Subgroups — C₂×#11720237 #13138092

Order 1728.

298 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×D₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720237	C₂	1	4	yes	maximal, direct product factor
5	C₂×#63543	—	3	1	no	maximal
6	C₂×#63371	—	3	1	no	maximal
7	C₂×C₄×D₃×C₉	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×C₃⋊C₈	C₄	1	1	yes	characteristic
10	D₃×C₈×C₉	C₄	1	2	yes	—
11	D₃×C₈×C₉	C₂²	1	2	yes	—
12	C₃⋊(C₉⋊C₁₆)	C₂²	1	2	yes	complemented
13	C₃×C₉⋊C₁₆	C₂²	1	2	yes	complemented
14	C₂×C₃×D₃×C₈	D₃	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	—	3	1	no	—
16	C₂×C₃×C₃⋊C₁₆	—	3	1	no	—
17	#63543	—	3	4	no	—
18	C₂²×C₈×C₉	—	3	1	no	—
19	#63371	—	3	4	no	—
20	C₂×C₉⋊C₁₆	—	3	1	no	—
21	C₂×C₉⋊C₁₆	—	6	1	no	—
22	C₂×C₉⋊C₁₆	D₃	1	1	yes	characteristic, complemented
23	C₂²×D₃×C₉	C₈	1	1	yes	characteristic
24	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₉×Dic₃	C₈	1	1	yes	characteristic
26	C₄×D₃×C₉	C₈	1	2	yes	—
27	C₄×D₃×C₉	C₂×C₄	1	2	yes	—
28	C₃×C₈×C₉	C₂×C₄	1	1	yes	characteristic
29	C₃×C₈×C₉	C₂³	1	1	yes	characteristic
30	C₉×C₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×#812	—	9	1	no	—
32	C₂×#798	—	9	1	no	—
33	C₂×C₃×C₄×D₃	Dic₃	1	1	yes	characteristic
34	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
35	C₂×C₃×C₃⋊C₈	Dic₃	1	1	yes	characteristic
36	C₃×D₃×C₈	Dic₃	1	2	yes	—
37	C₃×D₃×C₈	C₂×D₃	1	2	yes	—
38	C₂²×C₄×C₉	—	3	1	no	—
39	C₃⋊(C₃⋊C₁₆)	—	3	2	no	—
40	C₃×C₃⋊C₁₆	—	3	2	no	—
41	C₂×C₈×C₉	—	2	1	no	—
42	C₂×C₈×C₉	—	3	1	no	—
43	C₂×C₈×C₉	—	3	2	no	—
44	C₂×C₈×C₉	—	3	2	no	—
45	C₂×C₈×C₉	C₂×D₃	1	1	yes	characteristic
46	C₉⋊C₁₆	—	3	2	no	—
47	C₉⋊C₁₆	—	6	2	no	—
48	C₉⋊C₁₆	C₂×D₃	1	2	yes	complemented
49	C₂²×C₃×C₉	C₂×C₈	1	1	yes	characteristic
50	C₂×D₃×C₉	—	2	2	no	—
51	C₂×D₃×C₉	C₂×C₈	1	2	yes	—
52	C₃×C₄×C₉	C₂×C₈	1	1	yes	characteristic
53	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
54	C₉×Dic₃	C₂×C₈	1	2	yes	—
55	C₂²×C₃×C₈	—	3	1	no	—
56	C₂×D₃×C₈	D₉	1	1	yes	characteristic
57	C₂×C₃×C₁₆	—	9	1	no	—
58	#812	—	9	4	no	—
59	C₂×C₃⋊C₁₆	—	3	1	no	—
60	C₂×C₃⋊C₁₆	—	9	1	no	—
61	C₂×C₃⋊C₁₆	—	9	1	no	—
62	C₂×C₃⋊C₁₆	—	18	1	no	—
63	#798	—	9	4	no	—
64	C₂²×C₃×D₃	C₃⋊C₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
66	C₂×C₃×Dic₃	C₃⋊C₈	1	1	yes	characteristic
67	C₃×C₄×D₃	C₃⋊C₈	1	2	yes	—
68	C₃×C₄×D₃	C₂×Dic₃	1	2	yes	—
69	C₂³×C₉	—	3	1	no	—
70	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
71	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
72	C₃×C₃⋊C₈	C₂×Dic₃	1	2	yes	—
73	C₂×C₄×C₉	—	2	1	no	—
74	C₂×C₄×C₉	—	3	1	no	—
75	C₂×C₄×C₉	—	3	2	no	—
76	C₂×C₄×C₉	—	3	2	no	—
77	C₂×C₄×C₉	C₄×D₃	1	1	yes	characteristic
78	C₈×C₉	—	2	1	no	—
79	C₈×C₉	—	2	1	no	—
80	C₈×C₉	—	3	2	no	—
81	C₈×C₉	C₄×D₃	1	1	yes	characteristic
82	C₈×C₉	C₂²×D₃	1	1	yes	characteristic
83	C₂×M₃₂	—	27	1	no	Sylow
84	C₂×C₃×C₉	M₃₂	1	2	yes	complemented
85	C₂×C₃×C₉	C₂²×C₈	1	1	yes	characteristic
86	D₃×C₉	—	2	2	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂×C₄×D₃	Dic₉	1	1	yes	characteristic
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	3	1	no	—
91	C₂×C₃×C₈	—	3	2	no	—
92	C₂×C₃×C₈	—	3	2	no	—
93	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
94	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
95	C₂×C₃⋊C₈	Dic₉	1	1	yes	characteristic
96	D₃×C₈	Dic₉	1	2	yes	—
97	D₃×C₈	C₂×D₉	1	2	yes	—
98	C₃×C₁₆	—	9	2	no	—
99	C₃⋊C₁₆	—	3	2	no	—
100	C₃⋊C₁₆	—	9	2	no	—
101	C₃⋊C₁₆	—	9	2	no	—
102	C₃⋊C₁₆	—	18	2	no	—
103	C₂²×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
104	C₂×C₃×D₃	—	2	2	no	—
105	C₂×C₃×D₃	C₂×C₃⋊C₈	1	2	yes	—
106	C₃²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
107	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
108	C₃×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
109	C₂²×C₉	—	2	1	no	—
110	C₂²×C₉	—	3	2	no	—
111	C₂²×C₉	—	6	2	no	—
112	C₂²×C₉	D₃×C₈	1	1	yes	characteristic
113	C₄×C₉	—	2	1	no	—
114	C₄×C₉	—	2	1	no	—
115	C₄×C₉	—	3	2	no	—
116	C₄×C₉	D₃×C₈	1	1	yes	characteristic
117	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
118	C₂²×C₈	—	3	1	no	—
119	M₃₂	—	27	4	no	—
120	C₂×C₁₆	—	9	1	no	—
121	C₂×C₁₆	—	27	1	no	—
122	C₃×C₉	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	3	1	no	—
124	C₂²×D₃	C₉⋊C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	3	1	no	—
127	C₂×C₃×C₄	—	3	2	no	—
128	C₂×C₃×C₄	—	3	2	no	—
129	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
130	C₂×C₃×C₄	D₃×Dic₃	1	1	yes	characteristic
131	C₂×Dic₃	C₉⋊C₈	1	1	yes	characteristic
132	C₄×D₃	C₉⋊C₈	1	2	yes	—
133	C₄×D₃	C₂×Dic₉	1	2	yes	—
134	C₃×C₈	—	2	1	no	—
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	3	2	no	—
137	C₃×C₈	C₂×Dic₉	1	1	yes	characteristic
138	C₃×C₈	C₂²×D₉	1	1	yes	characteristic
139	C₃×C₈	D₃×Dic₃	1	1	yes	characteristic
140	C₃×C₈	C₂×D₃²	1	1	yes	characteristic
141	C₃⋊C₈	C₂×Dic₉	1	2	yes	—
142	C₂×C₃²	#812	1	2	yes	—
143	C₂×C₃²	C₂²×C₃⋊C₈	1	1	yes	characteristic
144	C₃×D₃	—	2	2	no	—
145	C₂×C₉	—	2	2	no	—
146	C₂×C₉	—	2	1	no	—
147	C₂×C₉	—	6	2	no	—
148	C₂×C₉	#798	1	2	yes	complemented
149	C₂×C₉	C₂×D₃×C₈	1	1	yes	characteristic
150	C₂²×C₄	—	3	1	no	—
151	C₂×C₈	—	3	1	no	—
152	C₂×C₈	—	3	2	no	—
153	C₂×C₈	—	3	2	no	—
154	C₂×C₈	D₃×D₉	1	1	yes	characteristic
155	C₁₆	—	9	2	no	—
156	C₁₆	—	27	2	no	—
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	3	2	no	—
159	C₂²×C₃	—	6	2	no	—
160	C₂²×C₃	C₂×C₉⋊C₈	1	1	yes	characteristic
161	C₂²×C₃	D₃×C₃⋊C₈	1	1	yes	characteristic
162	C₂×D₃	—	2	2	no	—
163	C₂×D₃	C₂×C₉⋊C₈	1	2	yes	—
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	1	no	—
166	C₃×C₄	—	3	2	no	—
167	C₃×C₄	C₂×C₉⋊C₈	1	1	yes	characteristic
168	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
169	C₃×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
170	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
171	Dic₃	C₂×C₉⋊C₈	1	2	yes	—
172	C₃²	C₂×#812	1	1	yes	characteristic
173	C₉	—	2	1	no	—
174	C₉	C₂×#798	1	1	yes	characteristic, complemented
175	C₂³	—	3	1	no	—
176	C₂×C₄	—	3	1	no	—
177	C₂×C₄	—	3	2	no	—
178	C₂×C₄	—	3	2	no	—
179	C₂×C₄	D₃×Dic₉	1	1	yes	characteristic
180	C₈	—	3	2	no	—
181	C₈	D₃×Dic₉	1	1	yes	characteristic
182	C₈	C₂×D₃×D₉	1	1	yes	characteristic
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	—	2	1	no	—
185	C₂×C₃	—	6	2	no	—
186	C₂×C₃	#63371	1	2	yes	complemented
187	C₂×C₃	C₂²×C₉⋊C₈	1	1	yes	characteristic
188	C₂×C₃	#63543	1	2	yes	—
189	C₂×C₃	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
190	D₃	—	2	2	no	—
191	C₂²	—	3	2	no	—
192	C₂²	—	6	2	no	—
193	C₂²	D₃×C₉⋊C₈	1	1	yes	characteristic
194	C₄	—	3	2	no	—
195	C₄	D₃×C₉⋊C₈	1	1	yes	characteristic
196	C₄	C₂×D₃×Dic₉	1	1	yes	characteristic
197	C₃	—	2	1	no	—
198	C₃	C₂×#63371	1	1	yes	characteristic, complemented
199	C₃	C₂×#63543	1	1	yes	characteristic
200	C₂	—	6	2	no	—
201	C₂	#11720237	1	2	yes	direct product factor
202	C₂	C₂×D₃×C₉⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	D₃×D₉
commutator	C₂×C₃×C₉	C₂²×C₈
Frattini	C₃×C₈	C₂×D₃²
Fitting	C₂×C₃×C₈×C₉	C₂²
socle	C₂²×C₃²	C₂×C₃⋊C₈
prime radical	C₂²×C₃×D₃	C₃⋊C₈
cosocle	C₃×C₈×C₉	C₂³
hypocenter	C₃×C₉	C₂×M₃₂
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊(C₉⋊C₁₆)	C₂	semidirect
3. C₂×C₃×C₉⋊C₁₆	C₂	semidirect
4. #11720237	C₂	direct (N × H)
12. C₃⋊(C₉⋊C₁₆)	C₂²	semidirect
13. C₃×C₉⋊C₁₆	C₂²	semidirect
22. C₂×C₉⋊C₁₆	D₃	semidirect
48. C₉⋊C₁₆	C₂×D₃	semidirect
84. C₂×C₃×C₉	M₃₂	semidirect
122. C₃×C₉	C₂×M₃₂	semidirect
148. C₂×C₉	#798	semidirect
174. C₉	C₂×#798	semidirect
186. C₂×C₃	#63371	semidirect
198. C₃	C₂×#63371	semidirect
201. C₂	#11720237	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×D₃×C₈×C₉	2
C₂×C₃⋊(C₉⋊C₁₆)	2
C₂×C₃×C₉⋊C₁₆	2
#11720237	2
C₂×#63543	3
C₂×#63371	3

Subgroups — C2×#11720237 #13138092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11720237 #13138092