Subgroups — #13139352

Order 1728.

534 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3667	C₂	1	1	yes	maximal, characteristic
4	#11720275	C₂	1	2	yes	maximal, complemented
5	#11720271	C₂	1	2	yes	maximal, complemented
6	#10591060	—	3	1	no	maximal
7	M₁₆×Dic₉	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
10	C₄×C₃⋊Dic₉	C₄	1	2	yes	—
11	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
13	C₉×#208	C₄	1	4	yes	complemented
14	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃×#3667	C₄	1	4	yes	complemented
16	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
17	C₂×C₄×C₃⋊Dic₃	—	3	1	no	—
18	C₂×C₃×#208	—	3	1	no	—
19	C₂×C₃×#208	D₃	1	1	yes	characteristic
20	#63555	—	3	2	no	—
21	#63553	—	3	2	no	—
22	C₂×C₉×M₁₆	—	3	1	no	—
23	C₂×C₄×Dic₉	—	3	1	no	—
24	C₂×C₄×Dic₉	—	6	1	no	—
25	C₂×#3667	D₃	1	1	yes	characteristic
26	#63375	—	3	2	no	—
27	C₈×Dic₉	—	3	2	no	—
28	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
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	C₂×C₃⋊Dic₉	—	2	4	no	—
34	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
35	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
36	C₉×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
37	C₃×C₉⋊C₈	C₂×C₄	1	4	yes	complemented
38	Dic₃×M₁₆	—	9	1	no	—
39	Dic₃×M₁₆	—	9	1	no	—
40	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₃⋊Dic₃	—	3	2	no	—
42	C₄×C₃⋊Dic₃	—	3	2	no	—
43	C₄×C₃⋊Dic₃	—	3	1	no	—
44	C₄×C₃⋊Dic₃	—	3	1	no	—
45	C₃×#208	—	3	4	no	—
46	C₃×#208	Dic₃	1	4	yes	—
47	C₂×C₃×C₃⋊C₈	—	3	2	no	—
48	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
51	C₂²×Dic₉	—	3	2	no	—
52	C₂²×Dic₉	—	6	2	no	—
53	C₉×M₁₆	—	3	4	no	—
54	C₂×C₈×C₉	—	3	2	no	—
55	C₄×Dic₉	—	3	2	no	—
56	C₄×Dic₉	—	3	1	no	—
57	C₄×Dic₉	—	3	1	no	—
58	C₄×Dic₉	—	6	2	no	—
59	C₄×Dic₉	—	6	1	no	—
60	C₄×Dic₉	—	6	1	no	—
61	#3667	Dic₃	1	4	yes	complemented
62	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
63	C₂²×C₃×C₉	—	2	2	no	—
64	C₂²×C₃×C₉	C₄²	1	2	yes	—
65	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₄²	1	2	yes	—
67	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₃⋊Dic₉	—	2	2	no	—
70	C₃⋊Dic₉	M₁₆	1	4	yes	complemented
71	C₂×C₃×M₁₆	—	3	1	no	—
72	C₂×C₃×M₁₆	—	9	1	no	—
73	C₂×C₄×Dic₃	—	9	1	no	—
74	C₂×C₄×Dic₃	—	9	1	no	—
75	C₂×C₄×Dic₃	—	18	1	no	—
76	C₂×#208	—	3	1	no	—
77	C₂×#208	D₉	1	1	yes	characteristic
78	#815	—	9	2	no	—
79	#815	—	9	2	no	—
80	C₈×Dic₃	—	9	2	no	—
81	C₈×Dic₃	—	9	2	no	—
82	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
84	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
86	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	3	2	no	—
88	C₂×C₃⋊Dic₃	—	3	2	no	—
89	C₂×C₃⋊Dic₃	—	6	4	no	—
90	C₂³×C₉	—	2	1	no	—
91	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
92	C₃×C₃⋊C₈	—	3	4	no	—
93	C₃×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
94	C₂×C₄×C₉	—	2	2	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	C₄×D₃	1	2	yes	—
99	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×Dic₃	1	2	yes	—
101	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
102	C₂×Dic₉	—	3	2	no	—
103	C₂×Dic₉	—	3	2	no	—
104	C₂×Dic₉	—	6	2	no	—
105	C₂×Dic₉	—	6	2	no	—
106	C₂×Dic₉	—	6	4	no	—
107	C₂×Dic₉	—	6	8	no	—
108	C₈×C₉	—	3	4	no	—
109	C₉⋊C₈	C₂×Dic₃	1	4	yes	complemented
110	C₄×M₁₆	—	27	1	no	Sylow
111	C₂×C₃×C₉	—	2	2	no	—
112	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
116	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
117	C₂²×Dic₃	—	9	2	no	—
118	C₂²×Dic₃	—	9	2	no	—
119	C₂²×Dic₃	—	18	2	no	—
120	C₃×M₁₆	—	3	4	no	—
121	C₃×M₁₆	—	9	4	no	—
122	C₂×C₃×C₈	—	3	2	no	—
123	C₂×C₃×C₈	—	9	2	no	—
124	C₄×Dic₃	—	9	2	no	—
125	C₄×Dic₃	—	9	1	no	—
126	C₄×Dic₃	—	9	1	no	—
127	C₄×Dic₃	—	9	2	no	—
128	C₄×Dic₃	—	9	1	no	—
129	C₄×Dic₃	—	9	1	no	—
130	C₄×Dic₃	—	18	2	no	—
131	C₄×Dic₃	—	18	1	no	—
132	C₄×Dic₃	—	18	1	no	—
133	#208	—	3	4	no	—
134	#208	Dic₉	1	4	yes	complemented
135	C₂×C₃⋊C₈	—	3	2	no	—
136	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
137	C₂²×C₃²	—	2	2	no	—
138	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
139	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
140	C₃²×C₄	C₄×Dic₃	1	2	yes	—
141	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
142	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
143	C₃⋊Dic₃	—	3	4	no	—
144	C₃⋊Dic₃	—	6	2	no	—
145	C₂²×C₉	—	2	4	no	—
146	C₂²×C₉	—	2	2	no	—
147	C₂²×C₉	—	2	2	no	—
148	C₂²×C₉	—	2	1	no	—
149	C₂²×C₉	C₄×Dic₃	1	2	yes	—
150	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
151	C₄×C₉	—	2	2	no	—
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	2	1	no	—
154	C₄×C₉	C₄×Dic₃	1	2	yes	—
155	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
156	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
157	Dic₉	—	3	4	no	—
158	Dic₉	—	6	4	no	—
159	Dic₉	—	6	4	no	—
160	Dic₉	—	6	2	no	—
161	C₂×M₁₆	—	3	1	no	—
162	C₂×M₁₆	—	9	1	no	—
163	C₂×C₄²	—	27	1	no	—
164	#97	—	27	2	no	—
165	C₄×C₈	—	27	2	no	—
166	C₃×C₉	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
167	C₂³×C₃	—	2	1	no	—
168	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
169	C₂³×C₃	#624	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	C₄×D₉	1	2	yes	—
175	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
177	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
179	C₂×C₃×C₄	D₃×Dic₃	1	2	yes	—
180	C₂×C₃×C₄	#624	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
182	C₂×Dic₃	—	9	2	no	—
183	C₂×Dic₃	—	9	2	no	—
184	C₂×Dic₃	—	9	2	no	—
185	C₂×Dic₃	—	9	2	no	—
186	C₂×Dic₃	—	18	2	no	—
187	C₂×Dic₃	—	18	2	no	—
188	C₂×Dic₃	—	18	4	no	—
189	C₂×Dic₃	—	18	4	no	—
190	C₂×Dic₃	—	18	8	no	—
191	C₃×C₈	—	3	4	no	—
192	C₃×C₈	—	9	4	no	—
193	C₃⋊C₈	—	3	4	no	—
194	C₃⋊C₈	C₂×Dic₉	1	4	yes	complemented
195	C₂×C₃²	—	2	2	no	—
196	C₂×C₃²	D₃×M₁₆	1	2	yes	—
197	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
198	C₂×C₉	—	2	4	no	—
199	C₂×C₉	—	2	2	no	—
200	C₂×C₉	—	2	2	no	—
201	C₂×C₉	—	2	1	no	—
202	C₂×C₉	D₃×M₁₆	1	2	yes	—
203	C₂×C₉	C₂×C₄×Dic₃	1	1	yes	characteristic
204	C₂²×C₄	—	27	2	no	—
205	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
206	M₁₆	—	3	4	no	—
207	M₁₆	—	9	4	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	—	9	2	no	—
210	C₄²	—	27	2	no	—
211	C₄²	—	27	1	no	—
212	C₄²	—	27	1	no	—
213	C₂²×C₃	—	2	4	no	—
214	C₂²×C₃	—	2	2	no	—
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	2	no	—
218	C₂²×C₃	C₄×Dic₉	1	2	yes	—
219	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
220	C₂²×C₃	Dic₃²	1	2	yes	—
221	C₂²×C₃	C₂×#624	1	1	yes	characteristic
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	C₄×Dic₉	1	2	yes	—
226	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
227	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
228	C₃×C₄	Dic₃²	1	2	yes	—
229	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
230	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
231	Dic₃	—	9	4	no	—
232	Dic₃	—	9	4	no	—
233	Dic₃	—	18	4	no	—
234	Dic₃	—	18	4	no	—
235	Dic₃	—	18	2	no	—
236	Dic₃	—	18	2	no	—
237	C₃²	Dic₃×M₁₆	1	1	yes	characteristic
238	C₉	—	2	1	no	—
239	C₉	Dic₃×M₁₆	1	1	yes	characteristic, complemented
240	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
241	C₂×C₄	—	27	2	no	—
242	C₂×C₄	—	27	2	no	—
243	C₂×C₄	—	54	4	no	—
244	C₂×C₄	Dic₃×D₉	1	2	yes	—
245	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
246	C₂×C₄	D₃×Dic₉	1	2	yes	—
247	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
248	C₈	—	3	4	no	—
249	C₈	—	9	4	no	—
250	C₂×C₃	—	2	4	no	—
251	C₂×C₃	—	2	2	no	—
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	M₁₆×D₉	1	2	yes	—
256	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
257	C₂×C₃	#63818	1	2	yes	—
258	C₂×C₃	C₂×Dic₃²	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	Dic₃×Dic₉	1	2	yes	—
261	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
262	C₄	—	27	4	no	—
263	C₄	—	54	2	no	—
264	C₄	Dic₃×Dic₉	1	2	yes	—
265	C₄	C₂×Dic₃×D₉	1	1	yes	characteristic
266	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
267	C₃	—	2	1	no	—
268	C₃	M₁₆×Dic₉	1	1	yes	characteristic, complemented
269	C₃	#10591060	1	1	yes	characteristic
270	C₂	—	2	2	no	—
271	C₂	#11720863	1	2	yes	—
272	C₂	C₂×Dic₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	C₄×M₁₆
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720275	C₂	semidirect
5. #11720271	C₂	semidirect
13. C₉×#208	C₄	semidirect
15. C₃×#3667	C₄	semidirect
36. C₉×C₃⋊C₈	C₂×C₄	semidirect
37. C₃×C₉⋊C₈	C₂×C₄	semidirect
61. #3667	Dic₃	semidirect
70. C₃⋊Dic₉	M₁₆	semidirect
109. C₉⋊C₈	C₂×Dic₃	semidirect
134. #208	Dic₉	semidirect
166. C₃×C₉	C₄×M₁₆	semidirect
194. C₃⋊C₈	C₂×Dic₉	semidirect
239. C₉	Dic₃×M₁₆	semidirect
268. C₃	M₁₆×Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃⋊Dic₉	2
C₂×C₉×#208	2
C₂×C₃×#3667	2
#11720275	2
#11720271	2
#10591060	3
M₁₆×Dic₉	3