Subgroups — C₃×#69076 #11797596

Order 1152.

315 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×#4901	C₂	1	2	yes	maximal, complemented
2	C₃×#4700	C₂	1	1	yes	maximal, characteristic
3	C₃²×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4508	C₂	1	2	yes	maximal, complemented
5	C₃×#4463	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2171	—	3	1	no	maximal
7	#69076	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃×#904	C₄	1	1	yes	characteristic
9	C₃×#901	C₂²	1	2	yes	—
10	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#819	C₄	1	1	yes	characteristic
14	C₃×#812	C₂²	1	2	yes	—
15	C₃×#805	C₄	1	2	yes	—
16	C₃×#798	—	2	2	no	—
17	C₃×D₃×C₁₆	—	2	2	no	—
18	C₃×#504	—	3	2	no	—
19	C₃×#443	—	3	1	no	—
20	#4901	C₂×C₃	1	2	yes	complemented
21	#4700	C₂×C₃	1	1	yes	characteristic
22	C₃×#364	—	3	1	no	—
23	C₃×#350	—	3	2	no	—
24	C₃×#346	—	2	1	no	—
25	C₃×#346	D₃	1	1	yes	characteristic, complemented
26	C₃×#346	C₂×C₃	1	1	yes	characteristic, complemented
27	#4508	C₂×C₃	1	2	yes	complemented
28	#4463	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#235	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
32	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₃×#208	C₂×C₄	1	2	yes	—
34	C₃×Dic₁₂	C₈	1	1	yes	characteristic
35	C₃×D₂₄	C₈	1	1	yes	characteristic
36	C₃×#204	C₈	1	2	yes	—
37	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
38	C₃×D₃×C₈	—	2	2	no	—
39	C₃²×C₁₆	—	2	1	no	—
40	C₃²×C₁₆	D₄	1	2	yes	—
41	C₃×C₃⋊C₁₆	—	2	2	no	—
42	#2171	—	3	1	no	Sylow
43	C₃×#135	—	3	1	no	—
44	C₃×#131	—	3	2	no	—
45	#904	C₃×C₄	1	1	yes	characteristic
46	#901	C₂²×C₃	1	2	yes	—
47	C₃×M₃₂	—	3	2	no	—
48	C₃×M₃₂	—	6	2	no	—
49	C₂×C₃×C₁₆	—	2	1	no	—
50	C₂×C₃×C₁₆	—	2	1	no	—
51	C₂×C₃×C₁₆	—	6	2	no	—
52	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
54	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
55	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
56	C₃×#108	—	3	1	no	—
57	C₃×#104	—	3	2	no	—
58	C₃×C₄×C₈	—	2	1	no	—
59	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	#819	C₃×C₄	1	1	yes	characteristic
62	#812	C₂²×C₃	1	2	yes	—
63	#805	C₃×C₄	1	2	yes	—
64	#798	—	2	2	no	—
65	D₃×C₁₆	—	2	2	no	—
66	C₂×C₃²×C₄	—	2	1	no	—
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₃×#67	—	2	2	no	—
69	C₃×D₁₂	C₂×C₈	1	2	yes	—
70	C₃×C₄×D₃	—	2	2	no	—
71	C₃×Dic₆	C₂×C₈	1	2	yes	—
72	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
73	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
74	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
75	C₃²×C₈	#41	1	1	yes	characteristic
76	C₃×C₃⋊C₈	—	2	2	no	—
77	#504	—	3	2	no	—
78	#443	—	3	1	no	—
79	#364	—	3	1	no	—
80	#350	—	3	2	no	—
81	#346	C₃×D₃	1	1	yes	characteristic, complemented
82	C₃×#41	—	3	2	no	—
83	#235	C₂×C₃×C₄	1	2	yes	—
84	C₃×Q₁₆	—	3	1	no	—
85	C₃×SD₁₆	—	3	2	no	—
86	C₃×D₈	—	3	1	no	—
87	C₃×M₁₆	—	3	2	no	—
88	C₃×M₁₆	—	6	2	no	—
89	C₂×C₃×C₈	—	2	1	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	6	2	no	—
92	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
96	C₃×C₄²	—	2	1	no	—
97	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
98	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
99	#208	C₂×C₃×C₄	1	2	yes	—
100	Dic₁₂	C₃×C₈	1	1	yes	characteristic
101	D₂₄	C₃×C₈	1	1	yes	characteristic
102	#204	C₃×C₈	1	2	yes	—
103	ΓL(1,𝔽₂₅)	—	2	2	no	—
104	D₃×C₈	—	2	2	no	—
105	C₃×C₁₆	—	2	2	no	—
106	C₃×C₁₆	—	2	1	no	—
107	C₃×C₁₆	—	2	1	no	—
108	C₃×C₁₆	—	2	2	no	—
109	C₃×C₁₆	—	6	2	no	—
110	C₃×C₁₆	D₁₂	1	2	yes	—
111	C₃×C₁₆	C₃×D₄	1	2	yes	—
112	C₃⋊C₁₆	—	2	2	no	—
113	C₂²×C₃²	#131	1	1	yes	characteristic
114	C₂×C₃×D₃	—	2	2	no	—
115	C₃²×C₄	—	4	1	no	—
116	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
117	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
118	C₃×Dic₃	—	2	2	no	—
119	#135	—	3	1	no	—
120	#131	—	3	2	no	—
121	M₃₂	—	3	2	no	—
122	M₃₂	—	6	2	no	—
123	C₂×C₁₆	—	6	2	no	—
124	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
126	#108	—	3	1	no	—
127	#104	—	3	2	no	—
128	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
129	C₃×Q₈	—	3	2	no	—
130	C₃×D₄	—	3	2	no	—
131	C₃×D₄	—	6	2	no	—
132	C₂×C₃×C₄	—	2	2	no	—
133	C₂×C₃×C₄	—	2	1	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	6	2	no	—
137	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
139	#67	—	2	2	no	—
140	D₁₂	C₂×C₃×C₈	1	2	yes	—
141	C₄×D₃	—	2	2	no	—
142	Dic₆	C₂×C₃×C₈	1	2	yes	—
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	—	2	1	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	—	2	1	no	—
147	C₃×C₈	—	6	2	no	—
148	C₃×C₈	D₃×C₈	1	1	yes	characteristic
149	C₃×C₈	D₃×C₈	1	1	yes	characteristic
150	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
151	C₃×C₈	C₂×C₃×C₈	1	1	yes	characteristic
152	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
153	C₃×C₈	#235	1	1	yes	characteristic
154	C₃×C₈	C₂×C₃×D₄	1	1	yes	characteristic
155	C₃×C₈	C₃×#41	1	1	yes	characteristic
156	C₃⋊C₈	—	2	2	no	—
157	C₂×C₃²	—	2	1	no	—
158	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
159	C₃×D₃	—	4	2	no	—
160	#41	—	3	2	no	—
161	Q₁₆	—	3	1	no	—
162	SD₁₆	—	3	2	no	—
163	D₈	—	3	1	no	—
164	M₁₆	—	3	2	no	—
165	M₁₆	—	6	2	no	—
166	C₂×C₈	—	6	2	no	—
167	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
168	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
169	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
170	C₁₆	—	2	1	no	—
171	C₁₆	—	6	2	no	—
172	C₁₆	C₃×D₁₂	1	2	yes	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	6	2	no	—
175	C₂²×C₃	#901	1	1	yes	characteristic
176	C₂²×C₃	C₃×#131	1	1	yes	characteristic
177	C₂×D₃	—	2	2	no	—
178	C₃×C₄	—	2	1	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	4	2	no	—
181	C₃×C₄	—	4	1	no	—
182	C₃×C₄	—	4	1	no	—
183	C₃×C₄	—	6	2	no	—
184	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
185	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
186	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
187	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
188	Dic₃	—	2	2	no	—
189	C₃²	#2171	1	1	yes	characteristic, complemented, Sylow
190	Q₈	—	3	2	no	—
191	D₄	—	3	2	no	—
192	D₄	—	6	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	6	2	no	—
195	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
196	C₈	—	6	2	no	—
197	C₈	C₃×D₃×C₈	1	1	yes	characteristic
198	C₈	C₃×D₃×C₈	1	1	yes	characteristic
199	C₈	C₂×C₃×D₁₂	1	1	yes	characteristic
200	C₈	C₃×#235	1	1	yes	characteristic
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	1	no	—
204	C₂×C₃	—	2	1	no	—
205	C₂×C₃	—	12	2	no	—
206	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
207	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
208	D₃	—	4	2	no	—
209	C₂²	—	6	2	no	—
210	C₂²	C₃×#901	1	1	yes	characteristic
211	C₄	—	4	1	no	—
212	C₄	—	6	2	no	—
213	C₄	C₃×C₄×D₁₂	1	1	yes	characteristic
214	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
215	C₃	—	2	1	no	—
216	C₃	#69076	1	1	yes	characteristic, direct product factor
217	C₃	C₃×#2171	1	1	yes	characteristic, complemented
218	C₂	—	2	1	no	—
219	C₂	—	12	2	no	—
220	C₂	C₃×C₈×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×#346	C₂
socle	C₂×C₃²	C₈×D₄
prime radical	C₃×#904	C₄
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#346	D₃
hypocenter	C₃	C₃×#2171
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₁₆ ◁ C₃×#346

Complements

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

N	H ≅ G/N	type
1. C₃×#4901	C₂	semidirect
3. C₃²×#346	C₂	semidirect
4. C₃×#4508	C₂	semidirect
5. C₃×#4463	C₂	semidirect
7. #69076	C₃	direct (N × H)
20. #4901	C₂×C₃	semidirect
25. C₃×#346	D₃	semidirect
26. C₃×#346	C₂×C₃	semidirect
27. #4508	C₂×C₃	semidirect
28. #4463	C₂×C₃	semidirect
81. #346	C₃×D₃	semidirect
189. C₃²	#2171	semidirect
216. C₃	#69076	direct (N × H)
217. C₃	C₃×#2171	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4901	2
C₃×#4700	2
C₃²×#346	2
C₃×#4508	2
C₃×#4463	2
C₃×#2171	3
#69076	3

Subgroups — C3×#69076 #11797596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69076 #11797596