Subgroups — C₂×C₃×#63377 #13143078

Order 1728.

466 conjugacy classes of subgroups.

The subgroup lattice has 214 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×C₄⋊Dic₉	C₂	1	2	yes	maximal
3	C₃×#63377	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#817	—	3	1	no	maximal
5	C₂×#63377	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×Dic₉	—	2	2	no	—
8	C₂×C₃×C₈×C₉	C₄	1	4	yes	complemented
9	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
10	C₃×C₄⋊Dic₉	—	2	2	no	—
11	C₃×C₄⋊Dic₉	C₂²	1	4	yes	—
12	C₂²×C₃²×C₈	D₃	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	—	3	2	no	—
14	C₃×#817	—	3	4	no	—
15	C₂²×C₈×C₉	—	2	1	no	—
16	C₂²×C₈×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂×C₄⋊Dic₉	C₂×C₃	1	2	yes	—
18	#63377	C₂×C₃	1	4	yes	direct product factor
19	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
20	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₉	Q₈	1	2	yes	—
23	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×C₃×Dic₉	—	2	4	no	—
25	C₂×C₃×Dic₉	—	4	4	no	—
26	C₃×C₈×C₉	C₂×C₄	1	4	yes	complemented
27	C₂×C₃×#106	—	9	1	no	—
28	C₂×#817	—	3	1	no	—
29	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
30	C₂²×C₃×Dic₃	—	6	2	no	—
31	C₂×C₃²×C₈	Dic₃	1	4	yes	—
32	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
33	C₃×C₄⋊Dic₃	—	3	4	no	—
34	C₃×C₄⋊Dic₃	—	6	2	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
37	C₂²×Dic₉	—	2	2	no	—
38	C₂×C₈×C₉	—	2	4	no	—
39	C₂×C₈×C₉	—	2	2	no	—
40	C₂×C₈×C₉	C₃×C₄	1	4	yes	complemented
41	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
42	C₄⋊Dic₉	—	2	2	no	—
43	C₄⋊Dic₉	C₂²×C₃	1	4	yes	—
44	C₂²×C₃×C₉	C₄⋊C₄	1	2	yes	—
45	C₂²×C₃×C₉	SD₁₆	1	4	yes	—
46	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
47	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
48	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
49	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
50	C₃×Dic₉	—	4	4	no	—
51	C₂²×C₃×C₈	—	2	1	no	—
52	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
53	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄⋊C₄	—	9	2	no	—
55	C₂×C₄⋊Dic₃	—	3	2	no	—
56	C₃×#106	—	9	4	no	—
57	#817	—	3	4	no	—
58	C₂³×C₃²	D₁₂	1	1	yes	characteristic
59	C₂×C₃²×C₄	Dic₆	1	2	yes	—
60	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
62	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
63	C₂×C₃×Dic₃	—	6	4	no	—
64	C₂×C₃×Dic₃	—	12	4	no	—
65	C₂³×C₉	—	2	1	no	—
66	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
67	C₃²×C₈	C₂×Dic₃	1	4	yes	—
68	C₂×C₄×C₉	—	2	2	no	—
69	C₂×C₄×C₉	—	2	1	no	—
70	C₂×C₄×C₉	—	2	2	no	—
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
74	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
75	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
76	C₂×Dic₉	—	2	4	no	—
77	C₂×Dic₉	—	4	4	no	—
78	C₈×C₉	—	2	4	no	—
79	C₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
80	C₂×#106	—	9	1	no	Sylow
81	C₂×C₃×C₉	#106	1	4	yes	complemented
82	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₉	C₂×SD₁₆	1	2	yes	—
84	C₂²×C₃×C₄	—	2	1	no	—
85	C₂²×C₃×C₄	—	18	2	no	—
86	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
88	C₂²×Dic₃	—	6	2	no	—
89	C₂×C₃×C₈	—	2	4	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	Dic₉	1	4	yes	complemented
92	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
93	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
94	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
95	C₃×C₄⋊C₄	—	9	4	no	—
96	C₃×C₄⋊C₄	—	18	2	no	—
97	C₄⋊Dic₃	—	3	4	no	—
98	C₄⋊Dic₃	—	6	2	no	—
99	C₂²×C₃²	#204	1	4	yes	—
100	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
101	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
102	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
103	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
104	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
105	C₃×Dic₃	—	12	4	no	—
106	C₂²×C₉	—	2	4	no	—
107	C₂²×C₉	—	2	2	no	—
108	C₂²×C₉	—	2	1	no	—
109	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
110	C₂²×C₉	C₃×SD₁₆	1	4	yes	—
111	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
112	C₄×C₉	—	2	2	no	—
113	C₄×C₉	—	2	1	no	—
114	C₄×C₉	—	2	1	no	—
115	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
116	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
118	Dic₉	—	4	4	no	—
119	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	9	2	no	—
121	#106	—	9	4	no	—
122	C₃×C₉	C₂×#106	1	1	yes	characteristic, complemented, Sylow
123	C₂³×C₃	—	2	1	no	—
124	C₂³×C₃	D₃₆	1	1	yes	characteristic
125	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	18	4	no	—
131	C₂×C₃×C₄	—	36	4	no	—
132	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
133	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
135	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
136	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
137	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
139	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
140	C₂×Dic₃	—	6	4	no	—
141	C₂×Dic₃	—	12	4	no	—
142	C₃×C₈	—	2	4	no	—
143	C₃×C₈	C₂×Dic₉	1	4	yes	complemented
144	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
145	C₂×C₃²	#817	1	4	yes	—
146	C₂×C₃²	C₂×#204	1	2	yes	—
147	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
148	C₂×C₉	—	2	2	no	—
149	C₂×C₉	—	2	1	no	—
150	C₂×C₉	—	2	4	no	—
151	C₂×C₉	C₃×#106	1	4	yes	complemented
152	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₉	C₂×C₃×SD₁₆	1	2	yes	—
154	C₂²×C₄	—	18	2	no	—
155	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
156	C₂×C₈	C₃×Dic₉	1	4	yes	complemented
157	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
158	C₄⋊C₄	—	9	4	no	—
159	C₄⋊C₄	—	18	2	no	—
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	4	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	#3664	1	4	yes	—
164	C₂²×C₃	C₄⋊Dic₉	1	2	yes	—
165	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
166	C₂²×C₃	C₃×#204	1	4	yes	—
167	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
168	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
169	C₃×C₄	—	2	1	no	—
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	2	no	—
172	C₃×C₄	—	36	4	no	—
173	C₃×C₄	C₄⋊Dic₉	1	2	yes	—
174	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
175	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
176	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
177	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
178	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
179	Dic₃	—	12	4	no	—
180	C₃²	C₂×#817	1	1	yes	characteristic
181	C₉	—	2	1	no	—
182	C₉	C₂×C₃×#106	1	1	yes	characteristic, complemented
183	C₂³	C₃×D₃₆	1	1	yes	characteristic
184	C₂×C₄	—	18	4	no	—
185	C₂×C₄	—	36	4	no	—
186	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
187	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
188	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
189	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
190	C₈	C₂×C₃×Dic₉	1	4	yes	complemented
191	C₂×C₃	—	2	2	no	—
192	C₂×C₃	—	2	1	no	—
193	C₂×C₃	—	2	4	no	—
194	C₂×C₃	#63377	1	4	yes	direct product factor
195	C₂×C₃	C₂×#3664	1	2	yes	—
196	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
197	C₂×C₃	C₃×#817	1	4	yes	—
198	C₂×C₃	C₂×C₃×#204	1	2	yes	—
199	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
200	C₂²	C₃×#3664	1	4	yes	—
201	C₂²	C₃×C₄⋊Dic₉	1	2	yes	—
202	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
203	C₄	—	36	4	no	—
204	C₄	C₃×C₄⋊Dic₉	1	2	yes	—
205	C₄	C₂×C₃×Dic₁₈	1	1	yes	characteristic
206	C₄	C₂²×C₃×Dic₉	1	1	yes	characteristic
207	C₃	—	2	1	no	—
208	C₃	C₂×#63377	1	1	yes	characteristic, direct product factor
209	C₃	C₂×C₃×#817	1	1	yes	characteristic
210	C₂	C₃×#63377	1	4	yes	direct product factor
211	C₂	C₂×C₃×#3664	1	2	yes	—
212	C₂	C₂×C₃×C₄⋊Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₃₆
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃×C₈×C₉	C₂
socle, prime radical	C₂³×C₃²	D₁₂
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₉
hypocenter	C₉	C₂×C₃×#106
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#63377	C₂	direct (N × H)
5. C₂×#63377	C₃	direct (N × H)
8. C₂×C₃×C₈×C₉	C₄	semidirect
18. #63377	C₂×C₃	direct (N × H)
26. C₃×C₈×C₉	C₂×C₄	semidirect
40. C₂×C₈×C₉	C₃×C₄	semidirect
79. C₈×C₉	C₂×C₃×C₄	semidirect
81. C₂×C₃×C₉	#106	semidirect
91. C₂×C₃×C₈	Dic₉	semidirect
122. C₃×C₉	C₂×#106	semidirect
143. C₃×C₈	C₂×Dic₉	semidirect
151. C₂×C₉	C₃×#106	semidirect
156. C₂×C₈	C₃×Dic₉	semidirect
182. C₉	C₂×C₃×#106	semidirect
190. C₈	C₂×C₃×Dic₉	semidirect
194. C₂×C₃	#63377	direct (N × H)
208. C₃	C₂×#63377	direct (N × H)
210. C₂	C₃×#63377	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×C₄⋊Dic₉	2
C₃×#63377	2
C₂×C₃×#817	3
C₂×#63377	3

Subgroups — C2×C3×#63377 #13143078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63377 #13143078