Subgroups — C₂×C₃×#63378 #13143079

Order 1728.

466 conjugacy classes of subgroups.

The subgroup lattice has 230 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×#63378	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×#818	—	3	1	no	maximal
5	C₂×#63378	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×#818	—	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	#63378	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×#107	—	9	1	no	—
28	C₂×#818	—	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	D₈	1	2	yes	—
46	C₂²×C₃×C₉	Q₁₆	1	2	yes	—
47	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
48	C₃×C₄×C₉	C₄⋊C₄	1	2	yes	—
49	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
50	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
51	C₃×Dic₉	—	4	4	no	—
52	C₂²×C₃×C₈	—	2	1	no	—
53	C₂²×C₃×C₈	D₉	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₃×D₃	1	1	yes	characteristic
55	C₂×C₃×C₄⋊C₄	—	9	2	no	—
56	C₂×C₄⋊Dic₃	—	3	2	no	—
57	C₃×#107	—	9	4	no	—
58	#818	—	3	4	no	—
59	C₂³×C₃²	D₁₂	1	1	yes	characteristic
60	C₂×C₃²×C₄	Dic₆	1	2	yes	—
61	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
63	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
64	C₂×C₃×Dic₃	—	6	4	no	—
65	C₂×C₃×Dic₃	—	12	4	no	—
66	C₂³×C₉	—	2	1	no	—
67	C₂³×C₉	C₃×D₄	1	1	yes	characteristic
68	C₃²×C₈	C₂×Dic₃	1	4	yes	—
69	C₂×C₄×C₉	—	2	2	no	—
70	C₂×C₄×C₉	—	2	1	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
74	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
75	C₂×C₄×C₉	C₃×Q₈	1	2	yes	—
76	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
77	C₂×Dic₉	—	2	4	no	—
78	C₂×Dic₉	—	4	4	no	—
79	C₈×C₉	—	2	4	no	—
80	C₈×C₉	C₂×C₃×C₄	1	4	yes	complemented
81	C₂×#107	—	9	1	no	Sylow
82	C₂×C₃×C₉	#107	1	4	yes	complemented
83	C₂×C₃×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
85	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	18	2	no	—
88	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
89	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
90	C₂²×Dic₃	—	6	2	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	—	2	2	no	—
93	C₂×C₃×C₈	Dic₉	1	4	yes	complemented
94	C₂×C₃×C₈	C₂×D₉	1	2	yes	—
95	C₂×C₃×C₈	C₃×Dic₃	1	4	yes	—
96	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
97	C₃×C₄⋊C₄	—	9	4	no	—
98	C₃×C₄⋊C₄	—	18	2	no	—
99	C₄⋊Dic₃	—	3	4	no	—
100	C₄⋊Dic₃	—	6	2	no	—
101	C₂²×C₃²	D₂₄	1	2	yes	—
102	C₂²×C₃²	Dic₁₂	1	2	yes	—
103	C₂²×C₃²	C₄⋊Dic₃	1	2	yes	—
104	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
105	C₃²×C₄	C₄⋊Dic₃	1	2	yes	—
106	C₃²×C₄	C₂×Dic₆	1	1	yes	characteristic
107	C₃²×C₄	C₂²×Dic₃	1	1	yes	characteristic
108	C₃×Dic₃	—	12	4	no	—
109	C₂²×C₉	—	2	2	no	—
110	C₂²×C₉	—	2	2	no	—
111	C₂²×C₉	—	2	2	no	—
112	C₂²×C₉	—	2	1	no	—
113	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
114	C₂²×C₉	C₃×D₈	1	2	yes	—
115	C₂²×C₉	C₃×Q₁₆	1	2	yes	—
116	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
117	C₄×C₉	—	2	2	no	—
118	C₄×C₉	—	2	1	no	—
119	C₄×C₉	—	2	1	no	—
120	C₄×C₉	C₃×C₄⋊C₄	1	2	yes	—
121	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
122	C₄×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
123	Dic₉	—	4	4	no	—
124	C₂²×C₈	C₃×D₉	1	1	yes	characteristic
125	C₂×C₄⋊C₄	—	9	2	no	—
126	#107	—	9	4	no	—
127	C₃×C₉	C₂×#107	1	1	yes	characteristic, complemented, Sylow
128	C₂³×C₃	—	2	1	no	—
129	C₂³×C₃	D₃₆	1	1	yes	characteristic
130	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
131	C₂×C₃×C₄	—	2	2	no	—
132	C₂×C₃×C₄	—	2	1	no	—
133	C₂×C₃×C₄	—	2	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	18	4	no	—
136	C₂×C₃×C₄	—	36	4	no	—
137	C₂×C₃×C₄	Dic₁₈	1	2	yes	—
138	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×Dic₉	1	2	yes	—
140	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
141	C₂×C₃×C₄	C₃×Dic₆	1	2	yes	—
142	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₃×Dic₃	1	2	yes	—
144	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
145	C₂×Dic₃	—	6	4	no	—
146	C₂×Dic₃	—	12	4	no	—
147	C₃×C₈	—	2	4	no	—
148	C₃×C₈	C₂×Dic₉	1	4	yes	complemented
149	C₃×C₈	C₂×C₃×Dic₃	1	4	yes	—
150	C₂×C₃²	#818	1	4	yes	—
151	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
152	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
153	C₂×C₃²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
154	C₂×C₉	—	2	1	no	—
155	C₂×C₉	—	2	1	no	—
156	C₂×C₉	—	2	1	no	—
157	C₂×C₉	—	2	4	no	—
158	C₂×C₉	C₃×#107	1	4	yes	complemented
159	C₂×C₉	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
161	C₂×C₉	C₂×C₃×Q₁₆	1	1	yes	characteristic
162	C₂²×C₄	—	18	2	no	—
163	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
164	C₂×C₈	C₃×Dic₉	1	4	yes	complemented
165	C₂×C₈	C₂×C₃×D₉	1	2	yes	—
166	C₄⋊C₄	—	9	4	no	—
167	C₄⋊C₄	—	18	2	no	—
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	—	2	2	no	—
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	Dic₃₆	1	2	yes	—
173	C₂²×C₃	D₇₂	1	2	yes	—
174	C₂²×C₃	C₄⋊Dic₉	1	2	yes	—
175	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
176	C₂²×C₃	C₃×D₂₄	1	2	yes	—
177	C₂²×C₃	C₃×Dic₁₂	1	2	yes	—
178	C₂²×C₃	C₃×C₄⋊Dic₃	1	2	yes	—
179	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	2	no	—
183	C₃×C₄	—	36	4	no	—
184	C₃×C₄	C₄⋊Dic₉	1	2	yes	—
185	C₃×C₄	C₂×Dic₁₈	1	1	yes	characteristic
186	C₃×C₄	C₂²×Dic₉	1	1	yes	characteristic
187	C₃×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
188	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
189	C₃×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
190	Dic₃	—	12	4	no	—
191	C₃²	C₂×#818	1	1	yes	characteristic
192	C₉	—	2	1	no	—
193	C₉	C₂×C₃×#107	1	1	yes	characteristic, complemented
194	C₂³	C₃×D₃₆	1	1	yes	characteristic
195	C₂×C₄	—	18	4	no	—
196	C₂×C₄	—	36	4	no	—
197	C₂×C₄	C₃×Dic₁₈	1	2	yes	—
198	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃×Dic₉	1	2	yes	—
200	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
201	C₈	C₂×C₃×Dic₉	1	4	yes	complemented
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	2	4	no	—
206	C₂×C₃	#63378	1	4	yes	direct product factor
207	C₂×C₃	C₂×Dic₃₆	1	1	yes	characteristic
208	C₂×C₃	C₂×D₇₂	1	1	yes	characteristic
209	C₂×C₃	C₂×C₄⋊Dic₉	1	1	yes	characteristic
210	C₂×C₃	C₃×#818	1	4	yes	—
211	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
212	C₂×C₃	C₂×C₃×Dic₁₂	1	1	yes	characteristic
213	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
214	C₂²	C₃×Dic₃₆	1	2	yes	—
215	C₂²	C₃×D₇₂	1	2	yes	—
216	C₂²	C₃×C₄⋊Dic₉	1	2	yes	—
217	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
218	C₄	—	36	4	no	—
219	C₄	C₃×C₄⋊Dic₉	1	2	yes	—
220	C₄	C₂×C₃×Dic₁₈	1	1	yes	characteristic
221	C₄	C₂²×C₃×Dic₉	1	1	yes	characteristic
222	C₃	—	2	1	no	—
223	C₃	C₂×#63378	1	1	yes	characteristic, direct product factor
224	C₃	C₂×C₃×#818	1	1	yes	characteristic
225	C₂	C₃×#63378	1	4	yes	direct product factor
226	C₂	C₂×C₃×Dic₃₆	1	1	yes	characteristic
227	C₂	C₂×C₃×D₇₂	1	1	yes	characteristic
228	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×#107
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×#63378	C₂	direct (N × H)
5. C₂×#63378	C₃	direct (N × H)
8. C₂×C₃×C₈×C₉	C₄	semidirect
18. #63378	C₂×C₃	direct (N × H)
26. C₃×C₈×C₉	C₂×C₄	semidirect
40. C₂×C₈×C₉	C₃×C₄	semidirect
80. C₈×C₉	C₂×C₃×C₄	semidirect
82. C₂×C₃×C₉	#107	semidirect
93. C₂×C₃×C₈	Dic₉	semidirect
127. C₃×C₉	C₂×#107	semidirect
148. C₃×C₈	C₂×Dic₉	semidirect
158. C₂×C₉	C₃×#107	semidirect
164. C₂×C₈	C₃×Dic₉	semidirect
193. C₉	C₂×C₃×#107	semidirect
201. C₈	C₂×C₃×Dic₉	semidirect
206. C₂×C₃	#63378	direct (N × H)
223. C₃	C₂×#63378	direct (N × H)
225. C₂	C₃×#63378	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×#63378	2
C₂×C₃×#818	3
C₂×#63378	3

Subgroups — C2×C3×#63378 #13143079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63378 #13143079