Subgroups — C₃×#69073 #11797317

Order 1152.

331 conjugacy classes of subgroups.

The subgroup lattice has 265 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃²×#346	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×(C₂×D₃)⋊C₁₆	C₂	1	2	yes	maximal
4	C₃×Dic₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₃×#4461	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#2163	—	3	1	no	maximal
7	#69073	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃×C₈	C₄	1	1	yes	characteristic
9	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄²×D₃	C₄	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₈×Dic₃	C₄	1	1	yes	characteristic
14	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₈	—	3	1	no	—
18	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
19	C₃×C₄⋊C₁₆	—	3	2	no	—
20	C₃×C₂²⋊C₁₆	—	3	2	no	—
21	C₃×#346	—	2	1	no	—
22	C₃×#346	—	3	1	no	—
23	C₃×#346	D₃	1	1	yes	characteristic, complemented
24	C₃×#346	C₂×C₃	1	1	yes	characteristic, complemented
25	(C₂×D₃)⋊C₁₆	C₂×C₃	1	2	yes	—
26	Dic₃⋊C₁₆	C₂×C₃	1	2	yes	complemented
27	#4461	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×C₄×D₃	C₈	1	1	yes	characteristic
29	C₂×C₃×C₄×D₃	C₈	1	1	yes	characteristic
30	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
33	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₈	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₈	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×D₃×C₈	—	2	2	no	—
40	C₃×D₃×C₈	—	2	2	no	—
41	C₃²×C₁₆	—	2	2	no	—
42	C₃×C₃⋊C₁₆	—	2	2	no	—
43	#2163	—	3	1	no	Sylow
44	C₂²×C₃×C₈	—	3	1	no	—
45	C₂²×C₃×C₈	—	3	1	no	—
46	C₂×C₃×C₄²	—	3	1	no	—
47	C₂×D₃×C₈	C₃×C₄	1	1	yes	characteristic
48	C₂×D₃×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₄²×D₃	C₃×C₄	1	1	yes	characteristic
50	C₂×C₃×C₁₆	—	2	2	no	—
51	C₂×C₃×C₁₆	—	3	2	no	—
52	C₂×C₃×C₁₆	C₂×D₃	1	2	yes	—
53	C₂×C₃×C₁₆	C₂²×C₃	1	2	yes	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	—	3	1	no	—
56	C₃×C₄×C₈	—	3	1	no	—
57	C₃×C₄×C₈	—	3	1	no	—
58	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₈×Dic₃	C₃×C₄	1	1	yes	characteristic
61	C₈×Dic₃	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃⋊C₁₆	C₂²×C₃	1	2	yes	—
63	C₄×C₃⋊C₈	C₃×C₄	1	1	yes	characteristic
64	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
71	C₃×C₄×D₃	—	2	2	no	—
72	C₃×C₄×D₃	—	2	1	no	—
73	C₃×C₄×D₃	—	2	1	no	—
74	C₃×C₄×D₃	—	2	2	no	—
75	C₃²×C₈	M₁₆	1	2	yes	—
76	C₃²×C₈	#41	1	2	yes	—
77	C₃×C₃⋊C₈	—	2	1	no	—
78	C₃×C₃⋊C₈	—	2	1	no	—
79	C₂×C₄×C₈	—	3	1	no	—
80	C₄⋊C₁₆	—	3	2	no	—
81	C₂²⋊C₁₆	—	3	2	no	—
82	#346	—	3	1	no	—
83	#346	C₃×D₃	1	1	yes	characteristic, complemented
84	C₂²×C₃×C₄	—	3	1	no	—
85	C₂²×C₃×C₄	—	3	1	no	—
86	C₂²×C₃×C₄	—	3	1	no	—
87	C₂×C₄×D₃	C₃×C₈	1	1	yes	characteristic
88	C₂×C₄×D₃	C₃×C₈	1	1	yes	characteristic
89	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	3	1	no	—
93	C₂×C₃×C₈	—	3	1	no	—
94	C₂×C₃×C₈	—	6	2	no	—
95	C₂×C₃×C₈	—	6	2	no	—
96	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
100	C₃×C₄²	—	2	1	no	—
101	C₃×C₄²	—	3	1	no	—
102	C₃×C₄²	—	3	1	no	—
103	C₃×C₄²	—	3	1	no	—
104	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
105	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
106	C₄×Dic₃	C₃×C₈	1	1	yes	characteristic
107	C₄×Dic₃	C₃×C₈	1	1	yes	characteristic
108	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
111	D₃×C₈	—	2	2	no	—
112	D₃×C₈	—	2	2	no	—
113	C₃×C₁₆	—	2	4	no	—
114	C₃×C₁₆	—	2	2	no	—
115	C₃×C₁₆	—	2	2	no	—
116	C₃×C₁₆	—	6	2	no	—
117	C₃⋊C₁₆	—	2	2	no	—
118	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₃×D₃	—	2	1	no	—
120	C₂×C₃×D₃	—	2	1	no	—
121	C₂×C₃×D₃	M₃₂	1	2	yes	—
122	C₃²×C₄	—	2	1	no	—
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	#117	1	1	yes	characteristic
125	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
126	C₃×Dic₃	—	2	1	no	—
127	C₃×Dic₃	—	2	1	no	—
128	C₃×Dic₃	M₃₂	1	2	yes	—
129	C₂²×C₈	—	3	1	no	—
130	C₂²×C₈	—	3	1	no	—
131	C₂×C₄²	—	3	1	no	—
132	C₂×C₁₆	—	3	2	no	—
133	C₂×C₁₆	C₂×C₃×D₃	1	2	yes	—
134	C₄×C₈	—	3	1	no	—
135	C₄×C₈	—	3	1	no	—
136	C₄×C₈	—	3	1	no	—
137	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
138	C₂³×C₃	—	3	1	no	—
139	C₂²×D₃	C₂×C₃×C₈	1	1	yes	characteristic
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	2	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	3	1	no	—
144	C₂×C₃×C₄	—	3	1	no	—
145	C₂×C₃×C₄	—	3	1	no	—
146	C₂×C₃×C₄	—	6	2	no	—
147	C₂×C₃×C₄	—	6	2	no	—
148	C₂×C₃×C₄	—	6	1	no	—
149	C₂×C₃×C₄	—	6	1	no	—
150	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
151	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
153	C₂×C₃×C₄	C₂×C₃×C₈	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
156	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
157	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
158	C₂×Dic₃	C₂×C₃×C₈	1	1	yes	characteristic
159	C₄×D₃	—	2	2	no	—
160	C₄×D₃	—	2	1	no	—
161	C₄×D₃	—	2	1	no	—
162	C₄×D₃	—	2	2	no	—
163	C₃×C₈	—	2	2	no	—
164	C₃×C₈	—	2	2	no	—
165	C₃×C₈	—	6	1	no	—
166	C₃×C₈	—	6	1	no	—
167	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
168	C₃×C₈	C₃×M₁₆	1	2	yes	—
169	C₃×C₈	#235	1	2	yes	—
170	C₃×C₈	C₃×#41	1	2	yes	—
171	C₃⋊C₈	—	2	1	no	—
172	C₃⋊C₈	—	2	1	no	—
173	C₂×C₃²	#431	1	1	yes	characteristic
174	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
175	C₂×C₃²	C₂×M₃₂	1	1	yes	characteristic
176	C₃×D₃	—	2	2	no	—
177	C₂²×C₄	—	3	1	no	—
178	C₂²×C₄	—	3	1	no	—
179	C₂²×C₄	—	3	1	no	—
180	C₂×C₈	—	3	1	no	—
181	C₂×C₈	—	3	1	no	—
182	C₂×C₈	—	6	2	no	—
183	C₂×C₈	—	6	2	no	—
184	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
185	C₂×C₈	C₂²×C₃×D₃	1	1	yes	characteristic
186	C₄²	—	3	1	no	—
187	C₄²	—	3	1	no	—
188	C₄²	—	3	1	no	—
189	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
190	C₁₆	—	2	2	no	—
191	C₁₆	—	6	2	no	—
192	C₂²×C₃	—	2	1	no	—
193	C₂²×C₃	—	3	2	no	—
194	C₂²×C₃	—	6	1	no	—
195	C₂²×C₃	—	6	1	no	—
196	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
197	C₂²×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
198	C₂×D₃	—	2	1	no	—
199	C₂×D₃	—	2	1	no	—
200	C₂×D₃	C₃×M₃₂	1	2	yes	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	1	no	—
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	3	2	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	—	4	1	no	—
210	C₃×C₄	—	6	1	no	—
211	C₃×C₄	—	6	1	no	—
212	C₃×C₄	#872	1	1	yes	characteristic
213	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
214	C₃×C₄	C₃×#117	1	1	yes	characteristic
215	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
216	Dic₃	—	2	1	no	—
217	Dic₃	—	2	1	no	—
218	Dic₃	C₃×M₃₂	1	2	yes	—
219	C₃²	#2163	1	1	yes	characteristic, complemented, Sylow
220	C₂³	—	3	1	no	—
221	C₂×C₄	—	3	1	no	—
222	C₂×C₄	—	3	1	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	—	6	2	no	—
226	C₂×C₄	—	6	1	no	—
227	C₂×C₄	—	6	1	no	—
228	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
229	C₂×C₄	C₃×D₃×C₈	1	1	yes	characteristic
230	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
231	C₈	—	6	1	no	—
232	C₈	—	6	1	no	—
233	C₈	C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
234	C₈	C₃×#235	1	2	yes	—
235	C₂×C₃	—	2	1	no	—
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	6	2	no	—
239	C₂×C₃	#4685	1	1	yes	characteristic
240	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
241	C₂×C₃	D₃×M₃₂	1	1	yes	characteristic
242	C₂×C₃	C₃×#431	1	1	yes	characteristic
243	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
244	C₂×C₃	C₂×C₃×M₃₂	1	1	yes	characteristic
245	D₃	—	2	2	no	—
246	C₂²	—	3	2	no	—
247	C₂²	—	6	1	no	—
248	C₂²	—	6	1	no	—
249	C₂²	C₂×C₃×D₃×C₈	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	3	2	no	—
253	C₄	—	6	1	no	—
254	C₄	—	6	1	no	—
255	C₄	C₃×#872	1	1	yes	characteristic
256	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
257	C₃	—	2	1	no	—
258	C₃	#69073	1	1	yes	characteristic, direct product factor
259	C₃	C₃×#2163	1	1	yes	characteristic, complemented
260	C₂	—	6	2	no	—
261	C₂	C₃×#4685	1	1	yes	characteristic
262	C₂	C₃×D₃×M₃₂	1	1	yes	characteristic
263	C₂	C₃×D₃×M₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₃
Fitting	C₃²×#346	C₂
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂²×C₃×D₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³×C₃
hypercenter	C₃×#346	D₃
hypocenter	C₃	C₃×#2163
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×#346

Complements

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

N	H ≅ G/N	type
2. C₃²×#346	C₂	semidirect
4. C₃×Dic₃⋊C₁₆	C₂	semidirect
5. C₃×#4461	C₂	semidirect
7. #69073	C₃	direct (N × H)
23. C₃×#346	D₃	semidirect
24. C₃×#346	C₂×C₃	semidirect
26. Dic₃⋊C₁₆	C₂×C₃	semidirect
27. #4461	C₂×C₃	semidirect
83. #346	C₃×D₃	semidirect
219. C₃²	#2163	semidirect
258. C₃	#69073	direct (N × H)
259. C₃	C₃×#2163	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₃×C₈	2
C₃²×#346	2
C₃×(C₂×D₃)⋊C₁₆	2
C₃×Dic₃⋊C₁₆	2
C₃×#4461	2
C₃×#2163	3
#69073	3

Subgroups — C3×#69073 #11797317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69073 #11797317