Subgroups — #13135136

Order 1728.

430 conjugacy classes of subgroups.

The subgroup lattice has 308 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×C₃×C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589155	—	3	1	no	maximal
5	#10588228	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×Dic₉	C₄	1	2	yes	—
8	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
9	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
10	C₂×C₃⋊(C₉⋊C₈)	C₄	1	2	yes	complemented
11	C₃×C₄×Dic₉	—	2	1	no	—
12	C₃×C₄×Dic₉	—	2	1	no	—
13	C₂×C₉×C₃⋊C₈	—	2	1	no	—
14	C₂×C₉×C₃⋊C₈	—	2	1	no	—
15	C₂×C₉×C₃⋊C₈	C₄	1	2	yes	complemented
16	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
17	C₂×C₃×C₄×Dic₃	—	3	1	no	—
18	C₂²×C₃×C₃⋊C₈	D₃	1	1	yes	characteristic
19	C₂²×C₈×C₉	—	3	1	no	—
20	C₂×C₄×Dic₉	D₃	1	1	yes	characteristic
21	C₂²×C₉⋊C₈	—	3	1	no	—
22	C₂²×C₉⋊C₈	—	6	1	no	—
23	C₂³×C₃×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	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
30	C₂×C₃×Dic₉	—	2	2	no	—
31	C₂×C₃×Dic₉	—	2	2	no	—
32	C₂×C₃×Dic₉	C₈	1	4	yes	complemented
33	C₃⋊(C₉⋊C₈)	—	2	2	no	—
34	C₉×C₃⋊C₈	—	2	2	no	—
35	#4550	—	9	1	no	—
36	#4524	—	9	1	no	—
37	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂²×C₃×Dic₃	—	3	2	no	—
39	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
40	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
42	C₃×C₄×Dic₃	—	6	1	no	—
43	C₃×C₄×Dic₃	—	6	1	no	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃×C₃⋊C₈	Dic₃	1	2	yes	—
47	C₂²×C₄×C₉	—	2	1	no	—
48	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
49	C₂²×Dic₉	Dic₃	1	2	yes	—
50	C₂×C₈×C₉	—	3	2	no	—
51	C₂×C₈×C₉	—	6	1	no	—
52	C₂×C₈×C₉	—	6	1	no	—
53	C₄×Dic₉	—	2	1	no	—
54	C₄×Dic₉	—	2	1	no	—
55	C₂×C₉⋊C₈	—	3	2	no	—
56	C₂×C₉⋊C₈	—	6	2	no	—
57	C₂×C₉⋊C₈	—	6	2	no	—
58	C₂×C₉⋊C₈	—	6	2	no	—
59	C₂×C₉⋊C₈	—	6	1	no	—
60	C₂×C₉⋊C₈	—	6	1	no	—
61	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×C₈	1	2	yes	—
65	C₂²×C₃×C₉	M₁₆	1	2	yes	—
66	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
70	C₃×Dic₉	—	2	4	no	—
71	C₂²×C₃×C₈	—	3	1	no	—
72	C₂×C₃×C₄²	—	9	1	no	—
73	C₂×C₄×Dic₃	—	3	1	no	—
74	C₂²×C₃⋊C₈	—	9	1	no	—
75	C₂²×C₃⋊C₈	—	9	1	no	—
76	C₂²×C₃⋊C₈	—	18	1	no	—
77	C₂²×C₃⋊C₈	D₉	1	1	yes	characteristic
78	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
83	C₂×C₃²×C₄	#67	1	1	yes	characteristic
84	C₂×C₃²×C₄	#67	1	1	yes	characteristic
85	C₂×C₃×Dic₃	—	3	4	no	—
86	C₂×C₃×Dic₃	—	6	2	no	—
87	C₂×C₃×Dic₃	—	6	2	no	—
88	C₂³×C₉	—	2	1	no	—
89	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
90	C₃⋊(C₃⋊C₈)	—	6	2	no	—
91	C₃×C₃⋊C₈	—	2	2	no	—
92	C₂×C₄×C₉	—	2	1	no	—
93	C₂×C₄×C₉	—	2	1	no	—
94	C₂×C₄×C₉	—	2	1	no	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	1	no	—
97	C₂×C₄×C₉	—	2	1	no	—
98	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
99	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
101	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
102	C₂×C₄×C₉	#67	1	1	yes	characteristic
103	C₂×C₄×C₉	#67	1	1	yes	characteristic
104	C₂×Dic₉	—	2	2	no	—
105	C₂×Dic₉	—	2	2	no	—
106	C₂×Dic₉	C₃⋊C₈	1	4	yes	complemented
107	C₈×C₉	—	6	2	no	—
108	C₉⋊C₈	—	6	4	no	—
109	C₉⋊C₈	—	6	2	no	—
110	#336	—	27	1	no	Sylow
111	C₂×C₃×C₉	#95	1	1	yes	characteristic
112	C₂×C₃×C₉	C₄×C₈	1	1	yes	characteristic
113	C₂×C₃×C₉	#97	1	1	yes	characteristic
114	C₂×C₃×C₉	C₂²⋊C₈	1	2	yes	—
115	C₂×C₃×C₉	C₄⋊C₈	1	2	yes	—
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	—	9	2	no	—
118	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
119	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
120	C₂²×Dic₃	—	3	2	no	—
121	C₂×C₃×C₈	—	3	2	no	—
122	C₂×C₃×C₈	—	6	1	no	—
123	C₂×C₃×C₈	—	6	1	no	—
124	C₃×C₄²	—	18	1	no	—
125	C₃×C₄²	—	18	1	no	—
126	C₄×Dic₃	—	6	1	no	—
127	C₄×Dic₃	—	6	1	no	—
128	C₂×C₃⋊C₈	—	2	1	no	—
129	C₂×C₃⋊C₈	—	2	1	no	—
130	C₂×C₃⋊C₈	—	9	2	no	—
131	C₂×C₃⋊C₈	—	9	2	no	—
132	C₂×C₃⋊C₈	—	18	2	no	—
133	C₂×C₃⋊C₈	—	18	2	no	—
134	C₂×C₃⋊C₈	—	18	2	no	—
135	C₂×C₃⋊C₈	—	18	1	no	—
136	C₂×C₃⋊C₈	—	18	1	no	—
137	C₂×C₃⋊C₈	—	18	1	no	—
138	C₂×C₃⋊C₈	—	18	1	no	—
139	C₂×C₃⋊C₈	Dic₉	1	2	yes	complemented
140	C₂²×C₃²	D₃×C₈	1	2	yes	—
141	C₂²×C₃²	ΓL(1,𝔽₂₅)	1	2	yes	—
142	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
143	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
145	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
146	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
147	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
148	C₃²×C₄	#217	1	1	yes	characteristic
149	C₃×Dic₃	—	6	4	no	—
150	C₂²×C₉	—	2	2	no	—
151	C₂²×C₉	—	2	2	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	2	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	C₂×C₃⋊C₈	1	2	yes	—
156	C₂²×C₉	#208	1	2	yes	—
157	C₂²×C₉	C₄×Dic₃	1	1	yes	characteristic
158	C₂²×C₉	C₄⋊Dic₃	1	1	yes	characteristic
159	C₂²×C₉	#217	1	1	yes	characteristic
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	—	2	1	no	—
163	C₄×C₉	—	2	1	no	—
164	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
165	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
166	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
168	Dic₉	—	2	4	no	—
169	C₂²×C₈	—	3	1	no	—
170	C₂²×C₈	—	27	1	no	—
171	C₂×C₄²	—	9	1	no	—
172	C₃×C₉	#336	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
175	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	9	4	no	—
183	C₂×C₃×C₄	—	18	2	no	—
184	C₂×C₃×C₄	—	18	2	no	—
185	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
186	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
187	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
189	C₂×C₃×C₄	#611	1	1	yes	characteristic
190	C₂×C₃×C₄	#611	1	1	yes	characteristic
191	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
192	C₂×C₃×C₄	#624	1	1	yes	characteristic
193	C₂×C₃×C₄	#625	1	1	yes	characteristic
194	C₂×C₃×C₄	#626	1	1	yes	characteristic
195	C₂×C₃×C₄	#626	1	1	yes	characteristic
196	C₂×C₃×C₄	#627	1	1	yes	characteristic
197	C₂×Dic₃	—	3	4	no	—
198	C₂×Dic₃	—	6	2	no	—
199	C₂×Dic₃	—	6	2	no	—
200	C₃×C₈	—	6	2	no	—
201	C₃⋊C₈	—	2	2	no	—
202	C₃⋊C₈	—	18	4	no	—
203	C₃⋊C₈	—	18	2	no	—
204	C₃⋊C₈	—	18	2	no	—
205	C₂×C₃²	C₈×Dic₃	1	1	yes	characteristic
206	C₂×C₃²	Dic₃⋊C₈	1	2	yes	—
207	C₂×C₃²	#815	1	1	yes	characteristic
208	C₂×C₃²	(C₂×D₃)⋊C₈	1	2	yes	—
209	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	2	2	no	—
213	C₂×C₉	—	2	2	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	C₄×C₃⋊C₈	1	1	yes	characteristic
216	C₂×C₉	#803	1	1	yes	characteristic
217	C₂×C₉	C₄⋊(C₃⋊C₈)	1	2	yes	—
218	C₂×C₉	#830	1	2	yes	—
219	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
220	C₂²×C₄	—	9	2	no	—
221	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
222	C₂×C₈	—	3	2	no	—
223	C₂×C₈	—	6	1	no	—
224	C₂×C₈	—	6	1	no	—
225	C₂×C₈	—	27	2	no	—
226	C₂×C₈	—	54	1	no	—
227	C₂×C₈	—	54	1	no	—
228	C₄²	—	18	1	no	—
229	C₄²	—	18	1	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	C₈×D₉	1	2	yes	—
236	C₂²×C₃	#3663	1	2	yes	—
237	C₂²×C₃	C₄×Dic₉	1	1	yes	characteristic
238	C₂²×C₃	Dic₉⋊C₄	1	1	yes	characteristic
239	C₂²×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
240	C₂²×C₃	D₃×C₃⋊C₈	1	2	yes	—
241	C₂²×C₃	#3711	1	2	yes	—
242	C₂²×C₃	Dic₃²	1	1	yes	characteristic
243	C₂²×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
244	C₂²×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	18	4	no	—
250	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
251	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
252	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
253	C₃×C₄	#3676	1	1	yes	characteristic
254	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
255	C₃×C₄	#3722	1	1	yes	characteristic
256	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
257	C₃×C₄	#3724	1	1	yes	characteristic
258	Dic₃	—	6	4	no	—
259	C₃²	#4550	1	1	yes	characteristic
260	C₉	—	2	1	no	—
261	C₉	#4524	1	1	yes	characteristic, complemented
262	C₂³	Dic₃×D₉	1	1	yes	characteristic
263	C₂×C₄	—	9	4	no	—
264	C₂×C₄	—	18	2	no	—
265	C₂×C₄	—	18	2	no	—
266	C₂×C₄	#6189	1	1	yes	characteristic
267	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
268	C₂×C₄	#6192	1	1	yes	characteristic
269	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
270	C₂×C₄	#6194	1	1	yes	characteristic
271	C₂×C₄	#6195	1	1	yes	characteristic
272	C₈	—	6	2	no	—
273	C₈	—	54	2	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	2	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	2	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	C₈×Dic₉	1	1	yes	characteristic
280	C₂×C₃	Dic₉⋊C₈	1	2	yes	—
281	C₂×C₃	#63375	1	1	yes	characteristic
282	C₂×C₃	(C₂×D₉)⋊C₈	1	2	yes	—
283	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
284	C₂×C₃	Dic₃×C₃⋊C₈	1	1	yes	characteristic
285	C₂×C₃	#63554	1	1	yes	characteristic
286	C₂×C₃	(C₂×D₃)⋊(C₃⋊C₈)	1	2	yes	—
287	C₂×C₃	Dic₃⋊(C₃⋊C₈)	1	2	yes	—
288	C₂×C₃	#63579	1	1	yes	characteristic
289	C₂²	D₉×C₃⋊C₈	1	2	yes	—
290	C₂²	#88413	1	2	yes	—
291	C₂²	Dic₃×Dic₉	1	1	yes	characteristic
292	C₂²	Dic₉⋊Dic₃	1	1	yes	characteristic
293	C₂²	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
294	C₄	—	18	4	no	—
295	C₄	#88442	1	1	yes	characteristic
296	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
297	C₄	#88445	1	1	yes	characteristic
298	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
299	C₃	—	2	1	no	—
300	C₃	#10588228	1	1	yes	characteristic, complemented
301	C₃	#10589155	1	1	yes	characteristic
302	C₂	C₃⋊C₈×Dic₉	1	1	yes	characteristic
303	C₂	Dic₉⋊(C₃⋊C₈)	1	2	yes	—
304	C₂	#11720274	1	1	yes	characteristic
305	C₂	(C₂×D₉)⋊(C₃⋊C₈)	1	2	yes	—
306	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	D₃×D₉
commutator	C₂×C₃×C₉	C₄×C₈
Frattini	C₂²×C₃×C₄	D₃²
Fitting, cosocle	C₂²×C₃×C₄×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₄×D₃
hypocenter	C₃×C₉	#336
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
10. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
15. C₂×C₉×C₃⋊C₈	C₄	semidirect
32. C₂×C₃×Dic₉	C₈	semidirect
106. C₂×Dic₉	C₃⋊C₈	semidirect
139. C₂×C₃⋊C₈	Dic₉	semidirect
172. C₃×C₉	#336	semidirect
261. C₉	#4524	semidirect
300. C₃	#10588228	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×C₃×C₄×Dic₉	2
C₂²×C₉×C₃⋊C₈	2
#10589155	3
#10588228	3