Subgroups — #13135890

Order 1728.

450 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#88532	C₂	1	1	yes	maximal, characteristic
3	C₂×#88532	C₂	1	1	yes	maximal, characteristic
4	#10589399	—	3	1	no	maximal
5	#10588209	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₃×C₉×C₄⋊C₄	—	2	1	no	—
8	C₃×C₉×C₂²⋊C₄	C₄	1	2	yes	—
9	C₃×C₄²×C₉	—	2	1	no	—
10	#88532	—	2	1	no	—
11	#88532	—	2	1	no	—
12	#88532	—	2	1	no	—
13	#88532	C₄	1	2	yes	complemented
14	C₂×C₃⋊(C₉⋊C₈)	—	2	1	no	—
15	C₂×C₃⋊(C₉⋊C₈)	C₄	1	2	yes	complemented
16	C₃²×#117	D₃	1	1	yes	characteristic
17	C₂×#3748	—	3	1	no	—
18	C₂×#3748	—	3	1	no	—
19	C₉×#117	D₃	1	3	yes	—
20	C₂×#3667	—	3	3	no	—
21	C₂×#3667	—	3	3	no	—
22	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	—	2	2	no	—
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	C₃⋊(C₉⋊C₈)	—	2	2	no	—
31	C₃⋊(C₉⋊C₈)	—	2	2	no	—
32	#4533	—	9	3	no	—
33	#4533	—	9	1	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₃²×C₄⋊C₄	—	2	1	no	—
36	C₃²×C₂²⋊C₄	Dic₃	1	2	yes	—
37	C₃²×C₄²	—	2	1	no	—
38	#3748	—	3	2	no	—
39	#3748	—	6	1	no	—
40	#3748	—	6	1	no	—
41	#3748	—	6	1	no	—
42	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
43	C₂×C₃⋊(C₃⋊C₈)	—	6	1	no	—
44	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
45	C₉×C₄⋊C₄	—	2	3	no	—
46	C₉×C₂²⋊C₄	Dic₃	1	6	yes	—
47	C₄²×C₉	—	2	3	no	—
48	#3667	—	3	6	no	—
49	#3667	—	6	3	no	—
50	#3667	—	6	3	no	—
51	#3667	—	6	3	no	—
52	C₂×C₉⋊C₈	—	3	6	no	—
53	C₂×C₉⋊C₈	—	6	3	no	—
54	C₂²×C₃×C₉	—	4	1	no	—
55	C₂²×C₃×C₉	C₄²	1	1	yes	characteristic
56	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
58	C₃×C₄×C₉	—	4	2	no	—
59	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
62	C₃×C₄×C₉	C₄⋊C₄	1	1	yes	characteristic
63	C₃×#117	D₉	1	3	yes	—
64	C₃×#117	C₃⋊D₃	1	1	yes	characteristic
65	C₂×#208	—	9	3	no	—
66	C₂×#208	—	9	1	no	—
67	C₂×#208	—	9	3	no	—
68	C₂×#208	—	9	1	no	—
69	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	—	2	2	no	—
71	C₂×C₃²×C₄	Dic₆	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
73	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
74	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
75	C₂×C₃²×C₄	#67	1	1	yes	characteristic
76	C₂×C₃²×C₄	#67	1	1	yes	characteristic
77	C₂³×C₉	C₂×Dic₃	1	3	yes	—
78	C₃⋊(C₃⋊C₈)	—	6	2	no	—
79	C₃⋊(C₃⋊C₈)	—	6	2	no	—
80	C₂×C₄×C₉	—	2	6	no	—
81	C₂×C₄×C₉	Dic₆	1	3	yes	—
82	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
83	C₂×C₄×C₉	C₄×D₃	1	3	yes	—
84	C₂×C₄×C₉	D₁₂	1	3	yes	—
85	C₂×C₄×C₉	#67	1	3	yes	—
86	C₂×C₄×C₉	#67	1	3	yes	—
87	C₉⋊C₈	—	6	6	no	—
88	C₉⋊C₈	—	6	6	no	—
89	#344	—	27	1	no	Sylow
90	C₂×C₃×C₉	—	2	1	no	—
91	C₂×C₃×C₉	—	2	1	no	—
92	C₂×C₃×C₉	—	2	1	no	—
93	C₂×C₃×C₉	#95	1	1	yes	characteristic
94	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
95	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
96	C₃×C₄⋊C₄	—	2	3	no	—
97	C₃×C₄⋊C₄	—	2	1	no	—
98	C₃×C₂²⋊C₄	Dic₉	1	6	yes	—
99	C₃×C₂²⋊C₄	C₃⋊Dic₃	1	2	yes	—
100	C₃×C₄²	—	2	3	no	—
101	C₃×C₄²	—	2	1	no	—
102	#208	—	9	6	no	—
103	#208	—	9	2	no	—
104	#208	—	18	3	no	—
105	#208	—	18	3	no	—
106	#208	—	18	3	no	—
107	#208	—	18	1	no	—
108	#208	—	18	1	no	—
109	#208	—	18	1	no	—
110	C₂×C₃⋊C₈	—	9	6	no	—
111	C₂×C₃⋊C₈	—	9	2	no	—
112	C₂×C₃⋊C₈	—	18	3	no	—
113	C₂×C₃⋊C₈	—	18	1	no	—
114	C₂²×C₃²	—	4	1	no	—
115	C₂²×C₃²	C₄×Dic₃	1	1	yes	characteristic
116	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
117	C₂²×C₃²	#217	1	1	yes	characteristic
118	C₃²×C₄	—	4	2	no	—
119	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
120	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
121	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
122	C₃²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
123	C₂²×C₉	—	4	3	no	—
124	C₂²×C₉	C₄×Dic₃	1	3	yes	—
125	C₂²×C₉	C₄⋊Dic₃	1	3	yes	—
126	C₂²×C₉	#217	1	3	yes	—
127	C₄×C₉	—	4	6	no	—
128	C₄×C₉	Dic₃⋊C₄	1	3	yes	—
129	C₄×C₉	Dic₃⋊C₄	1	3	yes	—
130	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
131	C₄×C₉	(C₂×D₃)⋊C₄	1	3	yes	—
132	C₂×M₁₆	—	27	1	no	—
133	C₂×M₁₆	—	27	1	no	—
134	#117	C₃⋊D₉	1	1	yes	characteristic
135	C₃×C₉	#344	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₃	C₂×Dic₉	1	3	yes	—
137	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
138	C₂×C₃×C₄	—	2	6	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	Dic₁₈	1	3	yes	—
141	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
142	C₂×C₃×C₄	C₄×D₉	1	3	yes	—
143	C₂×C₃×C₄	D₃₆	1	3	yes	—
144	C₂×C₃×C₄	#611	1	3	yes	—
145	C₂×C₃×C₄	#611	1	3	yes	—
146	C₂×C₃×C₄	#634	1	1	yes	characteristic
147	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
148	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
149	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
150	C₂×C₃×C₄	#638	1	1	yes	characteristic
151	C₂×C₃×C₄	#638	1	1	yes	characteristic
152	C₃⋊C₈	—	18	6	no	—
153	C₃⋊C₈	—	18	6	no	—
154	C₃⋊C₈	—	18	2	no	—
155	C₃⋊C₈	—	18	2	no	—
156	C₂×C₃²	—	2	1	no	—
157	C₂×C₃²	—	2	1	no	—
158	C₂×C₃²	—	2	1	no	—
159	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
160	C₂×C₉	—	2	3	no	—
161	C₂×C₉	—	2	3	no	—
162	C₂×C₉	—	2	3	no	—
163	C₂×C₉	C₃⋊#95	1	3	yes	—
164	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
165	M₁₆	—	27	2	no	—
166	M₁₆	—	54	1	no	—
167	M₁₆	—	54	1	no	—
168	M₁₆	—	54	1	no	—
169	C₂×C₈	—	27	2	no	—
170	C₂×C₈	—	54	1	no	—
171	C₄⋊C₄	—	2	1	no	—
172	C₂²⋊C₄	C₃⋊Dic₉	1	2	yes	—
173	C₄²	—	2	1	no	—
174	C₂²×C₃	—	4	3	no	—
175	C₂²×C₃	—	4	1	no	—
176	C₂²×C₃	C₄×Dic₉	1	3	yes	—
177	C₂²×C₃	C₄⋊Dic₉	1	3	yes	—
178	C₂²×C₃	#3676	1	3	yes	—
179	C₂²×C₃	C₄×C₃⋊Dic₃	1	1	yes	characteristic
180	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
181	C₂²×C₃	#3757	1	1	yes	characteristic
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	4	6	no	—
184	C₃×C₄	Dic₉⋊C₄	1	3	yes	—
185	C₃×C₄	Dic₉⋊C₄	1	3	yes	—
186	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
187	C₃×C₄	(C₂×D₉)⋊C₄	1	3	yes	—
188	C₃×C₄	#3750	1	1	yes	characteristic
189	C₃×C₄	#3750	1	1	yes	characteristic
190	C₃×C₄	#3752	1	1	yes	characteristic
191	C₃×C₄	#3752	1	1	yes	characteristic
192	C₃²	#4533	1	1	yes	characteristic
193	C₉	#4533	1	3	yes	complemented
194	C₂³	C₂×C₃⋊Dic₉	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	#6226	1	1	yes	characteristic
197	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
198	C₂×C₄	C₄×C₃⋊D₉	1	1	yes	characteristic
199	C₂×C₄	C₄⋊(C₃⋊D₉)	1	1	yes	characteristic
200	C₂×C₄	#6233	1	1	yes	characteristic
201	C₂×C₄	#6233	1	1	yes	characteristic
202	C₈	—	54	2	no	—
203	C₈	—	54	2	no	—
204	C₂×C₃	—	2	3	no	—
205	C₂×C₃	—	2	3	no	—
206	C₂×C₃	—	2	3	no	—
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	C₉⋊#95	1	3	yes	—
211	C₂×C₃	#63658	1	1	yes	characteristic
212	C₂²	—	4	1	no	—
213	C₂²	C₄×C₃⋊Dic₉	1	1	yes	characteristic
214	C₂²	C₄⋊(C₃⋊Dic₉)	1	1	yes	characteristic
215	C₂²	#88551	1	1	yes	characteristic
216	C₄	—	4	2	no	—
217	C₄	#88534	1	1	yes	characteristic
218	C₄	#88534	1	1	yes	characteristic
219	C₄	#88536	1	1	yes	characteristic
220	C₄	#88536	1	1	yes	characteristic
221	C₃	#10588209	1	3	yes	complemented
222	C₃	#10589399	1	1	yes	characteristic
223	C₂	—	2	1	no	—
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	#11720572	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#88536
commutator	C₂²×C₃×C₉	C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃⋊D₃
Fitting	C₃×C₉×#117	C₂
socle	C₂×C₃²	C₃⋊#95
prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂²×C₃×C₄×C₉	C₂²
hypercenter	#117	C₃⋊D₉
hypocenter	C₃×C₉	#344
LCS	G ▷ C₂²×C₃×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
13. #88532	C₄	semidirect
15. C₂×C₃⋊(C₉⋊C₈)	C₄	semidirect
135. C₃×C₉	#344	semidirect
193. C₉	#4533	semidirect
221. C₃	#10588209	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#117	2
C₂×#88532	2
#10589399	3
#10588209	3