Subgroups — #13139359

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₃×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
4	Dic₉⋊(C₃⋊C₈)	C₂	1	4	yes	maximal, complemented
5	#10591117	—	3	1	no	maximal
6	#10588803	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊(C₉⋊C₈)	—	2	2	no	—
9	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	2	yes	—
10	C₃×#3676	C₄	1	2	yes	—
11	C₃×C₄⋊Dic₉	C₄	1	2	yes	—
12	C₃×C₄×Dic₉	C₂²	1	2	yes	—
13	C₉×#208	—	2	2	no	—
14	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₃⋊(C₃⋊C₈)	—	3	1	no	—
16	C₃×#926	—	3	1	no	—
17	C₂×C₃×#208	D₃	1	1	yes	characteristic
18	Dic₃⋊(C₃⋊C₈)	—	3	4	no	—
19	C₂×C₉×M₁₆	—	3	1	no	—
20	#63488	D₃	1	1	yes	characteristic
21	C₂²×C₉⋊C₈	—	3	1	no	—
22	C₂²×C₉⋊C₈	—	6	1	no	—
23	Dic₉⋊C₈	—	3	4	no	—
24	C₂³×C₃×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	2	yes	—
27	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
28	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
29	C₂×C₃×Dic₉	C₂×C₄	1	4	yes	—
30	C₃⋊(C₉⋊C₈)	—	2	2	no	—
31	C₉×C₃⋊C₈	—	2	2	no	—
32	#5119	—	9	1	no	—
33	#4999	—	9	1	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂×C₃⋊(C₃⋊C₈)	—	3	2	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	6	2	no	—
37	C₃×#217	—	3	2	no	—
38	C₃×C₄⋊Dic₃	—	3	2	no	—
39	C₃×C₄×Dic₃	—	3	2	no	—
40	C₃×#208	—	2	2	no	—
41	C₂×C₃×C₃⋊C₈	C₂×D₃	1	2	yes	—
42	C₂²×C₄×C₉	—	2	1	no	—
43	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
44	C₉×M₁₆	—	6	2	no	—
45	C₂×C₈×C₉	—	3	2	no	—
46	#3676	Dic₃	1	2	yes	—
47	C₄⋊Dic₉	Dic₃	1	2	yes	—
48	C₄×Dic₉	C₂×D₃	1	2	yes	—
49	C₂×C₉⋊C₈	—	3	2	no	—
50	C₂×C₉⋊C₈	—	6	2	no	—
51	C₂×C₉⋊C₈	—	6	4	no	—
52	C₂×C₉⋊C₈	—	6	2	no	—
53	C₂²×C₃×C₉	—	2	2	no	—
54	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
57	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
58	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
59	C₃×Dic₉	—	2	4	no	—
60	C₂×C₃×M₁₆	—	3	1	no	—
61	C₃×#117	—	9	1	no	—
62	#926	—	3	1	no	—
63	C₂×#208	D₉	1	1	yes	characteristic
64	C₂²×C₃⋊C₈	—	9	1	no	—
65	C₂²×C₃⋊C₈	—	9	1	no	—
66	C₂²×C₃⋊C₈	—	18	1	no	—
67	Dic₃⋊C₈	—	9	4	no	—
68	C₄⋊(C₃⋊C₈)	—	9	4	no	—
69	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
70	C₂×C₃²×C₄	Dic₆	1	2	yes	—
71	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
72	C₂×C₃²×C₄	#67	1	2	yes	—
73	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₃×Dic₃	—	3	4	no	—
75	C₂³×C₉	—	2	1	no	—
76	C₂³×C₉	C₂×Dic₃	1	1	yes	characteristic
77	C₃⋊(C₃⋊C₈)	—	6	2	no	—
78	C₃×C₃⋊C₈	—	2	2	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	2	2	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	1	no	—
83	C₂×C₄×C₉	Dic₆	1	2	yes	—
84	C₂×C₄×C₉	D₁₂	1	2	yes	—
85	C₂×C₄×C₉	C₂×Dic₃	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
87	C₂×Dic₉	C₂×Dic₃	1	4	yes	—
88	C₈×C₉	—	6	2	no	—
89	C₉⋊C₈	—	6	4	no	—
90	C₉⋊C₈	—	6	2	no	—
91	#424	—	27	1	no	Sylow
92	C₂×C₃×C₉	—	2	2	no	—
93	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₉	#131	1	2	yes	—
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
97	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
98	C₃×M₁₆	—	6	2	no	—
99	C₂×C₃×C₈	—	3	2	no	—
100	C₃×C₄⋊C₄	—	9	2	no	—
101	C₃×C₂²⋊C₄	—	9	2	no	—
102	C₃×C₄²	—	9	2	no	—
103	#217	—	3	2	no	—
104	C₄⋊Dic₃	—	3	2	no	—
105	C₄×Dic₃	—	3	2	no	—
106	#208	—	2	2	no	—
107	C₂×C₃⋊C₈	—	9	2	no	—
108	C₂×C₃⋊C₈	—	9	2	no	—
109	C₂×C₃⋊C₈	—	18	2	no	—
110	C₂×C₃⋊C₈	—	18	4	no	—
111	C₂×C₃⋊C₈	—	18	2	no	—
112	C₂×C₃⋊C₈	—	18	2	no	—
113	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
114	C₂²×C₃²	—	2	2	no	—
115	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
116	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
117	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
118	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
119	C₃²×C₄	C₂×#67	1	1	yes	characteristic
120	C₃×Dic₃	—	6	4	no	—
121	C₂²×C₉	—	2	4	no	—
122	C₂²×C₉	—	2	2	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	1	no	—
125	C₂²×C₉	C₄⋊Dic₃	1	2	yes	—
126	C₂²×C₉	C₂²×Dic₃	1	1	yes	characteristic
127	C₄×C₉	—	2	2	no	—
128	C₄×C₉	—	2	1	no	—
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	C₄⋊Dic₃	1	2	yes	—
131	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
132	C₄×C₉	C₂×D₁₂	1	1	yes	characteristic
133	Dic₉	—	2	4	no	—
134	C₂×M₁₆	—	3	1	no	—
135	C₂²×C₈	—	27	1	no	—
136	#117	—	9	1	no	—
137	C₄⋊C₈	—	27	4	no	—
138	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
141	C₂³×C₃	D₃×Dic₃	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	2	no	—
143	C₂×C₃×C₄	—	2	1	no	—
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	9	4	no	—
147	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
148	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
149	C₂×C₃×C₄	#611	1	2	yes	—
150	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
151	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
152	C₂×C₃×C₄	#626	1	2	yes	—
153	C₂×C₃×C₄	#627	1	2	yes	—
154	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
155	C₂×Dic₃	—	3	4	no	—
156	C₃×C₈	—	6	2	no	—
157	C₃⋊C₈	—	2	2	no	—
158	C₃⋊C₈	—	18	4	no	—
159	C₃⋊C₈	—	18	2	no	—
160	C₃⋊C₈	—	18	2	no	—
161	C₂×C₃²	—	2	2	no	—
162	C₂×C₃²	#907	1	2	yes	—
163	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
164	C₂×C₉	—	2	4	no	—
165	C₂×C₉	—	2	2	no	—
166	C₂×C₉	—	2	2	no	—
167	C₂×C₉	—	2	1	no	—
168	C₂×C₉	C₂×C₄⋊Dic₃	1	1	yes	characteristic
169	C₂×C₉	#948	1	2	yes	—
170	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
171	M₁₆	—	6	2	no	—
172	C₂×C₈	—	3	2	no	—
173	C₂×C₈	—	27	2	no	—
174	C₂×C₈	—	54	2	no	—
175	C₄⋊C₄	—	9	2	no	—
176	C₂²⋊C₄	—	9	2	no	—
177	C₄²	—	9	2	no	—
178	C₂²×C₃	—	2	4	no	—
179	C₂²×C₃	—	2	2	no	—
180	C₂²×C₃	—	2	2	no	—
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	Dic₉⋊C₄	1	2	yes	—
184	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
185	C₂²×C₃	Dic₃⋊Dic₃	1	2	yes	—
186	C₂²×C₃	C₂×D₃×Dic₃	1	1	yes	characteristic
187	C₃×C₄	—	2	1	no	—
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	2	2	no	—
190	C₃×C₄	—	18	4	no	—
191	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
192	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
193	C₃×C₄	C₂×#611	1	1	yes	characteristic
194	C₃×C₄	Dic₃⋊Dic₃	1	2	yes	—
195	C₃×C₄	C₂×#626	1	1	yes	characteristic
196	C₃×C₄	C₂×#627	1	1	yes	characteristic
197	Dic₃	—	6	4	no	—
198	C₃²	#5119	1	1	yes	characteristic
199	C₉	—	2	1	no	—
200	C₉	#4999	1	1	yes	characteristic, complemented
201	C₂³	Dic₃×D₉	1	1	yes	characteristic
202	C₂×C₄	—	9	4	no	—
203	C₂×C₄	#6189	1	2	yes	—
204	C₂×C₄	Dic₃×D₉	1	1	yes	characteristic
205	C₂×C₄	#6195	1	2	yes	—
206	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
207	C₈	—	6	2	no	—
208	C₈	—	54	2	no	—
209	C₂×C₃	—	2	4	no	—
210	C₂×C₃	—	2	2	no	—
211	C₂×C₃	—	2	2	no	—
212	C₂×C₃	—	2	1	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	#63469	1	2	yes	—
215	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
216	C₂×C₃	#63815	1	2	yes	—
217	C₂×C₃	C₂×Dic₃⋊Dic₃	1	1	yes	characteristic
218	C₂²	—	2	2	no	—
219	C₂²	Dic₉⋊Dic₃	1	2	yes	—
220	C₂²	C₂×Dic₃×D₉	1	1	yes	characteristic
221	C₄	—	18	4	no	—
222	C₄	Dic₉⋊Dic₃	1	2	yes	—
223	C₄	C₂×#6189	1	1	yes	characteristic
224	C₄	C₂×#6195	1	1	yes	characteristic
225	C₃	—	2	1	no	—
226	C₃	#10588803	1	1	yes	characteristic, complemented
227	C₃	#10591117	1	1	yes	characteristic
228	C₂	—	2	2	no	—
229	C₂	#11720859	1	2	yes	—
230	C₂	C₂×Dic₉⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₃×D₉
commutator	C₂²×C₃×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂²×C₃×C₄×C₉	C₂²
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂³×C₃²	C₄×D₃
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂²×C₄	D₃×D₉
hypocenter	C₃×C₉	#424
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. Dic₉⋊(C₃⋊C₈)	C₂	semidirect
138. C₃×C₉	#424	semidirect
200. C₉	#4999	semidirect
226. C₃	#10588803	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₉⋊C₈)	2
C₃×#63488	2
C₂×C₉×#208	2
Dic₉⋊(C₃⋊C₈)	2
#10591117	3
#10588803	3