Subgroups — C₃²×D₃×#127 #13179092

Order 1728.

3420 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×D₃×D₄	C₂	1	6	yes	maximal, complemented
2	C₃³×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#940	C₂	1	6	yes	maximal, complemented
4	C₃²×#874	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃²×C₄²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃²×#127	—	3	1	no	maximal
7	C₃×D₃×#127	C₃	1	4	yes	maximal, direct product factor
8	C₂³×C₃²×D₃	C₂²	1	4	yes	—
9	C₂×C₃³×D₄	C₂²	1	6	yes	complemented
10	C₂×C₃²×#67	C₂²	1	12	yes	complemented
11	C₃²×D₃×D₄	—	2	24	no	—
12	C₂×C₃²×D₁₂	C₂²	1	6	yes	complemented
13	C₂×C₃²×C₄×D₃	C₂²	1	3	yes	—
14	C₃³×C₄²	C₂²	1	1	yes	characteristic, complemented
15	C₃²×C₄×Dic₃	C₂²	1	3	yes	complemented
16	C₂²×C₃²×D₄	—	3	6	no	—
17	C₂×C₃×D₃×D₄	C₂×C₃	1	24	yes	complemented
18	C₃²×#127	—	2	4	no	—
19	C₃²×#127	—	3	6	no	—
20	C₃²×#127	—	3	1	no	—
21	C₃²×#127	D₃	1	1	yes	characteristic, direct product factor
22	C₃²×#127	C₂×C₃	1	4	yes	complemented
23	C₂×C₃²×C₄²	—	3	1	no	—
24	C₃×#940	C₂×C₃	1	24	yes	complemented
25	C₃×#874	C₂×C₃	1	4	yes	complemented
26	C₃×C₄²×D₃	C₂×C₃	1	4	yes	complemented
27	C₂³×C₃³	C₂³	1	4	yes	—
28	C₂²×C₃²×D₃	—	2	24	no	—
29	C₂²×C₃²×D₃	C₂³	1	4	yes	—
30	C₂²×C₃²×D₃	C₂³	1	1	yes	characteristic
31	C₃³×D₄	—	2	12	no	—
32	C₂×C₃³×C₄	C₂³	1	3	yes	—
33	C₃²×#67	—	2	24	no	—
34	C₂×C₃²×Dic₃	C₂³	1	3	yes	—
35	C₃²×D₁₂	—	2	12	no	—
36	C₃²×C₄×D₃	D₄	1	12	yes	complemented
37	C₂×C₃×#127	—	3	4	no	—
38	D₃×#127	C₃²	1	1	yes	characteristic, direct product factor
39	C₂⁴×C₃²	—	3	4	no	—
40	C₂³×C₃×D₃	C₂²×C₃	1	16	yes	—
41	C₂×C₃²×D₄	—	2	24	no	—
42	C₂×C₃²×D₄	—	3	12	no	—
43	C₂×C₃²×D₄	—	3	6	no	—
44	C₂×C₃²×D₄	—	6	24	no	—
45	C₂×C₃²×D₄	C₂×D₃	1	6	yes	complemented
46	C₂×C₃²×D₄	C₂²×C₃	1	24	yes	complemented
47	C₂²×C₃²×C₄	—	3	3	no	—
48	C₂×C₃×#67	C₂²×C₃	1	48	yes	complemented
49	C₃×D₃×D₄	—	2	96	no	—
50	C₂×C₃×D₁₂	C₂²×C₃	1	24	yes	complemented
51	C₂×C₃×C₄×D₃	C₂²×C₃	1	12	yes	—
52	C₃²×C₄²	—	2	4	no	—
53	C₃²×C₄²	—	3	3	no	—
54	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic, complemented
55	C₃²×C₄²	C₂²×C₃	1	4	yes	complemented
56	C₃×C₄×Dic₃	C₂²×C₃	1	12	yes	complemented
57	C₂²×C₃³	—	2	12	no	—
58	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
59	C₂×C₃²×D₃	—	2	12	no	—
60	C₂×C₃²×D₃	—	4	16	no	—
61	C₂×C₃²×D₃	C₂×D₄	1	6	yes	—
62	C₃³×C₄	C₂×D₄	1	6	yes	complemented
63	C₃²×Dic₃	C₂×D₄	1	6	yes	complemented
64	C₂²×C₃×D₄	—	3	24	no	—
65	C₂×D₃×D₄	C₂×C₃²	1	6	yes	complemented
66	C₃×#127	—	2	4	no	—
67	C₃×#127	—	3	24	no	—
68	C₃×#127	—	3	4	no	—
69	C₃×#127	C₃×D₃	1	4	yes	direct product factor
70	C₃×#127	C₂×C₃²	1	1	yes	characteristic, complemented
71	C₂×C₃×C₄²	—	3	4	no	—
72	#940	C₂×C₃²	1	6	yes	complemented
73	#874	C₂×C₃²	1	1	yes	characteristic, complemented
74	C₄²×D₃	C₂×C₃²	1	1	yes	characteristic, complemented
75	C₂³×C₃²	—	2	16	no	—
76	C₂³×C₃²	—	3	1	no	—
77	C₂³×C₃²	—	3	4	no	—
78	C₂³×C₃²	—	6	24	no	—
79	C₂³×C₃²	C₂²×D₃	1	4	yes	—
80	C₂³×C₃²	C₂³×C₃	1	16	yes	—
81	C₂²×C₃×D₃	—	2	96	no	—
82	C₂²×C₃×D₃	C₂³×C₃	1	16	yes	—
83	C₂²×C₃×D₃	C₂³×C₃	1	4	yes	—
84	C₃²×D₄	—	2	12	no	—
85	C₃²×D₄	—	2	48	no	—
86	C₃²×D₄	—	4	48	no	—
87	C₃²×D₄	—	6	24	no	—
88	C₃²×D₄	—	6	12	no	—
89	C₂×C₃²×C₄	—	2	12	no	—
90	C₂×C₃²×C₄	—	3	3	no	—
91	C₂×C₃²×C₄	—	3	12	no	—
92	C₂×C₃²×C₄	C₂²×D₃	1	3	yes	—
93	C₂×C₃²×C₄	C₂³×C₃	1	12	yes	—
94	C₃×#67	—	2	96	no	—
95	C₂×C₃×Dic₃	C₂³×C₃	1	12	yes	—
96	C₃×D₁₂	—	2	48	no	—
97	C₃×C₄×D₃	C₃×D₄	1	48	yes	complemented
98	C₂×#127	—	3	1	no	Sylow
99	C₂×C₃³	—	4	4	no	—
100	C₂×C₃³	C₂²×D₄	1	3	yes	—
101	C₃²×D₃	—	4	4	no	—
102	C₃²×D₃	#127	1	4	yes	direct product factor
103	C₂⁴×C₃	—	3	16	no	—
104	C₂³×D₃	C₂²×C₃²	1	4	yes	—
105	C₂×C₃×D₄	—	2	24	no	—
106	C₂×C₃×D₄	—	3	48	no	—
107	C₂×C₃×D₄	—	3	24	no	—
108	C₂×C₃×D₄	—	6	96	no	—
109	C₂×C₃×D₄	C₂×C₃×D₃	1	24	yes	complemented
110	C₂×C₃×D₄	C₂²×C₃²	1	6	yes	complemented
111	C₂²×C₃×C₄	—	3	12	no	—
112	C₂×#67	C₂²×C₃²	1	12	yes	complemented
113	D₃×D₄	—	2	24	no	—
114	C₂×D₁₂	C₂²×C₃²	1	6	yes	complemented
115	C₂×C₄×D₃	C₂²×C₃²	1	3	yes	—
116	C₃×C₄²	—	2	4	no	—
117	C₃×C₄²	—	3	12	no	—
118	C₃×C₄²	C₂×C₃×D₃	1	4	yes	complemented
119	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic, complemented
120	C₄×Dic₃	C₂²×C₃²	1	3	yes	complemented
121	C₂²×C₃²	—	2	12	no	—
122	C₂²×C₃²	—	2	4	no	—
123	C₂²×C₃²	—	2	48	no	—
124	C₂²×C₃²	—	3	6	no	—
125	C₂²×C₃²	—	4	48	no	—
126	C₂²×C₃²	—	6	12	no	—
127	C₂²×C₃²	—	12	16	no	—
128	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
129	C₂²×C₃²	C₂⁴×C₃	1	4	yes	—
130	C₂×C₃×D₃	—	2	48	no	—
131	C₂×C₃×D₃	—	4	64	no	—
132	C₂×C₃×D₃	C₂×C₃×D₄	1	24	yes	—
133	C₃²×C₄	—	2	24	no	—
134	C₃²×C₄	—	3	6	no	—
135	C₃²×C₄	D₃×D₄	1	6	yes	complemented
136	C₃²×C₄	C₂×C₃×D₄	1	24	yes	complemented
137	C₃×Dic₃	C₂×C₃×D₄	1	24	yes	complemented
138	C₂²×D₄	—	3	6	no	—
139	#127	—	3	6	no	—
140	#127	—	3	1	no	—
141	#127	C₃²×D₃	1	1	yes	characteristic, direct product factor
142	C₂×C₄²	—	3	1	no	—
143	C₃³	C₂×#127	1	1	yes	characteristic, complemented, Sylow
144	C₂³×C₃	—	2	16	no	—
145	C₂³×C₃	—	3	4	no	—
146	C₂³×C₃	—	3	16	no	—
147	C₂³×C₃	—	6	96	no	—
148	C₂³×C₃	C₂²×C₃×D₃	1	16	yes	—
149	C₂³×C₃	C₂³×C₃²	1	4	yes	—
150	C₂²×D₃	—	2	24	no	—
151	C₂²×D₃	C₂³×C₃²	1	4	yes	—
152	C₂²×D₃	C₂³×C₃²	1	1	yes	characteristic
153	C₃×D₄	—	2	48	no	—
154	C₃×D₄	—	2	12	no	—
155	C₃×D₄	—	4	48	no	—
156	C₃×D₄	—	6	96	no	—
157	C₃×D₄	—	6	48	no	—
158	C₂×C₃×C₄	—	2	12	no	—
159	C₂×C₃×C₄	—	3	12	no	—
160	C₂×C₃×C₄	—	3	48	no	—
161	C₂×C₃×C₄	C₂²×C₃×D₃	1	12	yes	—
162	C₂×C₃×C₄	C₂³×C₃²	1	3	yes	—
163	#67	—	2	24	no	—
164	C₂×Dic₃	C₂³×C₃²	1	3	yes	—
165	D₁₂	—	2	12	no	—
166	C₄×D₃	C₃²×D₄	1	12	yes	complemented
167	C₂×C₃²	—	2	12	no	—
168	C₂×C₃²	—	3	4	no	—
169	C₂×C₃²	—	4	4	no	—
170	C₂×C₃²	—	4	16	no	—
171	C₂×C₃²	—	8	16	no	—
172	C₂×C₃²	—	12	4	no	—
173	C₂×C₃²	C₂×D₃×D₄	1	3	yes	—
174	C₂×C₃²	C₂²×C₃×D₄	1	12	yes	—
175	C₃×D₃	—	4	16	no	—
176	C₃×D₃	C₃×#127	1	16	yes	direct product factor
177	C₂⁴	—	3	4	no	—
178	C₂×D₄	—	3	12	no	—
179	C₂×D₄	—	3	6	no	—
180	C₂×D₄	—	6	24	no	—
181	C₂×D₄	C₂×C₃²×D₃	1	6	yes	complemented
182	C₂²×C₄	—	3	3	no	—
183	C₄²	—	3	3	no	—
184	C₄²	C₂×C₃²×D₃	1	1	yes	characteristic, complemented
185	C₂²×C₃	—	2	48	no	—
186	C₂²×C₃	—	2	12	no	—
187	C₂²×C₃	—	2	4	no	—
188	C₂²×C₃	—	3	24	no	—
189	C₂²×C₃	—	4	48	no	—
190	C₂²×C₃	—	6	48	no	—
191	C₂²×C₃	—	12	64	no	—
192	C₂²×C₃	C₂³×C₃×D₃	1	4	yes	—
193	C₂²×C₃	C₂⁴×C₃²	1	1	yes	characteristic
194	C₂×D₃	—	2	12	no	—
195	C₂×D₃	—	4	16	no	—
196	C₂×D₃	C₂×C₃²×D₄	1	6	yes	—
197	C₃×C₄	—	2	24	no	—
198	C₃×C₄	—	3	24	no	—
199	C₃×C₄	C₃×D₃×D₄	1	24	yes	complemented
200	C₃×C₄	C₂×C₃²×D₄	1	6	yes	complemented
201	Dic₃	C₂×C₃²×D₄	1	6	yes	complemented
202	C₃²	—	2	4	no	—
203	C₃²	D₃×#127	1	1	yes	characteristic, direct product factor
204	C₃²	C₂×C₃×#127	1	4	yes	complemented
205	C₂³	—	3	1	no	—
206	C₂³	—	3	4	no	—
207	C₂³	—	6	24	no	—
208	C₂³	C₂²×C₃²×D₃	1	4	yes	—
209	D₄	—	2	12	no	—
210	D₄	—	6	24	no	—
211	D₄	—	6	12	no	—
212	C₂×C₄	—	3	3	no	—
213	C₂×C₄	—	3	12	no	—
214	C₂×C₄	C₂²×C₃²×D₃	1	3	yes	—
215	C₂×C₃	—	2	12	no	—
216	C₂×C₃	—	3	16	no	—
217	C₂×C₃	—	4	16	no	—
218	C₂×C₃	—	4	4	no	—
219	C₂×C₃	—	8	16	no	—
220	C₂×C₃	—	12	16	no	—
221	C₂×C₃	C₂×C₃×D₃×D₄	1	12	yes	—
222	C₂×C₃	C₂²×C₃²×D₄	1	3	yes	—
223	D₃	—	4	4	no	—
224	D₃	C₃²×#127	1	4	yes	direct product factor
225	C₂²	—	2	12	no	—
226	C₂²	—	3	6	no	—
227	C₂²	—	6	12	no	—
228	C₂²	—	12	16	no	—
229	C₂²	C₂³×C₃²×D₃	1	1	yes	characteristic
230	C₄	—	3	6	no	—
231	C₄	C₃²×D₃×D₄	1	6	yes	complemented
232	C₃	—	2	4	no	—
233	C₃	C₃×D₃×#127	1	4	yes	direct product factor
234	C₃	C₂×C₃²×#127	1	1	yes	characteristic, complemented
235	C₂	—	3	4	no	—
236	C₂	—	4	4	no	—
237	C₂	—	12	4	no	—
238	C₂	C₂×C₃²×D₃×D₄	1	3	yes	—

	subgroup	G/H
center	C₂²×C₃²	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃²
Frattini	C₂²	C₂³×C₃²×D₃
Fitting	C₃³×#127	C₂
socle	C₂²×C₃³	C₂⁴
hypercenter	C₃²×#127	D₃
hypocenter	C₃	C₂×C₃²×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×#127

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×D₃×D₄	C₂	semidirect
2. C₃³×#127	C₂	semidirect
3. C₃²×#940	C₂	semidirect
4. C₃²×#874	C₂	semidirect
5. C₃²×C₄²×D₃	C₂	semidirect
7. C₃×D₃×#127	C₃	direct (N × H)
9. C₂×C₃³×D₄	C₂²	semidirect
10. C₂×C₃²×#67	C₂²	semidirect
12. C₂×C₃²×D₁₂	C₂²	semidirect
14. C₃³×C₄²	C₂²	semidirect
15. C₃²×C₄×Dic₃	C₂²	semidirect
17. C₂×C₃×D₃×D₄	C₂×C₃	semidirect
21. C₃²×#127	D₃	direct (N × H)
22. C₃²×#127	C₂×C₃	semidirect
24. C₃×#940	C₂×C₃	semidirect
25. C₃×#874	C₂×C₃	semidirect
26. C₃×C₄²×D₃	C₂×C₃	semidirect
36. C₃²×C₄×D₃	D₄	semidirect
38. D₃×#127	C₃²	direct (N × H)
45. C₂×C₃²×D₄	C₂×D₃	semidirect
46. C₂×C₃²×D₄	C₂²×C₃	semidirect
48. C₂×C₃×#67	C₂²×C₃	semidirect
50. C₂×C₃×D₁₂	C₂²×C₃	semidirect
54. C₃²×C₄²	C₂×D₃	semidirect
55. C₃²×C₄²	C₂²×C₃	semidirect
56. C₃×C₄×Dic₃	C₂²×C₃	semidirect
62. C₃³×C₄	C₂×D₄	semidirect
63. C₃²×Dic₃	C₂×D₄	semidirect
65. C₂×D₃×D₄	C₂×C₃²	semidirect
69. C₃×#127	C₃×D₃	direct (N × H)
70. C₃×#127	C₂×C₃²	semidirect
72. #940	C₂×C₃²	semidirect
73. #874	C₂×C₃²	semidirect
74. C₄²×D₃	C₂×C₃²	semidirect
97. C₃×C₄×D₃	C₃×D₄	semidirect
102. C₃²×D₃	#127	direct (N × H)
109. C₂×C₃×D₄	C₂×C₃×D₃	semidirect
110. C₂×C₃×D₄	C₂²×C₃²	semidirect
112. C₂×#67	C₂²×C₃²	semidirect
114. C₂×D₁₂	C₂²×C₃²	semidirect
118. C₃×C₄²	C₂×C₃×D₃	semidirect
119. C₃×C₄²	C₂²×C₃²	semidirect
120. C₄×Dic₃	C₂²×C₃²	semidirect
135. C₃²×C₄	D₃×D₄	semidirect
136. C₃²×C₄	C₂×C₃×D₄	semidirect
137. C₃×Dic₃	C₂×C₃×D₄	semidirect
141. #127	C₃²×D₃	direct (N × H)
143. C₃³	C₂×#127	semidirect
166. C₄×D₃	C₃²×D₄	semidirect
176. C₃×D₃	C₃×#127	direct (N × H)
181. C₂×D₄	C₂×C₃²×D₃	semidirect
184. C₄²	C₂×C₃²×D₃	semidirect
199. C₃×C₄	C₃×D₃×D₄	semidirect
200. C₃×C₄	C₂×C₃²×D₄	semidirect
201. Dic₃	C₂×C₃²×D₄	semidirect
203. C₃²	D₃×#127	direct (N × H)
204. C₃²	C₂×C₃×#127	semidirect
224. D₃	C₃²×#127	direct (N × H)
231. C₄	C₃²×D₃×D₄	semidirect
233. C₃	C₃×D₃×#127	direct (N × H)
234. C₃	C₂×C₃²×#127	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×D₃×D₄	2
C₃³×#127	2
C₃²×#940	2
C₃²×#874	2
C₃²×C₄²×D₃	2
C₂×C₃²×#127	3
C₃×D₃×#127	3

Subgroups — C32×D3×#127 #13179092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×#127 #13179092