Subgroups — C₃×#68047 #11773559

Order 1152.

461 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4685	C₂	1	2	yes	maximal
2	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4454	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1450	—	3	1	no	maximal
7	#68047	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃×C₈	C₄	1	2	yes	—
9	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
10	C₃²×C₄×C₈	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×(C₂×D₃)⋊C₈	C₄	1	4	yes	—
13	C₃×Dic₃⋊C₈	C₄	1	4	yes	—
14	C₃×C₈×Dic₃	C₄	1	2	yes	—
15	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×#431	—	3	2	no	—
18	C₂×C₃×C₄×C₈	—	3	1	no	—
19	#4685	C₂×C₃	1	2	yes	—
20	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
21	C₃×#322	—	3	2	no	—
22	C₃×C₈²	—	2	1	no	—
23	C₃×C₈²	—	3	1	no	—
24	C₃×C₈²	D₃	1	1	yes	characteristic, complemented
25	C₃×C₈²	C₂×C₃	1	1	yes	characteristic, complemented
26	#4454	C₂×C₃	1	2	yes	complemented
27	C₈×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	C₃²×C₄²	C₂³	1	1	yes	characteristic
33	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
37	C₃×D₃×C₈	C₈	1	8	yes	complemented
38	#1450	—	3	1	no	Sylow
39	C₂²×C₃×C₈	—	3	2	no	—
40	C₂×C₃×C₄²	—	3	1	no	—
41	C₂×D₃×C₈	C₃×C₄	1	2	yes	—
42	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
43	C₃×C₄⋊C₈	—	3	4	no	—
44	C₃×C₂²⋊C₈	—	3	4	no	—
45	C₃×C₄×C₈	—	2	1	no	—
46	C₃×C₄×C₈	—	2	2	no	—
47	C₃×C₄×C₈	—	3	2	no	—
48	C₃×C₄×C₈	—	3	1	no	—
49	C₃×C₄×C₈	—	3	2	no	—
50	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
51	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
52	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
54	(C₂×D₃)⋊C₈	C₃×C₄	1	4	yes	—
55	Dic₃⋊C₈	C₃×C₄	1	4	yes	—
56	C₈×Dic₃	C₃×C₄	1	2	yes	—
57	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
58	C₄×C₃⋊C₈	C₂²×C₃	1	2	yes	—
59	C₂²×C₃×D₃	C₄²	1	1	yes	characteristic
60	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
62	C₂×C₃×Dic₃	C₄²	1	2	yes	—
63	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
64	C₃×C₄×D₃	—	2	4	no	—
65	C₃×C₄×D₃	C₂×C₈	1	4	yes	—
66	C₃²×C₈	—	2	4	no	—
67	C₃²×C₈	C₂×C₈	1	4	yes	—
68	C₃×C₃⋊C₈	—	2	4	no	—
69	C₃×C₃⋊C₈	C₂×C₈	1	4	yes	—
70	#431	—	3	2	no	—
71	C₂×C₄×C₈	—	3	1	no	—
72	#322	—	3	2	no	—
73	C₈²	—	3	1	no	—
74	C₈²	C₃×D₃	1	1	yes	characteristic, complemented
75	C₂²×C₃×C₄	—	3	1	no	—
76	C₂²×C₃×C₄	—	3	2	no	—
77	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
78	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₃×C₈	—	2	4	no	—
80	C₂×C₃×C₈	—	2	2	no	—
81	C₂×C₃×C₈	—	3	2	no	—
82	C₂×C₃×C₈	—	3	8	no	—
83	C₂×C₃×C₈	—	3	4	no	—
84	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
85	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
86	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
88	C₃×C₄²	—	2	1	no	—
89	C₃×C₄²	—	3	1	no	—
90	C₃×C₄²	—	3	2	no	—
91	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
92	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
93	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
94	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
96	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
97	D₃×C₈	C₃×C₈	1	8	yes	complemented
98	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
99	C₂×C₃×D₃	—	2	2	no	—
100	C₂×C₃×D₃	C₄×C₈	1	2	yes	—
101	C₃²×C₄	C₂²×C₈	1	2	yes	—
102	C₃²×C₄	#131	1	4	yes	—
103	C₃×Dic₃	—	2	2	no	—
104	C₃×Dic₃	C₄×C₈	1	2	yes	—
105	C₂²×C₈	—	3	2	no	—
106	C₂×C₄²	—	3	1	no	—
107	C₄⋊C₈	—	3	4	no	—
108	C₂²⋊C₈	—	3	4	no	—
109	C₄×C₈	—	3	2	no	—
110	C₄×C₈	—	3	2	no	—
111	C₄×C₈	—	3	1	no	—
112	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
113	C₄×C₈	C₂×C₃×D₃	1	2	yes	—
114	C₂³×C₃	—	3	1	no	—
115	C₂²×D₃	C₃×C₄²	1	1	yes	characteristic
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	2	no	—
118	C₂×C₃×C₄	—	3	1	no	—
119	C₂×C₃×C₄	—	3	4	no	—
120	C₂×C₃×C₄	—	3	2	no	—
121	C₂×C₃×C₄	—	6	4	no	—
122	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
124	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
126	C₂×Dic₃	C₃×C₄²	1	2	yes	—
127	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
128	C₄×D₃	—	2	4	no	—
129	C₄×D₃	C₂×C₃×C₈	1	4	yes	—
130	C₃×C₈	—	2	8	no	—
131	C₃×C₈	—	2	4	no	—
132	C₃×C₈	—	2	4	no	—
133	C₃×C₈	—	2	4	no	—
134	C₃×C₈	—	3	4	no	—
135	C₃×C₈	—	6	4	no	—
136	C₃×C₈	D₃×C₈	1	4	yes	—
137	C₃×C₈	C₂×C₃×C₈	1	4	yes	—
138	C₃⋊C₈	—	2	4	no	—
139	C₃⋊C₈	C₂×C₃×C₈	1	4	yes	—
140	C₂×C₃²	C₂×C₄×C₈	1	1	yes	characteristic
141	C₂×C₃²	#405	1	2	yes	—
142	C₃×D₃	—	2	2	no	—
143	C₂²×C₄	—	3	1	no	—
144	C₂²×C₄	—	3	2	no	—
145	C₂×C₈	—	3	2	no	—
146	C₂×C₈	—	3	8	no	—
147	C₂×C₈	—	3	4	no	—
148	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
149	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
150	C₄²	—	3	1	no	—
151	C₄²	—	3	2	no	—
152	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
153	C₂²×C₃	—	2	1	no	—
154	C₂²×C₃	—	3	2	no	—
155	C₂²×C₃	—	6	2	no	—
156	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
157	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
158	C₂×D₃	—	2	2	no	—
159	C₂×D₃	C₃×C₄×C₈	1	2	yes	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	—	2	4	no	—
162	C₃×C₄	—	3	2	no	—
163	C₃×C₄	—	6	2	no	—
164	C₃×C₄	C₂×D₃×C₈	1	2	yes	—
165	C₃×C₄	#901	1	4	yes	—
166	C₃×C₄	C₂²×C₃×C₈	1	2	yes	—
167	C₃×C₄	C₃×#131	1	4	yes	—
168	Dic₃	—	2	2	no	—
169	Dic₃	C₃×C₄×C₈	1	2	yes	—
170	C₃²	#1450	1	1	yes	characteristic, complemented, Sylow
171	C₂³	—	3	1	no	—
172	C₂×C₄	—	3	1	no	—
173	C₂×C₄	—	3	4	no	—
174	C₂×C₄	—	3	2	no	—
175	C₂×C₄	—	6	4	no	—
176	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
177	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
178	C₈	—	2	4	no	—
179	C₈	—	3	4	no	—
180	C₈	—	6	4	no	—
181	C₈	C₃×D₃×C₈	1	4	yes	—
182	C₂×C₃	—	2	1	no	—
183	C₂×C₃	—	2	2	no	—
184	C₂×C₃	—	6	2	no	—
185	C₂×C₃	C₄×D₃×C₈	1	1	yes	characteristic
186	C₂×C₃	#4689	1	2	yes	—
187	C₂×C₃	C₂×C₃×C₄×C₈	1	1	yes	characteristic
188	C₂×C₃	C₃×#405	1	2	yes	—
189	D₃	—	2	2	no	—
190	C₂²	—	3	2	no	—
191	C₂²	—	6	2	no	—
192	C₂²	C₃×C₄²×D₃	1	1	yes	characteristic
193	C₄	—	3	2	no	—
194	C₄	—	6	2	no	—
195	C₄	C₂×C₃×D₃×C₈	1	2	yes	—
196	C₄	C₃×#901	1	4	yes	—
197	C₃	—	2	1	no	—
198	C₃	#68047	1	1	yes	characteristic, direct product factor
199	C₃	C₃×#1450	1	1	yes	characteristic, complemented
200	C₂	—	6	2	no	—
201	C₂	C₃×C₄×D₃×C₈	1	1	yes	characteristic
202	C₂	C₃×#4689	1	2	yes	—

	subgroup	G/H
center	C₃×C₄×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₃×C₄×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×C₈²	C₂
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂²×C₃×D₃	C₄²
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×C₈²	D₃
hypocenter	C₃	C₃×#1450
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄×C₈ ◁ C₃×C₈²

Complements

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

N	H ≅ G/N	type
3. C₃²×C₈²	C₂	semidirect
4. C₃×#4454	C₂	semidirect
5. C₃×C₈×C₃⋊C₈	C₂	semidirect
7. #68047	C₃	direct (N × H)
24. C₃×C₈²	D₃	semidirect
25. C₃×C₈²	C₂×C₃	semidirect
26. #4454	C₂×C₃	semidirect
27. C₈×C₃⋊C₈	C₂×C₃	semidirect
37. C₃×D₃×C₈	C₈	semidirect
74. C₈²	C₃×D₃	semidirect
97. D₃×C₈	C₃×C₈	semidirect
170. C₃²	#1450	semidirect
198. C₃	#68047	direct (N × H)
199. C₃	C₃×#1450	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4685	2
C₃×C₄×D₃×C₈	2
C₃²×C₈²	2
C₃×#4454	2
C₃×C₈×C₃⋊C₈	2
C₃×#1450	3
#68047	3

Subgroups — C3×#68047 #11773559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68047 #11773559