Subgroups — C₃×#68072 #11773645

Order 1152.

413 conjugacy classes of subgroups.

The subgroup lattice has 286 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×#4685	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×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1453	—	3	1	no	maximal
9	#68072	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
11	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
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×Dic₃⋊C₈	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×#431	—	3	1	no	—
25	C₃×C₄×M₁₆	—	3	1	no	—
26	C₄×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
27	#4685	C₂×C₃	1	1	yes	characteristic
28	#4685	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	D₃	1	1	yes	characteristic, complemented
33	C₃×#322	C₂×C₃	1	1	yes	characteristic, complemented
34	C₃×C₈²	—	3	1	no	—
35	#4454	C₂×C₃	1	1	yes	characteristic, complemented
36	#4454	C₂×C₃	1	1	yes	characteristic, complemented
37	C₈×C₃⋊C₈	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	#1453	—	3	1	no	Sylow
53	C₂×C₃×M₁₆	—	3	2	no	—
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	—
56	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
57	C₃×C₄⋊C₈	—	3	2	no	—
58	C₃×C₄⋊C₈	—	3	2	no	—
59	C₃×C₂²⋊C₈	—	3	2	no	—
60	C₃×C₂²⋊C₈	—	3	2	no	—
61	C₃×#97	—	3	2	no	—
62	C₃×C₄×C₈	—	2	1	no	—
63	C₃×C₄×C₈	—	2	1	no	—
64	C₃×C₄×C₈	—	2	1	no	—
65	C₃×C₄×C₈	—	3	1	no	—
66	C₃×C₄×C₈	—	3	1	no	—
67	C₃×C₄×C₈	—	3	1	no	—
68	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
69	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
70	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
74	(C₂×D₃)⋊C₈	C₃×C₄	1	2	yes	—
75	(C₂×D₃)⋊C₈	C₃×C₄	1	2	yes	—
76	#815	C₃×C₄	1	2	yes	—
77	Dic₃⋊C₈	C₃×C₄	1	2	yes	—
78	Dic₃⋊C₈	C₃×C₄	1	2	yes	—
79	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
80	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
81	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
82	C₂²×C₃×D₃	C₄²	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂²×C₄	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×C₃×Dic₃	C₄²	1	1	yes	characteristic
87	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
88	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
89	C₃×C₄×D₃	—	2	2	no	—
90	C₃×C₄×D₃	—	2	2	no	—
91	C₃×C₄×D₃	—	2	2	no	—
92	C₃²×C₈	—	2	2	no	—
93	C₃²×C₈	—	2	2	no	—
94	C₃²×C₈	—	2	2	no	—
95	C₃×C₃⋊C₈	—	2	2	no	—
96	C₃×C₃⋊C₈	—	2	2	no	—
97	C₃×C₃⋊C₈	M₁₆	1	4	yes	—
98	#431	—	3	1	no	—
99	#431	—	3	1	no	—
100	C₄×M₁₆	—	3	1	no	—
101	#322	—	3	1	no	—
102	#322	—	3	1	no	—
103	#322	C₃×D₃	1	1	yes	characteristic, complemented
104	C₈²	—	3	1	no	—
105	C₂²×C₃×C₄	—	3	1	no	—
106	C₂²×C₃×C₄	—	3	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
110	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
111	C₃×M₁₆	—	6	4	no	—
112	C₂×C₃×C₈	—	2	2	no	—
113	C₂×C₃×C₈	—	2	2	no	—
114	C₂×C₃×C₈	—	2	2	no	—
115	C₂×C₃×C₈	—	3	2	no	—
116	C₂×C₃×C₈	—	3	2	no	—
117	C₂×C₃×C₈	—	3	2	no	—
118	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
119	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
120	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
121	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
122	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
123	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
124	C₃×C₄²	—	2	1	no	—
125	C₃×C₄²	—	3	1	no	—
126	C₃×C₄²	—	3	1	no	—
127	C₃×C₄²	—	3	1	no	—
128	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
129	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
130	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
131	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
132	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
134	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
135	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
136	ΓL(1,𝔽₂₅)	—	2	4	no	—
137	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₃×D₃	—	2	1	no	—
139	C₂×C₃×D₃	—	2	1	no	—
140	C₂×C₃×D₃	—	2	1	no	—
141	C₃²×C₄	C₂×M₁₆	1	2	yes	—
142	C₃²×C₄	#131	1	2	yes	—
143	C₃²×C₄	#131	1	2	yes	—
144	C₃×Dic₃	—	2	1	no	—
145	C₃×Dic₃	—	2	1	no	—
146	C₃×Dic₃	—	2	1	no	—
147	C₂×M₁₆	—	3	2	no	—
148	C₂×C₄²	—	3	1	no	—
149	C₄⋊C₈	—	3	2	no	—
150	C₄⋊C₈	—	3	2	no	—
151	C₂²⋊C₈	—	3	2	no	—
152	C₂²⋊C₈	—	3	2	no	—
153	#97	—	3	2	no	—
154	C₄×C₈	—	3	1	no	—
155	C₄×C₈	—	3	1	no	—
156	C₄×C₈	—	3	1	no	—
157	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
158	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
159	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
160	C₂³×C₃	—	3	1	no	—
161	C₂²×D₃	C₃×C₄²	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	3	1	no	—
166	C₂×C₃×C₄	—	3	1	no	—
167	C₂×C₃×C₄	—	3	1	no	—
168	C₂×C₃×C₄	—	6	2	no	—
169	C₂×C₃×C₄	—	6	2	no	—
170	C₂×C₃×C₄	—	6	2	no	—
171	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
178	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
179	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
180	C₄×D₃	—	2	2	no	—
181	C₄×D₃	—	2	2	no	—
182	C₄×D₃	—	2	2	no	—
183	C₃×C₈	—	2	4	no	—
184	C₃×C₈	—	2	2	no	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	—	2	2	no	—
188	C₃×C₈	—	2	2	no	—
189	C₃×C₈	—	2	2	no	—
190	C₃×C₈	—	3	4	no	—
191	C₃×C₈	—	4	2	no	—
192	C₃×C₈	—	4	2	no	—
193	C₃×C₈	—	6	2	no	—
194	C₃×C₈	—	6	2	no	—
195	C₃⋊C₈	—	2	2	no	—
196	C₃⋊C₈	—	2	2	no	—
197	C₃⋊C₈	C₃×M₁₆	1	4	yes	—
198	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
199	C₂×C₃²	#405	1	1	yes	characteristic
200	C₂×C₃²	#405	1	1	yes	characteristic
201	C₃×D₃	—	4	1	no	—
202	C₂²×C₄	—	3	1	no	—
203	C₂²×C₄	—	3	1	no	—
204	C₂²×C₄	—	3	1	no	—
205	M₁₆	—	6	4	no	—
206	C₂×C₈	—	3	2	no	—
207	C₂×C₈	—	3	2	no	—
208	C₂×C₈	—	3	2	no	—
209	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
210	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
211	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
212	C₄²	—	3	1	no	—
213	C₄²	—	3	1	no	—
214	C₄²	—	3	1	no	—
215	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	6	1	no	—
218	C₂²×C₃	—	6	1	no	—
219	C₂²×C₃	—	6	1	no	—
220	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
221	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
222	C₂×D₃	—	2	1	no	—
223	C₂×D₃	—	2	1	no	—
224	C₂×D₃	—	2	1	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	6	1	no	—
229	C₃×C₄	—	6	1	no	—
230	C₃×C₄	—	6	1	no	—
231	C₃×C₄	#901	1	2	yes	—
232	C₃×C₄	D₃×M₁₆	1	2	yes	—
233	C₃×C₄	#907	1	2	yes	—
234	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
235	C₃×C₄	C₃×#131	1	2	yes	—
236	C₃×C₄	C₃×#131	1	2	yes	—
237	Dic₃	—	2	1	no	—
238	Dic₃	—	2	1	no	—
239	Dic₃	—	2	1	no	—
240	C₃²	#1453	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	3	1	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	—	3	1	no	—
244	C₂×C₄	—	3	1	no	—
245	C₂×C₄	—	6	2	no	—
246	C₂×C₄	—	6	2	no	—
247	C₂×C₄	—	6	2	no	—
248	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	2	no	—
253	C₈	—	2	2	no	—
254	C₈	—	3	4	no	—
255	C₈	—	6	2	no	—
256	C₈	—	6	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	12	1	no	—
261	C₂×C₃	#4689	1	1	yes	characteristic
262	C₂×C₃	#4707	1	1	yes	characteristic
263	C₂×C₃	#4708	1	1	yes	characteristic
264	C₂×C₃	C₃×C₄×M₁₆	1	1	yes	characteristic
265	C₂×C₃	C₃×#405	1	1	yes	characteristic
266	C₂×C₃	C₃×#405	1	1	yes	characteristic
267	D₃	—	4	1	no	—
268	C₂²	—	6	1	no	—
269	C₂²	—	6	1	no	—
270	C₂²	—	6	1	no	—
271	C₂²	C₃×C₄²×D₃	1	1	yes	characteristic
272	C₄	—	6	1	no	—
273	C₄	—	6	1	no	—
274	C₄	—	6	1	no	—
275	C₄	C₃×#901	1	2	yes	—
276	C₄	C₃×D₃×M₁₆	1	2	yes	—
277	C₄	C₃×#907	1	2	yes	—
278	C₃	—	2	1	no	—
279	C₃	#68072	1	1	yes	characteristic, direct product factor
280	C₃	C₃×#1453	1	1	yes	characteristic, complemented
281	C₂	—	12	1	no	—
282	C₂	C₃×#4689	1	1	yes	characteristic
283	C₂	C₃×#4707	1	1	yes	characteristic
284	C₂	C₃×#4708	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×#1453
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×C₈×C₃⋊C₈	C₂	semidirect
9. #68072	C₃	direct (N × H)
32. C₃×#322	D₃	semidirect
33. C₃×#322	C₂×C₃	semidirect
35. #4454	C₂×C₃	semidirect
36. #4454	C₂×C₃	semidirect
37. C₈×C₃⋊C₈	C₂×C₃	semidirect
103. #322	C₃×D₃	semidirect
240. C₃²	#1453	semidirect
279. C₃	#68072	direct (N × H)
280. C₃	C₃×#1453	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
C₃²×#322	2
C₃×#4454	2
C₃×C₈×C₃⋊C₈	2
C₃×#1453	3
#68072	3

Subgroups — C3×#68072 #11773645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68072 #11773645