Subgroups — C₂²×C₃×#3748 #13179532

Order 1728.

2732 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3748	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂²×#3748	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂²×C₃×#208	—	3	4	no	maximal
6	C₂⁴×C₃³	C₄	1	1	yes	characteristic
7	C₂²×C₃³×C₄	C₄	1	6	yes	—
8	C₂²×C₃³×C₄	C₄	1	1	yes	characteristic
9	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂²×C₃³×C₄	C₂²	1	6	yes	—
11	C₃×#3748	C₂²	1	16	yes	direct product factor
12	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	12	yes	complemented
13	C₂³×C₃²×C₄	—	2	4	no	—
14	C₂³×C₃²×C₄	D₃	1	4	yes	—
15	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
16	C₂×#3748	C₂×C₃	1	12	yes	direct product factor
17	C₂²×C₃⋊(C₃⋊C₈)	C₂×C₃	1	2	yes	complemented
18	C₂×C₃×#208	—	3	48	no	—
19	C₂²×C₃×C₃⋊C₈	—	3	8	no	—
20	C₂³×C₃³	—	2	4	no	—
21	C₂³×C₃³	C₂×C₄	1	6	yes	—
22	C₂³×C₃³	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃³×C₄	C₂×C₄	1	12	yes	—
24	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
25	C₂×C₃³×C₄	C₂×C₄	1	6	yes	—
26	C₂×C₃³×C₄	C₂³	1	3	yes	—
27	C₂×C₃³×C₄	C₂³	1	4	yes	—
28	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	8	yes	complemented
29	C₂²×C₃×M₁₆	—	9	1	no	—
30	C₂²×#208	—	3	4	no	—
31	C₂⁴×C₃²	—	2	4	no	—
32	C₂⁴×C₃²	Dic₃	1	4	yes	—
33	C₂⁴×C₃²	C₃×C₄	1	1	yes	characteristic
34	C₂²×C₃²×C₄	—	2	24	no	—
35	C₂²×C₃²×C₄	—	2	4	no	—
36	C₂²×C₃²×C₄	—	2	4	no	—
37	C₂²×C₃²×C₄	—	2	24	no	—
38	C₂²×C₃²×C₄	Dic₃	1	24	yes	—
39	C₂²×C₃²×C₄	Dic₃	1	4	yes	—
40	C₂²×C₃²×C₄	C₃×C₄	1	6	yes	—
41	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
42	C₂²×C₃²×C₄	C₂×D₃	1	24	yes	—
43	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
44	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
45	C₂²×C₃²×C₄	C₂²×C₃	1	6	yes	—
46	#3748	C₂²×C₃	1	16	yes	direct product factor
47	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	12	yes	complemented
48	C₃×#208	—	3	64	no	—
49	C₂×C₃×C₃⋊C₈	—	3	48	no	—
50	C₂²×C₃³	—	2	12	no	—
51	C₂²×C₃³	M₁₆	1	4	yes	complemented
52	C₂²×C₃³	C₂²×C₄	1	4	yes	—
53	C₂²×C₃³	C₂²×C₄	1	3	yes	—
54	C₃³×C₄	C₂²×C₄	1	3	yes	—
55	C₃³×C₄	C₂²×C₄	1	4	yes	—
56	C₃³×C₄	C₂⁴	1	1	yes	characteristic
57	C₂³×C₃×C₄	—	2	4	no	—
58	C₂³×C₃×C₄	C₃×D₃	1	4	yes	—
59	C₂³×C₃×C₄	C₃⋊D₃	1	1	yes	characteristic
60	C₂×C₃×M₁₆	—	9	12	no	—
61	C₂²×C₃×C₈	—	9	2	no	—
62	C₂×#208	—	3	48	no	—
63	C₂²×C₃⋊C₈	—	3	8	no	—
64	C₂³×C₃²	—	2	32	no	—
65	C₂³×C₃²	—	2	16	no	—
66	C₂³×C₃²	—	2	4	no	—
67	C₂³×C₃²	—	2	24	no	—
68	C₂³×C₃²	—	2	4	no	—
69	C₂³×C₃²	C₂×Dic₃	1	24	yes	—
70	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
71	C₂³×C₃²	C₂×C₃×C₄	1	6	yes	—
72	C₂³×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
73	C₂×C₃²×C₄	—	2	48	no	—
74	C₂×C₃²×C₄	—	2	12	no	—
75	C₂×C₃²×C₄	—	2	24	no	—
76	C₂×C₃²×C₄	—	2	12	no	—
77	C₂×C₃²×C₄	—	2	16	no	—
78	C₂×C₃²×C₄	C₂×Dic₃	1	48	yes	—
79	C₂×C₃²×C₄	C₂×Dic₃	1	24	yes	—
80	C₂×C₃²×C₄	C₂×Dic₃	1	12	yes	—
81	C₂×C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
82	C₂×C₃²×C₄	C₂×C₃×C₄	1	3	yes	—
83	C₂×C₃²×C₄	C₂×C₃×C₄	1	6	yes	—
84	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
85	C₂×C₃²×C₄	C₂²×D₃	1	16	yes	—
86	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
87	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
88	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	8	yes	complemented
89	C₃×C₃⋊C₈	—	3	32	no	—
90	C₂²×M₁₆	—	9	1	no	Sylow
91	C₂×C₃³	—	2	4	no	—
92	C₂×C₃³	C₂×M₁₆	1	6	yes	complemented
93	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
94	C₂⁴×C₃	—	2	4	no	—
95	C₂⁴×C₃	C₃×Dic₃	1	4	yes	—
96	C₂⁴×C₃	C₃⋊Dic₃	1	1	yes	characteristic
97	C₂²×C₃×C₄	—	2	24	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	4	no	—
100	C₂²×C₃×C₄	—	2	24	no	—
101	C₂²×C₃×C₄	C₃×Dic₃	1	24	yes	—
102	C₂²×C₃×C₄	C₃×Dic₃	1	4	yes	—
103	C₂²×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₃⋊Dic₃	1	6	yes	—
105	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
106	C₂²×C₃×C₄	C₂×C₃×D₃	1	24	yes	—
107	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	6	yes	—
108	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₃×M₁₆	—	9	16	no	—
110	C₂×C₃×C₈	—	9	12	no	—
111	#208	—	3	64	no	—
112	C₂×C₃⋊C₈	—	3	48	no	—
113	C₂²×C₃²	—	2	96	no	—
114	C₂²×C₃²	—	2	48	no	—
115	C₂²×C₃²	—	2	12	no	—
116	C₂²×C₃²	—	2	16	no	—
117	C₂²×C₃²	—	2	16	no	—
118	C₂²×C₃²	—	2	12	no	—
119	C₂²×C₃²	#208	1	16	yes	complemented
120	C₂²×C₃²	C₃×M₁₆	1	4	yes	complemented
121	C₂²×C₃²	C₂²×Dic₃	1	16	yes	—
122	C₂²×C₃²	C₂²×Dic₃	1	12	yes	—
123	C₂²×C₃²	C₂²×C₃×C₄	1	4	yes	—
124	C₂²×C₃²	C₂²×C₃×C₄	1	3	yes	—
125	C₃²×C₄	—	2	4	no	—
126	C₃²×C₄	—	2	12	no	—
127	C₃²×C₄	—	2	16	no	—
128	C₃²×C₄	C₂²×Dic₃	1	12	yes	—
129	C₃²×C₄	C₂²×Dic₃	1	16	yes	—
130	C₃²×C₄	C₂²×C₃×C₄	1	3	yes	—
131	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
132	C₃²×C₄	C₂³×D₃	1	4	yes	—
133	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
134	C₂³×C₄	C₃×C₃⋊D₃	1	1	yes	characteristic
135	C₂×M₁₆	—	9	12	no	—
136	C₂²×C₈	—	9	2	no	—
137	C₃³	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₃	—	2	32	no	—
139	C₂³×C₃	—	2	16	no	—
140	C₂³×C₃	—	2	4	no	—
141	C₂³×C₃	—	2	24	no	—
142	C₂³×C₃	—	2	4	no	—
143	C₂³×C₃	C₂×C₃×Dic₃	1	24	yes	—
144	C₂³×C₃	C₂×C₃×Dic₃	1	4	yes	—
145	C₂³×C₃	C₂×C₃⋊Dic₃	1	6	yes	—
146	C₂³×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
147	C₂×C₃×C₄	—	2	48	no	—
148	C₂×C₃×C₄	—	2	12	no	—
149	C₂×C₃×C₄	—	2	24	no	—
150	C₂×C₃×C₄	—	2	12	no	—
151	C₂×C₃×C₄	—	2	16	no	—
152	C₂×C₃×C₄	C₂×C₃×Dic₃	1	48	yes	—
153	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
154	C₂×C₃×C₄	C₂×C₃×Dic₃	1	24	yes	—
155	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	12	yes	—
156	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	6	yes	—
157	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	3	yes	—
158	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
159	C₂×C₃×C₄	C₂²×C₃×D₃	1	16	yes	—
160	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
161	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
162	C₃×C₈	—	9	8	no	—
163	C₃⋊C₈	—	3	32	no	—
164	C₂×C₃²	—	2	32	no	—
165	C₂×C₃²	—	2	4	no	—
166	C₂×C₃²	—	2	16	no	—
167	C₂×C₃²	—	2	24	no	—
168	C₂×C₃²	—	2	4	no	—
169	C₂×C₃²	C₂×#208	1	24	yes	complemented
170	C₂×C₃²	C₂×C₃×M₁₆	1	6	yes	complemented
171	C₂×C₃²	C₂³×Dic₃	1	4	yes	—
172	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
173	C₂⁴	C₃×C₃⋊Dic₃	1	1	yes	characteristic
174	C₂²×C₄	C₃×C₃⋊Dic₃	1	6	yes	—
175	C₂²×C₄	C₃×C₃⋊Dic₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	6	yes	—
178	M₁₆	—	9	16	no	—
179	C₂×C₈	—	9	12	no	—
180	C₂²×C₃	—	2	96	no	—
181	C₂²×C₃	—	2	48	no	—
182	C₂²×C₃	—	2	12	no	—
183	C₂²×C₃	—	2	16	no	—
184	C₂²×C₃	—	2	16	no	—
185	C₂²×C₃	—	2	12	no	—
186	C₂²×C₃	C₃×#208	1	16	yes	complemented
187	C₂²×C₃	#3748	1	4	yes	direct product factor
188	C₂²×C₃	C₂²×C₃×Dic₃	1	16	yes	—
189	C₂²×C₃	C₂²×C₃×Dic₃	1	12	yes	—
190	C₂²×C₃	C₂²×C₃⋊Dic₃	1	4	yes	—
191	C₂²×C₃	C₂²×C₃⋊Dic₃	1	3	yes	—
192	C₃×C₄	—	2	4	no	—
193	C₃×C₄	—	2	12	no	—
194	C₃×C₄	—	2	16	no	—
195	C₃×C₄	C₂²×C₃×Dic₃	1	12	yes	—
196	C₃×C₄	C₂²×C₃×Dic₃	1	16	yes	—
197	C₃×C₄	C₂²×C₃⋊Dic₃	1	4	yes	—
198	C₃×C₄	C₂²×C₃⋊Dic₃	1	3	yes	—
199	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
200	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
201	C₃²	—	2	4	no	—
202	C₃²	C₂²×#208	1	4	yes	complemented
203	C₃²	C₂²×C₃×M₁₆	1	1	yes	characteristic, complemented
204	C₂³	—	2	4	no	—
205	C₂³	C₂×C₃×C₃⋊Dic₃	1	6	yes	—
206	C₂³	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
207	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	12	yes	—
208	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	3	yes	—
209	C₂×C₄	C₂×C₃×C₃⋊Dic₃	1	6	yes	—
210	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	3	yes	—
211	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	4	yes	—
212	C₈	—	9	8	no	—
213	C₂×C₃	—	2	32	no	—
214	C₂×C₃	—	2	16	no	—
215	C₂×C₃	—	2	4	no	—
216	C₂×C₃	—	2	24	no	—
217	C₂×C₃	—	2	4	no	—
218	C₂×C₃	C₂×C₃×#208	1	24	yes	complemented
219	C₂×C₃	C₂×#3748	1	6	yes	direct product factor
220	C₂×C₃	C₂³×C₃×Dic₃	1	4	yes	—
221	C₂×C₃	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
222	C₂²	—	2	12	no	—
223	C₂²	C₃×#3748	1	4	yes	direct product factor
224	C₂²	C₂²×C₃×C₃⋊Dic₃	1	4	yes	—
225	C₂²	C₂²×C₃×C₃⋊Dic₃	1	3	yes	—
226	C₄	C₂²×C₃×C₃⋊Dic₃	1	3	yes	—
227	C₄	C₂²×C₃×C₃⋊Dic₃	1	4	yes	—
228	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
229	C₃	—	2	4	no	—
230	C₃	C₂²×C₃×#208	1	4	yes	complemented
231	C₃	C₂²×#3748	1	1	yes	characteristic, direct product factor
232	C₂	—	2	4	no	—
233	C₂	C₂×C₃×#3748	1	6	yes	direct product factor
234	C₂	C₂³×C₃×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting	C₂³×C₃³×C₄	C₂
socle	C₂³×C₃³	C₂×C₄
prime radical	C₂⁴×C₃³	C₄
cosocle	C₃²×C₄	C₂⁴×C₃
hypercenter	C₂³×C₃×C₄	C₃⋊D₃
hypocenter	C₃²	C₂²×C₃×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂³×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#3748	C₂	direct (N × H)
3. C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
4. C₂²×#3748	C₃	direct (N × H)
11. C₃×#3748	C₂²	direct (N × H)
12. C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	semidirect
16. C₂×#3748	C₂×C₃	direct (N × H)
17. C₂²×C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
28. C₃×C₃⋊(C₃⋊C₈)	C₂³	semidirect
46. #3748	C₂²×C₃	direct (N × H)
47. C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	semidirect
51. C₂²×C₃³	M₁₆	semidirect
88. C₃⋊(C₃⋊C₈)	C₂³×C₃	semidirect
92. C₂×C₃³	C₂×M₁₆	semidirect
119. C₂²×C₃²	#208	semidirect
120. C₂²×C₃²	C₃×M₁₆	semidirect
137. C₃³	C₂²×M₁₆	semidirect
169. C₂×C₃²	C₂×#208	semidirect
170. C₂×C₃²	C₂×C₃×M₁₆	semidirect
186. C₂²×C₃	C₃×#208	semidirect
187. C₂²×C₃	#3748	direct (N × H)
202. C₃²	C₂²×#208	semidirect
203. C₃²	C₂²×C₃×M₁₆	semidirect
218. C₂×C₃	C₂×C₃×#208	semidirect
219. C₂×C₃	C₂×#3748	direct (N × H)
223. C₂²	C₃×#3748	direct (N × H)
230. C₃	C₂²×C₃×#208	semidirect
231. C₃	C₂²×#3748	direct (N × H)
233. C₂	C₂×C₃×#3748	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃³×C₄	2
C₂×C₃×#3748	2
C₂²×C₃×C₃⋊(C₃⋊C₈)	2
C₂²×#3748	3
C₂²×C₃×#208	3

Subgroups — C22×C3×#3748 #13179532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#3748 #13179532