Subgroups — C₃×#71511 #11797842

Order 1152.

334 conjugacy classes of subgroups.

The subgroup lattice has 229 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#508	C₂	1	1	yes	maximal, characteristic
2	C₃×#5196	C₂	1	2	yes	maximal, complemented
3	C₃×#4567	C₂	1	2	yes	maximal, complemented
4	C₃×#4506	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2178	—	3	1	no	maximal
7	#71511	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃²×#135	C₂²	1	2	yes	—
9	C₃×#948	—	2	2	no	—
10	C₃²×Q₃₂	C₄	1	1	yes	characteristic, complemented
11	C₃²×SD₃₂	C₄	1	2	yes	complemented
12	C₃²×D₁₆	C₄	1	1	yes	characteristic, complemented
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×#837	—	2	2	no	—
15	C₃×#819	C₂²	1	2	yes	—
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₃×#508	—	2	1	no	—
19	C₃×#508	D₃	1	1	yes	characteristic
20	C₃×#508	C₂×C₃	1	1	yes	characteristic
21	C₃×#443	—	3	2	no	—
22	#5196	C₂×C₃	1	2	yes	complemented
23	C₃×#368	—	3	1	no	—
24	C₃×#359	—	3	2	no	—
25	C₃×C₄×C₁₆	—	3	1	no	—
26	#4567	C₂×C₃	1	2	yes	complemented
27	#4506	C₂×C₃	1	1	yes	characteristic, complemented
28	Dic₃×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃²×#41	—	2	2	no	—
30	C₃²×Q₁₆	C₂×C₄	1	2	yes	—
31	C₃²×SD₁₆	—	2	2	no	—
32	C₃²×D₈	C₂×C₄	1	2	yes	—
33	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
34	C₃×C₄×Dic₃	D₄	1	1	yes	characteristic
35	C₃×#208	—	2	2	no	—
36	C₃×#208	—	4	2	no	—
37	C₂×C₃×C₃⋊C₈	—	4	2	no	—
38	C₂×C₃×C₃⋊C₈	D₄	1	1	yes	characteristic
39	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
40	C₃²×C₁₆	C₂×C₄	1	1	yes	characteristic
41	C₃×C₃⋊C₁₆	—	2	1	no	—
42	#2178	—	3	1	no	Sylow
43	C₃×#135	—	2	2	no	—
44	C₃×#135	C₂×D₃	1	2	yes	—
45	C₃×#135	C₂²×C₃	1	2	yes	—
46	C₃×#131	—	6	2	no	—
47	#948	—	2	2	no	—
48	C₃×Q₃₂	—	2	1	no	—
49	C₃×Q₃₂	Dic₃	1	1	yes	characteristic, complemented
50	C₃×Q₃₂	C₃×C₄	1	1	yes	characteristic, complemented
51	C₃×SD₃₂	—	2	2	no	—
52	C₃×SD₃₂	Dic₃	1	2	yes	complemented
53	C₃×SD₃₂	C₃×C₄	1	2	yes	complemented
54	C₃×D₁₆	—	2	1	no	—
55	C₃×D₁₆	Dic₃	1	1	yes	characteristic, complemented
56	C₃×D₁₆	C₃×C₄	1	1	yes	characteristic, complemented
57	C₂×C₃×C₁₆	—	2	1	no	—
58	C₂×C₃×C₁₆	—	3	1	no	—
59	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
60	C₂×C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
61	C₃×#108	—	3	2	no	—
62	C₃×#104	—	6	2	no	—
63	C₃×C₄×C₈	—	3	1	no	—
64	#837	—	2	2	no	—
65	#819	C₂²×C₃	1	2	yes	—
66	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic
67	C₂×C₃⋊C₁₆	C₂²×C₃	1	1	yes	characteristic
68	C₃²×Q₈	—	2	2	no	—
69	C₃²×D₄	—	2	2	no	—
70	C₃²×D₄	—	4	2	no	—
71	C₂×C₃²×C₄	—	4	2	no	—
72	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×Dic₃	—	2	1	no	—
74	C₃²×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₃²×C₈	#41	1	1	yes	characteristic
76	C₃×C₃⋊C₈	—	4	2	no	—
77	C₃×C₃⋊C₈	D₈	1	1	yes	characteristic
78	C₃×C₃⋊C₈	D₈	1	1	yes	characteristic
79	#508	C₃×D₃	1	1	yes	characteristic
80	#443	—	3	2	no	—
81	#368	—	3	1	no	—
82	#359	—	3	2	no	—
83	C₄×C₁₆	—	3	1	no	—
84	C₃×#41	—	2	2	no	—
85	C₃×#41	—	2	2	no	—
86	C₃×#41	—	4	2	no	—
87	C₃×Q₁₆	—	2	2	no	—
88	C₃×Q₁₆	C₂×Dic₃	1	2	yes	—
89	C₃×Q₁₆	C₂×C₃×C₄	1	2	yes	—
90	C₃×SD₁₆	—	2	2	no	—
91	C₃×SD₁₆	—	2	2	no	—
92	C₃×SD₁₆	—	4	2	no	—
93	C₃×D₈	—	2	2	no	—
94	C₃×D₈	C₂×Dic₃	1	2	yes	—
95	C₃×D₈	C₂×C₃×C₄	1	2	yes	—
96	C₃×M₁₆	—	6	2	no	—
97	C₃×M₁₆	—	12	2	no	—
98	C₂×C₃×C₈	—	2	1	no	—
99	C₂×C₃×C₈	—	3	1	no	—
100	C₂×C₃×C₈	—	12	2	no	—
101	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
102	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
103	C₃×C₄²	—	3	1	no	—
104	C₄×Dic₃	C₃×D₄	1	1	yes	characteristic
105	#208	—	2	2	no	—
106	#208	—	4	2	no	—
107	C₂×C₃⋊C₈	—	4	2	no	—
108	C₂×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
109	C₃×C₁₆	—	2	1	no	—
110	C₃×C₁₆	—	2	1	no	—
111	C₃×C₁₆	—	6	1	no	—
112	C₃×C₁₆	C₂×Dic₃	1	1	yes	characteristic
113	C₃×C₁₆	C₂×Dic₃	1	1	yes	characteristic
114	C₃×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
115	C₃×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
116	C₃⋊C₁₆	—	2	1	no	—
117	C₂²×C₃²	—	4	2	no	—
118	C₂²×C₃²	#135	1	1	yes	characteristic
119	C₃²×C₄	—	4	2	no	—
120	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
121	C₃²×C₄	C₂×D₈	1	1	yes	characteristic
122	C₃×Dic₃	—	2	2	no	—
123	#135	C₂×C₃×D₃	1	2	yes	—
124	#131	—	6	2	no	—
125	Q₃₂	C₃×Dic₃	1	1	yes	characteristic, complemented
126	SD₃₂	C₃×Dic₃	1	2	yes	complemented
127	D₁₆	C₃×Dic₃	1	1	yes	characteristic, complemented
128	C₂×C₁₆	—	3	1	no	—
129	C₂×C₁₆	C₂×C₃×D₃	1	1	yes	characteristic
130	#108	—	3	2	no	—
131	#104	—	6	2	no	—
132	C₄×C₈	—	3	1	no	—
133	C₃×Q₈	—	2	2	no	—
134	C₃×Q₈	—	2	2	no	—
135	C₃×Q₈	—	4	2	no	—
136	C₃×D₄	—	2	2	no	—
137	C₃×D₄	—	2	2	no	—
138	C₃×D₄	—	4	2	no	—
139	C₃×D₄	—	4	2	no	—
140	C₃×D₄	—	4	2	no	—
141	C₃×D₄	—	8	2	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	4	2	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	6	1	no	—
146	C₂×C₃×C₄	—	8	2	no	—
147	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
149	C₂×Dic₃	—	2	1	no	—
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	3	1	no	—
153	C₃×C₈	—	3	1	no	—
154	C₃×C₈	—	12	2	no	—
155	C₃×C₈	#237	1	1	yes	characteristic
156	C₃×C₈	C₂²×Dic₃	1	1	yes	characteristic
157	C₃×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₃×C₈	C₃×#41	1	1	yes	characteristic
159	C₃⋊C₈	—	4	2	no	—
160	C₃⋊C₈	C₃×D₈	1	1	yes	characteristic
161	C₃⋊C₈	C₃×D₈	1	1	yes	characteristic
162	C₂×C₃²	—	2	1	no	—
163	C₂×C₃²	—	8	2	no	—
164	C₂×C₃²	C₄×D₈	1	1	yes	characteristic
165	#41	—	2	2	no	—
166	Q₁₆	C₂×C₃×Dic₃	1	2	yes	—
167	SD₁₆	—	2	2	no	—
168	D₈	C₂×C₃×Dic₃	1	2	yes	—
169	M₁₆	—	6	2	no	—
170	M₁₆	—	12	2	no	—
171	C₂×C₈	—	3	1	no	—
172	C₂×C₈	—	12	2	no	—
173	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
174	C₄²	—	3	1	no	—
175	C₁₆	—	6	1	no	—
176	C₁₆	C₂×C₃×Dic₃	1	1	yes	characteristic
177	C₁₆	C₂×C₃×Dic₃	1	1	yes	characteristic
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	—	4	2	no	—
181	C₂²×C₃	—	8	2	no	—
182	C₂²×C₃	#912	1	1	yes	characteristic
183	C₂²×C₃	C₃×#135	1	1	yes	characteristic
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	2	1	no	—
186	C₃×C₄	—	4	2	no	—
187	C₃×C₄	—	4	2	no	—
188	C₃×C₄	—	6	2	no	—
189	C₃×C₄	—	8	2	no	—
190	C₃×C₄	D₃×D₈	1	1	yes	characteristic
191	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
192	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
193	C₃×C₄	C₂×C₃×D₈	1	1	yes	characteristic
194	Dic₃	—	2	2	no	—
195	C₃²	#2178	1	1	yes	characteristic, complemented, Sylow
196	Q₈	—	2	2	no	—
197	D₄	—	2	2	no	—
198	D₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	6	1	no	—
201	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
202	C₈	—	3	1	no	—
203	C₈	—	3	1	no	—
204	C₈	—	12	2	no	—
205	C₈	C₃×#237	1	1	yes	characteristic
206	C₈	C₂²×C₃×Dic₃	1	1	yes	characteristic
207	C₂×C₃	—	2	1	no	—
208	C₂×C₃	—	2	1	no	—
209	C₂×C₃	—	2	1	no	—
210	C₂×C₃	—	4	1	no	—
211	C₂×C₃	—	8	2	no	—
212	C₂×C₃	—	8	2	no	—
213	C₂×C₃	—	16	2	no	—
214	C₂×C₃	Dic₃×D₈	1	1	yes	characteristic
215	C₂×C₃	C₃×C₄×D₈	1	1	yes	characteristic
216	C₂²	—	4	2	no	—
217	C₂²	C₃×#912	1	1	yes	characteristic
218	C₄	—	4	2	no	—
219	C₄	—	6	2	no	—
220	C₄	C₃×D₃×D₈	1	1	yes	characteristic
221	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
222	C₃	—	2	1	no	—
223	C₃	#71511	1	1	yes	characteristic, direct product factor
224	C₃	C₃×#2178	1	1	yes	characteristic, complemented
225	C₂	—	2	1	no	—
226	C₂	—	8	2	no	—
227	C₂	C₃×Dic₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	D₃×D₈
commutator	C₃×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting, prime radical	C₃²×#508	C₂
socle	C₂×C₃²	C₄×D₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#508	D₃
hypocenter	C₃	C₃×#2178
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈ ◁ C₃×#508

Complements

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

N	H ≅ G/N	type
2. C₃×#5196	C₂	semidirect
3. C₃×#4567	C₂	semidirect
4. C₃×#4506	C₂	semidirect
5. C₃×Dic₃×C₁₆	C₂	semidirect
7. #71511	C₃	direct (N × H)
10. C₃²×Q₃₂	C₄	semidirect
11. C₃²×SD₃₂	C₄	semidirect
12. C₃²×D₁₆	C₄	semidirect
22. #5196	C₂×C₃	semidirect
26. #4567	C₂×C₃	semidirect
27. #4506	C₂×C₃	semidirect
28. Dic₃×C₁₆	C₂×C₃	semidirect
49. C₃×Q₃₂	Dic₃	semidirect
50. C₃×Q₃₂	C₃×C₄	semidirect
52. C₃×SD₃₂	Dic₃	semidirect
53. C₃×SD₃₂	C₃×C₄	semidirect
55. C₃×D₁₆	Dic₃	semidirect
56. C₃×D₁₆	C₃×C₄	semidirect
125. Q₃₂	C₃×Dic₃	semidirect
126. SD₃₂	C₃×Dic₃	semidirect
127. D₁₆	C₃×Dic₃	semidirect
195. C₃²	#2178	semidirect
223. C₃	#71511	direct (N × H)
224. C₃	C₃×#2178	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#508	2
C₃×#5196	2
C₃×#4567	2
C₃×#4506	2
C₃×Dic₃×C₁₆	2
C₃×#2178	3
#71511	3

Subgroups — C3×#71511 #11797842

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#71511 #11797842