Subgroups — C₃×#10588226 #13135644

Order 1728.

274 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₉⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×#4549	—	3	1	no	maximal
4	#10588226	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂²×C₃×C₄×C₉	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈×C₉	—	2	2	no	—
7	C₂×C₃×C₈×C₉	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₃×C₉⋊C₁₆	—	2	2	no	—
10	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
11	C₂×C₃×C₃⋊C₁₆	—	3	2	no	—
12	C₂²×C₈×C₉	—	2	1	no	—
13	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
14	C₂×C₉⋊C₁₆	C₂×C₃	1	2	yes	complemented
15	C₂³×C₃×C₉	C₈	1	1	yes	characteristic
16	C₂×C₃×C₄×C₉	—	2	1	no	—
17	C₂×C₃×C₄×C₉	—	2	1	no	—
18	C₂×C₃×C₄×C₉	C₈	1	1	yes	characteristic
19	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
20	C₃×C₈×C₉	—	2	1	no	—
21	C₃×C₈×C₉	D₄	1	2	yes	—
22	C₃×C₂²⋊C₁₆	—	9	1	no	—
23	#4549	—	3	1	no	—
24	C₂²×C₃²×C₄	Dic₃	1	1	yes	characteristic
25	C₂×C₃²×C₈	—	2	2	no	—
26	C₂×C₃²×C₈	Dic₃	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
28	C₂²×C₄×C₉	—	2	1	no	—
29	C₂²×C₄×C₉	C₃×C₄	1	1	yes	characteristic
30	C₃×C₃⋊C₁₆	—	6	2	no	—
31	C₂×C₈×C₉	—	2	4	no	—
32	C₂×C₈×C₉	—	2	1	no	—
33	C₂×C₈×C₉	—	2	1	no	—
34	C₂×C₈×C₉	—	2	2	no	—
35	C₂×C₈×C₉	C₃×C₄	1	1	yes	characteristic
36	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₉⋊C₁₆	—	2	2	no	—
38	C₂²×C₃×C₉	—	2	1	no	—
39	C₂²×C₃×C₉	—	2	1	no	—
40	C₂²×C₃×C₉	C₁₆	1	2	yes	complemented
41	C₂²×C₃×C₉	C₂×C₈	1	1	yes	characteristic
42	C₃×C₄×C₉	—	2	1	no	—
43	C₃×C₄×C₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₃×C₄×C₉	M₁₆	1	1	yes	characteristic
45	C₂²×C₃×C₈	—	2	1	no	—
46	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
47	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
48	C₂×C₃×C₁₆	—	9	2	no	—
49	C₂×C₃⋊C₁₆	—	3	2	no	—
50	C₂³×C₃²	C₃⋊C₈	1	1	yes	characteristic
51	C₂×C₃²×C₄	—	2	1	no	—
52	C₂×C₃²×C₄	—	2	1	no	—
53	C₂×C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
54	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
55	C₂³×C₉	—	2	1	no	—
56	C₂³×C₉	C₃×C₈	1	1	yes	characteristic
57	C₃²×C₈	—	2	1	no	—
58	C₃²×C₈	#67	1	2	yes	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	2	no	—
61	C₂×C₄×C₉	—	2	1	no	—
62	C₂×C₄×C₉	—	2	1	no	—
63	C₂×C₄×C₉	—	2	1	no	—
64	C₂×C₄×C₉	—	2	1	no	—
65	C₂×C₄×C₉	C₃×C₈	1	1	yes	characteristic
66	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
67	C₈×C₉	—	2	2	no	—
68	C₈×C₉	—	2	1	no	—
69	C₈×C₉	—	2	2	no	—
70	C₈×C₉	C₃×D₄	1	2	yes	—
71	C₂²⋊C₁₆	—	9	1	no	Sylow
72	C₂×C₃×C₉	—	2	2	no	—
73	C₂×C₃×C₉	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₃×C₉	C₂×C₁₆	1	1	yes	characteristic
75	C₂×C₃×C₉	M₃₂	1	1	yes	characteristic
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	Dic₉	1	1	yes	characteristic
78	C₂²×C₃×C₄	C₃×Dic₃	1	1	yes	characteristic
79	C₂×C₃×C₈	—	2	4	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	1	no	—
83	C₂×C₃×C₈	—	2	1	no	—
84	C₂×C₃×C₈	Dic₉	1	1	yes	characteristic
85	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
86	C₂×C₃×C₈	C₃×Dic₃	1	1	yes	characteristic
87	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
88	C₃×C₁₆	—	18	2	no	—
89	C₃⋊C₁₆	—	6	2	no	—
90	C₂²×C₃²	—	2	1	no	—
91	C₂²×C₃²	—	2	1	no	—
92	C₂²×C₃²	C₃⋊C₁₆	1	2	yes	—
93	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
94	C₃²×C₄	—	2	1	no	—
95	C₃²×C₄	#208	1	1	yes	characteristic
96	C₃²×C₄	#217	1	1	yes	characteristic
97	C₂²×C₉	—	2	2	no	—
98	C₂²×C₉	—	2	2	no	—
99	C₂²×C₉	—	2	1	no	—
100	C₂²×C₉	—	2	1	no	—
101	C₂²×C₉	—	2	2	no	—
102	C₂²×C₉	—	2	1	no	—
103	C₂²×C₉	C₃×C₁₆	1	2	yes	complemented
104	C₂²×C₉	C₂×C₃×C₈	1	1	yes	characteristic
105	C₄×C₉	—	2	2	no	—
106	C₄×C₉	—	2	1	no	—
107	C₄×C₉	—	2	1	no	—
108	C₄×C₉	—	2	1	no	—
109	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₉	C₃×M₁₆	1	1	yes	characteristic
111	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
112	C₂×C₁₆	—	9	2	no	—
113	C₃×C₉	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₃	—	2	1	no	—
115	C₂³×C₃	C₉⋊C₈	1	1	yes	characteristic
116	C₂³×C₃	C₃×C₃⋊C₈	1	1	yes	characteristic
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	2	2	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	C₉⋊C₈	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
127	C₂×C₃×C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
129	C₃×C₈	—	2	2	no	—
130	C₃×C₈	—	2	1	no	—
131	C₃×C₈	—	2	1	no	—
132	C₃×C₈	—	2	2	no	—
133	C₃×C₈	#611	1	2	yes	—
134	C₃×C₈	C₃×#67	1	2	yes	—
135	C₂×C₃²	—	2	2	no	—
136	C₂×C₃²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
137	C₂×C₃²	#812	1	1	yes	characteristic
138	C₂×C₃²	#830	1	1	yes	characteristic
139	C₂×C₉	—	2	4	no	—
140	C₂×C₉	—	2	2	no	—
141	C₂×C₉	—	2	1	no	—
142	C₂×C₉	—	2	1	no	—
143	C₂×C₉	—	2	1	no	—
144	C₂×C₉	C₃×C₂²⋊C₈	1	1	yes	characteristic
145	C₂×C₉	C₂×C₃×C₁₆	1	1	yes	characteristic
146	C₂×C₉	C₃×M₃₂	1	1	yes	characteristic
147	C₂²×C₄	C₃×Dic₉	1	1	yes	characteristic
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	C₃×Dic₉	1	1	yes	characteristic
150	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
151	C₁₆	—	18	2	no	—
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	—	2	2	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	—	2	2	no	—
159	C₂²×C₃	—	2	1	no	—
160	C₂²×C₃	C₉⋊C₁₆	1	2	yes	complemented
161	C₂²×C₃	C₂×C₉⋊C₈	1	1	yes	characteristic
162	C₂²×C₃	C₃×C₃⋊C₁₆	1	2	yes	—
163	C₂²×C₃	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	2	1	no	—
167	C₃×C₄	—	2	1	no	—
168	C₃×C₄	—	2	1	no	—
169	C₃×C₄	#3667	1	1	yes	characteristic
170	C₃×C₄	#3676	1	1	yes	characteristic
171	C₃×C₄	C₃×#208	1	1	yes	characteristic
172	C₃×C₄	C₃×#217	1	1	yes	characteristic
173	C₃²	#4549	1	1	yes	characteristic
174	C₉	—	2	1	no	—
175	C₉	C₃×C₂²⋊C₁₆	1	1	yes	characteristic, complemented
176	C₂³	C₃×C₉⋊C₈	1	1	yes	characteristic
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	C₃×C₉⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
181	C₈	—	2	1	no	—
182	C₈	C₃×#611	1	2	yes	—
183	C₂×C₃	—	2	4	no	—
184	C₂×C₃	—	2	2	no	—
185	C₂×C₃	—	2	2	no	—
186	C₂×C₃	—	2	1	no	—
187	C₂×C₃	—	2	1	no	—
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	C₂×C₉⋊C₁₆	1	1	yes	characteristic
190	C₂×C₃	#63371	1	1	yes	characteristic
191	C₂×C₃	#63389	1	1	yes	characteristic
192	C₂×C₃	C₂×C₃×C₃⋊C₁₆	1	1	yes	characteristic
193	C₂×C₃	C₃×#812	1	1	yes	characteristic
194	C₂×C₃	C₃×#830	1	1	yes	characteristic
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	C₃×C₉⋊C₁₆	1	2	yes	complemented
198	C₂²	C₂×C₃×C₉⋊C₈	1	1	yes	characteristic
199	C₄	—	2	1	no	—
200	C₄	C₃×#3667	1	1	yes	characteristic
201	C₄	C₃×#3676	1	1	yes	characteristic
202	C₃	—	2	1	no	—
203	C₃	#10588226	1	1	yes	characteristic, direct product factor
204	C₃	C₃×#4549	1	1	yes	characteristic
205	C₂	—	2	2	no	—
206	C₂	C₂×C₃×C₉⋊C₁₆	1	1	yes	characteristic
207	C₂	C₃×#63371	1	1	yes	characteristic
208	C₂	C₃×#63389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₉
commutator	C₂×C₉	C₂×C₃×C₁₆
Frattini	C₂×C₃×C₈	C₂×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×C₃⋊C₈
prime radical	C₂³×C₃²	C₃⋊C₈
cosocle	C₂×C₈×C₉	C₂²×C₃
hypercenter	C₂²×C₃×C₈	D₉
hypocenter	C₉	C₃×C₂²⋊C₁₆
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₉⋊C₁₆	C₂	semidirect
4. #10588226	C₃	direct (N × H)
14. C₂×C₉⋊C₁₆	C₂×C₃	semidirect
40. C₂²×C₃×C₉	C₁₆	semidirect
103. C₂²×C₉	C₃×C₁₆	semidirect
113. C₃×C₉	C₂²⋊C₁₆	semidirect
160. C₂²×C₃	C₉⋊C₁₆	semidirect
175. C₉	C₃×C₂²⋊C₁₆	semidirect
197. C₂²	C₃×C₉⋊C₁₆	semidirect
203. C₃	#10588226	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₈×C₉	2
C₂×C₃×C₉⋊C₁₆	2
C₃×#4549	3
#10588226	3

Subgroups — C3×#10588226 #13135644

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588226 #13135644