Subgroups — #13139388

Order 1728.

374 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88532	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	#11720280	C₂	1	4	yes	maximal, complemented
5	#10591141	—	3	1	no	maximal
6	#10588800	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	#88532	—	2	1	no	—
9	#88532	C₂²	1	2	yes	—
10	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
11	C₉×#208	C₄	1	2	yes	—
12	C₉×#208	C₄	1	2	yes	—
13	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
14	C₃×#3667	—	2	1	no	—
15	C₃×#3667	C₂²	1	2	yes	—
16	C₂×C₃×C₉⋊C₈	—	2	1	no	—
17	C₂×#3748	—	3	1	no	—
18	C₂×C₃×#208	—	3	1	no	—
19	C₂×C₃×#208	D₃	1	1	yes	characteristic
20	#63577	—	3	4	no	—
21	C₂×C₉×M₁₆	—	3	1	no	—
22	C₂×#3667	—	3	1	no	—
23	C₂×#3667	—	6	1	no	—
24	C₂×#3667	D₃	1	1	yes	characteristic
25	#63372	—	3	4	no	—
26	C₂³×C₃×C₉	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	Q₈	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
33	C₃⋊(C₉⋊C₈)	—	2	2	no	—
34	C₉×C₃⋊C₈	C₂×C₄	1	4	yes	—
35	C₃×C₉⋊C₈	—	2	2	no	—
36	#5125	—	9	1	no	—
37	#5124	—	9	1	no	—
38	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
39	#3748	—	3	2	no	—
40	#3748	—	6	1	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
42	C₃×#208	—	3	2	no	—
43	C₃×#208	—	6	1	no	—
44	C₃×#208	Dic₃	1	2	yes	—
45	C₃×#208	Dic₃	1	2	yes	—
46	C₂×C₃×C₃⋊C₈	—	6	1	no	—
47	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
50	C₉×M₁₆	—	3	2	no	—
51	C₉×M₁₆	—	3	2	no	—
52	C₂×C₈×C₉	—	3	2	no	—
53	#3667	—	2	1	no	—
54	#3667	—	3	2	no	—
55	#3667	—	6	2	no	—
56	#3667	—	6	2	no	—
57	#3667	—	6	1	no	—
58	#3667	C₂×D₃	1	2	yes	—
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂×C₉⋊C₈	—	6	2	no	—
61	C₂×C₉⋊C₈	—	6	1	no	—
62	C₂²×C₃×C₉	—	4	1	no	—
63	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
66	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
68	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×M₁₆	—	3	1	no	—
71	C₂×C₃×M₁₆	—	9	1	no	—
72	C₂×#208	—	3	1	no	—
73	C₂×#208	—	9	1	no	—
74	C₂×#208	—	9	1	no	—
75	C₂×#208	—	18	1	no	—
76	C₂×#208	D₉	1	1	yes	characteristic
77	#822	—	9	4	no	—
78	#819	—	9	4	no	—
79	C₂³×C₃²	Dic₆	1	1	yes	characteristic
80	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
81	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
82	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
86	C₂³×C₉	—	2	1	no	—
87	C₂³×C₉	Dic₆	1	1	yes	characteristic
88	C₃⋊(C₃⋊C₈)	—	6	2	no	—
89	C₃×C₃⋊C₈	—	6	2	no	—
90	C₃×C₃⋊C₈	C₂×Dic₃	1	4	yes	—
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	—	2	1	no	—
97	C₂×C₄×C₉	Dic₆	1	1	yes	characteristic
98	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
99	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
100	C₂×C₄×C₉	#67	1	1	yes	characteristic
101	C₂×C₄×C₉	#67	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
103	C₈×C₉	—	3	4	no	—
104	C₉⋊C₈	—	2	2	no	—
105	C₉⋊C₈	—	6	4	no	—
106	C₉⋊C₈	—	6	2	no	—
107	#430	—	27	1	no	Sylow
108	C₂×C₃×C₉	—	2	1	no	—
109	C₂×C₃×C₉	—	2	1	no	—
110	C₂×C₃×C₉	—	2	1	no	—
111	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
114	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
115	C₃×M₁₆	—	3	2	no	—
116	C₃×M₁₆	—	3	2	no	—
117	C₃×M₁₆	—	9	2	no	—
118	C₃×M₁₆	—	18	1	no	—
119	C₂×C₃×C₈	—	3	2	no	—
120	C₂×C₃×C₈	—	18	1	no	—
121	#208	—	3	2	no	—
122	#208	—	6	1	no	—
123	#208	—	9	2	no	—
124	#208	—	9	2	no	—
125	#208	—	18	2	no	—
126	#208	—	18	2	no	—
127	#208	—	18	1	no	—
128	#208	—	18	1	no	—
129	#208	Dic₉	1	2	yes	—
130	#208	Dic₉	1	2	yes	—
131	C₂×C₃⋊C₈	—	6	1	no	—
132	C₂×C₃⋊C₈	—	18	2	no	—
133	C₂×C₃⋊C₈	—	18	1	no	—
134	C₂×C₃⋊C₈	—	18	1	no	—
135	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
136	C₂²×C₃²	—	4	1	no	—
137	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
138	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
139	C₂²×C₃²	C₂×Dic₆	1	1	yes	characteristic
140	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
141	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
142	C₃²×C₄	C₂×D₁₂	1	1	yes	characteristic
143	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	—	2	1	no	—
146	C₂²×C₉	—	2	1	no	—
147	C₂²×C₉	—	4	1	no	—
148	C₂²×C₉	—	4	2	no	—
149	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
150	C₂²×C₉	Dic₃⋊C₄	1	1	yes	characteristic
151	C₂²×C₉	C₂×Dic₆	1	1	yes	characteristic
152	C₄×C₉	—	2	1	no	—
153	C₄×C₉	—	2	1	no	—
154	C₄×C₉	—	2	1	no	—
155	C₄×C₉	—	2	1	no	—
156	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
157	C₄×C₉	Dic₃⋊C₄	1	1	yes	characteristic
158	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
159	C₄×C₉	C₂×#67	1	1	yes	characteristic
160	C₂×M₁₆	—	3	1	no	—
161	C₂×M₁₆	—	9	1	no	—
162	C₂×M₁₆	—	27	1	no	—
163	#108	—	27	4	no	—
164	C₃×C₉	#430	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	Dic₁₈	1	1	yes	characteristic
167	C₂³×C₃	#627	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	1	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	Dic₁₈	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
176	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
180	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
181	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
182	C₂×C₃×C₄	#626	1	1	yes	characteristic
183	C₂×C₃×C₄	#626	1	1	yes	characteristic
184	C₂×C₃×C₄	#627	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
186	C₃×C₈	—	3	4	no	—
187	C₃×C₈	—	18	2	no	—
188	C₃⋊C₈	—	6	2	no	—
189	C₃⋊C₈	—	18	4	no	—
190	C₃⋊C₈	—	18	2	no	—
191	C₃⋊C₈	—	18	2	no	—
192	C₃⋊C₈	C₂×Dic₉	1	4	yes	—
193	C₂×C₃²	—	2	1	no	—
194	C₂×C₃²	—	2	1	no	—
195	C₂×C₃²	—	2	1	no	—
196	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
197	C₂×C₉	—	2	2	no	—
198	C₂×C₉	—	2	1	no	—
199	C₂×C₉	—	2	1	no	—
200	C₂×C₉	—	2	1	no	—
201	C₂×C₉	—	2	1	no	—
202	C₂×C₉	—	4	1	no	—
203	C₂×C₉	—	4	1	no	—
204	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
206	M₁₆	—	3	2	no	—
207	M₁₆	—	3	2	no	—
208	M₁₆	—	9	2	no	—
209	M₁₆	—	18	1	no	—
210	M₁₆	—	27	2	no	—
211	M₁₆	—	54	1	no	—
212	C₂×C₈	—	3	2	no	—
213	C₂×C₈	—	18	1	no	—
214	C₂×C₈	—	54	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	4	1	no	—
219	C₂²×C₃	—	4	1	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	C₄⋊Dic₉	1	1	yes	characteristic
222	C₂²×C₃	C₄⋊Dic₉	1	1	yes	characteristic
223	C₂²×C₃	C₂×Dic₁₈	1	1	yes	characteristic
224	C₂²×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
225	C₂²×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
226	C₂²×C₃	C₂×#627	1	1	yes	characteristic
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
232	C₃×C₄	C₄⋊Dic₉	1	1	yes	characteristic
233	C₃×C₄	C₂×D₃₆	1	1	yes	characteristic
234	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
235	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
236	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
237	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
238	C₃×C₄	C₂×#626	1	1	yes	characteristic
239	C₃²	#5125	1	1	yes	characteristic
240	C₉	—	2	1	no	—
241	C₉	#5124	1	1	yes	characteristic, complemented
242	C₂³	#6189	1	1	yes	characteristic
243	C₂×C₄	#6189	1	1	yes	characteristic
244	C₂×C₄	#6192	1	1	yes	characteristic
245	C₂×C₄	#6192	1	1	yes	characteristic
246	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
247	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
248	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
249	C₈	—	3	4	no	—
250	C₈	—	18	2	no	—
251	C₈	—	54	2	no	—
252	C₂×C₃	—	2	2	no	—
253	C₂×C₃	—	2	1	no	—
254	C₂×C₃	—	2	1	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	4	1	no	—
261	C₂×C₃	—	4	1	no	—
262	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
263	C₂×C₃	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
264	C₂²	—	4	1	no	—
265	C₂²	Dic₃⋊Dic₉	1	1	yes	characteristic
266	C₂²	Dic₃⋊Dic₉	1	1	yes	characteristic
267	C₂²	C₂×#6189	1	1	yes	characteristic
268	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
269	C₄	Dic₃⋊Dic₉	1	1	yes	characteristic
270	C₄	C₂×#6192	1	1	yes	characteristic
271	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
272	C₃	—	2	1	no	—
273	C₃	#10588800	1	1	yes	characteristic, complemented
274	C₃	#10591141	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	C₂×Dic₃⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#6192
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂×C₃²	C₂×C₄⋊Dic₃
prime radical	C₂³×C₃²	Dic₆
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#430
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11720280	C₂	semidirect
164. C₃×C₉	#430	semidirect
241. C₉	#5124	semidirect
273. C₃	#10588800	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88532	2
C₂×C₉×#208	2
C₂×C₃×#3667	2
#11720280	2
#10591141	3
#10588800	3