Subgroups — #11720331

Order 864.

282 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	C₂²×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×Dic₉	C₂	1	1	yes	maximal, characteristic
4	#63579	—	3	1	no	maximal
5	C₉⋊#95	—	3	1	no	maximal
6	C₂³×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊Dic₉	—	2	1	no	—
8	C₂×C₃⋊Dic₉	—	2	1	no	—
9	C₂×C₃⋊Dic₉	C₄	1	2	yes	complemented
10	C₂×C₉×Dic₃	—	2	1	no	—
11	C₂×C₉×Dic₃	—	2	1	no	—
12	C₂×C₉×Dic₃	C₄	1	2	yes	complemented
13	C₂×C₃×Dic₉	—	2	1	no	—
14	C₂×C₃×Dic₉	—	2	1	no	—
15	C₂×C₃×Dic₉	C₄	1	2	yes	complemented
16	C₂²×C₃⋊Dic₃	—	3	1	no	—
17	C₂²×C₃×Dic₃	—	3	1	no	—
18	C₂²×C₃×Dic₃	D₃	1	1	yes	characteristic
19	C₂²×C₄×C₉	—	3	1	no	—
20	C₂²×Dic₉	—	3	1	no	—
21	C₂²×Dic₉	—	6	1	no	—
22	C₂²×Dic₉	D₃	1	1	yes	characteristic
23	C₂²×C₃×C₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₉	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₉	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₉	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₉	Q₈	1	1	yes	characteristic
30	C₃⋊Dic₉	—	2	2	no	—
31	C₉×Dic₃	—	2	2	no	—
32	C₃×Dic₉	—	2	2	no	—
33	C₃⋊#95	—	9	1	no	—
34	C₃⋊#95	—	9	1	no	—
35	C₂³×C₃²	C₂×D₃	1	1	yes	characteristic
36	C₂×C₃⋊Dic₃	—	3	2	no	—
37	C₂×C₃⋊Dic₃	—	6	1	no	—
38	C₂×C₃⋊Dic₃	—	6	1	no	—
39	C₂×C₃×Dic₃	—	2	1	no	—
40	C₂×C₃×Dic₃	—	2	1	no	—
41	C₂×C₃×Dic₃	—	3	2	no	—
42	C₂×C₃×Dic₃	—	6	1	no	—
43	C₂×C₃×Dic₃	—	6	1	no	—
44	C₂×C₃×Dic₃	Dic₃	1	2	yes	—
45	C₂³×C₉	—	2	1	no	—
46	C₂³×C₉	C₂×D₃	1	1	yes	characteristic
47	C₂×C₄×C₉	—	3	2	no	—
48	C₂×C₄×C₉	—	6	1	no	—
49	C₂×C₄×C₉	—	6	1	no	—
50	C₂×Dic₉	—	2	1	no	—
51	C₂×Dic₉	—	2	1	no	—
52	C₂×Dic₉	—	3	2	no	—
53	C₂×Dic₉	—	6	2	no	—
54	C₂×Dic₉	—	6	2	no	—
55	C₂×Dic₉	—	6	2	no	—
56	C₂×Dic₉	—	6	1	no	—
57	C₂×Dic₉	—	6	1	no	—
58	C₂×Dic₉	Dic₃	1	2	yes	complemented
59	C₂×C₃×C₉	C₄²	1	1	yes	characteristic
60	C₂×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₃×C₄	—	3	1	no	—
67	C₂²×C₃×C₄	—	9	1	no	—
68	C₂²×Dic₃	—	3	1	no	—
69	C₂²×Dic₃	—	9	1	no	—
70	C₂²×Dic₃	—	9	1	no	—
71	C₂²×Dic₃	—	18	1	no	—
72	C₂²×Dic₃	D₉	1	1	yes	characteristic
73	C₂²×C₃²	Dic₆	1	1	yes	characteristic
74	C₂²×C₃²	C₄×D₃	1	1	yes	characteristic
75	C₂²×C₃²	C₄×D₃	1	1	yes	characteristic
76	C₂²×C₃²	D₁₂	1	1	yes	characteristic
77	C₂²×C₃²	C₂×Dic₃	1	1	yes	characteristic
78	C₂²×C₃²	#67	1	1	yes	characteristic
79	C₂²×C₃²	#67	1	1	yes	characteristic
80	C₃⋊Dic₃	—	6	2	no	—
81	C₃×Dic₃	—	2	2	no	—
82	C₃×Dic₃	—	6	2	no	—
83	C₂²×C₉	—	2	1	no	—
84	C₂²×C₉	—	2	1	no	—
85	C₂²×C₉	—	2	1	no	—
86	C₂²×C₉	—	2	1	no	—
87	C₂²×C₉	—	2	1	no	—
88	C₂²×C₉	—	2	1	no	—
89	C₂²×C₉	—	2	1	no	—
90	C₂²×C₉	Dic₆	1	1	yes	characteristic
91	C₂²×C₉	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₉	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₉	D₁₂	1	1	yes	characteristic
94	C₂²×C₉	C₂×Dic₃	1	1	yes	characteristic
95	C₂²×C₉	#67	1	1	yes	characteristic
96	C₂²×C₉	#67	1	1	yes	characteristic
97	C₄×C₉	—	6	2	no	—
98	Dic₉	—	2	2	no	—
99	Dic₉	—	6	4	no	—
100	Dic₉	—	6	2	no	—
101	#95	—	27	1	no	Sylow
102	C₃×C₉	#95	1	1	yes	characteristic, complemented, Sylow
103	C₂³×C₃	—	2	1	no	—
104	C₂³×C₃	C₂×D₉	1	1	yes	characteristic
105	C₂³×C₃	D₃²	1	1	yes	characteristic
106	C₂×C₃×C₄	—	3	2	no	—
107	C₂×C₃×C₄	—	6	1	no	—
108	C₂×C₃×C₄	—	6	1	no	—
109	C₂×C₃×C₄	—	9	2	no	—
110	C₂×C₃×C₄	—	18	1	no	—
111	C₂×C₃×C₄	—	18	1	no	—
112	C₂×Dic₃	—	2	1	no	—
113	C₂×Dic₃	—	2	1	no	—
114	C₂×Dic₃	—	3	2	no	—
115	C₂×Dic₃	—	6	1	no	—
116	C₂×Dic₃	—	6	1	no	—
117	C₂×Dic₃	—	9	2	no	—
118	C₂×Dic₃	—	9	2	no	—
119	C₂×Dic₃	—	18	2	no	—
120	C₂×Dic₃	—	18	2	no	—
121	C₂×Dic₃	—	18	2	no	—
122	C₂×Dic₃	—	18	1	no	—
123	C₂×Dic₃	—	18	1	no	—
124	C₂×Dic₃	—	18	1	no	—
125	C₂×Dic₃	—	18	1	no	—
126	C₂×Dic₃	Dic₉	1	2	yes	complemented
127	C₂×C₃²	C₄×Dic₃	1	1	yes	characteristic
128	C₂×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
129	C₂×C₃²	Dic₃⋊C₄	1	1	yes	characteristic
130	C₂×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
131	C₂×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
132	C₂×C₃²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
133	C₂×C₃²	#217	1	1	yes	characteristic
134	C₂×C₉	—	2	1	no	—
135	C₂×C₉	—	2	1	no	—
136	C₂×C₉	—	2	1	no	—
137	C₂×C₉	—	2	1	no	—
138	C₂×C₉	—	2	1	no	—
139	C₂×C₉	—	2	1	no	—
140	C₂×C₉	—	2	1	no	—
141	C₂×C₉	C₄×Dic₃	1	1	yes	characteristic
142	C₂×C₉	Dic₃⋊C₄	1	1	yes	characteristic
143	C₂×C₉	Dic₃⋊C₄	1	1	yes	characteristic
144	C₂×C₉	C₄⋊Dic₃	1	1	yes	characteristic
145	C₂×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
146	C₂×C₉	(C₂×D₃)⋊C₄	1	1	yes	characteristic
147	C₂×C₉	#217	1	1	yes	characteristic
148	C₂²×C₄	—	3	1	no	—
149	C₂²×C₄	—	9	1	no	—
150	C₂²×C₄	—	27	1	no	—
151	C₂²×C₃	—	2	1	no	—
152	C₂²×C₃	—	2	1	no	—
153	C₂²×C₃	—	2	1	no	—
154	C₂²×C₃	—	2	1	no	—
155	C₂²×C₃	—	2	1	no	—
156	C₂²×C₃	—	2	1	no	—
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	Dic₁₈	1	1	yes	characteristic
159	C₂²×C₃	C₄×D₉	1	1	yes	characteristic
160	C₂²×C₃	C₄×D₉	1	1	yes	characteristic
161	C₂²×C₃	D₃₆	1	1	yes	characteristic
162	C₂²×C₃	C₂×Dic₉	1	1	yes	characteristic
163	C₂²×C₃	#611	1	1	yes	characteristic
164	C₂²×C₃	#611	1	1	yes	characteristic
165	C₂²×C₃	D₃×Dic₃	1	1	yes	characteristic
166	C₂²×C₃	D₃×Dic₃	1	1	yes	characteristic
167	C₂²×C₃	#624	1	1	yes	characteristic
168	C₂²×C₃	#625	1	1	yes	characteristic
169	C₂²×C₃	#626	1	1	yes	characteristic
170	C₂²×C₃	#626	1	1	yes	characteristic
171	C₂²×C₃	#627	1	1	yes	characteristic
172	C₃×C₄	—	6	2	no	—
173	C₃×C₄	—	18	2	no	—
174	Dic₃	—	2	2	no	—
175	Dic₃	—	6	2	no	—
176	Dic₃	—	18	4	no	—
177	Dic₃	—	18	2	no	—
178	Dic₃	—	18	2	no	—
179	C₃²	C₃⋊#95	1	1	yes	characteristic
180	C₉	—	2	1	no	—
181	C₉	C₃⋊#95	1	1	yes	characteristic, complemented
182	C₂³	D₃×D₉	1	1	yes	characteristic
183	C₂×C₄	—	3	2	no	—
184	C₂×C₄	—	6	1	no	—
185	C₂×C₄	—	6	1	no	—
186	C₂×C₄	—	9	2	no	—
187	C₂×C₄	—	18	1	no	—
188	C₂×C₄	—	18	1	no	—
189	C₂×C₄	—	27	2	no	—
190	C₂×C₄	—	54	1	no	—
191	C₂×C₄	—	54	1	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	1	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	1	no	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	2	1	no	—
198	C₂×C₃	—	2	1	no	—
199	C₂×C₃	C₄×Dic₉	1	1	yes	characteristic
200	C₂×C₃	Dic₉⋊C₄	1	1	yes	characteristic
201	C₂×C₃	Dic₉⋊C₄	1	1	yes	characteristic
202	C₂×C₃	C₄⋊Dic₉	1	1	yes	characteristic
203	C₂×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
204	C₂×C₃	(C₂×D₉)⋊C₄	1	1	yes	characteristic
205	C₂×C₃	#3676	1	1	yes	characteristic
206	C₂×C₃	Dic₃²	1	1	yes	characteristic
207	C₂×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
208	C₂×C₃	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
209	C₂×C₃	#3722	1	1	yes	characteristic
210	C₂×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
211	C₂×C₃	Dic₃⋊Dic₃	1	1	yes	characteristic
212	C₂×C₃	#3724	1	1	yes	characteristic
213	C₂²	#6189	1	1	yes	characteristic
214	C₂²	Dic₃×D₉	1	1	yes	characteristic
215	C₂²	(C₃⋊D₉)⋊C₄	1	1	yes	characteristic
216	C₂²	#6192	1	1	yes	characteristic
217	C₂²	D₃×Dic₉	1	1	yes	characteristic
218	C₂²	#6194	1	1	yes	characteristic
219	C₂²	#6195	1	1	yes	characteristic
220	C₄	—	6	2	no	—
221	C₄	—	18	2	no	—
222	C₄	—	54	2	no	—
223	C₃	—	2	1	no	—
224	C₃	C₉⋊#95	1	1	yes	characteristic, complemented
225	C₃	#63579	1	1	yes	characteristic
226	C₂	Dic₃×Dic₉	1	1	yes	characteristic
227	C₂	Dic₉⋊Dic₃	1	1	yes	characteristic
228	C₂	#88442	1	1	yes	characteristic
229	C₂	Dic₃⋊Dic₉	1	1	yes	characteristic
230	C₂	(C₂×D₉)⋊Dic₃	1	1	yes	characteristic
231	C₂	#88445	1	1	yes	characteristic
232	C₂	(C₂×D₃)⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₃×D₉
commutator	C₂×C₃×C₉	C₄²
Frattini	C₂³×C₃	D₃²
Fitting, cosocle	C₂³×C₃×C₉	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₃
hypocenter	C₃×C₉	#95
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
9. C₂×C₃⋊Dic₉	C₄	semidirect
12. C₂×C₉×Dic₃	C₄	semidirect
15. C₂×C₃×Dic₉	C₄	semidirect
58. C₂×Dic₉	Dic₃	semidirect
102. C₃×C₉	#95	semidirect
126. C₂×Dic₃	Dic₉	semidirect
181. C₉	C₃⋊#95	semidirect
224. C₃	C₉⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊Dic₉	2
C₂²×C₉×Dic₃	2
C₂²×C₃×Dic₉	2
#63579	3
C₉⋊#95	3