Subgroups — #13144797

Order 1728.

950 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	2	yes	maximal, complemented
3	#11720571	C₂	1	4	yes	maximal, complemented
4	#10592906	—	3	1	no	maximal
5	#10588887	—	3	3	no	maximal
6	C₂⁴×C₃×C₉	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	—	2	4	no	—
8	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	#88532	—	2	4	no	—
13	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	4	yes	complemented
14	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
15	C₂×#3748	—	3	2	no	—
16	C₂³×C₄×C₉	D₃	1	3	yes	—
17	#63657	—	3	4	no	—
18	C₂×#3667	—	3	6	no	—
19	#63389	—	3	12	no	—
20	C₂³×C₃×C₉	—	2	2	no	—
21	C₂³×C₃×C₉	—	2	4	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₃×C₉	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₄×C₉	—	2	2	no	—
25	C₂×C₃×C₄×C₉	—	2	4	no	—
26	C₂×C₃×C₄×C₉	—	2	4	no	—
27	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	4	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₃⋊(C₉⋊C₈)	—	2	4	no	—
32	#5207	—	9	3	no	—
33	#5207	—	9	1	no	—
34	C₂⁴×C₃²	Dic₃	1	1	yes	characteristic
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
37	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
38	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
39	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₂⁴×C₉	Dic₃	1	3	yes	—
41	#3748	—	6	4	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	4	no	—
43	C₂²×C₄×C₉	—	2	12	no	—
44	C₂²×C₄×C₉	Dic₃	1	3	yes	—
45	C₂²×C₄×C₉	Dic₃	1	6	yes	—
46	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
47	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
48	#3667	—	6	12	no	—
49	C₂×C₉⋊C₈	—	3	12	no	—
50	C₂²×C₃×C₉	—	2	8	no	—
51	C₂²×C₃×C₉	—	2	2	no	—
52	C₂²×C₃×C₉	—	2	2	no	—
53	C₂²×C₃×C₉	—	2	2	no	—
54	C₂²×C₃×C₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃×C₉	M₁₆	1	4	yes	complemented
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄×C₉	—	2	2	no	—
58	C₃×C₄×C₉	C₂²⋊C₄	1	2	yes	—
59	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
60	C₂³×C₃×C₄	D₉	1	3	yes	—
61	C₂³×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
62	C₂×#208	—	9	6	no	—
63	C₂×#208	—	9	2	no	—
64	#830	—	9	12	no	—
65	#830	—	9	4	no	—
66	C₂³×C₃²	—	2	4	no	—
67	C₂³×C₃²	—	2	2	no	—
68	C₂³×C₃²	C₂×Dic₃	1	2	yes	—
69	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	—	2	4	no	—
72	C₂×C₃²×C₄	—	2	4	no	—
73	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
74	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
75	C₂×C₃²×C₄	#67	1	4	yes	—
76	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂³×C₉	—	2	12	no	—
78	C₂³×C₉	—	2	6	no	—
79	C₂³×C₉	C₂×Dic₃	1	6	yes	—
80	C₂³×C₉	C₂×Dic₃	1	3	yes	—
81	C₃⋊(C₃⋊C₈)	—	6	4	no	—
82	C₂×C₄×C₉	—	2	6	no	—
83	C₂×C₄×C₉	—	2	12	no	—
84	C₂×C₄×C₉	—	2	12	no	—
85	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
86	C₂×C₄×C₉	C₂×Dic₃	1	3	yes	—
87	C₂×C₄×C₉	#67	1	12	yes	—
88	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
89	C₉⋊C₈	—	6	12	no	—
90	#407	—	27	1	no	Sylow
91	C₂×C₃×C₉	—	2	2	no	—
92	C₂×C₃×C₉	—	2	4	no	—
93	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
95	C₂⁴×C₃	Dic₉	1	3	yes	—
96	C₂⁴×C₃	C₃⋊Dic₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	—	2	12	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	Dic₉	1	3	yes	—
100	C₂²×C₃×C₄	Dic₉	1	6	yes	—
101	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
102	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
103	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₃⋊Dic₃	1	2	yes	—
105	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
106	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
107	#208	—	18	12	no	—
108	#208	—	18	4	no	—
109	C₂×C₃⋊C₈	—	9	12	no	—
110	C₂×C₃⋊C₈	—	9	4	no	—
111	C₂²×C₃²	—	2	8	no	—
112	C₂²×C₃²	—	2	2	no	—
113	C₂²×C₃²	—	2	2	no	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	#208	1	4	yes	—
116	C₂²×C₃²	#217	1	2	yes	—
117	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	#217	1	2	yes	—
120	C₃²×C₄	C₂×#67	1	2	yes	—
121	C₂²×C₉	—	2	24	no	—
122	C₂²×C₉	—	2	6	no	—
123	C₂²×C₉	—	2	6	no	—
124	C₂²×C₉	—	2	6	no	—
125	C₂²×C₉	#208	1	12	yes	complemented
126	C₂²×C₉	#217	1	6	yes	—
127	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
128	C₄×C₉	—	2	6	no	—
129	C₄×C₉	#217	1	6	yes	—
130	C₄×C₉	C₂×#67	1	6	yes	—
131	C₂³×C₄	C₃⋊D₉	1	1	yes	characteristic
132	C₂×M₁₆	—	27	2	no	—
133	C₂²⋊C₈	—	27	4	no	—
134	C₃×C₉	#407	1	1	yes	characteristic, complemented, Sylow
135	C₂³×C₃	—	2	12	no	—
136	C₂³×C₃	—	2	6	no	—
137	C₂³×C₃	—	2	4	no	—
138	C₂³×C₃	—	2	2	no	—
139	C₂³×C₃	C₂×Dic₉	1	6	yes	—
140	C₂³×C₃	C₂×Dic₉	1	3	yes	—
141	C₂³×C₃	C₂×C₃⋊Dic₃	1	2	yes	—
142	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
143	C₂×C₃×C₄	—	2	6	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	12	no	—
146	C₂×C₃×C₄	—	2	12	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
150	C₂×C₃×C₄	C₂×Dic₉	1	3	yes	—
151	C₂×C₃×C₄	#611	1	12	yes	—
152	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
153	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
154	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
155	C₂×C₃×C₄	#638	1	4	yes	—
156	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
157	C₃⋊C₈	—	18	12	no	—
158	C₃⋊C₈	—	18	4	no	—
159	C₂×C₃²	—	2	4	no	—
160	C₂×C₃²	—	2	2	no	—
161	C₂×C₃²	C₂×#208	1	2	yes	—
162	C₂×C₃²	C₂×#217	1	1	yes	characteristic
163	C₂×C₉	—	2	12	no	—
164	C₂×C₉	—	2	6	no	—
165	C₂×C₉	C₂×#208	1	6	yes	—
166	C₂×C₉	C₂×#217	1	3	yes	—
167	C₂⁴	C₃⋊Dic₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	C₃⋊Dic₉	1	1	yes	characteristic
170	C₂²×C₄	C₃⋊Dic₉	1	2	yes	—
171	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
172	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
173	M₁₆	—	54	4	no	—
174	C₂×C₈	—	27	4	no	—
175	C₂²×C₃	—	2	8	no	—
176	C₂²×C₃	—	2	24	no	—
177	C₂²×C₃	—	2	6	no	—
178	C₂²×C₃	—	2	2	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	6	no	—
182	C₂²×C₃	—	2	6	no	—
183	C₂²×C₃	#3667	1	12	yes	complemented
184	C₂²×C₃	#3676	1	6	yes	—
185	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
186	C₂²×C₃	#3748	1	4	yes	—
187	C₂²×C₃	#3757	1	2	yes	—
188	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
189	C₃×C₄	—	2	6	no	—
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	#3676	1	6	yes	—
192	C₃×C₄	C₂×#611	1	6	yes	—
193	C₃×C₄	#3757	1	2	yes	—
194	C₃×C₄	C₂×#638	1	2	yes	—
195	C₃²	#5207	1	1	yes	characteristic
196	C₉	#5207	1	3	yes	complemented
197	C₂³	—	2	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	C₂×C₃⋊Dic₉	1	2	yes	—
200	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
205	C₂×C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
206	C₂×C₄	#6233	1	4	yes	—
207	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
208	C₈	—	54	4	no	—
209	C₂×C₃	—	2	4	no	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	—	2	12	no	—
212	C₂×C₃	—	2	6	no	—
213	C₂×C₃	C₂×#3667	1	6	yes	—
214	C₂×C₃	C₂×#3676	1	3	yes	—
215	C₂×C₃	C₂×#3748	1	2	yes	—
216	C₂×C₃	C₂×#3757	1	1	yes	characteristic
217	C₂²	—	2	8	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	#88532	1	4	yes	complemented
222	C₂²	#88551	1	2	yes	—
223	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	#88551	1	2	yes	—
226	C₄	C₂×#6233	1	2	yes	—
227	C₃	#10588887	1	3	yes	complemented
228	C₃	#10592906	1	1	yes	characteristic
229	C₂	—	2	4	no	—
230	C₂	—	2	2	no	—
231	C₂	C₂×#88532	1	2	yes	—
232	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	C₂³×C₃×C₄×C₉	C₂
socle	C₂²×C₃²	C₂²×Dic₃
prime radical	C₂⁴×C₃²	Dic₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂³×C₄	C₃⋊D₉
hypocenter	C₃×C₉	#407
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#88532	C₂	semidirect
3. #11720571	C₂	semidirect
13. C₂×C₃⋊(C₉⋊C₈)	C₂²	semidirect
55. C₂²×C₃×C₉	M₁₆	semidirect
125. C₂²×C₉	#208	semidirect
134. C₃×C₉	#407	semidirect
183. C₂²×C₃	#3667	semidirect
196. C₉	#5207	semidirect
221. C₂²	#88532	semidirect
227. C₃	#10588887	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂×#88532	2
#11720571	2
#10592906	3
#10588887	3