Subgroups — C₂×C₃×#63362 #13142365

Order 1728.

546 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₉⋊C₈	C₂	1	2	yes	maximal
3	C₃×#63362	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#803	—	3	1	no	maximal
5	C₂×#63362	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×C₄×C₉	C₄	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₉	C₄	1	2	yes	—
9	C₃×C₄²×C₉	C₂²	1	2	yes	—
10	C₂×C₃×C₉⋊C₈	C₄	1	8	yes	complemented
11	C₂×C₃×C₉⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
13	C₂²×C₃×C₃⋊C₈	—	3	2	no	—
14	C₃×#803	—	3	4	no	—
15	C₂×C₄²×C₉	—	2	1	no	—
16	C₂×C₄²×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₉⋊C₈	C₂×C₃	1	2	yes	—
18	#63362	C₂×C₃	1	4	yes	direct product factor
19	C₂³×C₃×C₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	—	2	4	no	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₃×C₉⋊C₈	C₂×C₄	1	8	yes	complemented
26	C₂×C₃×#97	—	9	1	no	—
27	C₂×#803	—	3	1	no	—
28	C₂²×C₃²×C₄	Dic₃	1	2	yes	—
29	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
30	C₃²×C₄²	Dic₃	1	2	yes	—
31	C₃²×C₄²	C₂×D₃	1	2	yes	—
32	C₂×C₃×C₃⋊C₈	—	3	8	no	—
33	C₂×C₃×C₃⋊C₈	—	3	4	no	—
34	C₂²×C₄×C₉	—	2	2	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	C₃×C₄	1	2	yes	—
37	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
38	C₄²×C₉	—	2	2	no	—
39	C₄²×C₉	—	2	2	no	—
40	C₄²×C₉	C₃×C₄	1	2	yes	—
41	C₄²×C₉	C₂²×C₃	1	2	yes	—
42	C₂×C₉⋊C₈	C₃×C₄	1	8	yes	complemented
43	C₂×C₉⋊C₈	C₂²×C₃	1	4	yes	—
44	C₂²×C₃×C₉	C₄²	1	2	yes	—
45	C₂²×C₃×C₉	M₁₆	1	4	yes	—
46	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
47	C₃×C₄×C₉	—	2	4	no	—
48	C₃×C₄×C₉	C₄²	1	2	yes	—
49	C₃×C₄×C₉	C₂²×C₄	1	2	yes	—
50	C₂²×C₃×C₈	—	9	2	no	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	D₉	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
54	C₂²×C₃⋊C₈	—	3	2	no	—
55	C₃×#97	—	9	4	no	—
56	#803	—	3	4	no	—
57	C₂³×C₃²	C₂×Dic₃	1	1	yes	characteristic
58	C₂×C₃²×C₄	—	2	4	no	—
59	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
60	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
61	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₂³×C₉	—	2	1	no	—
64	C₂³×C₉	C₂×C₃×C₄	1	1	yes	characteristic
65	C₃×C₃⋊C₈	—	3	8	no	—
66	C₂×C₄×C₉	—	2	8	no	—
67	C₂×C₄×C₉	—	2	4	no	—
68	C₂×C₄×C₉	—	2	4	no	—
69	C₂×C₄×C₉	—	2	4	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
73	C₂×C₄×C₉	C₂×C₃×C₄	1	4	yes	—
74	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
76	C₉⋊C₈	C₂×C₃×C₄	1	8	yes	complemented
77	C₂×#97	—	9	1	no	Sylow
78	C₂×C₃×C₉	#97	1	4	yes	complemented
79	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₃×C₉	C₂×M₁₆	1	2	yes	—
81	C₂²×C₃×C₄	—	2	2	no	—
82	C₂²×C₃×C₄	—	2	1	no	—
83	C₂²×C₃×C₄	Dic₉	1	2	yes	—
84	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
85	C₂²×C₃×C₄	C₃×Dic₃	1	2	yes	—
86	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
87	C₂×C₃×C₈	—	9	8	no	—
88	C₂×C₃×C₈	—	9	4	no	—
89	C₃×C₄²	—	2	2	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	Dic₉	1	2	yes	—
92	C₃×C₄²	C₂×D₉	1	2	yes	—
93	C₃×C₄²	C₃×Dic₃	1	2	yes	—
94	C₃×C₄²	C₂×C₃×D₃	1	2	yes	—
95	C₂×C₃⋊C₈	—	3	8	no	—
96	C₂×C₃⋊C₈	—	3	4	no	—
97	C₂²×C₃²	#208	1	4	yes	—
98	C₂²×C₃²	C₄×Dic₃	1	2	yes	—
99	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
100	C₃²×C₄	—	2	4	no	—
101	C₃²×C₄	C₄×Dic₃	1	2	yes	—
102	C₃²×C₄	C₂×C₄×D₃	1	2	yes	—
103	C₂²×C₉	—	2	4	no	—
104	C₂²×C₉	—	2	2	no	—
105	C₂²×C₉	—	2	1	no	—
106	C₂²×C₉	C₃×C₄²	1	2	yes	—
107	C₂²×C₉	C₃×M₁₆	1	4	yes	—
108	C₂²×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
109	C₄×C₉	—	2	8	no	—
110	C₄×C₉	—	2	4	no	—
111	C₄×C₉	—	2	2	no	—
112	C₄×C₉	—	2	2	no	—
113	C₄×C₉	C₃×C₄²	1	2	yes	—
114	C₄×C₉	C₂²×C₃×C₄	1	2	yes	—
115	C₂²×C₈	—	9	2	no	—
116	C₂×C₄²	C₃×D₉	1	1	yes	characteristic
117	#97	—	9	4	no	—
118	C₃×C₉	C₂×#97	1	1	yes	characteristic, complemented, Sylow
119	C₂³×C₃	—	2	1	no	—
120	C₂³×C₃	C₂×Dic₉	1	1	yes	characteristic
121	C₂³×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
122	C₂×C₃×C₄	—	2	8	no	—
123	C₂×C₃×C₄	—	2	4	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	2	4	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	C₄×D₉	1	4	yes	—
130	C₂×C₃×C₄	C₂×Dic₉	1	4	yes	—
131	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
132	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
133	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
134	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
135	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
137	C₃×C₈	—	9	8	no	—
138	C₃⋊C₈	—	3	8	no	—
139	C₂×C₃²	#803	1	4	yes	—
140	C₂×C₃²	C₂×#208	1	2	yes	—
141	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
142	C₂×C₉	—	2	2	no	—
143	C₂×C₉	—	2	1	no	—
144	C₂×C₉	—	2	4	no	—
145	C₂×C₉	C₃×#97	1	4	yes	complemented
146	C₂×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
147	C₂×C₉	C₂×C₃×M₁₆	1	2	yes	—
148	C₂²×C₄	C₃×Dic₉	1	2	yes	—
149	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
150	C₂×C₈	—	9	8	no	—
151	C₂×C₈	—	9	4	no	—
152	C₄²	C₃×Dic₉	1	2	yes	—
153	C₄²	C₂×C₃×D₉	1	2	yes	—
154	C₂²×C₃	—	2	1	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	2	2	no	—
157	C₂²×C₃	#3667	1	4	yes	—
158	C₂²×C₃	C₄×Dic₉	1	2	yes	—
159	C₂²×C₃	C₂²×Dic₉	1	1	yes	characteristic
160	C₂²×C₃	C₃×#208	1	4	yes	—
161	C₂²×C₃	C₃×C₄×Dic₃	1	2	yes	—
162	C₂²×C₃	C₂²×C₃×Dic₃	1	1	yes	characteristic
163	C₃×C₄	—	2	8	no	—
164	C₃×C₄	—	2	4	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	2	2	no	—
167	C₃×C₄	—	2	2	no	—
168	C₃×C₄	C₄×Dic₉	1	2	yes	—
169	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
170	C₃×C₄	C₃×C₄×Dic₃	1	2	yes	—
171	C₃×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
172	C₃²	C₂×#803	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	C₂×C₃×#97	1	1	yes	characteristic, complemented
175	C₂³	C₂×C₃×Dic₉	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	C₃×C₄×D₉	1	4	yes	—
178	C₂×C₄	C₂×C₃×Dic₉	1	4	yes	—
179	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
181	C₈	—	9	8	no	—
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	—	2	1	no	—
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	#63362	1	4	yes	direct product factor
186	C₂×C₃	C₂×#3667	1	2	yes	—
187	C₂×C₃	C₂×C₄×Dic₉	1	1	yes	characteristic
188	C₂×C₃	C₃×#803	1	4	yes	—
189	C₂×C₃	C₂×C₃×#208	1	2	yes	—
190	C₂×C₃	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
191	C₂²	C₃×#3667	1	4	yes	—
192	C₂²	C₃×C₄×Dic₉	1	2	yes	—
193	C₂²	C₂²×C₃×Dic₉	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	C₃×C₄×Dic₉	1	2	yes	—
196	C₄	C₂×C₃×C₄×D₉	1	2	yes	—
197	C₃	—	2	1	no	—
198	C₃	C₂×#63362	1	1	yes	characteristic, direct product factor
199	C₃	C₂×C₃×#803	1	1	yes	characteristic
200	C₂	C₃×#63362	1	4	yes	direct product factor
201	C₂	C₂×C₃×#3667	1	2	yes	—
202	C₂	C₂×C₃×C₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×D₉
commutator	C₂×C₉	C₂×C₃×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂×Dic₃
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂×C₃×C₄²	D₉
hypocenter	C₉	C₂×C₃×#97
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄²

Complements

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

N	H ≅ G/N	type
3. C₃×#63362	C₂	direct (N × H)
5. C₂×#63362	C₃	direct (N × H)
10. C₂×C₃×C₉⋊C₈	C₄	semidirect
18. #63362	C₂×C₃	direct (N × H)
25. C₃×C₉⋊C₈	C₂×C₄	semidirect
42. C₂×C₉⋊C₈	C₃×C₄	semidirect
76. C₉⋊C₈	C₂×C₃×C₄	semidirect
78. C₂×C₃×C₉	#97	semidirect
118. C₃×C₉	C₂×#97	semidirect
145. C₂×C₉	C₃×#97	semidirect
174. C₉	C₂×C₃×#97	semidirect
185. C₂×C₃	#63362	direct (N × H)
198. C₃	C₂×#63362	direct (N × H)
200. C₂	C₃×#63362	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂²×C₃×C₉⋊C₈	2
C₃×#63362	2
C₂×C₃×#803	3
C₂×#63362	3

Subgroups — C2×C3×#63362 #13142365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63362 #13142365