Subgroups — #13139361

Order 1728.

418 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11721815	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#208	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720273	C₂	1	4	yes	maximal, complemented
5	#10591119	—	3	1	no	maximal
6	#10588803	—	3	1	no	maximal
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	#88551	C₄	1	2	yes	—
9	C₄⋊(C₃⋊Dic₉)	C₄	1	2	yes	—
10	C₄×C₃⋊Dic₉	C₂²	1	2	yes	—
11	C₉×#208	—	2	2	no	—
12	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃×C₉⋊C₈	—	2	2	no	—
14	C₂×C₃×C₉⋊C₈	C₂²	1	2	yes	—
15	#64135	—	3	1	no	—
16	C₂×C₃×#208	D₃	1	1	yes	characteristic
17	C₂²×C₃×C₃⋊C₈	—	3	1	no	—
18	#63572	—	3	4	no	—
19	C₂×C₉×M₁₆	—	3	1	no	—
20	#63488	—	3	1	no	—
21	#63488	—	6	1	no	—
22	C₂²×C₉⋊C₈	D₃	1	1	yes	characteristic
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	C₉×C₃⋊C₈	—	2	2	no	—
31	C₃×C₉⋊C₈	—	2	2	no	—
32	#5119	—	9	1	no	—
33	#5102	—	9	1	no	—
34	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
35	#3757	—	3	2	no	—
36	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
37	C₄×C₃⋊Dic₃	—	3	2	no	—
38	C₃×#208	—	2	2	no	—
39	C₂×C₃×C₃⋊C₈	—	3	2	no	—
40	C₂×C₃×C₃⋊C₈	—	6	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	—	3	2	no	—
47	#3676	—	6	2	no	—
48	C₄⋊Dic₉	—	3	2	no	—
49	C₄⋊Dic₉	—	6	2	no	—
50	C₄×Dic₉	—	3	2	no	—
51	C₄×Dic₉	—	6	2	no	—
52	C₂×C₉⋊C₈	—	2	2	no	—
53	C₂×C₉⋊C₈	C₂×D₃	1	2	yes	—
54	C₂²×C₃×C₉	—	2	2	no	—
55	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
56	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
58	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
59	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
60	C₃⋊Dic₉	—	2	4	no	—
61	C₂×C₃×M₁₆	—	3	1	no	—
62	C₂²×C₃×C₈	—	9	1	no	—
63	#926	—	9	1	no	—
64	#926	—	9	1	no	—
65	#926	—	18	1	no	—
66	C₂×#208	D₉	1	1	yes	characteristic
67	C₂²×C₃⋊C₈	—	3	1	no	—
68	Dic₃⋊C₈	—	9	4	no	—
69	Dic₃⋊C₈	—	9	4	no	—
70	C₂³×C₃²	C₄×D₃	1	1	yes	characteristic
71	C₂×C₃²×C₄	Dic₆	1	2	yes	—
72	C₂×C₃²×C₄	C₄×D₃	1	1	yes	characteristic
73	C₂×C₃²×C₄	#67	1	2	yes	—
74	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₃⋊Dic₃	—	3	4	no	—
76	C₂³×C₉	—	2	1	no	—
77	C₂³×C₉	C₄×D₃	1	1	yes	characteristic
78	C₃×C₃⋊C₈	—	2	2	no	—
79	C₃×C₃⋊C₈	—	6	2	no	—
80	C₂×C₄×C₉	—	2	1	no	—
81	C₂×C₄×C₉	—	2	2	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	1	no	—
84	C₂×C₄×C₉	Dic₆	1	2	yes	—
85	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
86	C₂×C₄×C₉	#67	1	2	yes	—
87	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
88	C₂×Dic₉	—	3	4	no	—
89	C₂×Dic₉	—	6	4	no	—
90	C₈×C₉	—	6	2	no	—
91	C₉⋊C₈	—	2	2	no	—
92	#424	—	27	1	no	Sylow
93	C₂×C₃×C₉	—	2	2	no	—
94	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₃×C₉	#131	1	2	yes	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
98	C₂²×C₃×C₄	D₃²	1	1	yes	characteristic
99	C₃×M₁₆	—	6	2	no	—
100	C₂×C₃×C₈	—	3	2	no	—
101	C₂×C₃×C₈	—	9	2	no	—
102	C₂×C₃×C₈	—	18	2	no	—
103	#217	—	9	2	no	—
104	#217	—	9	2	no	—
105	#217	—	18	2	no	—
106	C₄⋊Dic₃	—	9	2	no	—
107	C₄⋊Dic₃	—	9	2	no	—
108	C₄⋊Dic₃	—	18	2	no	—
109	C₄×Dic₃	—	9	2	no	—
110	C₄×Dic₃	—	9	2	no	—
111	C₄×Dic₃	—	18	2	no	—
112	#208	—	2	2	no	—
113	C₂×C₃⋊C₈	—	3	2	no	—
114	C₂×C₃⋊C₈	—	6	2	no	—
115	C₂×C₃⋊C₈	C₂×D₉	1	2	yes	—
116	C₂²×C₃²	—	2	2	no	—
117	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
118	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
119	C₃²×C₄	Dic₃⋊C₄	1	2	yes	—
120	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
121	C₃²×C₄	C₂×#67	1	1	yes	characteristic
122	C₃⋊Dic₃	—	6	4	no	—
123	C₂²×C₉	—	2	2	no	—
124	C₂²×C₉	—	2	2	no	—
125	C₂²×C₉	—	2	1	no	—
126	C₂²×C₉	—	4	2	no	—
127	C₂²×C₉	Dic₃⋊C₄	1	2	yes	—
128	C₂²×C₉	C₂×C₄×D₃	1	1	yes	characteristic
129	C₄×C₉	—	2	2	no	—
130	C₄×C₉	—	2	1	no	—
131	C₄×C₉	—	2	1	no	—
132	C₄×C₉	Dic₃⋊C₄	1	2	yes	—
133	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
134	C₄×C₉	C₂×#67	1	1	yes	characteristic
135	Dic₉	—	6	4	no	—
136	Dic₉	—	12	4	no	—
137	C₂×M₁₆	—	3	1	no	—
138	C₂²×C₈	—	9	1	no	—
139	#117	—	27	1	no	—
140	C₄⋊C₈	—	27	4	no	—
141	C₃×C₉	#424	1	1	yes	characteristic, complemented, Sylow
142	C₂³×C₃	—	2	1	no	—
143	C₂³×C₃	C₄×D₉	1	1	yes	characteristic
144	C₂³×C₃	#624	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	2	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	1	no	—
149	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
150	C₂×C₃×C₄	C₄×D₉	1	1	yes	characteristic
151	C₂×C₃×C₄	#611	1	2	yes	—
152	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
153	C₂×C₃×C₄	#624	1	1	yes	characteristic
154	C₂×C₃×C₄	#625	1	2	yes	—
155	C₂×C₃×C₄	#627	1	2	yes	—
156	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
157	C₂×Dic₃	—	9	4	no	—
158	C₂×Dic₃	—	9	4	no	—
159	C₂×Dic₃	—	18	4	no	—
160	C₃×C₈	—	6	2	no	—
161	C₃×C₈	—	18	2	no	—
162	C₃⋊C₈	—	2	2	no	—
163	C₃⋊C₈	—	6	2	no	—
164	C₂×C₃²	—	2	2	no	—
165	C₂×C₃²	#907	1	2	yes	—
166	C₂×C₃²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
167	C₂×C₉	—	2	2	no	—
168	C₂×C₉	—	2	2	no	—
169	C₂×C₉	—	2	1	no	—
170	C₂×C₉	—	4	2	no	—
171	C₂×C₉	#901	1	2	yes	—
172	C₂×C₉	C₂×Dic₃⋊C₄	1	1	yes	characteristic
173	C₂²×C₄	D₃×D₉	1	1	yes	characteristic
174	M₁₆	—	6	2	no	—
175	C₂×C₈	—	3	2	no	—
176	C₂×C₈	—	9	2	no	—
177	C₂×C₈	—	18	2	no	—
178	C₄⋊C₄	—	27	2	no	—
179	C₂²⋊C₄	—	27	2	no	—
180	C₄²	—	27	2	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	2	no	—
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	2	2	no	—
185	C₂²×C₃	—	4	2	no	—
186	C₂²×C₃	Dic₉⋊C₄	1	2	yes	—
187	C₂²×C₃	C₂×C₄×D₉	1	1	yes	characteristic
188	C₂²×C₃	#3724	1	2	yes	—
189	C₂²×C₃	C₂×#624	1	1	yes	characteristic
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	Dic₉⋊C₄	1	2	yes	—
194	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
195	C₃×C₄	C₂×#611	1	1	yes	characteristic
196	C₃×C₄	#3724	1	2	yes	—
197	C₃×C₄	C₂×#625	1	1	yes	characteristic
198	C₃×C₄	C₂×#627	1	1	yes	characteristic
199	Dic₃	—	18	4	no	—
200	Dic₃	—	18	4	no	—
201	Dic₃	—	36	4	no	—
202	C₃²	#5119	1	1	yes	characteristic
203	C₉	—	2	1	no	—
204	C₉	#5102	1	1	yes	characteristic, complemented
205	C₂³	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
206	C₂×C₄	—	27	4	no	—
207	C₂×C₄	#6189	1	2	yes	—
208	C₂×C₄	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
209	C₂×C₄	#6194	1	2	yes	—
210	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
211	C₈	—	6	2	no	—
212	C₈	—	18	2	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	2	no	—
217	C₂×C₃	—	4	2	no	—
218	C₂×C₃	#63469	1	2	yes	—
219	C₂×C₃	C₂×Dic₉⋊C₄	1	1	yes	characteristic
220	C₂×C₃	#63817	1	2	yes	—
221	C₂×C₃	C₂×#3724	1	1	yes	characteristic
222	C₂²	—	2	2	no	—
223	C₂²	#88442	1	2	yes	—
224	C₂²	C₂×(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
225	C₄	—	54	4	no	—
226	C₄	#88442	1	2	yes	—
227	C₄	C₂×#6189	1	1	yes	characteristic
228	C₄	C₂×#6194	1	1	yes	characteristic
229	C₃	—	2	1	no	—
230	C₃	#10588803	1	1	yes	characteristic, complemented
231	C₃	#10591119	1	1	yes	characteristic
232	C₂	—	2	2	no	—
233	C₂	#11720861	1	2	yes	—
234	C₂	C₂×#88442	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. #11720273	C₂	semidirect
141. C₃×C₉	#424	semidirect
204. C₉	#5102	semidirect
230. C₃	#10588803	semidirect

Maximal subgroups

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

subgroup	index
#11721815	2
C₂×C₉×#208	2
C₂²×C₃×C₉⋊C₈	2
#11720273	2
#10591119	3
#10588803	3