Subgroups — #13135141

Order 1728.

386 conjugacy classes of subgroups.

The subgroup lattice has 306 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	#10589168	—	3	1	no	maximal
5	#10588234	—	3	1	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊Dic₉	C₄	1	2	yes	—
8	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
9	C₄⋊(C₃⋊Dic₉)	—	2	1	no	—
10	C₉×#208	—	2	1	no	—
11	C₉×#208	C₄	1	2	yes	complemented
12	C₂×C₉×C₃⋊C₈	—	2	1	no	—
13	C₃×#3667	—	2	1	no	—
14	C₃×#3667	C₄	1	2	yes	complemented
15	C₂×C₃×C₉⋊C₈	—	2	1	no	—
16	C₂×C₄⋊(C₃⋊Dic₃)	—	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×C₄⋊Dic₉	—	3	1	no	—
21	C₂×C₄⋊Dic₉	—	6	1	no	—
22	C₂×#3667	D₃	1	1	yes	characteristic
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	—	4	2	no	—
32	C₉×C₃⋊C₈	—	2	2	no	—
33	C₃×C₉⋊C₈	—	2	2	no	—
34	#4554	—	9	1	no	—
35	#4554	—	9	1	no	—
36	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₃⋊Dic₃	—	3	2	no	—
38	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
39	C₄⋊(C₃⋊Dic₃)	—	6	1	no	—
40	C₃×#208	—	2	1	no	—
41	C₃×#208	—	3	2	no	—
42	C₃×#208	—	6	1	no	—
43	C₃×#208	Dic₃	1	2	yes	—
44	C₂×C₃×C₃⋊C₈	—	2	1	no	—
45	C₂×C₃×C₃⋊C₈	—	6	1	no	—
46	C₂²×C₄×C₉	—	2	1	no	—
47	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
48	C₂²×Dic₉	—	3	2	no	—
49	C₂²×Dic₉	—	6	2	no	—
50	C₉×M₁₆	—	3	2	no	—
51	C₉×M₁₆	—	6	1	no	—
52	C₂×C₈×C₉	—	6	1	no	—
53	C₄⋊Dic₉	—	6	2	no	—
54	C₄⋊Dic₉	—	6	2	no	—
55	C₄⋊Dic₉	—	6	1	no	—
56	C₄⋊Dic₉	—	6	1	no	—
57	#3667	—	2	1	no	—
58	#3667	Dic₃	1	2	yes	complemented
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂²×C₃×C₉	—	2	1	no	—
61	C₂²×C₃×C₉	—	2	1	no	—
62	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₄⋊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	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
69	C₃⋊Dic₉	—	4	2	no	—
70	C₂×C₃×M₁₆	—	3	1	no	—
71	C₂×C₃×M₁₆	—	9	1	no	—
72	C₂×C₄⋊Dic₃	—	9	1	no	—
73	C₂×C₄⋊Dic₃	—	9	1	no	—
74	C₂×C₄⋊Dic₃	—	18	1	no	—
75	C₂×#208	—	3	1	no	—
76	C₂×#208	D₉	1	1	yes	characteristic
77	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
78	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
80	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
82	C₂×C₃²×C₄	#67	1	1	yes	characteristic
83	C₂×C₃²×C₄	#67	1	1	yes	characteristic
84	C₂×C₃⋊Dic₃	—	6	2	no	—
85	C₂×C₃⋊Dic₃	—	12	2	no	—
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
88	C₃×C₃⋊C₈	—	2	2	no	—
89	C₃×C₃⋊C₈	—	6	2	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	1	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	Dic₆	1	1	yes	characteristic
97	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
98	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
100	C₂×C₄×C₉	#67	1	1	yes	characteristic
101	C₂×C₄×C₉	#67	1	1	yes	characteristic
102	C₂×Dic₉	—	6	4	no	—
103	C₂×Dic₉	—	6	2	no	—
104	C₂×Dic₉	—	12	4	no	—
105	C₂×Dic₉	—	12	2	no	—
106	C₈×C₉	—	6	2	no	—
107	C₉⋊C₈	—	2	2	no	—
108	#343	—	27	1	no	Sylow
109	C₂×C₃×C₉	—	2	1	no	—
110	C₂×C₃×C₉	—	2	1	no	—
111	C₂×C₃×C₉	#95	1	1	yes	characteristic
112	C₂×C₃×C₉	#100	1	1	yes	characteristic
113	C₂×C₃×C₉	#101	1	1	yes	characteristic
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	2	no	—
121	C₃×M₁₆	—	6	1	no	—
122	C₃×M₁₆	—	9	2	no	—
123	C₃×M₁₆	—	18	1	no	—
124	C₂×C₃×C₈	—	6	1	no	—
125	C₂×C₃×C₈	—	18	1	no	—
126	C₄⋊Dic₃	—	18	2	no	—
127	C₄⋊Dic₃	—	18	2	no	—
128	C₄⋊Dic₃	—	18	1	no	—
129	C₄⋊Dic₃	—	18	1	no	—
130	C₄⋊Dic₃	—	18	1	no	—
131	C₄⋊Dic₃	—	18	1	no	—
132	#208	—	2	1	no	—
133	#208	—	3	2	no	—
134	#208	—	6	1	no	—
135	#208	Dic₉	1	2	yes	complemented
136	C₂×C₃⋊C₈	—	2	1	no	—
137	C₂×C₃⋊C₈	—	6	1	no	—
138	C₂²×C₃²	—	2	1	no	—
139	C₂²×C₃²	—	2	1	no	—
140	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
141	C₂²×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
142	C₂²×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
143	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
144	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
145	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
146	C₃²×C₄	#217	1	1	yes	characteristic
147	C₃⋊Dic₃	—	12	2	no	—
148	C₂²×C₉	—	2	2	no	—
149	C₂²×C₉	—	2	2	no	—
150	C₂²×C₉	—	2	1	no	—
151	C₂²×C₉	—	2	1	no	—
152	C₂²×C₉	—	2	1	no	—
153	C₂²×C₉	—	2	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	C₄×Dic₃	1	1	yes	characteristic
156	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
157	C₂²×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
158	C₄×C₉	—	2	1	no	—
159	C₄×C₉	—	2	1	no	—
160	C₄×C₉	—	2	1	no	—
161	C₄×C₉	—	2	1	no	—
162	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
163	C₄×C₉	C₄⋊Dic₃	1	1	yes	characteristic
164	C₄×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
165	C₄×C₉	#217	1	1	yes	characteristic
166	Dic₉	—	12	4	no	—
167	Dic₉	—	12	2	no	—
168	C₂×M₁₆	—	3	1	no	—
169	C₂×M₁₆	—	9	1	no	—
170	C₂×C₄⋊C₄	—	27	1	no	—
171	C₃×C₉	#343	1	1	yes	characteristic, complemented, Sylow
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
174	C₂³×C₃	#624	1	1	yes	characteristic
175	C₂×C₃×C₄	—	2	1	no	—
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	Dic₁₈	1	1	yes	characteristic
182	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
183	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
185	C₂×C₃×C₄	#611	1	1	yes	characteristic
186	C₂×C₃×C₄	#611	1	1	yes	characteristic
187	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
188	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
189	C₂×C₃×C₄	#625	1	1	yes	characteristic
190	C₂×C₃×C₄	#626	1	1	yes	characteristic
191	C₂×C₃×C₄	#626	1	1	yes	characteristic
192	C₂×C₃×C₄	#627	1	1	yes	characteristic
193	C₂×Dic₃	—	18	4	no	—
194	C₂×Dic₃	—	18	2	no	—
195	C₂×Dic₃	—	18	2	no	—
196	C₂×Dic₃	—	36	4	no	—
197	C₂×Dic₃	—	36	2	no	—
198	C₂×Dic₃	—	36	2	no	—
199	C₃×C₈	—	6	2	no	—
200	C₃×C₈	—	18	2	no	—
201	C₃⋊C₈	—	2	2	no	—
202	C₃⋊C₈	—	6	2	no	—
203	C₂×C₃²	—	2	1	no	—
204	C₂×C₃²	—	2	1	no	—
205	C₂×C₃²	#823	1	1	yes	characteristic
206	C₂×C₃²	#824	1	1	yes	characteristic
207	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
208	C₂×C₉	—	2	2	no	—
209	C₂×C₉	—	2	2	no	—
210	C₂×C₉	—	2	1	no	—
211	C₂×C₉	—	2	1	no	—
212	C₂×C₉	—	2	1	no	—
213	C₂×C₉	—	2	1	no	—
214	C₂×C₉	—	2	1	no	—
215	C₂×C₉	#823	1	1	yes	characteristic
216	C₂×C₉	#824	1	1	yes	characteristic
217	C₂×C₉	C₃⋊#95	1	1	yes	characteristic
218	C₂²×C₄	—	27	2	no	—
219	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
220	M₁₆	—	3	2	no	—
221	M₁₆	—	6	1	no	—
222	M₁₆	—	9	2	no	—
223	M₁₆	—	18	1	no	—
224	C₂×C₈	—	6	1	no	—
225	C₂×C₈	—	18	1	no	—
226	C₄⋊C₄	—	54	1	no	—
227	C₄⋊C₄	—	54	1	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	1	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	1	no	—
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	2	1	no	—
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	Dic₃²	1	1	yes	characteristic
241	C₂²×C₃	#3722	1	1	yes	characteristic
242	C₂²×C₃	#3724	1	1	yes	characteristic
243	C₃×C₄	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	1	no	—
247	C₃×C₄	Dic₉⋊C₄	1	1	yes	characteristic
248	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
249	C₃×C₄	(C₂×D₉)⋊C₄	1	1	yes	characteristic
250	C₃×C₄	#3676	1	1	yes	characteristic
251	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
252	C₃×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
253	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
254	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
255	Dic₃	—	36	4	no	—
256	Dic₃	—	36	2	no	—
257	Dic₃	—	36	2	no	—
258	C₃²	#4554	1	1	yes	characteristic
259	C₉	—	2	1	no	—
260	C₉	#4554	1	1	yes	characteristic, complemented
261	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
262	C₂×C₄	—	54	2	no	—
263	C₂×C₄	—	108	2	no	—
264	C₂×C₄	#6189	1	1	yes	characteristic
265	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
266	C₂×C₄	#6192	1	1	yes	characteristic
267	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
268	C₂×C₄	#6194	1	1	yes	characteristic
269	C₂×C₄	#6195	1	1	yes	characteristic
270	C₈	—	6	2	no	—
271	C₈	—	18	2	no	—
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	2	2	no	—
274	C₂×C₃	—	2	1	no	—
275	C₂×C₃	—	2	1	no	—
276	C₂×C₃	—	2	1	no	—
277	C₂×C₃	—	2	1	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	#63382	1	1	yes	characteristic
282	C₂×C₃	#63383	1	1	yes	characteristic
283	C₂×C₃	C₉⋊#95	1	1	yes	characteristic
284	C₂×C₃	#63559	1	1	yes	characteristic
285	C₂×C₃	#63561	1	1	yes	characteristic
286	C₂×C₃	#63579	1	1	yes	characteristic
287	C₂²	—	2	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	Dic₃×Dic₉	1	1	yes	characteristic
290	C₂²	#88442	1	1	yes	characteristic
291	C₂²	#88445	1	1	yes	characteristic
292	C₄	—	108	2	no	—
293	C₄	Dic₉⋊Dic₃	1	1	yes	characteristic
294	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
295	C₄	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
296	C₄	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic
297	C₃	—	2	1	no	—
298	C₃	#10588234	1	1	yes	characteristic, complemented
299	C₃	#10589168	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	#11720288	1	1	yes	characteristic
303	C₂	#11720290	1	1	yes	characteristic
304	C₂	#11720331	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88445
commutator	C₂²×C₃×C₉	C₄²
Frattini	C₂²×C₃×C₄	D₃²
Fitting, cosocle	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	(C₂×D₃)⋊C₄
prime radical	C₂³×C₃²	C₄×D₃
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#343
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
11. C₉×#208	C₄	semidirect
14. C₃×#3667	C₄	semidirect
58. #3667	Dic₃	semidirect
135. #208	Dic₉	semidirect
171. C₃×C₉	#343	semidirect
260. C₉	#4554	semidirect
298. C₃	#10588234	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
#10589168	3
#10588234	3