Subgroups — C₂²×C₃×#3757 #13179590

Order 1728.

7036 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃³	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×#3757	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#3757	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂²×C₃×#217	—	3	4	no	maximal
6	C₂⁴×C₃³	—	2	8	no	—
7	C₂⁴×C₃³	C₄	1	8	yes	complemented
8	C₂⁴×C₃³	C₂²	1	6	yes	—
9	C₂⁴×C₃³	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₃⋊Dic₃	—	2	8	no	—
11	C₂²×C₃×C₃⋊Dic₃	C₂²	1	12	yes	complemented
12	C₃×#3757	C₂²	1	16	yes	direct product factor
13	C₂⁵×C₃²	—	2	4	no	—
14	C₂⁵×C₃²	D₃	1	4	yes	—
15	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic
16	C₂³×C₃⋊Dic₃	C₂×C₃	1	2	yes	complemented
17	C₂³×C₃×Dic₃	—	3	8	no	—
18	C₂×#3757	C₂×C₃	1	12	yes	direct product factor
19	C₂×C₃×#217	—	3	48	no	—
20	C₂³×C₃³	—	2	32	no	—
21	C₂³×C₃³	—	2	24	no	—
22	C₂³×C₃³	C₂×C₄	1	4	yes	—
23	C₂³×C₃³	C₂×C₄	1	24	yes	complemented
24	C₂³×C₃³	D₄	1	8	yes	—
25	C₂³×C₃³	C₂³	1	4	yes	—
26	C₂³×C₃³	C₂³	1	3	yes	—
27	C₂×C₃×C₃⋊Dic₃	—	2	24	no	—
28	C₂×C₃×C₃⋊Dic₃	C₂³	1	8	yes	complemented
29	C₂²×C₃×C₂²⋊C₄	—	9	1	no	—
30	C₂²×#217	—	3	4	no	—
31	C₂⁴×C₃²	—	2	64	no	—
32	C₂⁴×C₃²	—	2	32	no	—
33	C₂⁴×C₃²	—	2	24	no	—
34	C₂⁴×C₃²	—	2	4	no	—
35	C₂⁴×C₃²	—	2	32	no	—
36	C₂⁴×C₃²	—	2	8	no	—
37	C₂⁴×C₃²	Dic₃	1	32	yes	complemented
38	C₂⁴×C₃²	C₃×C₄	1	8	yes	complemented
39	C₂⁴×C₃²	C₂×D₃	1	24	yes	—
40	C₂⁴×C₃²	C₂×D₃	1	4	yes	—
41	C₂⁴×C₃²	C₂²×C₃	1	6	yes	—
42	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃⋊Dic₃	—	2	8	no	—
44	C₂²×C₃⋊Dic₃	C₂²×C₃	1	12	yes	complemented
45	C₂²×C₃×Dic₃	—	3	48	no	—
46	C₂²×C₃×Dic₃	—	6	32	no	—
47	#3757	C₂²×C₃	1	16	yes	direct product factor
48	C₃×#217	—	3	64	no	—
49	C₂²×C₃³	—	2	48	no	—
50	C₂²×C₃³	—	2	8	no	—
51	C₂²×C₃³	C₂²⋊C₄	1	16	yes	complemented
52	C₂²×C₃³	C₂²×C₄	1	6	yes	—
53	C₂²×C₃³	C₂²×C₄	1	8	yes	complemented
54	C₂²×C₃³	C₂×D₄	1	12	yes	—
55	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
56	C₃×C₃⋊Dic₃	—	2	8	no	—
57	C₂⁵×C₃	—	2	4	no	—
58	C₂⁵×C₃	C₃×D₃	1	4	yes	—
59	C₂⁵×C₃	C₃⋊D₃	1	1	yes	characteristic
60	C₂³×C₃×C₄	—	9	2	no	—
61	C₂³×Dic₃	—	3	8	no	—
62	C₂×C₃×C₂²⋊C₄	—	9	12	no	—
63	C₂×#217	—	3	48	no	—
64	C₂³×C₃²	—	2	256	no	—
65	C₂³×C₃²	—	2	192	no	—
66	C₂³×C₃²	—	2	128	no	—
67	C₂³×C₃²	—	2	32	no	—
68	C₂³×C₃²	—	2	96	no	—
69	C₂³×C₃²	—	2	24	no	—
70	C₂³×C₃²	—	2	16	no	—
71	C₂³×C₃²	—	2	96	no	—
72	C₂³×C₃²	—	2	32	no	—
73	C₂³×C₃²	—	2	16	no	—
74	C₂³×C₃²	—	2	12	no	—
75	C₂³×C₃²	C₂×Dic₃	1	16	yes	—
76	C₂³×C₃²	C₂×Dic₃	1	96	yes	complemented
77	C₂³×C₃²	#67	1	32	yes	—
78	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
79	C₂³×C₃²	C₂×C₃×C₄	1	24	yes	complemented
80	C₂³×C₃²	C₃×D₄	1	8	yes	—
81	C₂³×C₃²	C₂²×D₃	1	16	yes	—
82	C₂³×C₃²	C₂²×D₃	1	12	yes	—
83	C₂³×C₃²	C₂³×C₃	1	4	yes	—
84	C₂³×C₃²	C₂³×C₃	1	3	yes	—
85	C₂×C₃⋊Dic₃	—	2	24	no	—
86	C₂×C₃⋊Dic₃	C₂³×C₃	1	8	yes	complemented
87	C₂×C₃×Dic₃	—	3	32	no	—
88	C₂×C₃×Dic₃	—	6	96	no	—
89	C₂²×C₂²⋊C₄	—	9	1	no	Sylow
90	C₂×C₃³	—	2	8	no	—
91	C₂×C₃³	C₂×C₂²⋊C₄	1	12	yes	complemented
92	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₃³	C₂²×D₄	1	2	yes	—
94	C₂⁴×C₃	—	2	64	no	—
95	C₂⁴×C₃	—	2	32	no	—
96	C₂⁴×C₃	—	2	8	no	—
97	C₂⁴×C₃	—	2	32	no	—
98	C₂⁴×C₃	—	2	24	no	—
99	C₂⁴×C₃	—	2	4	no	—
100	C₂⁴×C₃	C₃×Dic₃	1	32	yes	complemented
101	C₂⁴×C₃	C₃⋊Dic₃	1	8	yes	complemented
102	C₂⁴×C₃	C₂×C₃×D₃	1	24	yes	—
103	C₂⁴×C₃	C₂×C₃×D₃	1	4	yes	—
104	C₂⁴×C₃	C₂×C₃⋊D₃	1	6	yes	—
105	C₂⁴×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	—	9	12	no	—
107	C₂²×C₃×C₄	—	18	8	no	—
108	C₂²×Dic₃	—	3	48	no	—
109	C₂²×Dic₃	—	6	32	no	—
110	C₃×C₂²⋊C₄	—	9	16	no	—
111	#217	—	3	64	no	—
112	C₂²×C₃²	—	2	384	no	—
113	C₂²×C₃²	—	2	64	no	—
114	C₂²×C₃²	—	2	192	no	—
115	C₂²×C₃²	—	2	48	no	—
116	C₂²×C₃²	—	2	32	no	—
117	C₂²×C₃²	—	2	8	no	—
118	C₂²×C₃²	—	2	4	no	—
119	C₂²×C₃²	—	2	64	no	—
120	C₂²×C₃²	—	2	24	no	—
121	C₂²×C₃²	—	2	32	no	—
122	C₂²×C₃²	—	2	48	no	—
123	C₂²×C₃²	#217	1	64	yes	complemented
124	C₂²×C₃²	C₃×C₂²⋊C₄	1	16	yes	complemented
125	C₂²×C₃²	C₂²×Dic₃	1	24	yes	—
126	C₂²×C₃²	C₂²×Dic₃	1	32	yes	complemented
127	C₂²×C₃²	C₂×#67	1	48	yes	—
128	C₂²×C₃²	C₂²×C₃×C₄	1	6	yes	—
129	C₂²×C₃²	C₂²×C₃×C₄	1	8	yes	complemented
130	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
131	C₂²×C₃²	C₂³×D₃	1	4	yes	—
132	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
133	C₃⋊Dic₃	—	2	8	no	—
134	C₃×Dic₃	—	6	32	no	—
135	C₂⁵	C₃×C₃⋊D₃	1	1	yes	characteristic
136	C₂³×C₄	—	9	2	no	—
137	C₂×C₂²⋊C₄	—	9	12	no	—
138	C₃³	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₃	—	2	256	no	—
140	C₂³×C₃	—	2	192	no	—
141	C₂³×C₃	—	2	128	no	—
142	C₂³×C₃	—	2	32	no	—
143	C₂³×C₃	—	2	96	no	—
144	C₂³×C₃	—	2	24	no	—
145	C₂³×C₃	—	2	16	no	—
146	C₂³×C₃	—	2	96	no	—
147	C₂³×C₃	—	2	32	no	—
148	C₂³×C₃	—	2	16	no	—
149	C₂³×C₃	—	2	12	no	—
150	C₂³×C₃	C₂×C₃×Dic₃	1	16	yes	—
151	C₂³×C₃	C₂×C₃×Dic₃	1	96	yes	complemented
152	C₂³×C₃	C₃×#67	1	32	yes	—
153	C₂³×C₃	C₂×C₃⋊Dic₃	1	4	yes	—
154	C₂³×C₃	C₂×C₃⋊Dic₃	1	24	yes	complemented
155	C₂³×C₃	#638	1	8	yes	—
156	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
157	C₂³×C₃	C₂²×C₃×D₃	1	12	yes	—
158	C₂³×C₃	C₂²×C₃⋊D₃	1	4	yes	—
159	C₂³×C₃	C₂²×C₃⋊D₃	1	3	yes	—
160	C₂×C₃×C₄	—	9	8	no	—
161	C₂×C₃×C₄	—	18	24	no	—
162	C₂×Dic₃	—	3	32	no	—
163	C₂×Dic₃	—	6	96	no	—
164	C₂×C₃²	—	2	64	no	—
165	C₂×C₃²	—	2	32	no	—
166	C₂×C₃²	—	2	8	no	—
167	C₂×C₃²	—	2	48	no	—
168	C₂×C₃²	—	2	4	no	—
169	C₂×C₃²	—	2	8	no	—
170	C₂×C₃²	C₂×#217	1	48	yes	complemented
171	C₂×C₃²	C₂×C₃×C₂²⋊C₄	1	12	yes	complemented
172	C₂×C₃²	C₂³×Dic₃	1	4	yes	—
173	C₂×C₃²	C₂²×#67	1	8	yes	—
174	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
175	C₂×C₃²	C₂²×C₃×D₄	1	2	yes	—
176	C₂⁴	—	2	8	no	—
177	C₂⁴	C₃×C₃⋊Dic₃	1	8	yes	complemented
178	C₂⁴	C₂×C₃×C₃⋊D₃	1	6	yes	—
179	C₂⁴	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
180	C₂²×C₄	—	9	12	no	—
181	C₂²×C₄	—	18	8	no	—
182	C₂²⋊C₄	—	9	16	no	—
183	C₂²×C₃	—	2	64	no	—
184	C₂²×C₃	—	2	384	no	—
185	C₂²×C₃	—	2	192	no	—
186	C₂²×C₃	—	2	32	no	—
187	C₂²×C₃	—	2	8	no	—
188	C₂²×C₃	—	2	48	no	—
189	C₂²×C₃	—	2	64	no	—
190	C₂²×C₃	—	2	4	no	—
191	C₂²×C₃	—	2	24	no	—
192	C₂²×C₃	—	2	32	no	—
193	C₂²×C₃	—	2	48	no	—
194	C₂²×C₃	C₃×#217	1	64	yes	complemented
195	C₂²×C₃	#3757	1	16	yes	direct product factor
196	C₂²×C₃	C₂²×C₃×Dic₃	1	24	yes	—
197	C₂²×C₃	C₂²×C₃×Dic₃	1	32	yes	complemented
198	C₂²×C₃	C₂×C₃×#67	1	48	yes	—
199	C₂²×C₃	C₂²×C₃⋊Dic₃	1	6	yes	—
200	C₂²×C₃	C₂²×C₃⋊Dic₃	1	8	yes	complemented
201	C₂²×C₃	C₂×#638	1	12	yes	—
202	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
203	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
204	C₃×C₄	—	18	8	no	—
205	Dic₃	—	6	32	no	—
206	C₃²	—	2	4	no	—
207	C₃²	C₂²×#217	1	4	yes	complemented
208	C₃²	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented
209	C₂³	—	2	32	no	—
210	C₂³	—	2	24	no	—
211	C₂³	C₂×C₃×C₃⋊Dic₃	1	4	yes	—
212	C₂³	C₂×C₃×C₃⋊Dic₃	1	24	yes	complemented
213	C₂³	C₃×#638	1	8	yes	—
214	C₂³	C₂²×C₃×C₃⋊D₃	1	4	yes	—
215	C₂³	C₂²×C₃×C₃⋊D₃	1	3	yes	—
216	C₂×C₄	—	9	8	no	—
217	C₂×C₄	—	18	24	no	—
218	C₂×C₃	—	2	64	no	—
219	C₂×C₃	—	2	32	no	—
220	C₂×C₃	—	2	8	no	—
221	C₂×C₃	—	2	4	no	—
222	C₂×C₃	—	2	8	no	—
223	C₂×C₃	—	2	48	no	—
224	C₂×C₃	C₂×C₃×#217	1	48	yes	complemented
225	C₂×C₃	C₂×#3757	1	12	yes	direct product factor
226	C₂×C₃	C₂³×C₃×Dic₃	1	4	yes	—
227	C₂×C₃	C₂²×C₃×#67	1	8	yes	—
228	C₂×C₃	C₂³×C₃⋊Dic₃	1	1	yes	characteristic
229	C₂×C₃	C₂²×#638	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	48	no	—
232	C₂²	C₃×#3757	1	16	yes	direct product factor
233	C₂²	C₂²×C₃×C₃⋊Dic₃	1	6	yes	—
234	C₂²	C₂²×C₃×C₃⋊Dic₃	1	8	yes	complemented
235	C₂²	C₂×C₃×#638	1	12	yes	—
236	C₂²	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
237	C₄	—	18	8	no	—
238	C₃	—	2	4	no	—
239	C₃	C₂²×C₃×#217	1	4	yes	complemented
240	C₃	C₂²×#3757	1	1	yes	characteristic, direct product factor
241	C₂	—	2	8	no	—
242	C₂	C₂×C₃×#3757	1	12	yes	direct product factor
243	C₂	C₂³×C₃×C₃⋊Dic₃	1	1	yes	characteristic
244	C₂	C₂²×C₃×#638	1	2	yes	—

	subgroup	G/H
center	C₂⁴×C₃	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×C₃⋊D₃
Fitting, prime radical	C₂⁵×C₃³	C₂
socle	C₂⁴×C₃³	C₂²
cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₂⁵×C₃	C₃⋊D₃
hypocenter	C₃²	C₂²×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁵×C₃

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×C₃⋊Dic₃	C₂	semidirect
3. C₂×C₃×#3757	C₂	direct (N × H)
4. C₂²×#3757	C₃	direct (N × H)
7. C₂⁴×C₃³	C₄	semidirect
11. C₂²×C₃×C₃⋊Dic₃	C₂²	semidirect
12. C₃×#3757	C₂²	direct (N × H)
16. C₂³×C₃⋊Dic₃	C₂×C₃	semidirect
18. C₂×#3757	C₂×C₃	direct (N × H)
23. C₂³×C₃³	C₂×C₄	semidirect
28. C₂×C₃×C₃⋊Dic₃	C₂³	semidirect
37. C₂⁴×C₃²	Dic₃	semidirect
38. C₂⁴×C₃²	C₃×C₄	semidirect
44. C₂²×C₃⋊Dic₃	C₂²×C₃	semidirect
47. #3757	C₂²×C₃	direct (N × H)
51. C₂²×C₃³	C₂²⋊C₄	semidirect
53. C₂²×C₃³	C₂²×C₄	semidirect
76. C₂³×C₃²	C₂×Dic₃	semidirect
79. C₂³×C₃²	C₂×C₃×C₄	semidirect
86. C₂×C₃⋊Dic₃	C₂³×C₃	semidirect
91. C₂×C₃³	C₂×C₂²⋊C₄	semidirect
100. C₂⁴×C₃	C₃×Dic₃	semidirect
101. C₂⁴×C₃	C₃⋊Dic₃	semidirect
123. C₂²×C₃²	#217	semidirect
124. C₂²×C₃²	C₃×C₂²⋊C₄	semidirect
126. C₂²×C₃²	C₂²×Dic₃	semidirect
129. C₂²×C₃²	C₂²×C₃×C₄	semidirect
138. C₃³	C₂²×C₂²⋊C₄	semidirect
151. C₂³×C₃	C₂×C₃×Dic₃	semidirect
154. C₂³×C₃	C₂×C₃⋊Dic₃	semidirect
170. C₂×C₃²	C₂×#217	semidirect
171. C₂×C₃²	C₂×C₃×C₂²⋊C₄	semidirect
177. C₂⁴	C₃×C₃⋊Dic₃	semidirect
194. C₂²×C₃	C₃×#217	semidirect
195. C₂²×C₃	#3757	direct (N × H)
197. C₂²×C₃	C₂²×C₃×Dic₃	semidirect
200. C₂²×C₃	C₂²×C₃⋊Dic₃	semidirect
207. C₃²	C₂²×#217	semidirect
208. C₃²	C₂²×C₃×C₂²⋊C₄	semidirect
212. C₂³	C₂×C₃×C₃⋊Dic₃	semidirect
224. C₂×C₃	C₂×C₃×#217	semidirect
225. C₂×C₃	C₂×#3757	direct (N × H)
232. C₂²	C₃×#3757	direct (N × H)
234. C₂²	C₂²×C₃×C₃⋊Dic₃	semidirect
239. C₃	C₂²×C₃×#217	semidirect
240. C₃	C₂²×#3757	direct (N × H)
242. C₂	C₂×C₃×#3757	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃³	2
C₂³×C₃×C₃⋊Dic₃	2
C₂×C₃×#3757	2
C₂²×#3757	3
C₂²×C₃×#217	3

Subgroups — C22×C3×#3757 #13179590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#3757 #13179590