Subgroups — C₂²×C₃×#3754 #13179551

Order 1728.

2732 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×#634	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3754	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#3754	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃×#214	—	3	4	no	maximal
7	C₂⁴×C₃³	—	2	1	no	—
8	C₂×C₃³×D₄	—	2	3	no	—
9	C₂×C₃³×D₄	C₂²	1	6	yes	—
10	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₃⋊Dic₃	—	2	1	no	—
12	C₂×C₃×#634	—	2	3	no	—
13	C₂×C₃×#634	C₂²	1	6	yes	complemented
14	C₃×#3754	C₂²	1	16	yes	direct product factor
15	C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	1	6	yes	complemented
16	C₂²×C₃²×D₄	—	2	4	no	—
17	C₂²×C₃²×D₄	D₃	1	4	yes	—
18	C₂²×C₃²×D₄	C₂×C₃	1	1	yes	characteristic
19	C₂²×#634	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂²×C₃×Dic₆	—	3	4	no	—
21	C₂×#3754	C₂×C₃	1	12	yes	direct product factor
22	C₂²×C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×#214	—	3	48	no	—
24	C₂²×C₃×C₃⋊C₈	—	3	4	no	—
25	C₂³×C₃³	—	2	6	no	—
26	C₂³×C₃³	—	4	4	no	—
27	C₂³×C₃³	D₄	1	1	yes	characteristic
28	C₃³×D₄	—	2	6	no	—
29	C₃³×D₄	C₂³	1	4	yes	—
30	C₂×C₃³×C₄	D₄	1	3	yes	—
31	C₂×C₃³×C₄	C₂³	1	3	yes	—
32	C₂×C₃×C₃⋊Dic₃	—	2	6	no	—
33	C₃×#634	—	2	6	no	—
34	C₃×#634	C₂³	1	4	yes	complemented
35	C₃×C₃⋊(C₃⋊C₈)	C₂³	1	4	yes	complemented
36	C₂²×C₃×SD₁₆	—	9	1	no	—
37	C₂²×#214	—	3	4	no	—
38	C₂⁴×C₃²	—	2	8	no	—
39	C₂⁴×C₃²	—	2	4	no	—
40	C₂⁴×C₃²	—	2	1	no	—
41	C₂×C₃²×D₄	—	2	24	no	—
42	C₂×C₃²×D₄	—	2	24	no	—
43	C₂×C₃²×D₄	—	2	12	no	—
44	C₂×C₃²×D₄	—	2	3	no	—
45	C₂×C₃²×D₄	C₂×D₃	1	24	yes	—
46	C₂×C₃²×D₄	C₂²×C₃	1	6	yes	—
47	C₂²×C₃²×C₄	—	2	4	no	—
48	C₂²×C₃²×C₄	C₂×D₃	1	4	yes	—
49	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₃⋊Dic₃	—	2	1	no	—
51	C₂×#634	—	2	3	no	—
52	C₂×#634	C₂²×C₃	1	6	yes	complemented
53	C₂²×C₃×Dic₃	—	6	4	no	—
54	C₂×C₃×Dic₆	—	3	24	no	—
55	C₂×C₃×Dic₆	—	6	12	no	—
56	#3754	C₂²×C₃	1	16	yes	direct product factor
57	C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	1	6	yes	complemented
58	C₃×#214	—	3	64	no	—
59	C₂×C₃×C₃⋊C₈	—	3	24	no	—
60	C₂²×C₃³	—	2	4	no	—
61	C₂²×C₃³	—	4	12	no	—
62	C₂²×C₃³	SD₁₆	1	4	yes	complemented
63	C₂²×C₃³	C₂×D₄	1	3	yes	—
64	C₃³×C₄	C₂×D₄	1	3	yes	—
65	C₃³×C₄	C₂⁴	1	1	yes	characteristic
66	C₃×C₃⋊Dic₃	—	2	4	no	—
67	C₂²×C₃×Q₈	—	9	1	no	—
68	C₂²×C₃×D₄	—	2	4	no	—
69	C₂²×C₃×D₄	C₃×D₃	1	4	yes	—
70	C₂²×C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
71	C₂²×Dic₆	—	3	4	no	—
72	C₂×C₃×SD₁₆	—	9	12	no	—
73	C₂²×C₃×C₈	—	9	1	no	—
74	C₂×#214	—	3	48	no	—
75	C₂²×C₃⋊C₈	—	3	4	no	—
76	C₂³×C₃²	—	2	48	no	—
77	C₂³×C₃²	—	2	24	no	—
78	C₂³×C₃²	—	2	6	no	—
79	C₂³×C₃²	—	2	4	no	—
80	C₂³×C₃²	—	4	32	no	—
81	C₂³×C₃²	—	4	16	no	—
82	C₂³×C₃²	—	4	4	no	—
83	C₂³×C₃²	#67	1	4	yes	—
84	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
85	C₃²×D₄	—	2	48	no	—
86	C₃²×D₄	—	2	24	no	—
87	C₃²×D₄	—	2	6	no	—
88	C₃²×D₄	—	2	16	no	—
89	C₃²×D₄	C₂²×D₃	1	16	yes	—
90	C₃²×D₄	C₂³×C₃	1	4	yes	—
91	C₂×C₃²×C₄	—	2	12	no	—
92	C₂×C₃²×C₄	—	2	12	no	—
93	C₂×C₃²×C₄	#67	1	12	yes	—
94	C₂×C₃²×C₄	C₃×D₄	1	3	yes	—
95	C₂×C₃²×C₄	C₂²×D₃	1	12	yes	—
96	C₂×C₃²×C₄	C₂³×C₃	1	3	yes	—
97	C₂×C₃⋊Dic₃	—	2	6	no	—
98	#634	—	2	6	no	—
99	#634	C₂³×C₃	1	4	yes	complemented
100	C₂×C₃×Dic₃	—	6	24	no	—
101	C₃×Dic₆	—	3	16	no	—
102	C₃×Dic₆	—	6	24	no	—
103	C₃⋊(C₃⋊C₈)	C₂³×C₃	1	4	yes	complemented
104	C₃×C₃⋊C₈	—	3	16	no	—
105	C₂²×SD₁₆	—	9	1	no	Sylow
106	C₂×C₃³	—	4	4	no	—
107	C₂×C₃³	C₂×SD₁₆	1	6	yes	complemented
108	C₂×C₃³	C₂²×D₄	1	1	yes	characteristic
109	C₂⁴×C₃	—	2	8	no	—
110	C₂⁴×C₃	—	2	4	no	—
111	C₂⁴×C₃	—	2	1	no	—
112	C₂×C₃×Q₈	—	9	6	no	—
113	C₂×C₃×Q₈	—	18	3	no	—
114	C₂×C₃×D₄	—	2	24	no	—
115	C₂×C₃×D₄	—	2	12	no	—
116	C₂×C₃×D₄	—	2	3	no	—
117	C₂×C₃×D₄	—	2	24	no	—
118	C₂×C₃×D₄	C₂×C₃×D₃	1	24	yes	—
119	C₂×C₃×D₄	C₂×C₃⋊D₃	1	6	yes	—
120	C₂²×C₃×C₄	—	2	4	no	—
121	C₂²×C₃×C₄	—	18	1	no	—
122	C₂²×C₃×C₄	C₂×C₃×D₃	1	4	yes	—
123	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
124	C₂²×Dic₃	—	6	4	no	—
125	C₂×Dic₆	—	3	24	no	—
126	C₂×Dic₆	—	6	12	no	—
127	C₃×SD₁₆	—	9	16	no	—
128	C₂×C₃×C₈	—	9	6	no	—
129	#214	—	3	64	no	—
130	C₂×C₃⋊C₈	—	3	24	no	—
131	C₂²×C₃²	—	2	32	no	—
132	C₂²×C₃²	—	2	16	no	—
133	C₂²×C₃²	—	2	4	no	—
134	C₂²×C₃²	—	2	16	no	—
135	C₂²×C₃²	—	2	12	no	—
136	C₂²×C₃²	—	4	96	no	—
137	C₂²×C₃²	—	4	48	no	—
138	C₂²×C₃²	—	4	12	no	—
139	C₂²×C₃²	#214	1	16	yes	complemented
140	C₂²×C₃²	C₃×SD₁₆	1	4	yes	complemented
141	C₂²×C₃²	C₂×#67	1	12	yes	—
142	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
143	C₃²×C₄	—	2	4	no	—
144	C₃²×C₄	—	2	12	no	—
145	C₃²×C₄	C₂×#67	1	12	yes	—
146	C₃²×C₄	C₂×C₃×D₄	1	3	yes	—
147	C₃²×C₄	C₂³×D₃	1	4	yes	—
148	C₃²×C₄	C₂⁴×C₃	1	1	yes	characteristic
149	C₃⋊Dic₃	—	2	4	no	—
150	C₃×Dic₃	—	6	16	no	—
151	C₂²×Q₈	—	9	1	no	—
152	C₂²×D₄	C₃×C₃⋊D₃	1	1	yes	characteristic
153	C₂×SD₁₆	—	9	12	no	—
154	C₂²×C₈	—	9	1	no	—
155	C₃³	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	2	48	no	—
157	C₂³×C₃	—	2	24	no	—
158	C₂³×C₃	—	2	6	no	—
159	C₂³×C₃	—	2	4	no	—
160	C₂³×C₃	—	4	32	no	—
161	C₂³×C₃	—	4	16	no	—
162	C₂³×C₃	—	4	4	no	—
163	C₂³×C₃	C₃×#67	1	4	yes	—
164	C₂³×C₃	#638	1	1	yes	characteristic
165	C₃×Q₈	—	9	4	no	—
166	C₃×Q₈	—	18	6	no	—
167	C₃×D₄	—	2	48	no	—
168	C₃×D₄	—	2	24	no	—
169	C₃×D₄	—	2	6	no	—
170	C₃×D₄	—	2	16	no	—
171	C₃×D₄	C₂²×C₃×D₃	1	16	yes	—
172	C₃×D₄	C₂²×C₃⋊D₃	1	4	yes	—
173	C₂×C₃×C₄	—	2	12	no	—
174	C₂×C₃×C₄	—	2	12	no	—
175	C₂×C₃×C₄	—	18	6	no	—
176	C₂×C₃×C₄	C₃×#67	1	12	yes	—
177	C₂×C₃×C₄	#638	1	3	yes	—
178	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
179	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	3	yes	—
180	C₂×Dic₃	—	6	24	no	—
181	Dic₆	—	3	16	no	—
182	Dic₆	—	6	24	no	—
183	C₃×C₈	—	9	4	no	—
184	C₃⋊C₈	—	3	16	no	—
185	C₂×C₃²	—	2	24	no	—
186	C₂×C₃²	—	2	4	no	—
187	C₂×C₃²	—	4	32	no	—
188	C₂×C₃²	—	4	4	no	—
189	C₂×C₃²	—	4	16	no	—
190	C₂×C₃²	C₂×#214	1	24	yes	complemented
191	C₂×C₃²	C₂×C₃×SD₁₆	1	6	yes	complemented
192	C₂×C₃²	C₂²×#67	1	4	yes	—
193	C₂×C₃²	C₂²×C₃×D₄	1	1	yes	characteristic
194	C₂⁴	—	2	1	no	—
195	C₂×Q₈	—	9	6	no	—
196	C₂×Q₈	—	18	3	no	—
197	C₂×D₄	—	2	3	no	—
198	C₂×D₄	C₂×C₃×C₃⋊D₃	1	6	yes	—
199	C₂²×C₄	—	18	1	no	—
200	C₂²×C₄	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
201	SD₁₆	—	9	16	no	—
202	C₂×C₈	—	9	6	no	—
203	C₂²×C₃	—	2	32	no	—
204	C₂²×C₃	—	2	16	no	—
205	C₂²×C₃	—	2	4	no	—
206	C₂²×C₃	—	2	16	no	—
207	C₂²×C₃	—	2	12	no	—
208	C₂²×C₃	—	4	96	no	—
209	C₂²×C₃	—	4	48	no	—
210	C₂²×C₃	—	4	12	no	—
211	C₂²×C₃	C₃×#214	1	16	yes	complemented
212	C₂²×C₃	#3754	1	4	yes	direct product factor
213	C₂²×C₃	C₂×C₃×#67	1	12	yes	—
214	C₂²×C₃	C₂×#638	1	3	yes	—
215	C₃×C₄	—	2	4	no	—
216	C₃×C₄	—	2	12	no	—
217	C₃×C₄	—	18	4	no	—
218	C₃×C₄	C₂×C₃×#67	1	12	yes	—
219	C₃×C₄	C₂×#638	1	3	yes	—
220	C₃×C₄	C₂³×C₃×D₃	1	4	yes	—
221	C₃×C₄	C₂³×C₃⋊D₃	1	1	yes	characteristic
222	Dic₃	—	6	16	no	—
223	C₃²	—	2	4	no	—
224	C₃²	C₂²×#214	1	4	yes	complemented
225	C₃²	C₂²×C₃×SD₁₆	1	1	yes	characteristic, complemented
226	C₂³	—	2	6	no	—
227	C₂³	—	4	4	no	—
228	C₂³	C₃×#638	1	1	yes	characteristic
229	Q₈	—	9	4	no	—
230	Q₈	—	18	6	no	—
231	D₄	—	2	6	no	—
232	D₄	C₂²×C₃×C₃⋊D₃	1	4	yes	—
233	C₂×C₄	—	18	6	no	—
234	C₂×C₄	C₃×#638	1	3	yes	—
235	C₂×C₄	C₂²×C₃×C₃⋊D₃	1	3	yes	—
236	C₈	—	9	4	no	—
237	C₂×C₃	—	2	24	no	—
238	C₂×C₃	—	2	4	no	—
239	C₂×C₃	—	4	32	no	—
240	C₂×C₃	—	4	16	no	—
241	C₂×C₃	—	4	4	no	—
242	C₂×C₃	C₂×C₃×#214	1	24	yes	complemented
243	C₂×C₃	C₂×#3754	1	6	yes	direct product factor
244	C₂×C₃	C₂²×C₃×#67	1	4	yes	—
245	C₂×C₃	C₂²×#638	1	1	yes	characteristic
246	C₂²	—	2	4	no	—
247	C₂²	—	4	12	no	—
248	C₂²	C₃×#3754	1	4	yes	direct product factor
249	C₂²	C₂×C₃×#638	1	3	yes	—
250	C₄	—	18	4	no	—
251	C₄	C₂×C₃×#638	1	3	yes	—
252	C₄	C₂³×C₃×C₃⋊D₃	1	1	yes	characteristic
253	C₃	—	2	4	no	—
254	C₃	C₂²×C₃×#214	1	4	yes	complemented
255	C₃	C₂²×#3754	1	1	yes	characteristic, direct product factor
256	C₂	—	4	4	no	—
257	C₂	C₂×C₃×#3754	1	6	yes	direct product factor
258	C₂	C₂²×C₃×#638	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	#638
commutator, cosocle	C₃²×C₄	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×C₃⋊D₃
Fitting, prime radical	C₂²×C₃³×D₄	C₂
socle	C₂³×C₃³	D₄
hypercenter	C₂²×C₃×D₄	C₃⋊D₃
hypocenter	C₃²	C₂²×C₃×SD₁₆
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×#634	C₂	semidirect
3. C₂×C₃×#3754	C₂	direct (N × H)
4. C₂²×C₃×C₃⋊(C₃⋊C₈)	C₂	semidirect
5. C₂²×#3754	C₃	direct (N × H)
13. C₂×C₃×#634	C₂²	semidirect
14. C₃×#3754	C₂²	direct (N × H)
15. C₂×C₃×C₃⋊(C₃⋊C₈)	C₂²	semidirect
19. C₂²×#634	C₂×C₃	semidirect
21. C₂×#3754	C₂×C₃	direct (N × H)
22. C₂²×C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
34. C₃×#634	C₂³	semidirect
35. C₃×C₃⋊(C₃⋊C₈)	C₂³	semidirect
52. C₂×#634	C₂²×C₃	semidirect
56. #3754	C₂²×C₃	direct (N × H)
57. C₂×C₃⋊(C₃⋊C₈)	C₂²×C₃	semidirect
62. C₂²×C₃³	SD₁₆	semidirect
99. #634	C₂³×C₃	semidirect
103. C₃⋊(C₃⋊C₈)	C₂³×C₃	semidirect
107. C₂×C₃³	C₂×SD₁₆	semidirect
139. C₂²×C₃²	#214	semidirect
140. C₂²×C₃²	C₃×SD₁₆	semidirect
155. C₃³	C₂²×SD₁₆	semidirect
190. C₂×C₃²	C₂×#214	semidirect
191. C₂×C₃²	C₂×C₃×SD₁₆	semidirect
211. C₂²×C₃	C₃×#214	semidirect
212. C₂²×C₃	#3754	direct (N × H)
224. C₃²	C₂²×#214	semidirect
225. C₃²	C₂²×C₃×SD₁₆	semidirect
242. C₂×C₃	C₂×C₃×#214	semidirect
243. C₂×C₃	C₂×#3754	direct (N × H)
248. C₂²	C₃×#3754	direct (N × H)
254. C₃	C₂²×C₃×#214	semidirect
255. C₃	C₂²×#3754	direct (N × H)
257. C₂	C₂×C₃×#3754	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×D₄	2
C₂²×C₃×#634	2
C₂×C₃×#3754	2
C₂²×C₃×C₃⋊(C₃⋊C₈)	2
C₂²×#3754	3
C₂²×C₃×#214	3

Subgroups — C22×C3×#3754 #13179551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#3754 #13179551