Subgroups — C₃×Dic₃×#127 #11802529

Order 1152.

2097 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₄×Dic₃	C₂	1	6	yes	maximal, complemented
3	C₃×#5221	C₂	1	6	yes	maximal, complemented
4	C₃×#4935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×#127	—	3	1	no	maximal
7	Dic₃×#127	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×D₄	C₂²	1	6	yes	—
9	C₂³×C₃×Dic₃	C₂²	1	4	yes	—
10	C₃²×#127	C₄	1	8	yes	complemented
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃×#217	C₂²	1	12	yes	—
13	C₃×D₄×Dic₃	—	2	24	no	—
14	C₂×C₃×C₄⋊Dic₃	C₂²	1	6	yes	—
15	C₂×C₃×C₄×Dic₃	C₂²	1	3	yes	—
16	C₂×C₃×C₄×Dic₃	C₂²	1	3	yes	—
17	C₂×C₃×#127	—	2	1	no	—
18	C₂×C₃×#127	D₃	1	1	yes	characteristic
19	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×C₄×D₄	—	3	6	no	—
21	C₂×D₄×Dic₃	C₂×C₃	1	6	yes	complemented
22	C₃×#390	—	3	6	no	—
23	C₃×#384	—	3	1	no	—
24	C₃×C₄³	—	3	1	no	—
25	#5221	C₂×C₃	1	6	yes	complemented
26	#4935	C₂×C₃	1	1	yes	characteristic, complemented
27	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂⁴×C₃²	C₂³	1	4	yes	—
29	C₂×C₃²×D₄	—	2	12	no	—
30	C₂×C₃²×D₄	—	2	12	no	—
31	C₂×C₃²×D₄	C₂×C₄	1	24	yes	complemented
32	C₂²×C₃²×C₄	C₂³	1	3	yes	—
33	C₂²×C₃×Dic₃	—	2	24	no	—
34	C₂²×C₃×Dic₃	C₂³	1	4	yes	—
35	C₂²×C₃×Dic₃	C₂³	1	3	yes	—
36	C₂²×C₃×Dic₃	C₂³	1	1	yes	characteristic
37	C₃²×C₄²	C₂×C₄	1	4	yes	complemented
38	C₃×#217	—	2	24	no	—
39	C₃×C₄⋊Dic₃	—	2	12	no	—
40	C₃×C₄×Dic₃	—	2	6	no	—
41	C₃×C₄×Dic₃	D₄	1	12	yes	complemented
42	C₄×#127	—	3	1	no	Sylow
43	C₂²×C₃×D₄	—	2	6	no	—
44	C₂²×C₃×D₄	C₂×D₃	1	6	yes	—
45	C₂²×C₃×D₄	C₂²×C₃	1	6	yes	—
46	C₂³×C₃×C₄	—	3	4	no	—
47	C₂³×Dic₃	C₂²×C₃	1	4	yes	—
48	C₃×#127	—	2	8	no	—
49	C₃×#127	Dic₃	1	8	yes	direct product factor
50	C₃×#127	C₃×C₄	1	8	yes	complemented
51	C₃×C₄×D₄	—	6	24	no	—
52	C₂×C₃×C₄⋊C₄	—	3	6	no	—
53	C₂×C₃×C₂²⋊C₄	—	3	12	no	—
54	C₂×C₃×C₄²	—	2	1	no	—
55	C₂×C₃×C₄²	—	3	3	no	—
56	C₂×C₃×C₄²	—	3	3	no	—
57	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
59	C₂×#217	C₂²×C₃	1	12	yes	—
60	D₄×Dic₃	—	2	24	no	—
61	C₂×C₄⋊Dic₃	C₂²×C₃	1	6	yes	—
62	C₂×C₄×Dic₃	C₂²×C₃	1	3	yes	—
63	C₂×C₄×Dic₃	C₂²×C₃	1	3	yes	—
64	C₂³×C₃²	—	2	12	no	—
65	C₂³×C₃²	—	2	12	no	—
66	C₂³×C₃²	C₂²×C₄	1	8	yes	—
67	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
68	C₃²×D₄	—	2	48	no	—
69	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
70	C₂×C₃²×C₄	C₂×D₄	1	6	yes	—
71	C₂×C₃²×C₄	#41	1	6	yes	—
72	C₂×C₃×Dic₃	—	2	6	no	—
73	C₂×C₃×Dic₃	—	2	12	no	—
74	C₂×C₃×Dic₃	—	4	16	no	—
75	C₂×C₃×Dic₃	C₂×D₄	1	6	yes	—
76	C₂×C₃×Dic₃	C₂×D₄	1	6	yes	—
77	C₂×#127	C₃×D₃	1	1	yes	characteristic
78	C₂×C₄×D₄	—	3	6	no	—
79	#390	—	3	6	no	—
80	#384	—	3	1	no	—
81	C₄³	—	3	1	no	—
82	C₂⁴×C₃	—	2	4	no	—
83	C₂⁴×C₃	C₂²×D₃	1	4	yes	—
84	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
85	C₂×C₃×D₄	—	2	12	no	—
86	C₂×C₃×D₄	—	2	12	no	—
87	C₂×C₃×D₄	—	2	12	no	—
88	C₂×C₃×D₄	—	2	12	no	—
89	C₂×C₃×D₄	—	2	24	no	—
90	C₂×C₃×D₄	—	4	12	no	—
91	C₂×C₃×D₄	—	4	12	no	—
92	C₂×C₃×D₄	C₂×Dic₃	1	24	yes	complemented
93	C₂×C₃×D₄	C₂×C₃×C₄	1	24	yes	complemented
94	C₂²×C₃×C₄	—	2	3	no	—
95	C₂²×C₃×C₄	—	3	1	no	—
96	C₂²×C₃×C₄	—	3	3	no	—
97	C₂²×C₃×C₄	—	3	4	no	—
98	C₂²×C₃×C₄	—	6	24	no	—
99	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
100	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
101	C₂²×Dic₃	—	2	24	no	—
102	C₂²×Dic₃	C₂³×C₃	1	4	yes	—
103	C₂²×Dic₃	C₂³×C₃	1	3	yes	—
104	C₂²×Dic₃	C₂³×C₃	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	6	12	no	—
106	C₃×C₂²⋊C₄	—	6	24	no	—
107	C₃×C₄²	—	2	4	no	—
108	C₃×C₄²	—	3	12	no	—
109	C₃×C₄²	—	6	6	no	—
110	C₃×C₄²	C₂×Dic₃	1	4	yes	complemented
111	C₃×C₄²	C₂×C₃×C₄	1	4	yes	complemented
112	#217	—	2	24	no	—
113	C₄⋊Dic₃	—	2	12	no	—
114	C₄×Dic₃	—	2	6	no	—
115	C₄×Dic₃	C₃×D₄	1	12	yes	complemented
116	C₂²×C₃²	—	2	24	no	—
117	C₂²×C₃²	—	4	4	no	—
118	C₂²×C₃²	—	4	12	no	—
119	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
120	C₂²×C₃²	C₂²×D₄	1	3	yes	—
121	C₂²×C₃²	C₂×#41	1	3	yes	—
122	C₃²×C₄	C₄×D₄	1	12	yes	complemented
123	C₃×Dic₃	—	2	6	no	—
124	C₃×Dic₃	—	4	4	no	—
125	C₃×Dic₃	#127	1	4	yes	direct product factor
126	C₂²×D₄	C₂×C₃×D₃	1	6	yes	—
127	C₂³×C₄	—	3	4	no	—
128	#127	C₃×Dic₃	1	8	yes	direct product factor
129	C₄×D₄	—	6	24	no	—
130	C₂×C₄⋊C₄	—	3	6	no	—
131	C₂×C₂²⋊C₄	—	3	12	no	—
132	C₂×C₄²	—	3	3	no	—
133	C₂×C₄²	—	3	3	no	—
134	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
135	C₂³×C₃	—	2	12	no	—
136	C₂³×C₃	—	2	12	no	—
137	C₂³×C₃	—	2	12	no	—
138	C₂³×C₃	—	2	12	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	—	2	8	no	—
141	C₂³×C₃	—	4	12	no	—
142	C₂³×C₃	—	4	12	no	—
143	C₂³×C₃	C₂²×Dic₃	1	8	yes	—
144	C₂³×C₃	C₂²×C₃×C₄	1	8	yes	—
145	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
146	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
147	C₃×D₄	—	2	48	no	—
148	C₃×D₄	—	2	48	no	—
149	C₃×D₄	—	4	48	no	—
150	C₂×C₃×C₄	—	2	6	no	—
151	C₂×C₃×C₄	—	2	6	no	—
152	C₂×C₃×C₄	—	2	6	no	—
153	C₂×C₃×C₄	—	3	6	no	—
154	C₂×C₃×C₄	—	3	6	no	—
155	C₂×C₃×C₄	—	6	6	no	—
156	C₂×C₃×C₄	—	6	12	no	—
157	C₂×C₃×C₄	—	12	16	no	—
158	C₂×C₃×C₄	D₃×D₄	1	6	yes	—
159	C₂×C₃×C₄	#237	1	6	yes	—
160	C₂×C₃×C₄	C₂²×Dic₃	1	6	yes	—
161	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
162	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
163	C₂×C₃×C₄	C₃×#41	1	6	yes	—
164	C₂×Dic₃	—	2	6	no	—
165	C₂×Dic₃	—	2	12	no	—
166	C₂×Dic₃	—	4	16	no	—
167	C₂×Dic₃	C₂×C₃×D₄	1	6	yes	—
168	C₂×Dic₃	C₂×C₃×D₄	1	6	yes	—
169	C₂×C₃²	—	4	8	no	—
170	C₂×C₃²	C₂×C₄×D₄	1	3	yes	—
171	C₂×C₃²	C₂×#127	1	1	yes	characteristic
172	C₂×C₃²	#532	1	3	yes	—
173	C₂⁴	C₂²×C₃×D₃	1	4	yes	—
174	C₂×D₄	—	2	12	no	—
175	C₂×D₄	—	2	12	no	—
176	C₂×D₄	C₂×C₃×Dic₃	1	24	yes	complemented
177	C₂²×C₄	—	3	1	no	—
178	C₂²×C₄	—	3	3	no	—
179	C₂²×C₄	—	3	4	no	—
180	C₂²×C₄	—	6	24	no	—
181	C₂²×C₄	C₂²×C₃×D₃	1	3	yes	—
182	C₄⋊C₄	—	6	12	no	—
183	C₂²⋊C₄	—	6	24	no	—
184	C₄²	—	3	12	no	—
185	C₄²	—	6	6	no	—
186	C₄²	C₂×C₃×Dic₃	1	4	yes	complemented
187	C₂²×C₃	—	2	24	no	—
188	C₂²×C₃	—	2	24	no	—
189	C₂²×C₃	—	2	3	no	—
190	C₂²×C₃	—	2	3	no	—
191	C₂²×C₃	—	2	1	no	—
192	C₂²×C₃	—	4	24	no	—
193	C₂²×C₃	—	4	4	no	—
194	C₂²×C₃	—	4	12	no	—
195	C₂²×C₃	—	4	4	no	—
196	C₂²×C₃	—	4	12	no	—
197	C₂²×C₃	—	8	4	no	—
198	C₂²×C₃	—	8	12	no	—
199	C₂²×C₃	C₂×D₃×D₄	1	3	yes	—
200	C₂²×C₃	C₂×#237	1	3	yes	—
201	C₂²×C₃	C₂³×Dic₃	1	1	yes	characteristic
202	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
203	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
204	C₂²×C₃	C₂×C₃×#41	1	3	yes	—
205	C₃×C₄	—	2	12	no	—
206	C₃×C₄	—	3	4	no	—
207	C₃×C₄	—	6	6	no	—
208	C₃×C₄	—	12	4	no	—
209	C₃×C₄	D₄×Dic₃	1	12	yes	complemented
210	C₃×C₄	C₃×C₄×D₄	1	12	yes	complemented
211	Dic₃	—	2	6	no	—
212	Dic₃	—	4	4	no	—
213	Dic₃	C₃×#127	1	4	yes	direct product factor
214	C₃²	C₄×#127	1	1	yes	characteristic, complemented, Sylow
215	C₂³	—	2	12	no	—
216	C₂³	—	2	12	no	—
217	C₂³	C₂²×C₃×Dic₃	1	8	yes	—
218	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
219	D₄	—	2	48	no	—
220	C₂×C₄	—	3	6	no	—
221	C₂×C₄	—	3	6	no	—
222	C₂×C₄	—	6	6	no	—
223	C₂×C₄	—	6	12	no	—
224	C₂×C₄	—	12	16	no	—
225	C₂×C₄	C₃×D₃×D₄	1	6	yes	—
226	C₂×C₄	C₃×#237	1	6	yes	—
227	C₂×C₄	C₂²×C₃×Dic₃	1	6	yes	—
228	C₂×C₃	—	2	3	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	3	no	—
231	C₂×C₃	—	4	8	no	—
232	C₂×C₃	—	4	8	no	—
233	C₂×C₃	—	8	8	no	—
234	C₂×C₃	D₃×#127	1	1	yes	characteristic
235	C₂×C₃	#5716	1	3	yes	—
236	C₂×C₃	C₂×D₄×Dic₃	1	3	yes	—
237	C₂×C₃	C₂×C₃×C₄×D₄	1	3	yes	—
238	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
239	C₂×C₃	C₃×#532	1	3	yes	—
240	C₂²	—	2	24	no	—
241	C₂²	—	4	4	no	—
242	C₂²	—	4	12	no	—
243	C₂²	C₂×C₃×D₃×D₄	1	3	yes	—
244	C₂²	C₂×C₃×#237	1	3	yes	—
245	C₂²	C₂³×C₃×Dic₃	1	1	yes	characteristic
246	C₄	—	3	4	no	—
247	C₄	—	6	6	no	—
248	C₄	—	12	4	no	—
249	C₄	C₃×D₄×Dic₃	1	12	yes	complemented
250	C₃	—	2	1	no	—
251	C₃	Dic₃×#127	1	1	yes	characteristic, direct product factor
252	C₃	C₃×C₄×#127	1	1	yes	characteristic, complemented
253	C₂	—	4	8	no	—
254	C₂	C₃×D₃×#127	1	1	yes	characteristic
255	C₂	C₃×#5716	1	3	yes	—
256	C₂	C₂×C₃×D₄×Dic₃	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₃
Fitting, prime radical	C₂×C₃²×#127	C₂
socle	C₂³×C₃²	C₂⁴
cosocle	C₂³×C₃	C₂⁴×C₃
hypercenter	C₂×C₃×#127	D₃
hypocenter	C₃	C₃×C₄×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×D₄×Dic₃	C₂	semidirect
3. C₃×#5221	C₂	semidirect
4. C₃×#4935	C₂	semidirect
5. C₃×C₄²×Dic₃	C₂	semidirect
7. Dic₃×#127	C₃	direct (N × H)
10. C₃²×#127	C₄	semidirect
21. C₂×D₄×Dic₃	C₂×C₃	semidirect
25. #5221	C₂×C₃	semidirect
26. #4935	C₂×C₃	semidirect
27. C₄²×Dic₃	C₂×C₃	semidirect
31. C₂×C₃²×D₄	C₂×C₄	semidirect
37. C₃²×C₄²	C₂×C₄	semidirect
41. C₃×C₄×Dic₃	D₄	semidirect
49. C₃×#127	Dic₃	direct (N × H)
50. C₃×#127	C₃×C₄	semidirect
92. C₂×C₃×D₄	C₂×Dic₃	semidirect
93. C₂×C₃×D₄	C₂×C₃×C₄	semidirect
110. C₃×C₄²	C₂×Dic₃	semidirect
111. C₃×C₄²	C₂×C₃×C₄	semidirect
115. C₄×Dic₃	C₃×D₄	semidirect
122. C₃²×C₄	C₄×D₄	semidirect
125. C₃×Dic₃	#127	direct (N × H)
128. #127	C₃×Dic₃	direct (N × H)
176. C₂×D₄	C₂×C₃×Dic₃	semidirect
186. C₄²	C₂×C₃×Dic₃	semidirect
209. C₃×C₄	D₄×Dic₃	semidirect
210. C₃×C₄	C₃×C₄×D₄	semidirect
213. Dic₃	C₃×#127	direct (N × H)
214. C₃²	C₄×#127	semidirect
249. C₄	C₃×D₄×Dic₃	semidirect
251. C₃	Dic₃×#127	direct (N × H)
252. C₃	C₃×C₄×#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×#127	2
C₂×C₃×D₄×Dic₃	2
C₃×#5221	2
C₃×#4935	2
C₃×C₄²×Dic₃	2
C₃×C₄×#127	3
Dic₃×#127	3

Subgroups — C3×Dic3×#127 #11802529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₃×#127 #11802529