Subgroups — #13135148

Order 1728.

346 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	C₂²×C₃⋊(C₉⋊C₈)	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	#10589165	—	3	1	no	maximal
5	#10588231	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
8	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
9	C₂×C₃⋊(C₉⋊C₈)	C₄	1	2	yes	—
10	C₉×#208	—	2	1	no	—
11	C₉×#208	C₄	1	2	yes	—
12	C₂×C₉×C₃⋊C₈	—	2	1	no	—
13	C₃×#3667	—	2	1	no	—
14	C₃×#3667	C₄	1	2	yes	—
15	C₂×C₃×C₉⋊C₈	—	2	1	no	—
16	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
17	C₂×C₃×#208	—	3	1	no	—
18	C₂×C₃×#208	D₃	1	1	yes	characteristic
19	C₂×C₉×M₁₆	—	3	1	no	—
20	C₂×#3667	D₃	1	1	yes	characteristic
21	C₂²×C₉⋊C₈	—	3	1	no	—
22	C₂²×C₉⋊C₈	—	6	1	no	—
23	C₂³×C₃×C₉	Q₈	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	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₃⋊(C₉⋊C₈)	—	2	2	no	—
31	C₉×C₃⋊C₈	—	2	2	no	—
32	C₃×C₉⋊C₈	—	2	2	no	—
33	#4558	—	9	1	no	—
34	#4558	—	9	1	no	—
35	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
36	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
39	C₃×#208	—	2	1	no	—
40	C₃×#208	—	3	2	no	—
41	C₃×#208	—	6	1	no	—
42	C₃×#208	Dic₃	1	2	yes	—
43	C₂×C₃×C₃⋊C₈	—	2	1	no	—
44	C₂×C₃×C₃⋊C₈	—	6	1	no	—
45	C₂²×C₄×C₉	—	2	1	no	—
46	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
47	C₉×M₁₆	—	3	2	no	—
48	C₉×M₁₆	—	6	1	no	—
49	C₂×C₈×C₉	—	6	1	no	—
50	#3667	—	2	1	no	—
51	#3667	Dic₃	1	2	yes	—
52	C₂×C₉⋊C₈	—	2	1	no	—
53	C₂×C₉⋊C₈	—	3	2	no	—
54	C₂×C₉⋊C₈	—	6	2	no	—
55	C₂×C₉⋊C₈	—	6	2	no	—
56	C₂×C₉⋊C₈	—	6	2	no	—
57	C₂×C₉⋊C₈	—	6	1	no	—
58	C₂×C₉⋊C₈	—	6	1	no	—
59	C₂²×C₃×C₉	—	2	2	no	—
60	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
63	C₃×C₄×C₉	C₄²	1	1	yes	characteristic
64	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃×M₁₆	—	3	1	no	—
68	C₂×C₃×M₁₆	—	9	1	no	—
69	C₂×#208	—	3	1	no	—
70	C₂×#208	D₉	1	1	yes	characteristic
71	C₂²×C₃⋊C₈	—	9	1	no	—
72	C₂²×C₃⋊C₈	—	9	1	no	—
73	C₂²×C₃⋊C₈	—	18	1	no	—
74	C₂³×C₃²	Dic₆	1	1	yes	characteristic
75	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
77	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
78	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	#67	1	1	yes	characteristic
80	C₂×C₃²×C₄	#67	1	1	yes	characteristic
81	C₂³×C₉	—	2	1	no	—
82	C₂³×C₉	Dic₆	1	1	yes	characteristic
83	C₃⋊(C₃⋊C₈)	—	6	2	no	—
84	C₃×C₃⋊C₈	—	2	2	no	—
85	C₃×C₃⋊C₈	—	6	2	no	—
86	C₂×C₄×C₉	—	2	1	no	—
87	C₂×C₄×C₉	—	2	1	no	—
88	C₂×C₄×C₉	—	2	1	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
93	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
94	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₄×C₉	#67	1	1	yes	characteristic
97	C₂×C₄×C₉	#67	1	1	yes	characteristic
98	C₈×C₉	—	6	2	no	—
99	C₉⋊C₈	—	2	2	no	—
100	C₉⋊C₈	—	6	4	no	—
101	C₉⋊C₈	—	6	2	no	—
102	#341	—	27	1	no	Sylow
103	C₂×C₃×C₉	—	2	1	no	—
104	C₂×C₃×C₉	—	2	1	no	—
105	C₂×C₃×C₉	#95	1	1	yes	characteristic
106	C₂×C₃×C₉	#108	1	2	yes	—
107	C₂²×C₃×C₄	—	2	1	no	—
108	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
109	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
110	C₃×M₁₆	—	3	2	no	—
111	C₃×M₁₆	—	6	1	no	—
112	C₃×M₁₆	—	9	2	no	—
113	C₃×M₁₆	—	18	1	no	—
114	C₂×C₃×C₈	—	6	1	no	—
115	C₂×C₃×C₈	—	18	1	no	—
116	#208	—	2	1	no	—
117	#208	—	3	2	no	—
118	#208	—	6	1	no	—
119	#208	Dic₉	1	2	yes	—
120	C₂×C₃⋊C₈	—	2	1	no	—
121	C₂×C₃⋊C₈	—	6	1	no	—
122	C₂×C₃⋊C₈	—	9	2	no	—
123	C₂×C₃⋊C₈	—	9	2	no	—
124	C₂×C₃⋊C₈	—	18	2	no	—
125	C₂×C₃⋊C₈	—	18	2	no	—
126	C₂×C₃⋊C₈	—	18	2	no	—
127	C₂×C₃⋊C₈	—	18	1	no	—
128	C₂×C₃⋊C₈	—	18	1	no	—
129	C₂×C₃⋊C₈	—	18	1	no	—
130	C₂×C₃⋊C₈	—	18	1	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
133	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
134	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
135	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
136	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
137	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
138	C₃²×C₄	#217	1	1	yes	characteristic
139	C₂²×C₉	—	2	4	no	—
140	C₂²×C₉	—	2	2	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	—	2	1	no	—
143	C₂²×C₉	—	2	1	no	—
144	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
145	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
146	C₂²×C₉	C₄⋊Dic₃	1	1	yes	characteristic
147	C₄×C₉	—	2	1	no	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	2	1	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	C₄×Dic₃	1	1	yes	characteristic
152	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
153	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
154	C₄×C₉	#217	1	1	yes	characteristic
155	C₂×M₁₆	—	3	1	no	—
156	C₂×M₁₆	—	9	1	no	—
157	C₂²×C₈	—	27	1	no	—
158	C₃×C₉	#341	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	Dic₁₈	1	1	yes	characteristic
161	C₂³×C₃	#627	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
169	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
170	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
172	C₂×C₃×C₄	#611	1	1	yes	characteristic
173	C₂×C₃×C₄	#611	1	1	yes	characteristic
174	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	#624	1	1	yes	characteristic
177	C₂×C₃×C₄	#625	1	1	yes	characteristic
178	C₂×C₃×C₄	#626	1	1	yes	characteristic
179	C₂×C₃×C₄	#626	1	1	yes	characteristic
180	C₃×C₈	—	6	2	no	—
181	C₃×C₈	—	18	2	no	—
182	C₃⋊C₈	—	2	2	no	—
183	C₃⋊C₈	—	6	2	no	—
184	C₃⋊C₈	—	18	4	no	—
185	C₃⋊C₈	—	18	2	no	—
186	C₃⋊C₈	—	18	2	no	—
187	C₂×C₃²	—	2	1	no	—
188	C₂×C₃²	—	2	1	no	—
189	C₂×C₃²	#822	1	2	yes	—
190	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
191	C₂×C₉	—	2	2	no	—
192	C₂×C₉	—	2	2	no	—
193	C₂×C₉	—	2	1	no	—
194	C₂×C₉	—	2	1	no	—
195	C₂×C₉	—	2	2	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	#822	1	2	yes	—
198	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
199	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
200	M₁₆	—	3	2	no	—
201	M₁₆	—	6	1	no	—
202	M₁₆	—	9	2	no	—
203	M₁₆	—	18	1	no	—
204	C₂×C₈	—	6	1	no	—
205	C₂×C₈	—	18	1	no	—
206	C₂×C₈	—	27	2	no	—
207	C₂×C₈	—	54	1	no	—
208	C₂×C₈	—	54	1	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
216	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
217	C₂²×C₃	C₄⋊Dic₉	1	1	yes	characteristic
218	C₂²×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
219	C₂²×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
220	C₂²×C₃	#3724	1	1	yes	characteristic
221	C₃×C₄	—	2	1	no	—
222	C₃×C₄	—	2	1	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	C₄×Dic₉	1	1	yes	characteristic
226	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
227	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
228	C₃×C₄	#3676	1	1	yes	characteristic
229	C₃×C₄	Dic₃²	1	1	yes	characteristic
230	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
231	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
232	C₃×C₄	#3722	1	1	yes	characteristic
233	C₃²	#4558	1	1	yes	characteristic
234	C₉	—	2	1	no	—
235	C₉	#4558	1	1	yes	characteristic, complemented
236	C₂³	#6189	1	1	yes	characteristic
237	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
238	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	#6192	1	1	yes	characteristic
240	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
241	C₂×C₄	#6194	1	1	yes	characteristic
242	C₂×C₄	#6195	1	1	yes	characteristic
243	C₈	—	6	2	no	—
244	C₈	—	18	2	no	—
245	C₈	—	54	2	no	—
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	—	2	1	no	—
250	C₂×C₃	—	2	1	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	#63381	1	2	yes	—
255	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
256	C₂×C₃	#63578	1	2	yes	—
257	C₂×C₃	#63579	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	Dic₉⋊Dic₃	1	1	yes	characteristic
260	C₂²	#88442	1	1	yes	characteristic
261	C₂²	Dic₃⋊Dic₉	1	1	yes	characteristic
262	C₄	Dic₃×Dic₉	1	1	yes	characteristic
263	C₄	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
264	C₄	#88445	1	1	yes	characteristic
265	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
266	C₃	—	2	1	no	—
267	C₃	#10588231	1	1	yes	characteristic, complemented
268	C₃	#10589165	1	1	yes	characteristic
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	#11720286	1	2	yes	—
272	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6194
commutator	C₃×C₄×C₉	C₄²
Frattini	C₂²×C₃×C₄	D₃²
Fitting, cosocle	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	Dic₃⋊C₄
prime radical	C₂³×C₃²	Dic₆
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#341
LCS	G ▷ C₃×C₄×C₉ ▷ 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
158. C₃×C₉	#341	semidirect
235. C₉	#4558	semidirect
267. C₃	#10588231	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₉⋊C₈)	2
C₂×C₉×#208	2
C₂×C₃×#3667	2
#10589165	3
#10588231	3