Subgroups — C₃×#69749 #11783036

Order 1152.

737 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×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#815	C₂	1	2	yes	maximal
3	C₂×C₃×C₈×Dic₃	C₂	1	2	yes	maximal
4	C₃×#4932	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#803	C₂	1	1	yes	maximal, characteristic
6	C₃×#1726	—	3	1	no	maximal
7	#69749	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃²×C₄×C₈	C₄	1	4	yes	complemented
14	C₃×C₃⋊#95	C₄	1	4	yes	—
15	C₃×#815	C₄	1	8	yes	complemented
16	C₃×C₈×Dic₃	C₄	1	8	yes	complemented
17	C₃×#803	C₄	1	4	yes	complemented
18	C₂×C₃×#97	—	3	2	no	—
19	C₂×C₃×#97	—	3	1	no	—
20	C₂×C₃×C₄×C₈	—	2	1	no	—
21	C₂×C₃×C₄×C₈	—	3	2	no	—
22	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
23	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#376	—	3	1	no	—
25	C₂×#815	C₂×C₃	1	2	yes	—
26	C₂×C₈×Dic₃	C₂×C₃	1	2	yes	—
27	#4932	C₂×C₃	1	1	yes	characteristic
28	C₂×#803	C₂×C₃	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	C₂×C₄	1	4	yes	—
32	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
33	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
34	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
36	C₃²×C₄²	C₂×C₄	1	2	yes	—
37	C₃²×C₄²	C₂×C₄	1	2	yes	—
38	C₃×C₄×Dic₃	C₂×C₄	1	4	yes	—
39	C₃×C₄×Dic₃	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
41	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
42	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
43	#1726	—	3	1	no	Sylow
44	C₂²×C₃×C₈	—	2	1	no	—
45	C₂²×C₃×C₈	—	2	1	no	—
46	C₂²×C₃×C₈	—	3	2	no	—
47	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	—	3	2	no	—
53	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₄×Dic₃	C₂²×C₃	1	2	yes	—
56	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
57	C₃×#97	—	3	8	no	—
58	C₃×#97	—	3	4	no	—
59	C₃×C₄×C₈	—	2	4	no	—
60	C₃×C₄×C₈	—	3	8	no	—
61	C₃×C₄×C₈	Dic₃	1	4	yes	complemented
62	C₃×C₄×C₈	C₃×C₄	1	4	yes	complemented
63	C₃×#95	—	3	4	no	—
64	C₃⋊#95	C₃×C₄	1	4	yes	—
65	#815	C₃×C₄	1	8	yes	complemented
66	C₈×Dic₃	C₃×C₄	1	8	yes	complemented
67	#803	C₃×C₄	1	4	yes	complemented
68	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃²×C₄	—	2	4	no	—
70	C₂×C₃²×C₄	C₄²	1	4	yes	—
71	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
73	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
74	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×Dic₃	—	2	8	no	—
76	C₂×C₃×Dic₃	C₄²	1	8	yes	—
77	C₃²×C₈	C₄²	1	4	yes	—
78	C₃²×C₈	C₄²	1	4	yes	—
79	C₃×C₃⋊C₈	C₄²	1	8	yes	—
80	C₂×#97	—	3	2	no	—
81	C₂×#97	—	3	1	no	—
82	C₂×C₄×C₈	—	3	2	no	—
83	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
84	#376	—	3	1	no	—
85	C₂²×C₃×C₄	—	2	2	no	—
86	C₂²×C₃×C₄	—	2	1	no	—
87	C₂²×C₃×C₄	—	3	4	no	—
88	C₂²×C₃×C₄	C₄×D₃	1	2	yes	—
89	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
92	C₂²×Dic₃	C₂×C₃×C₄	1	4	yes	—
93	C₂×C₃×C₈	—	2	4	no	—
94	C₂×C₃×C₈	—	2	4	no	—
95	C₂×C₃×C₈	—	2	2	no	—
96	C₂×C₃×C₈	—	2	2	no	—
97	C₂×C₃×C₈	—	3	4	no	—
98	C₂×C₃×C₈	—	3	4	no	—
99	C₂×C₃×C₈	—	3	4	no	—
100	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
101	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
102	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
103	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
104	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
105	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
106	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
108	C₃×C₄²	—	2	2	no	—
109	C₃×C₄²	—	2	2	no	—
110	C₃×C₄²	—	3	4	no	—
111	C₃×C₄²	—	3	4	no	—
112	C₃×C₄²	C₄×D₃	1	2	yes	—
113	C₃×C₄²	C₂×Dic₃	1	2	yes	—
114	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
115	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
116	C₄×Dic₃	C₂×C₃×C₄	1	4	yes	—
117	C₄×Dic₃	C₂×C₃×C₄	1	4	yes	—
118	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
119	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
120	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
121	C₂²×C₃²	C₂×C₄²	1	2	yes	—
122	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
123	C₂²×C₃²	#131	1	4	yes	—
124	C₃²×C₄	—	2	4	no	—
125	C₃²×C₄	C₂×C₄²	1	2	yes	—
126	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
127	C₃²×C₄	C₂×C₄²	1	1	yes	characteristic
128	C₃×Dic₃	—	2	8	no	—
129	C₂²×C₈	—	3	2	no	—
130	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
132	C₂×C₄²	—	3	2	no	—
133	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
134	#97	—	3	8	no	—
135	#97	—	3	4	no	—
136	C₄×C₈	—	3	8	no	—
137	C₄×C₈	C₃×Dic₃	1	4	yes	complemented
138	#95	—	3	4	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂³×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	—	2	8	no	—
143	C₂×C₃×C₄	—	2	4	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	2	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	3	8	no	—
151	C₂×C₃×C₄	—	6	8	no	—
152	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
153	C₂×C₃×C₄	C₃×C₄²	1	4	yes	—
154	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
156	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
157	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
159	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
160	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
161	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×Dic₃	—	2	8	no	—
163	C₂×Dic₃	C₃×C₄²	1	8	yes	—
164	C₃×C₈	—	2	4	no	—
165	C₃×C₈	—	2	4	no	—
166	C₃×C₈	—	3	8	no	—
167	C₃×C₈	C₄×Dic₃	1	4	yes	—
168	C₃×C₈	C₄×Dic₃	1	4	yes	—
169	C₃×C₈	C₃×C₄²	1	4	yes	—
170	C₃×C₈	C₃×C₄²	1	4	yes	—
171	C₃⋊C₈	C₃×C₄²	1	8	yes	—
172	C₂×C₃²	C₄³	1	1	yes	characteristic
173	C₂×C₃²	#405	1	4	yes	—
174	C₂×C₃²	#405	1	2	yes	—
175	C₂²×C₄	—	3	4	no	—
176	C₂²×C₄	C₃×C₄×D₃	1	2	yes	—
177	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
178	C₂×C₈	—	3	4	no	—
179	C₂×C₈	—	3	4	no	—
180	C₂×C₈	—	3	4	no	—
181	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
182	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
183	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
184	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
185	C₄²	—	3	4	no	—
186	C₄²	—	3	4	no	—
187	C₄²	C₃×C₄×D₃	1	2	yes	—
188	C₄²	C₂×C₃×Dic₃	1	2	yes	—
189	C₂²×C₃	—	2	4	no	—
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	2	2	no	—
192	C₂²×C₃	C₄²×D₃	1	2	yes	—
193	C₂²×C₃	#901	1	4	yes	—
194	C₂²×C₃	C₂×C₄×Dic₃	1	1	yes	characteristic
195	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
196	C₂²×C₃	C₂×C₃×C₄²	1	2	yes	—
197	C₂²×C₃	C₃×#131	1	4	yes	—
198	C₃×C₄	—	2	8	no	—
199	C₃×C₄	—	2	4	no	—
200	C₃×C₄	—	2	4	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	2	no	—
204	C₃×C₄	—	6	8	no	—
205	C₃×C₄	C₄²×D₃	1	2	yes	—
206	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
207	C₃×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
208	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
209	C₃×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
210	C₃×C₄	C₂×C₃×C₄²	1	2	yes	—
211	Dic₃	—	2	8	no	—
212	C₃²	#1726	1	1	yes	characteristic, complemented, Sylow
213	C₂³	C₂×C₃×C₄×D₃	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	3	8	no	—
216	C₂×C₄	—	6	8	no	—
217	C₂×C₄	C₃×C₄×Dic₃	1	4	yes	—
218	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
219	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
220	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
222	C₈	—	3	8	no	—
223	C₈	C₃×C₄×Dic₃	1	4	yes	—
224	C₈	C₃×C₄×Dic₃	1	4	yes	—
225	C₂×C₃	—	2	1	no	—
226	C₂×C₃	—	2	4	no	—
227	C₂×C₃	—	2	2	no	—
228	C₂×C₃	#4689	1	4	yes	—
229	C₂×C₃	C₄²×Dic₃	1	1	yes	characteristic
230	C₂×C₃	#5101	1	2	yes	—
231	C₂×C₃	C₃×C₄³	1	1	yes	characteristic
232	C₂×C₃	C₃×#405	1	4	yes	—
233	C₂×C₃	C₃×#405	1	2	yes	—
234	C₂²	C₃×C₄²×D₃	1	2	yes	—
235	C₂²	C₃×#901	1	4	yes	—
236	C₂²	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
237	C₄	—	2	4	no	—
238	C₄	—	6	8	no	—
239	C₄	C₃×C₄²×D₃	1	2	yes	—
240	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
241	C₄	C₂×C₃×C₄×Dic₃	1	1	yes	characteristic
242	C₃	—	2	1	no	—
243	C₃	#69749	1	1	yes	characteristic, direct product factor
244	C₃	C₃×#1726	1	1	yes	characteristic, complemented
245	C₂	C₃×#4689	1	4	yes	—
246	C₂	C₃×C₄²×Dic₃	1	1	yes	characteristic
247	C₂	C₃×#5101	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₈	C₂×D₃
commutator	C₂×C₃	C₃×C₄³
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#1726
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
7. #69749	C₃	direct (N × H)
13. C₃²×C₄×C₈	C₄	semidirect
15. C₃×#815	C₄	semidirect
16. C₃×C₈×Dic₃	C₄	semidirect
17. C₃×#803	C₄	semidirect
61. C₃×C₄×C₈	Dic₃	semidirect
62. C₃×C₄×C₈	C₃×C₄	semidirect
65. #815	C₃×C₄	semidirect
66. C₈×Dic₃	C₃×C₄	semidirect
67. #803	C₃×C₄	semidirect
137. C₄×C₈	C₃×Dic₃	semidirect
212. C₃²	#1726	semidirect
243. C₃	#69749	direct (N × H)
244. C₃	C₃×#1726	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×C₃×#815	2
C₂×C₃×C₈×Dic₃	2
C₃×#4932	2
C₂×C₃×#803	2
C₃×#1726	3
#69749	3

Subgroups — C3×#69749 #11783036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69749 #11783036