Subgroups — C₃×#68070 #11773591

Order 1152.

449 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₃×#4685	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×D₃×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4454	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1451	—	3	1	no	maximal
9	#68070	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
11	C₂×C₃×D₃×C₈	C₄	1	2	yes	—
12	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
17	C₃×#815	C₄	1	2	yes	—
18	C₃×Dic₃⋊C₈	C₄	1	2	yes	—
19	C₃×C₈×Dic₃	C₄	1	2	yes	—
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×#431	—	3	1	no	—
24	C₃×C₄×M₁₆	—	3	1	no	—
25	C₂×C₃×C₄×C₈	—	3	1	no	—
26	C₄×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
27	#4685	C₂×C₃	1	1	yes	characteristic
28	C₄×D₃×C₈	C₂×C₃	1	1	yes	characteristic
29	C₃×#322	—	2	1	no	—
30	C₃×#322	—	3	1	no	—
31	C₃×#322	—	3	1	no	—
32	C₃×#322	—	3	1	no	—
33	C₃×#322	D₃	1	1	yes	characteristic, complemented
34	C₃×#322	C₂×C₃	1	1	yes	characteristic, complemented
35	#4454	C₂×C₃	1	1	yes	characteristic, complemented
36	#4454	C₂×C₃	1	1	yes	characteristic, complemented
37	#4454	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
44	C₃²×C₄²	C₂³	1	1	yes	characteristic
45	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
52	C₃×D₃×C₈	—	2	4	no	—
53	#1451	—	3	1	no	Sylow
54	C₂×C₃×M₁₆	—	3	2	no	—
55	C₂²×C₃×C₈	—	3	2	no	—
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	—
58	C₂×D₃×C₈	C₃×C₄	1	2	yes	—
59	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
60	C₃×C₄⋊C₈	—	3	2	no	—
61	C₃×C₂²⋊C₈	—	3	2	no	—
62	C₃×#97	—	3	2	no	—
63	C₃×C₄×C₈	—	2	1	no	—
64	C₃×C₄×C₈	—	2	1	no	—
65	C₃×C₄×C₈	—	2	1	no	—
66	C₃×C₄×C₈	—	3	2	no	—
67	C₃×C₄×C₈	—	3	1	no	—
68	C₃×C₄×C₈	—	3	1	no	—
69	C₃×C₄×C₈	—	3	1	no	—
70	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
76	(C₂×D₃)⋊C₈	C₃×C₄	1	2	yes	—
77	#815	C₃×C₄	1	2	yes	—
78	Dic₃⋊C₈	C₃×C₄	1	2	yes	—
79	C₈×Dic₃	C₃×C₄	1	2	yes	—
80	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
83	C₂²×C₃×D₃	C₄²	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
89	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
90	C₃×C₄×D₃	—	2	2	no	—
91	C₃×C₄×D₃	—	2	2	no	—
92	C₃×C₄×D₃	M₁₆	1	4	yes	—
93	C₃²×C₈	—	2	2	no	—
94	C₃²×C₈	—	2	2	no	—
95	C₃²×C₈	—	2	2	no	—
96	C₃×C₃⋊C₈	—	2	2	no	—
97	C₃×C₃⋊C₈	—	2	2	no	—
98	C₃×C₃⋊C₈	M₁₆	1	4	yes	complemented
99	#431	—	3	1	no	—
100	C₄×M₁₆	—	3	1	no	—
101	C₂×C₄×C₈	—	3	1	no	—
102	#322	—	3	1	no	—
103	#322	—	3	1	no	—
104	#322	—	3	1	no	—
105	#322	C₃×D₃	1	1	yes	characteristic, complemented
106	C₂²×C₃×C₄	—	3	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂²×C₃×C₄	—	3	1	no	—
109	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
111	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
112	C₃×M₁₆	—	6	4	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	2	2	no	—
116	C₂×C₃×C₈	—	3	2	no	—
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₂×C₃×C₈	—	6	4	no	—
120	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
121	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
122	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
123	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
125	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	3	1	no	—
128	C₃×C₄²	—	3	1	no	—
129	C₃×C₄²	—	3	1	no	—
130	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
131	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
132	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
133	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
134	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
135	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
136	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
137	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
138	ΓL(1,𝔽₂₅)	—	2	4	no	—
139	D₃×C₈	—	2	4	no	—
140	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₃×D₃	—	2	1	no	—
142	C₂×C₃×D₃	—	2	1	no	—
143	C₂×C₃×D₃	#97	1	2	yes	—
144	C₃²×C₄	C₂×M₁₆	1	2	yes	—
145	C₃²×C₄	C₂×M₁₆	1	2	yes	—
146	C₃²×C₄	#131	1	2	yes	—
147	C₃×Dic₃	—	2	1	no	—
148	C₃×Dic₃	—	2	1	no	—
149	C₃×Dic₃	#97	1	2	yes	—
150	C₂×M₁₆	—	3	2	no	—
151	C₂²×C₈	—	3	2	no	—
152	C₂×C₄²	—	3	1	no	—
153	C₄⋊C₈	—	3	2	no	—
154	C₂²⋊C₈	—	3	2	no	—
155	#97	—	3	2	no	—
156	C₄×C₈	—	3	2	no	—
157	C₄×C₈	—	3	1	no	—
158	C₄×C₈	—	3	1	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
161	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
162	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
163	C₂³×C₃	—	3	1	no	—
164	C₂²×D₃	C₃×C₄²	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	3	1	no	—
169	C₂×C₃×C₄	—	3	4	no	—
170	C₂×C₃×C₄	—	3	1	no	—
171	C₂×C₃×C₄	—	3	1	no	—
172	C₂×C₃×C₄	—	6	2	no	—
173	C₂×C₃×C₄	—	6	2	no	—
174	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
180	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
181	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
182	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
183	C₄×D₃	—	2	2	no	—
184	C₄×D₃	—	2	2	no	—
185	C₄×D₃	C₃×M₁₆	1	4	yes	—
186	C₃×C₈	—	2	4	no	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	2	2	no	—
191	C₃×C₈	—	2	2	no	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	3	4	no	—
194	C₃×C₈	—	4	2	no	—
195	C₃×C₈	—	4	2	no	—
196	C₃×C₈	—	6	2	no	—
197	C₃×C₈	—	6	2	no	—
198	C₃⋊C₈	—	2	2	no	—
199	C₃⋊C₈	—	2	2	no	—
200	C₃⋊C₈	C₃×M₁₆	1	4	yes	complemented
201	C₂×C₃²	C₂×#97	1	1	yes	characteristic
202	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
203	C₂×C₃²	#405	1	1	yes	characteristic
204	C₃×D₃	—	2	2	no	—
205	C₂²×C₄	—	3	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₂²×C₄	—	3	1	no	—
208	M₁₆	—	6	4	no	—
209	C₂×C₈	—	3	2	no	—
210	C₂×C₈	—	3	2	no	—
211	C₂×C₈	—	3	2	no	—
212	C₂×C₈	—	6	4	no	—
213	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
214	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
215	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
216	C₄²	—	3	1	no	—
217	C₄²	—	3	1	no	—
218	C₄²	—	3	1	no	—
219	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	3	2	no	—
222	C₂²×C₃	—	6	1	no	—
223	C₂²×C₃	—	6	1	no	—
224	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
225	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
226	C₂×D₃	—	2	1	no	—
227	C₂×D₃	—	2	1	no	—
228	C₂×D₃	C₃×#97	1	2	yes	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	3	2	no	—
233	C₃×C₄	—	6	1	no	—
234	C₃×C₄	—	6	1	no	—
235	C₃×C₄	#901	1	2	yes	—
236	C₃×C₄	D₃×M₁₆	1	2	yes	—
237	C₃×C₄	D₃×M₁₆	1	2	yes	—
238	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
239	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
240	C₃×C₄	C₃×#131	1	2	yes	—
241	Dic₃	—	2	1	no	—
242	Dic₃	—	2	1	no	—
243	Dic₃	C₃×#97	1	2	yes	—
244	C₃²	#1451	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	3	1	no	—
246	C₂×C₄	—	3	1	no	—
247	C₂×C₄	—	3	4	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	6	2	no	—
251	C₂×C₄	—	6	2	no	—
252	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	3	4	no	—
259	C₈	—	6	2	no	—
260	C₈	—	6	2	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	6	2	no	—
265	C₂×C₃	#4689	1	1	yes	characteristic
266	C₂×C₃	D₃×#97	1	1	yes	characteristic
267	C₂×C₃	#4707	1	1	yes	characteristic
268	C₂×C₃	C₂×C₃×#97	1	1	yes	characteristic
269	C₂×C₃	C₃×C₄×M₁₆	1	1	yes	characteristic
270	C₂×C₃	C₃×#405	1	1	yes	characteristic
271	D₃	—	2	2	no	—
272	C₂²	—	3	2	no	—
273	C₂²	—	6	1	no	—
274	C₂²	—	6	1	no	—
275	C₂²	C₃×C₄²×D₃	1	1	yes	characteristic
276	C₄	—	3	2	no	—
277	C₄	—	6	1	no	—
278	C₄	—	6	1	no	—
279	C₄	C₃×#901	1	2	yes	—
280	C₄	C₃×D₃×M₁₆	1	2	yes	—
281	C₄	C₃×D₃×M₁₆	1	2	yes	—
282	C₃	—	2	1	no	—
283	C₃	#68070	1	1	yes	characteristic, direct product factor
284	C₃	C₃×#1451	1	1	yes	characteristic, complemented
285	C₂	—	6	2	no	—
286	C₂	C₃×#4689	1	1	yes	characteristic
287	C₂	C₃×D₃×#97	1	1	yes	characteristic
288	C₂	C₃×#4707	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄²	C₂²×D₃
commutator	C₂²×C₃	C₂×C₃×C₄²
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#322	C₂
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂²×C₃×D₃	C₄²
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#322	D₃
hypocenter	C₃	C₃×#1451
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄² ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
4. C₃²×#322	C₂	semidirect
5. C₃×#4454	C₂	semidirect
6. C₃×#4454	C₂	semidirect
7. C₃×#4454	C₂	semidirect
9. #68070	C₃	direct (N × H)
33. C₃×#322	D₃	semidirect
34. C₃×#322	C₂×C₃	semidirect
35. #4454	C₂×C₃	semidirect
36. #4454	C₂×C₃	semidirect
37. #4454	C₂×C₃	semidirect
98. C₃×C₃⋊C₈	M₁₆	semidirect
105. #322	C₃×D₃	semidirect
200. C₃⋊C₈	C₃×M₁₆	semidirect
244. C₃²	#1451	semidirect
283. C₃	#68070	direct (N × H)
284. C₃	C₃×#1451	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×ΓL(1,𝔽₂₅)	2
C₃×#4685	2
C₃×C₄×D₃×C₈	2
C₃²×#322	2
C₃×#4454	2
C₃×#1451	3
#68070	3

Subgroups — C3×#68070 #11773591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68070 #11773591