Subgroups — C₃×#74319 #11802517

Order 1152.

1941 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#893	C₂	1	6	yes	maximal, complemented
2	C₂×C₃×#874	C₂	1	1	yes	maximal, characteristic
3	C₃²×#384	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4988	C₂	1	6	yes	maximal, complemented
5	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×#127	—	3	1	no	maximal
7	#74319	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×D₁₂	C₂²	1	6	yes	—
9	C₂²×C₃×C₄×D₃	C₂²	1	4	yes	—
10	C₂×C₃²×C₄⋊C₄	C₂²	1	6	yes	—
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃×(C₂×D₃)⋊C₄	C₂²	1	12	yes	—
13	C₂×C₃×C₄×Dic₃	C₂²	1	3	yes	—
14	C₂×C₃×C₄×Dic₃	C₂²	1	3	yes	—
15	C₃×#893	—	2	24	no	—
16	C₃×#874	C₄	1	8	yes	complemented
17	C₂×C₃×#127	—	3	1	no	—
18	C₂×C₃×C₄×D₄	—	3	6	no	—
19	C₂×#893	C₂×C₃	1	6	yes	complemented
20	C₂×#874	C₂×C₃	1	1	yes	characteristic
21	C₃×#390	—	3	6	no	—
22	C₃×#384	—	2	1	no	—
23	C₃×#384	D₃	1	1	yes	characteristic, complemented
24	C₃×#384	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×C₄³	—	3	1	no	—
26	#4988	C₂×C₃	1	6	yes	complemented
27	C₄²×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂³×C₃×D₃	C₂³	1	4	yes	—
29	C₂²×C₃²×C₄	C₂³	1	4	yes	—
30	C₂²×C₃²×C₄	C₂³	1	3	yes	—
31	C₂²×C₃×Dic₃	C₂³	1	3	yes	—
32	C₂²×C₃×Dic₃	C₂³	1	1	yes	characteristic
33	C₂×C₃×D₁₂	—	2	12	no	—
34	C₂×C₃×D₁₂	—	2	12	no	—
35	C₂×C₃×D₁₂	C₂×C₄	1	24	yes	complemented
36	C₂×C₃×C₄×D₃	—	2	24	no	—
37	C₃²×C₄⋊C₄	—	2	12	no	—
38	C₃²×C₄²	C₂×C₄	1	4	yes	complemented
39	C₃×(C₂×D₃)⋊C₄	—	2	24	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	—	3	6	no	—
44	C₂³×C₃×C₄	—	3	4	no	—
45	C₂²×D₁₂	C₂²×C₃	1	6	yes	—
46	C₂²×C₄×D₃	C₂²×C₃	1	4	yes	—
47	C₃×#127	—	3	8	no	—
48	C₃×C₄×D₄	—	6	24	no	—
49	C₂×C₃×C₄⋊C₄	—	2	6	no	—
50	C₂×C₃×C₄⋊C₄	C₂×D₃	1	6	yes	—
51	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	6	yes	—
52	C₂×C₃×C₂²⋊C₄	—	3	12	no	—
53	C₂×C₃×C₄²	—	2	1	no	—
54	C₂×C₃×C₄²	—	3	3	no	—
55	C₂×C₃×C₄²	—	3	3	no	—
56	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
58	C₂×(C₂×D₃)⋊C₄	C₂²×C₃	1	12	yes	—
59	C₂×C₄×Dic₃	C₂²×C₃	1	3	yes	—
60	C₂×C₄×Dic₃	C₂²×C₃	1	3	yes	—
61	#893	—	2	24	no	—
62	#874	C₃×C₄	1	8	yes	complemented
63	C₂³×C₃²	C₂⁴	1	1	yes	characteristic
64	C₂²×C₃×D₃	—	2	12	no	—
65	C₂²×C₃×D₃	—	2	12	no	—
66	C₂²×C₃×D₃	C₂²×C₄	1	8	yes	—
67	C₂×C₃²×C₄	—	2	12	no	—
68	C₂×C₃²×C₄	C₂²×C₄	1	6	yes	—
69	C₂×C₃²×C₄	C₂×D₄	1	6	yes	—
70	C₂×C₃²×C₄	#41	1	6	yes	—
71	C₂×C₃×Dic₃	—	2	6	no	—
72	C₂×C₃×Dic₃	C₂×D₄	1	6	yes	—
73	C₂×C₃×Dic₃	C₂×D₄	1	6	yes	—
74	C₃×D₁₂	—	2	48	no	—
75	C₃×C₄×D₃	—	4	16	no	—
76	C₂×#127	—	3	1	no	—
77	C₂×C₄×D₄	—	3	6	no	—
78	#390	—	3	6	no	—
79	#384	C₃×D₃	1	1	yes	characteristic, complemented
80	C₄³	—	3	1	no	—
81	C₂⁴×C₃	—	3	4	no	—
82	C₂³×D₃	C₂³×C₃	1	4	yes	—
83	C₂×C₃×D₄	—	3	24	no	—
84	C₂×C₃×D₄	—	6	12	no	—
85	C₂×C₃×D₄	—	6	12	no	—
86	C₂²×C₃×C₄	—	2	3	no	—
87	C₂²×C₃×C₄	—	2	4	no	—
88	C₂²×C₃×C₄	—	3	1	no	—
89	C₂²×C₃×C₄	—	3	3	no	—
90	C₂²×C₃×C₄	—	6	24	no	—
91	C₂²×C₃×C₄	C₂²×D₃	1	4	yes	—
92	C₂²×C₃×C₄	C₂²×D₃	1	3	yes	—
93	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
94	C₂²×C₃×C₄	C₂³×C₃	1	3	yes	—
95	C₂²×Dic₃	C₂³×C₃	1	3	yes	—
96	C₂²×Dic₃	C₂³×C₃	1	1	yes	characteristic
97	C₂×D₁₂	—	2	12	no	—
98	C₂×D₁₂	—	2	12	no	—
99	C₂×D₁₂	C₂×C₃×C₄	1	24	yes	complemented
100	C₂×C₄×D₃	—	2	24	no	—
101	C₃×C₄⋊C₄	—	2	12	no	—
102	C₃×C₄⋊C₄	—	2	12	no	—
103	C₃×C₄⋊C₄	—	4	12	no	—
104	C₃×C₂²⋊C₄	—	6	24	no	—
105	C₃×C₄²	—	2	4	no	—
106	C₃×C₄²	—	3	12	no	—
107	C₃×C₄²	—	6	6	no	—
108	C₃×C₄²	C₄×D₃	1	4	yes	complemented
109	C₃×C₄²	C₂×C₃×C₄	1	4	yes	complemented
110	(C₂×D₃)⋊C₄	—	2	24	no	—
111	C₄×Dic₃	—	2	6	no	—
112	C₄×Dic₃	C₃×D₄	1	12	yes	complemented
113	C₂²×C₃²	C₂³×C₄	1	1	yes	characteristic
114	C₂²×C₃²	C₂²×D₄	1	3	yes	—
115	C₂²×C₃²	C₂×#41	1	3	yes	—
116	C₂×C₃×D₃	—	2	24	no	—
117	C₂×C₃×D₃	—	4	4	no	—
118	C₂×C₃×D₃	—	4	12	no	—
119	C₃²×C₄	—	4	4	no	—
120	C₃²×C₄	C₄×D₄	1	12	yes	complemented
121	C₃×Dic₃	—	2	6	no	—
122	C₃×Dic₃	#127	1	4	yes	complemented
123	C₂²×D₄	—	3	6	no	—
124	C₂³×C₄	—	3	4	no	—
125	#127	—	3	8	no	—
126	C₄×D₄	—	6	24	no	—
127	C₂×C₄⋊C₄	C₂×C₃×D₃	1	6	yes	—
128	C₂×C₂²⋊C₄	—	3	12	no	—
129	C₂×C₄²	—	3	3	no	—
130	C₂×C₄²	—	3	3	no	—
131	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
132	C₂³×C₃	—	2	1	no	—
133	C₂³×C₃	—	3	8	no	—
134	C₂³×C₃	—	6	12	no	—
135	C₂³×C₃	—	6	12	no	—
136	C₂³×C₃	C₂³×D₃	1	1	yes	characteristic
137	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
138	C₂²×D₃	—	2	12	no	—
139	C₂²×D₃	—	2	12	no	—
140	C₂²×D₃	C₂²×C₃×C₄	1	8	yes	—
141	C₃×D₄	—	6	48	no	—
142	C₂×C₃×C₄	—	2	12	no	—
143	C₂×C₃×C₄	—	2	12	no	—
144	C₂×C₃×C₄	—	2	6	no	—
145	C₂×C₃×C₄	—	2	6	no	—
146	C₂×C₃×C₄	—	2	6	no	—
147	C₂×C₃×C₄	—	3	6	no	—
148	C₂×C₃×C₄	—	3	6	no	—
149	C₂×C₃×C₄	—	4	12	no	—
150	C₂×C₃×C₄	—	6	6	no	—
151	C₂×C₃×C₄	—	12	16	no	—
152	C₂×C₃×C₄	C₂×C₄×D₃	1	6	yes	—
153	C₂×C₃×C₄	D₃×D₄	1	6	yes	—
154	C₂×C₃×C₄	#239	1	6	yes	—
155	C₂×C₃×C₄	C₂²×C₃×C₄	1	6	yes	—
156	C₂×C₃×C₄	C₂×C₃×D₄	1	6	yes	—
157	C₂×C₃×C₄	C₃×#41	1	6	yes	—
158	C₂×Dic₃	—	2	6	no	—
159	C₂×Dic₃	C₂×C₃×D₄	1	6	yes	—
160	C₂×Dic₃	C₂×C₃×D₄	1	6	yes	—
161	D₁₂	—	2	48	no	—
162	C₄×D₃	—	4	16	no	—
163	C₂×C₃²	C₂×C₄×D₄	1	3	yes	—
164	C₂×C₃²	C₂×#127	1	1	yes	characteristic
165	C₂×C₃²	#532	1	3	yes	—
166	C₃×D₃	—	4	8	no	—
167	C₂⁴	—	3	4	no	—
168	C₂×D₄	—	3	24	no	—
169	C₂×D₄	—	6	12	no	—
170	C₂×D₄	—	6	12	no	—
171	C₂²×C₄	—	3	1	no	—
172	C₂²×C₄	—	3	3	no	—
173	C₂²×C₄	—	6	24	no	—
174	C₂²×C₄	C₂²×C₃×D₃	1	4	yes	—
175	C₂²×C₄	C₂²×C₃×D₃	1	3	yes	—
176	C₄⋊C₄	—	2	12	no	—
177	C₂²⋊C₄	—	6	24	no	—
178	C₄²	—	3	12	no	—
179	C₄²	—	6	6	no	—
180	C₄²	C₃×C₄×D₃	1	4	yes	complemented
181	C₂²×C₃	—	2	1	no	—
182	C₂²×C₃	—	2	3	no	—
183	C₂²×C₃	—	2	3	no	—
184	C₂²×C₃	—	6	24	no	—
185	C₂²×C₃	—	12	4	no	—
186	C₂²×C₃	—	12	12	no	—
187	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
188	C₂²×C₃	C₂×D₃×D₄	1	3	yes	—
189	C₂²×C₃	C₂×#239	1	3	yes	—
190	C₂²×C₃	C₂³×C₃×C₄	1	1	yes	characteristic
191	C₂²×C₃	C₂²×C₃×D₄	1	3	yes	—
192	C₂²×C₃	C₂×C₃×#41	1	3	yes	—
193	C₂×D₃	—	2	24	no	—
194	C₂×D₃	—	4	4	no	—
195	C₂×D₃	—	4	12	no	—
196	C₃×C₄	—	2	12	no	—
197	C₃×C₄	—	3	4	no	—
198	C₃×C₄	—	4	4	no	—
199	C₃×C₄	—	4	4	no	—
200	C₃×C₄	—	6	6	no	—
201	C₃×C₄	—	8	4	no	—
202	C₃×C₄	#893	1	12	yes	complemented
203	C₃×C₄	C₃×C₄×D₄	1	12	yes	complemented
204	Dic₃	—	2	6	no	—
205	Dic₃	C₃×#127	1	4	yes	complemented
206	C₃²	C₄×#127	1	1	yes	characteristic, complemented, Sylow
207	C₂³	—	3	8	no	—
208	C₂³	—	6	12	no	—
209	C₂³	—	6	12	no	—
210	C₂³	C₂³×C₃×D₃	1	1	yes	characteristic
211	D₄	—	6	48	no	—
212	C₂×C₄	—	2	12	no	—
213	C₂×C₄	—	3	6	no	—
214	C₂×C₄	—	3	6	no	—
215	C₂×C₄	—	6	6	no	—
216	C₂×C₄	—	12	16	no	—
217	C₂×C₄	C₂×C₃×C₄×D₃	1	6	yes	—
218	C₂×C₄	C₃×D₃×D₄	1	6	yes	—
219	C₂×C₄	C₃×#239	1	6	yes	—
220	C₂×C₃	—	2	3	no	—
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	3	no	—
223	C₂×C₃	—	12	8	no	—
224	C₂×C₃	C₂×#893	1	3	yes	—
225	C₂×C₃	D₃×#127	1	1	yes	characteristic
226	C₂×C₃	#5725	1	3	yes	—
227	C₂×C₃	C₂×C₃×C₄×D₄	1	3	yes	—
228	C₂×C₃	C₂×C₃×#127	1	1	yes	characteristic
229	C₂×C₃	C₃×#532	1	3	yes	—
230	D₃	—	4	8	no	—
231	C₂²	—	6	24	no	—
232	C₂²	—	12	4	no	—
233	C₂²	—	12	12	no	—
234	C₂²	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
235	C₂²	C₂×C₃×D₃×D₄	1	3	yes	—
236	C₂²	C₂×C₃×#239	1	3	yes	—
237	C₄	—	3	4	no	—
238	C₄	—	4	4	no	—
239	C₄	—	6	6	no	—
240	C₄	C₃×#893	1	12	yes	complemented
241	C₃	—	2	1	no	—
242	C₃	#74319	1	1	yes	characteristic, direct product factor
243	C₃	C₃×C₄×#127	1	1	yes	characteristic, complemented
244	C₂	—	12	8	no	—
245	C₂	C₂×C₃×#893	1	3	yes	—
246	C₂	C₃×D₃×#127	1	1	yes	characteristic
247	C₂	C₃×#5725	1	3	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₃×C₄
Frattini	C₂³	C₂³×C₃×D₃
Fitting	C₃²×#384	C₂
socle	C₂³×C₃²	C₂⁴
prime radical	C₂×C₃×#874	C₂
cosocle	C₂³×C₃	C₂⁴×C₃
hypercenter	C₃×#384	D₃
hypocenter	C₃	C₃×C₄×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#384

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#893	C₂	semidirect
3. C₃²×#384	C₂	semidirect
4. C₃×#4988	C₂	semidirect
5. C₃×C₄²×Dic₃	C₂	semidirect
7. #74319	C₃	direct (N × H)
16. C₃×#874	C₄	semidirect
19. C₂×#893	C₂×C₃	semidirect
23. C₃×#384	D₃	semidirect
24. C₃×#384	C₂×C₃	semidirect
26. #4988	C₂×C₃	semidirect
27. C₄²×Dic₃	C₂×C₃	semidirect
35. C₂×C₃×D₁₂	C₂×C₄	semidirect
38. C₃²×C₄²	C₂×C₄	semidirect
41. C₃×C₄×Dic₃	D₄	semidirect
62. #874	C₃×C₄	semidirect
79. #384	C₃×D₃	semidirect
99. C₂×D₁₂	C₂×C₃×C₄	semidirect
108. C₃×C₄²	C₄×D₃	semidirect
109. C₃×C₄²	C₂×C₃×C₄	semidirect
112. C₄×Dic₃	C₃×D₄	semidirect
120. C₃²×C₄	C₄×D₄	semidirect
122. C₃×Dic₃	#127	semidirect
180. C₄²	C₃×C₄×D₃	semidirect
202. C₃×C₄	#893	semidirect
203. C₃×C₄	C₃×C₄×D₄	semidirect
205. Dic₃	C₃×#127	semidirect
206. C₃²	C₄×#127	semidirect
240. C₄	C₃×#893	semidirect
242. C₃	#74319	direct (N × H)
243. 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×C₃×#893	2
C₂×C₃×#874	2
C₃²×#384	2
C₃×#4988	2
C₃×C₄²×Dic₃	2
C₃×C₄×#127	3
#74319	3

Subgroups — C3×#74319 #11802517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#74319 #11802517