Subgroups — C₂×C₃×#63646 #13163144

Order 1728.

940 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3748	C₂	1	2	yes	maximal
3	C₃×#63646	C₂	1	4	yes	maximal, direct product factor
4	C₂×#63646	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×#819	—	3	4	no	maximal
6	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃³×C₈	C₄	1	2	yes	—
8	C₂×C₃³×C₈	C₄	1	2	yes	—
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₃×#3748	—	2	2	no	—
11	C₃×#3748	C₂²	1	4	yes	—
12	C₂×C₃×C₃⋊(C₃⋊C₈)	—	2	2	no	—
13	C₂²×C₃²×C₈	—	2	4	no	—
14	C₂²×C₃²×C₈	D₃	1	4	yes	—
15	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
16	C₂×#3748	C₂×C₃	1	2	yes	—
17	C₂×C₃×#208	—	3	8	no	—
18	#63646	C₂×C₃	1	4	yes	direct product factor
19	C₃×#819	—	3	16	no	—
20	C₂³×C₃³	Q₈	1	1	yes	characteristic
21	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃³×C₄	Q₈	1	1	yes	characteristic
26	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
27	C₃³×C₈	C₂×C₄	1	4	yes	—
28	C₃×C₃⋊(C₃⋊C₈)	—	2	4	no	—
29	C₂×C₃×#108	—	9	1	no	—
30	C₂×#819	—	3	4	no	—
31	C₂²×C₃²×C₄	—	2	4	no	—
32	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
33	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₂×C₃²×C₈	—	2	8	no	—
35	C₂×C₃²×C₈	—	2	8	no	—
36	C₂×C₃²×C₈	—	2	8	no	—
37	C₂×C₃²×C₈	Dic₃	1	8	yes	—
38	C₂×C₃²×C₈	Dic₃	1	8	yes	—
39	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
40	C₂×C₃²×C₈	C₃×C₄	1	2	yes	—
41	C₂×C₃²×C₈	C₂×D₃	1	8	yes	—
42	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
43	#3748	—	2	2	no	—
44	#3748	C₂²×C₃	1	4	yes	—
45	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
46	C₃×#208	—	3	16	no	—
47	C₃×#208	—	6	8	no	—
48	C₂×C₃×C₃⋊C₈	—	6	8	no	—
49	C₂²×C₃³	—	2	2	no	—
50	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₃³	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₃³	C₂×Q₈	1	1	yes	characteristic
53	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
54	C₃³×C₄	C₄⋊C₄	1	1	yes	characteristic
55	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₃³×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×M₁₆	—	9	2	no	—
58	C₂²×C₃×C₈	—	2	4	no	—
59	C₂²×C₃×C₈	C₃×D₃	1	4	yes	—
60	C₂²×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
61	C₂×#208	—	3	8	no	—
62	C₃×#108	—	9	4	no	—
63	#819	—	3	16	no	—
64	C₂³×C₃²	—	2	4	no	—
65	C₂³×C₃²	Dic₆	1	4	yes	—
66	C₂³×C₃²	C₃×Q₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	—	2	4	no	—
68	C₂×C₃²×C₄	—	2	4	no	—
69	C₂×C₃²×C₄	—	2	4	no	—
70	C₂×C₃²×C₄	—	2	4	no	—
71	C₂×C₃²×C₄	—	2	4	no	—
72	C₂×C₃²×C₄	—	2	4	no	—
73	C₂×C₃²×C₄	Dic₆	1	4	yes	—
74	C₂×C₃²×C₄	D₁₂	1	4	yes	—
75	C₂×C₃²×C₄	D₁₂	1	4	yes	—
76	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
77	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
78	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
80	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₃×Q₈	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
84	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
85	C₃²×C₈	—	2	16	no	—
86	C₃²×C₈	C₂×Dic₃	1	16	yes	—
87	C₃²×C₈	C₂×C₃×C₄	1	4	yes	—
88	C₃⋊(C₃⋊C₈)	—	2	4	no	—
89	C₃×C₃⋊C₈	—	6	16	no	—
90	C₂×#108	—	9	1	no	Sylow
91	C₂×C₃³	—	2	1	no	—
92	C₂×C₃³	—	2	1	no	—
93	C₂×C₃³	#108	1	2	yes	complemented
94	C₂×C₃³	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₃×C₄	—	2	4	no	—
96	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
97	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₃×M₁₆	—	9	4	no	—
99	C₃×M₁₆	—	18	2	no	—
100	C₂×C₃×C₈	—	2	8	no	—
101	C₂×C₃×C₈	—	2	8	no	—
102	C₂×C₃×C₈	—	2	8	no	—
103	C₂×C₃×C₈	—	18	2	no	—
104	C₂×C₃×C₈	C₃×Dic₃	1	8	yes	—
105	C₂×C₃×C₈	C₃×Dic₃	1	8	yes	—
106	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	—
107	C₂×C₃×C₈	C₃⋊Dic₃	1	2	yes	—
108	C₂×C₃×C₈	C₂×C₃×D₃	1	8	yes	—
109	C₂×C₃×C₈	C₂×C₃⋊D₃	1	2	yes	—
110	#208	—	3	16	no	—
111	#208	—	6	8	no	—
112	C₂×C₃⋊C₈	—	6	8	no	—
113	C₂²×C₃²	—	2	16	no	—
114	C₂²×C₃²	—	2	8	no	—
115	C₂²×C₃²	—	2	2	no	—
116	C₂²×C₃²	—	2	4	no	—
117	C₂²×C₃²	—	2	4	no	—
118	C₂²×C₃²	—	2	4	no	—
119	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
120	C₂²×C₃²	C₄⋊Dic₃	1	4	yes	—
121	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
122	C₂²×C₃²	C₃×C₄⋊C₄	1	1	yes	characteristic
123	C₂²×C₃²	C₂×Dic₆	1	4	yes	—
124	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
125	C₃²×C₄	—	2	4	no	—
126	C₃²×C₄	—	2	4	no	—
127	C₃²×C₄	—	2	4	no	—
128	C₃²×C₄	—	2	4	no	—
129	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
130	C₃²×C₄	C₄⋊Dic₃	1	4	yes	—
131	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
132	C₃²×C₄	C₃×C₄⋊C₄	1	1	yes	characteristic
133	C₃²×C₄	C₂×D₁₂	1	4	yes	—
134	C₃²×C₄	C₂²×Dic₃	1	4	yes	—
135	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×M₁₆	—	9	2	no	—
138	C₂²×C₈	C₃×C₃⋊D₃	1	1	yes	characteristic
139	#108	—	9	4	no	—
140	C₃³	C₂×#108	1	1	yes	characteristic, complemented, Sylow
141	C₂³×C₃	—	2	4	no	—
142	C₂³×C₃	C₃×Dic₆	1	4	yes	—
143	C₂³×C₃	#634	1	1	yes	characteristic
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	4	no	—
150	C₂×C₃×C₄	C₃×Dic₆	1	4	yes	—
151	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
152	C₂×C₃×C₄	C₃×D₁₂	1	4	yes	—
153	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
154	C₂×C₃×C₄	C₂×C₃×Dic₃	1	4	yes	—
155	C₂×C₃×C₄	#634	1	1	yes	characteristic
156	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
157	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
161	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
162	C₃×C₈	—	2	16	no	—
163	C₃×C₈	—	18	4	no	—
164	C₃×C₈	C₂×C₃×Dic₃	1	16	yes	—
165	C₃×C₈	C₂×C₃⋊Dic₃	1	4	yes	—
166	C₃⋊C₈	—	6	16	no	—
167	C₂×C₃²	—	2	8	no	—
168	C₂×C₃²	—	2	8	no	—
169	C₂×C₃²	—	2	1	no	—
170	C₂×C₃²	—	2	1	no	—
171	C₂×C₃²	—	2	4	no	—
172	C₂×C₃²	—	2	4	no	—
173	C₂×C₃²	—	2	4	no	—
174	C₂×C₃²	—	2	8	no	—
175	C₂×C₃²	#819	1	8	yes	complemented
176	C₂×C₃²	C₃×#108	1	2	yes	complemented
177	C₂×C₃²	C₂×C₄⋊Dic₃	1	4	yes	—
178	C₂×C₃²	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
180	M₁₆	—	9	4	no	—
181	M₁₆	—	18	2	no	—
182	C₂×C₈	—	18	2	no	—
183	C₂×C₈	C₃×C₃⋊Dic₃	1	2	yes	—
184	C₂×C₈	C₃×C₃⋊Dic₃	1	2	yes	—
185	C₂×C₈	C₂×C₃×C₃⋊D₃	1	2	yes	—
186	C₂²×C₃	—	2	16	no	—
187	C₂²×C₃	—	2	8	no	—
188	C₂²×C₃	—	2	2	no	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	2	4	no	—
192	C₂²×C₃	C₃×C₄⋊Dic₃	1	4	yes	—
193	C₂²×C₃	C₃×C₄⋊Dic₃	1	4	yes	—
194	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
195	C₂²×C₃	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
196	C₂²×C₃	C₂×C₃×Dic₆	1	4	yes	—
197	C₂²×C₃	C₂×#634	1	1	yes	characteristic
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	4	no	—
202	C₃×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
203	C₃×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
204	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
205	C₃×C₄	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
206	C₃×C₄	C₂×C₃×D₁₂	1	4	yes	—
207	C₃×C₄	C₂²×C₃×Dic₃	1	4	yes	—
208	C₃×C₄	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
209	C₃×C₄	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
210	C₃²	—	2	4	no	—
211	C₃²	C₂×#819	1	4	yes	complemented
212	C₃²	C₂×C₃×#108	1	1	yes	characteristic, complemented
213	C₂³	C₃×#634	1	1	yes	characteristic
214	C₂×C₄	C₃×#634	1	1	yes	characteristic
215	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
216	C₂×C₄	C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
217	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
219	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	1	yes	characteristic
220	C₈	—	18	4	no	—
221	C₈	C₂×C₃×C₃⋊Dic₃	1	4	yes	—
222	C₂×C₃	—	2	8	no	—
223	C₂×C₃	—	2	8	no	—
224	C₂×C₃	—	2	1	no	—
225	C₂×C₃	—	2	1	no	—
226	C₂×C₃	—	2	4	no	—
227	C₂×C₃	—	2	4	no	—
228	C₂×C₃	—	2	4	no	—
229	C₂×C₃	—	2	8	no	—
230	C₂×C₃	C₃×#819	1	8	yes	complemented
231	C₂×C₃	#63646	1	2	yes	direct product factor
232	C₂×C₃	C₂×C₃×C₄⋊Dic₃	1	4	yes	—
233	C₂×C₃	C₂×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
236	C₂²	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
237	C₂²	C₂×C₃×#634	1	1	yes	characteristic
238	C₄	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
239	C₄	C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
240	C₄	C₂×C₃×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
241	C₄	C₂²×C₃×C₃⋊Dic₃	1	1	yes	characteristic
242	C₃	—	2	4	no	—
243	C₃	C₂×C₃×#819	1	4	yes	complemented
244	C₃	C₂×#63646	1	1	yes	characteristic, direct product factor
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	C₃×#63646	1	2	yes	direct product factor
248	C₂	C₂×C₃×C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×C₃⋊D₃
Fitting	C₂²×C₃³×C₈	C₂
socle	C₂²×C₃³	C₄⋊C₄
prime radical	C₂³×C₃³	Q₈
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₂²×C₃×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×C₃×#108
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×#63646	C₂	direct (N × H)
4. C₂×#63646	C₃	direct (N × H)
18. #63646	C₂×C₃	direct (N × H)
93. C₂×C₃³	#108	semidirect
140. C₃³	C₂×#108	semidirect
175. C₂×C₃²	#819	semidirect
176. C₂×C₃²	C₃×#108	semidirect
211. C₃²	C₂×#819	semidirect
212. C₃²	C₂×C₃×#108	semidirect
230. C₂×C₃	C₃×#819	semidirect
231. C₂×C₃	#63646	direct (N × H)
243. C₃	C₂×C₃×#819	semidirect
244. C₃	C₂×#63646	direct (N × H)
247. C₂	C₃×#63646	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×C₃³×C₈	2
C₂×C₃×#3748	2
C₃×#63646	2
C₂×#63646	3
C₂×C₃×#819	3

Subgroups — C2×C3×#63646 #13163144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#63646 #13163144