Subgroups — #13144848

Order 1728.

790 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720571	C₂	1	4	yes	maximal, complemented
5	#10592933	—	3	1	no	maximal
6	#10588912	—	3	3	no	maximal
7	C₃×C₉×#41	—	2	2	no	—
8	C₃×C₉×#41	—	2	2	no	—
9	C₂×C₃×Q₈×C₉	C₄	1	1	yes	characteristic
10	C₂×C₃×D₄×C₉	C₄	1	2	yes	—
11	C₂×C₃×D₄×C₉	C₄	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
13	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
14	#88532	—	2	2	no	—
15	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
16	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₃²×#41	D₃	1	1	yes	characteristic
19	C₂×#3748	—	3	1	no	—
20	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
21	C₂×C₉×#41	D₃	1	3	yes	—
22	#63657	—	3	4	no	—
23	C₂×#3667	—	3	3	no	—
24	C₂²×C₉⋊C₈	—	3	3	no	—
25	#63389	—	3	12	no	—
26	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
27	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
28	C₃×Q₈×C₉	—	2	2	no	—
29	C₃×D₄×C₉	—	2	4	no	—
30	C₃×D₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	—	2	2	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
36	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₃⋊(C₉⋊C₈)	—	2	2	no	—
39	C₃⋊(C₉⋊C₈)	—	2	2	no	—
40	#5234	—	9	3	no	—
41	#5234	—	9	1	no	—
42	C₃²×#41	—	2	2	no	—
43	C₃²×#41	—	2	2	no	—
44	C₂×C₃²×Q₈	Dic₃	1	1	yes	characteristic
45	C₂×C₃²×D₄	Dic₃	1	2	yes	—
46	C₂×C₃²×D₄	Dic₃	1	1	yes	characteristic
47	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
48	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
49	#3748	—	6	2	no	—
50	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
51	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
52	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
53	C₉×#41	—	2	6	no	—
54	C₉×#41	—	2	6	no	—
55	C₂×Q₈×C₉	Dic₃	1	3	yes	—
56	C₂×D₄×C₉	Dic₃	1	6	yes	—
57	C₂×D₄×C₉	Dic₃	1	3	yes	—
58	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
59	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
60	#3667	—	6	6	no	—
61	C₂×C₉⋊C₈	—	3	6	no	—
62	C₂×C₉⋊C₈	—	3	6	no	—
63	C₂×C₉⋊C₈	—	6	6	no	—
64	C₂²×C₃×C₉	—	2	2	no	—
65	C₂²×C₃×C₉	—	2	2	no	—
66	C₂²×C₃×C₉	—	2	4	no	—
67	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄×C₉	—	2	2	no	—
70	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
71	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
72	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×#41	D₉	1	3	yes	—
74	C₂×C₃×#41	C₃⋊D₃	1	1	yes	characteristic
75	C₂×#208	—	9	3	no	—
76	C₂×#208	—	9	1	no	—
77	C₂²×C₃⋊C₈	—	9	3	no	—
78	C₂²×C₃⋊C₈	—	9	1	no	—
79	#830	—	9	12	no	—
80	#830	—	9	4	no	—
81	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
82	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
83	C₃²×Q₈	—	2	2	no	—
84	C₃²×D₄	—	2	4	no	—
85	C₃²×D₄	—	2	2	no	—
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	—	2	2	no	—
88	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
89	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
90	C₂×C₃²×C₄	#67	1	2	yes	—
91	C₂×C₃²×C₄	#67	1	2	yes	—
92	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
93	C₂³×C₉	C₂×Dic₃	1	6	yes	—
94	C₂³×C₉	C₂×Dic₃	1	3	yes	—
95	C₃⋊(C₃⋊C₈)	—	6	2	no	—
96	C₃⋊(C₃⋊C₈)	—	6	2	no	—
97	Q₈×C₉	—	2	6	no	—
98	D₄×C₉	—	2	12	no	—
99	D₄×C₉	—	2	6	no	—
100	C₂×C₄×C₉	—	2	6	no	—
101	C₂×C₄×C₉	—	2	6	no	—
102	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
103	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
104	C₂×C₄×C₉	#67	1	6	yes	—
105	C₂×C₄×C₉	#67	1	6	yes	—
106	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
107	C₉⋊C₈	—	6	6	no	—
108	C₉⋊C₈	—	6	6	no	—
109	#408	—	27	1	no	Sylow
110	C₂×C₃×C₉	—	2	2	no	—
111	C₂×C₃×C₉	—	4	2	no	—
112	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₃×C₉	#131	1	2	yes	—
114	C₃×#41	—	2	6	no	—
115	C₃×#41	—	2	6	no	—
116	C₃×#41	—	2	2	no	—
117	C₃×#41	—	2	2	no	—
118	C₂×C₃×Q₈	Dic₉	1	3	yes	—
119	C₂×C₃×Q₈	C₃⋊Dic₃	1	1	yes	characteristic
120	C₂×C₃×D₄	Dic₉	1	6	yes	—
121	C₂×C₃×D₄	Dic₉	1	3	yes	—
122	C₂×C₃×D₄	C₃⋊Dic₃	1	2	yes	—
123	C₂×C₃×D₄	C₃⋊Dic₃	1	1	yes	characteristic
124	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
125	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
126	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
127	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
128	#208	—	18	6	no	—
129	#208	—	18	2	no	—
130	C₂×C₃⋊C₈	—	9	6	no	—
131	C₂×C₃⋊C₈	—	9	6	no	—
132	C₂×C₃⋊C₈	—	9	2	no	—
133	C₂×C₃⋊C₈	—	9	2	no	—
134	C₂×C₃⋊C₈	—	18	6	no	—
135	C₂×C₃⋊C₈	—	18	2	no	—
136	C₂²×C₃²	—	2	2	no	—
137	C₂²×C₃²	—	2	4	no	—
138	C₂²×C₃²	—	2	2	no	—
139	C₂²×C₃²	#217	1	2	yes	—
140	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
141	C₃²×C₄	—	2	2	no	—
142	C₃²×C₄	#217	1	2	yes	—
143	C₃²×C₄	C₂×#67	1	1	yes	characteristic
144	C₃²×C₄	C₂×#67	1	1	yes	characteristic
145	C₂²×C₉	—	2	6	no	—
146	C₂²×C₉	—	2	12	no	—
147	C₂²×C₉	—	2	6	no	—
148	C₂²×C₉	#217	1	6	yes	—
149	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
150	C₄×C₉	—	2	6	no	—
151	C₄×C₉	#217	1	6	yes	—
152	C₄×C₉	C₂×#67	1	3	yes	—
153	C₄×C₉	C₂×#67	1	3	yes	—
154	C₂×#41	C₃⋊D₉	1	1	yes	characteristic
155	C₂×M₁₆	—	27	1	no	—
156	C₂²×C₈	—	27	1	no	—
157	C₂²⋊C₈	—	27	4	no	—
158	C₃×C₉	#408	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₃	C₂×Dic₉	1	6	yes	—
160	C₂³×C₃	C₂×Dic₉	1	3	yes	—
161	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
162	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
163	C₃×Q₈	—	2	6	no	—
164	C₃×Q₈	—	2	2	no	—
165	C₃×D₄	—	2	12	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	6	no	—
168	C₃×D₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	6	no	—
170	C₂×C₃×C₄	—	2	6	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
174	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
175	C₂×C₃×C₄	#611	1	6	yes	—
176	C₂×C₃×C₄	#611	1	6	yes	—
177	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
178	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
180	C₂×C₃×C₄	#638	1	2	yes	—
181	C₂×C₃×C₄	#638	1	2	yes	—
182	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
183	C₃⋊C₈	—	18	6	no	—
184	C₃⋊C₈	—	18	6	no	—
185	C₃⋊C₈	—	18	2	no	—
186	C₃⋊C₈	—	18	2	no	—
187	C₂×C₃²	—	2	2	no	—
188	C₂×C₃²	—	4	2	no	—
189	C₂×C₃²	#948	1	2	yes	—
190	C₂×C₃²	C₂×#217	1	1	yes	characteristic
191	C₂×C₉	—	2	6	no	—
192	C₂×C₉	—	4	6	no	—
193	C₂×C₉	#948	1	6	yes	—
194	C₂×C₉	C₂×#217	1	3	yes	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	C₃⋊Dic₉	1	1	yes	characteristic
198	C₂×D₄	C₃⋊Dic₉	1	2	yes	—
199	C₂×D₄	C₃⋊Dic₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
201	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
202	M₁₆	—	54	2	no	—
203	C₂×C₈	—	27	2	no	—
204	C₂×C₈	—	27	2	no	—
205	C₂×C₈	—	54	2	no	—
206	C₂²×C₃	—	2	6	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	12	no	—
209	C₂²×C₃	—	2	6	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	#3676	1	6	yes	—
213	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
214	C₂²×C₃	#3757	1	2	yes	—
215	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
216	C₃×C₄	—	2	6	no	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	#3676	1	6	yes	—
219	C₃×C₄	C₂×#611	1	3	yes	—
220	C₃×C₄	C₂×#611	1	3	yes	—
221	C₃×C₄	#3757	1	2	yes	—
222	C₃×C₄	C₂×#638	1	1	yes	characteristic
223	C₃×C₄	C₂×#638	1	1	yes	characteristic
224	C₃²	#5234	1	1	yes	characteristic
225	C₉	#5234	1	3	yes	complemented
226	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
227	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
234	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
235	C₂×C₄	#6233	1	2	yes	—
236	C₂×C₄	#6233	1	2	yes	—
237	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
238	C₈	—	54	2	no	—
239	C₈	—	54	2	no	—
240	C₂×C₃	—	2	2	no	—
241	C₂×C₃	—	2	6	no	—
242	C₂×C₃	—	4	2	no	—
243	C₂×C₃	—	4	6	no	—
244	C₂×C₃	#63510	1	6	yes	—
245	C₂×C₃	C₂×#3676	1	3	yes	—
246	C₂×C₃	#64157	1	2	yes	—
247	C₂×C₃	C₂×#3757	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	2	no	—
251	C₂²	#88551	1	2	yes	—
252	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	#88551	1	2	yes	—
255	C₄	C₂×#6233	1	1	yes	characteristic
256	C₄	C₂×#6233	1	1	yes	characteristic
257	C₃	#10588912	1	3	yes	complemented
258	C₃	#10592933	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	—	4	2	no	—
261	C₂	#11721865	1	2	yes	—
262	C₂	C₂×#88551	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₉×#41	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂×C₃²×#41	D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×#41	C₃⋊D₉
hypocenter	C₃×C₉	#408
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#88532	C₂	semidirect
3. C₂²×C₃⋊(C₉⋊C₈)	C₂	semidirect
4. #11720571	C₂	semidirect
158. C₃×C₉	#408	semidirect
225. C₉	#5234	semidirect
257. C₃	#10588912	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×#41	2
C₂×#88532	2
C₂²×C₃⋊(C₉⋊C₈)	2
#11720571	2
#10592933	3
#10588912	3