1	no	—
179	C₄²	—	27	2	no	—
180	C₄²	—	54	1	no	—
181	C₄²	—	54	1	no	—
182	C₂²×C₃	—	2	6	no	—
183	C₂²×C₃	—	2	2	no	—
184	C₂²×C₃	C₄⋊Dic₉	1	3	yes	—
185	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
186	C₂²×C₃	(C₂×D₉)⋊C₄	1	3	yes	—
187	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
188	C₂²×C₃	#3752	1	1	yes	characteristic
189	C₂²×C₃	#3752	1	1	yes	characteristic
190	C₃×C₄	C₄×Dic₉	1	3	yes	—
191	C₃×C₄	Dic₉⋊C₄	1	3	yes	—
192	C₃×C₄	Dic₉⋊C₄	1	3	yes	—
193	C₃×C₄	#3676	1	3	yes	—
194	C₃×C₄	C₄×C₃⋊Dic₃	1	1	yes	characteristic
195	C₃×C₄	#3750	1	1	yes	characteristic
196	C₃×C₄	#3750	1	1	yes	characteristic
197	C₃×C₄	#3757	1	1	yes	characteristic
198	Dic₃	—	18	4	no	—
199	Dic₃	—	18	12	no	—
200	Dic₃	—	36	6	no	—
201	Dic₃	—	36	2	no	—
202	C₃²	#4555	1	1	yes	characteristic
203	C₉	#4555	1	3	yes	complemented
204	C₂³	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
205	C₂×C₄	—	54	4	no	—
206	C₂×C₄	—	54	2	no	—
207	C₂×C₄	—	54	2	no	—
208	C₂×C₄	#6226	1	1	yes	characteristic
209	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
210	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
211	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
212	C₂×C₄	#6233	1	1	yes	characteristic
213	C₂×C₄	#6233	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₂×C₃	—	2	3	no	—
216	C₂×C₃	—	2	3	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	#63384	1	6	yes	—
220	C₂×C₃	C₉⋊#95	1	3	yes	—
221	C₂×C₃	#63652	1	2	yes	—
222	C₂×C₃	#63658	1	1	yes	characteristic
223	C₂²	—	2	2	no	—
224	C₂²	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
225	C₂²	#88536	1	1	yes	characteristic
226	C₂²	#88536	1	1	yes	characteristic
227	C₄	—	54	4	no	—
228	C₄	—	108	2	no	—
229	C₄	C₄×C₃⋊Dic₉	1	1	yes	characteristic
230	C₄	#88534	1	1	yes	characteristic
231	C₄	#88534	1	1	yes	characteristic
232	C₄	#88551	1	1	yes	characteristic
233	C₃	#10588235	1	3	yes	complemented
234	C₃	#10589421	1	1	yes	characteristic
235	C₂	—	2	1	no	—
236	C₂	—	2	1	no	—
237	C₂	#11720558	1	2	yes	—
238	C₂	#11720572	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6233
commutator	C₃×C₄×C₉	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₉×M₁₆	C₂
socle	C₂²×C₃²	(C₂×D₃)⋊C₄
prime radical	C₂³×C₃²	D₁₂
cosocle	C₂²×C₃×C₄×C₉	C₂²
hypercenter	C₂×M₁₆	C₃⋊D₉
hypocenter	C₃×C₉	#339
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
9. C₃×C₉×M₁₆	C₄	semidirect
12. C₄⋊(C₃⋊Dic₉)	C₄	semidirect
15. C₄×C₃⋊Dic₉	C₄	semidirect
48. C₉×M₁₆	Dic₃	semidirect
104. C₃×M₁₆	Dic₉	semidirect
141. C₃×C₉	#339	semidirect
175. M₁₆	C₃⋊Dic₉	semidirect
203. C₉	#4555	semidirect
233. C₃	#10588235	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×C₄×C₃⋊Dic₉	2
#10589421	3
#10588235	3