Subgroups — C₃×#68517 #11777305

Order 1152.

347 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4830	C₂	1	2	yes	maximal
2	C₃×C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic
3	C₃²×#334	C₂	1	1	yes	maximal, characteristic
4	C₃×#4484	C₂	1	2	yes	maximal
5	C₃×#4455	C₂	1	1	yes	maximal, characteristic
6	C₃×#1579	—	3	1	no	maximal
7	#68517	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×Dic₁₂	C₄	1	1	yes	characteristic
9	C₃×C₄×Dic₆	C₂²	1	2	yes	—
10	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×#818	C₄	1	1	yes	characteristic
13	C₃×#816	C₄	1	2	yes	—
14	C₃×Dic₃⋊C₈	—	2	2	no	—
15	C₃×C₈×Dic₃	—	2	2	no	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₈×Q₈	—	3	2	no	—
19	C₃×C₄×Q₁₆	—	3	1	no	—
20	#4830	C₂×C₃	1	2	yes	—
21	C₄×Dic₁₂	C₂×C₃	1	1	yes	characteristic
22	C₃×#334	—	2	1	no	—
23	C₃×#334	D₃	1	1	yes	characteristic
24	C₃×#334	C₂×C₃	1	1	yes	characteristic
25	C₃×Q₈⋊C₈	—	3	2	no	—
26	C₃×#322	—	3	1	no	—
27	#4484	C₂×C₃	1	2	yes	—
28	#4455	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	—	2	2	no	—
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₃²×C₄²	C₂³	1	1	yes	characteristic
34	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
35	C₃×Dic₃⋊C₄	—	2	2	no	—
36	C₃×C₄×Dic₃	—	2	2	no	—
37	C₂×C₃×C₃⋊C₈	—	2	1	no	—
38	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
39	C₃×Dic₁₂	C₈	1	4	yes	complemented
40	#1579	—	3	1	no	Sylow
41	C₂×C₃×Q₁₆	—	3	1	no	—
42	C₃×C₄×Q₈	—	3	2	no	—
43	C₂×Dic₁₂	C₃×C₄	1	1	yes	characteristic
44	C₄×Dic₆	C₂²×C₃	1	2	yes	—
45	C₃×#107	—	3	1	no	—
46	C₃×C₄⋊C₈	—	2	2	no	—
47	C₃×C₄⋊C₈	—	6	2	no	—
48	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
49	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
50	C₃×#103	—	3	2	no	—
51	C₃×C₄×C₈	—	2	1	no	—
52	C₃×C₄×C₈	—	3	1	no	—
53	C₃×C₄×C₈	—	3	1	no	—
54	C₃×C₄×C₈	—	6	2	no	—
55	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
57	#818	C₃×C₄	1	1	yes	characteristic
58	#816	C₃×C₄	1	2	yes	—
59	Dic₃⋊C₈	—	2	2	no	—
60	C₈×Dic₃	—	2	2	no	—
61	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
62	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃²×C₄	#41	1	1	yes	characteristic
66	C₂×C₃×Dic₃	—	2	2	no	—
67	C₂×C₃×Dic₃	—	2	2	no	—
68	C₃×Dic₆	—	2	2	no	—
69	C₃×Dic₆	C₂×C₈	1	4	yes	—
70	C₃²×C₈	—	2	1	no	—
71	C₃²×C₈	—	4	2	no	—
72	C₃²×C₈	C₂×C₈	1	2	yes	—
73	C₃×C₃⋊C₈	—	2	2	no	—
74	C₃×C₃⋊C₈	—	4	1	no	—
75	C₈×Q₈	—	3	2	no	—
76	C₄×Q₁₆	—	3	1	no	—
77	#334	C₃×D₃	1	1	yes	characteristic
78	Q₈⋊C₈	—	3	2	no	—
79	#322	—	3	1	no	—
80	C₂×C₃×Q₈	—	3	2	no	—
81	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
82	C₃×Q₁₆	—	3	4	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	—	2	2	no	—
85	C₂×C₃×C₈	—	2	1	no	—
86	C₂×C₃×C₈	—	2	1	no	—
87	C₂×C₃×C₈	—	3	2	no	—
88	C₂×C₃×C₈	—	4	2	no	—
89	C₂×C₃×C₈	—	6	1	no	—
90	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
91	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
92	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
94	C₃×C₄⋊C₄	—	3	2	no	—
95	C₃×C₄⋊C₄	—	6	2	no	—
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	—	6	2	no	—
98	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
99	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
100	C₄⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
101	Dic₃⋊C₄	—	2	2	no	—
102	C₄×Dic₃	—	2	2	no	—
103	C₂×C₃⋊C₈	—	2	1	no	—
104	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
105	Dic₁₂	C₃×C₈	1	4	yes	complemented
106	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
107	C₃²×C₄	—	2	1	no	—
108	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
109	C₃²×C₄	#131	1	1	yes	characteristic
110	C₃²×C₄	#137	1	2	yes	—
111	C₃×Dic₃	—	2	4	no	—
112	C₃×Dic₃	—	4	2	no	—
113	C₂×Q₁₆	—	3	1	no	—
114	C₄×Q₈	—	3	2	no	—
115	#107	—	3	1	no	—
116	C₄⋊C₈	—	6	2	no	—
117	C₄⋊C₈	C₂×C₃×D₃	1	2	yes	—
118	#103	—	3	2	no	—
119	C₄×C₈	—	3	1	no	—
120	C₄×C₈	—	3	1	no	—
121	C₄×C₈	—	6	2	no	—
122	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₃×Q₈	—	3	4	no	—
124	C₃×Q₈	—	6	2	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	6	2	no	—
129	C₂×C₃×C₄	—	6	2	no	—
130	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
131	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
132	C₂×C₃×C₄	#239	1	1	yes	characteristic
133	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
136	C₂×Dic₃	—	2	2	no	—
137	C₂×Dic₃	—	2	2	no	—
138	Dic₆	—	2	2	no	—
139	Dic₆	C₂×C₃×C₈	1	4	yes	—
140	C₃×C₈	—	2	1	no	—
141	C₃×C₈	—	2	1	no	—
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	4	1	no	—
144	C₃×C₈	—	4	2	no	—
145	C₃×C₈	—	4	2	no	—
146	C₃×C₈	—	6	2	no	—
147	C₃×C₈	—	8	2	no	—
148	C₃×C₈	—	12	1	no	—
149	C₃×C₈	D₃×C₈	1	2	yes	—
150	C₃×C₈	C₂×C₃×C₈	1	2	yes	—
151	C₃⋊C₈	—	2	2	no	—
152	C₃⋊C₈	—	4	1	no	—
153	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
154	C₂×C₃²	#441	1	1	yes	characteristic
155	C₂×C₃²	#444	1	1	yes	characteristic
156	C₂×Q₈	—	3	2	no	—
157	Q₁₆	—	3	4	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	3	2	no	—
160	C₂×C₈	—	6	1	no	—
161	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
162	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
163	C₄⋊C₄	—	3	2	no	—
164	C₄⋊C₄	—	6	2	no	—
165	C₄²	—	6	2	no	—
166	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	#893	1	1	yes	characteristic
169	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
170	C₃×C₄	—	2	1	no	—
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	2	1	no	—
173	C₃×C₄	—	2	1	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	4	1	no	—
176	C₃×C₄	—	6	4	no	—
177	C₃×C₄	—	12	2	no	—
178	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
179	C₃×C₄	#907	1	1	yes	characteristic
180	C₃×C₄	#915	1	2	yes	—
181	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
182	C₃×C₄	C₃×#131	1	1	yes	characteristic
183	C₃×C₄	C₃×#137	1	2	yes	—
184	Dic₃	—	2	4	no	—
185	Dic₃	—	4	2	no	—
186	C₃²	#1579	1	1	yes	characteristic, complemented, Sylow
187	Q₈	—	3	4	no	—
188	Q₈	—	6	2	no	—
189	C₂×C₄	—	6	2	no	—
190	C₂×C₄	—	6	2	no	—
191	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
192	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
193	C₂×C₄	C₃×#239	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	4	2	no	—
196	C₈	—	6	2	no	—
197	C₈	—	12	1	no	—
198	C₈	C₃×D₃×C₈	1	2	yes	—
199	C₂×C₃	—	2	1	no	—
200	C₂×C₃	—	2	1	no	—
201	C₂×C₃	—	2	1	no	—
202	C₂×C₃	#4834	1	1	yes	characteristic
203	C₂×C₃	#4853	1	1	yes	characteristic
204	C₂×C₃	#4894	1	1	yes	characteristic
205	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
206	C₂×C₃	C₃×#441	1	1	yes	characteristic
207	C₂×C₃	C₃×#444	1	1	yes	characteristic
208	C₂²	C₃×#893	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	6	4	no	—
211	C₄	—	12	2	no	—
212	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
213	C₄	C₃×#907	1	1	yes	characteristic
214	C₄	C₃×#915	1	2	yes	—
215	C₃	—	2	1	no	—
216	C₃	#68517	1	1	yes	characteristic, direct product factor
217	C₃	C₃×#1579	1	1	yes	characteristic, complemented
218	C₂	C₃×#4834	1	1	yes	characteristic
219	C₂	C₃×#4853	1	1	yes	characteristic
220	C₂	C₃×#4894	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#334	C₂
socle, prime radical	C₂²×C₃²	C₄×D₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#334	D₃
hypocenter	C₃	C₃×#1579
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#334

Complements

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

N	H ≅ G/N	type
7. #68517	C₃	direct (N × H)
39. C₃×Dic₁₂	C₈	semidirect
105. Dic₁₂	C₃×C₈	semidirect
186. C₃²	#1579	semidirect
216. C₃	#68517	direct (N × H)
217. C₃	C₃×#1579	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4830	2
C₃×C₄×Dic₁₂	2
C₃²×#334	2
C₃×#4484	2
C₃×#4455	2
C₃×#1579	3
#68517	3

Subgroups — C3×#68517 #11777305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68517 #11777305