Subgroups — C₃×#5011 #10592108

Order 576.

295 conjugacy classes of subgroups.

The subgroup lattice has 263 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#830	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#830	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#803	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#435	—	3	1	no	maximal
9	#5011	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃²×D₄	C₄	1	1	yes	characteristic
11	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
13	C₃²×C₄⋊C₄	C₄	1	1	yes	characteristic
14	C₃²×C₂²⋊C₄	C₄	1	1	yes	characteristic
15	C₃²×C₂²⋊C₄	C₄	1	1	yes	characteristic
16	C₃²×C₄²	C₂²	1	1	yes	characteristic
17	C₃×#208	—	2	2	no	—
18	C₂×C₃×C₃⋊C₈	—	2	2	no	—
19	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃×M₁₆	—	3	1	no	—
24	C₂²×C₃×C₈	—	3	1	no	—
25	C₃×C₄×D₄	—	2	1	no	—
26	C₃×C₄×D₄	D₃	1	1	yes	characteristic
27	C₃×C₄×D₄	C₂×C₃	1	1	yes	characteristic
28	C₂×#208	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂²×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×C₄⋊C₈	—	3	1	no	—
31	C₃×C₂²⋊C₈	—	3	1	no	—
32	C₃×C₂²⋊C₈	—	3	1	no	—
33	C₃×#97	—	3	1	no	—
34	#830	C₂×C₃	1	1	yes	characteristic, complemented
35	#830	C₂×C₃	1	1	yes	characteristic, complemented
36	C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
37	#803	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
40	C₃²×D₄	—	2	2	no	—
41	C₂×C₃²×C₄	—	2	1	no	—
42	C₂×C₃²×C₄	—	2	1	no	—
43	C₂×C₃²×C₄	—	2	1	no	—
44	C₂×C₃²×C₄	—	2	1	no	—
45	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
50	C₃×C₃⋊C₈	—	2	1	no	—
51	C₃×C₃⋊C₈	—	2	1	no	—
52	C₃×C₃⋊C₈	—	2	1	no	—
53	C₃×C₃⋊C₈	D₄	1	2	yes	complemented
54	#435	—	3	1	no	Sylow
55	C₂×C₃×D₄	—	2	1	no	—
56	C₂×C₃×D₄	Dic₃	1	1	yes	characteristic
57	C₂×C₃×D₄	C₃×C₄	1	1	yes	characteristic
58	C₂²×C₃×C₄	—	2	1	no	—
59	C₂²×C₃×C₄	—	2	1	no	—
60	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
64	C₃×M₁₆	—	6	2	no	—
65	C₂×C₃×C₈	—	3	1	no	—
66	C₂×C₃×C₈	—	3	1	no	—
67	C₂×C₃×C₈	—	3	1	no	—
68	C₂×C₃×C₈	—	3	1	no	—
69	C₂×C₃×C₈	—	6	2	no	—
70	C₃×C₄⋊C₄	—	2	1	no	—
71	C₃×C₄⋊C₄	Dic₃	1	1	yes	characteristic
72	C₃×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
73	C₃×C₂²⋊C₄	—	2	1	no	—
74	C₃×C₂²⋊C₄	—	2	1	no	—
75	C₃×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
76	C₃×C₂²⋊C₄	Dic₃	1	1	yes	characteristic
77	C₃×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
78	C₃×C₂²⋊C₄	C₃×C₄	1	1	yes	characteristic
79	C₃×C₄²	—	2	1	no	—
80	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
81	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
82	#208	—	2	2	no	—
83	C₂×C₃⋊C₈	—	2	2	no	—
84	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
88	C₂²×C₃²	—	2	1	no	—
89	C₂²×C₃²	—	2	1	no	—
90	C₂²×C₃²	—	2	1	no	—
91	C₂²×C₃²	—	2	1	no	—
92	C₂²×C₃²	—	2	1	no	—
93	C₂²×C₃²	M₁₆	1	2	yes	complemented
94	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
95	C₃²×C₄	—	2	1	no	—
96	C₃²×C₄	—	2	1	no	—
97	C₃²×C₄	—	2	1	no	—
98	C₃²×C₄	—	2	1	no	—
99	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
100	C₃²×C₄	#41	1	1	yes	characteristic
101	C₂×M₁₆	—	3	1	no	—
102	C₂²×C₈	—	3	1	no	—
103	C₄×D₄	C₃×D₃	1	1	yes	characteristic
104	C₄⋊C₈	—	3	1	no	—
105	C₂²⋊C₈	—	3	1	no	—
106	C₂²⋊C₈	—	3	1	no	—
107	#97	—	3	1	no	—
108	C₂³×C₃	—	2	1	no	—
109	C₂³×C₃	—	2	1	no	—
110	C₂³×C₃	C₂×Dic₃	1	1	yes	characteristic
111	C₂³×C₃	C₂×Dic₃	1	1	yes	characteristic
112	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
114	C₃×D₄	—	2	4	no	—
115	C₃×D₄	—	2	2	no	—
116	C₃×D₄	—	2	2	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	1	no	—
121	C₂×C₃×C₄	—	2	1	no	—
122	C₂×C₃×C₄	—	2	1	no	—
123	C₂×C₃×C₄	—	2	1	no	—
124	C₂×C₃×C₄	—	2	1	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	1	no	—
127	C₂×C₃×C₄	—	2	1	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	4	1	no	—
131	C₂×C₃×C₄	—	4	1	no	—
132	C₂×C₃×C₄	—	4	1	no	—
133	C₂×C₃×C₄	—	4	1	no	—
134	C₂×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
136	C₂×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
137	C₂×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
144	C₃×C₈	—	3	2	no	—
145	C₃×C₈	—	6	1	no	—
146	C₃×C₈	—	6	1	no	—
147	C₃×C₈	—	6	1	no	—
148	C₃⋊C₈	—	2	1	no	—
149	C₃⋊C₈	—	2	1	no	—
150	C₃⋊C₈	—	2	1	no	—
151	C₃⋊C₈	C₃×D₄	1	2	yes	complemented
152	C₂×C₃²	—	2	2	no	—
153	C₂×C₃²	—	4	1	no	—
154	C₂×C₃²	C₄×D₄	1	1	yes	characteristic
155	C₂×C₃²	C₂×M₁₆	1	1	yes	characteristic
156	C₂×C₃²	#131	1	1	yes	characteristic
157	C₂×D₄	C₃×Dic₃	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
159	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
160	M₁₆	—	6	2	no	—
161	C₂×C₈	—	3	1	no	—
162	C₂×C₈	—	3	1	no	—
163	C₂×C₈	—	3	1	no	—
164	C₂×C₈	—	3	1	no	—
165	C₂×C₈	—	6	2	no	—
166	C₄⋊C₄	C₃×Dic₃	1	1	yes	characteristic
167	C₂²⋊C₄	C₃×Dic₃	1	1	yes	characteristic
168	C₂²⋊C₄	C₃×Dic₃	1	1	yes	characteristic
169	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
170	C₂²×C₃	—	2	1	no	—
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	—	2	1	no	—
174	C₂²×C₃	—	2	2	no	—
175	C₂²×C₃	—	2	1	no	—
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	2	2	no	—
182	C₂²×C₃	—	2	1	no	—
183	C₂²×C₃	—	4	1	no	—
184	C₂²×C₃	—	4	1	no	—
185	C₂²×C₃	—	4	1	no	—
186	C₂²×C₃	—	4	1	no	—
187	C₂²×C₃	#208	1	2	yes	complemented
188	C₂²×C₃	C₃×M₁₆	1	2	yes	complemented
189	C₂²×C₃	C₂²×Dic₃	1	1	yes	characteristic
190	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
191	C₃×C₄	—	2	2	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	1	no	—
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	2	1	no	—
200	C₃×C₄	—	2	1	no	—
201	C₃×C₄	—	2	1	no	—
202	C₃×C₄	—	4	1	no	—
203	C₃×C₄	—	4	1	no	—
204	C₃×C₄	—	4	1	no	—
205	C₃×C₄	D₃×D₄	1	1	yes	characteristic
206	C₃×C₄	#237	1	1	yes	characteristic
207	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
208	C₃×C₄	C₃×#41	1	1	yes	characteristic
209	C₃²	#435	1	1	yes	characteristic, complemented, Sylow
210	C₂³	C₂×C₃×Dic₃	1	1	yes	characteristic
211	C₂³	C₂×C₃×Dic₃	1	1	yes	characteristic
212	D₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
219	C₂×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
222	C₈	—	3	2	no	—
223	C₈	—	6	1	no	—
224	C₈	—	6	1	no	—
225	C₈	—	6	1	no	—
226	C₂×C₃	—	2	2	no	—
227	C₂×C₃	—	2	2	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	2	1	no	—
231	C₂×C₃	—	4	2	no	—
232	C₂×C₃	—	4	2	no	—
233	C₂×C₃	—	4	1	no	—
234	C₂×C₃	—	4	1	no	—
235	C₂×C₃	C₂×#208	1	1	yes	characteristic
236	C₂×C₃	D₄×Dic₃	1	1	yes	characteristic
237	C₂×C₃	#948	1	1	yes	characteristic
238	C₂×C₃	C₃×C₄×D₄	1	1	yes	characteristic
239	C₂×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
240	C₂×C₃	C₃×#131	1	1	yes	characteristic
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	1	no	—
246	C₂²	C₃×#208	1	2	yes	complemented
247	C₂²	C₂²×C₃×Dic₃	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	C₃×D₃×D₄	1	1	yes	characteristic
253	C₄	C₃×#237	1	1	yes	characteristic
254	C₃	—	2	1	no	—
255	C₃	#5011	1	1	yes	characteristic, direct product factor
256	C₃	C₃×#435	1	1	yes	characteristic, complemented
257	C₂	—	2	2	no	—
258	C₂	—	4	1	no	—
259	C₂	C₂×C₃×#208	1	1	yes	characteristic
260	C₂	C₃×D₄×Dic₃	1	1	yes	characteristic
261	C₂	C₃×#948	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₃²×C₄×D₄	C₂
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂×C₃²×D₄	C₄
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄×D₄	D₃
hypocenter	C₃	C₃×#435
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#208	C₂	semidirect
3. C₂²×C₃×C₃⋊C₈	C₂	semidirect
4. C₃×#830	C₂	semidirect
5. C₃×#830	C₂	semidirect
6. C₃×C₄⋊(C₃⋊C₈)	C₂	semidirect
7. C₃×#803	C₂	semidirect
9. #5011	C₃	direct (N × H)
28. C₂×#208	C₂×C₃	semidirect
29. C₂²×C₃⋊C₈	C₂×C₃	semidirect
34. #830	C₂×C₃	semidirect
35. #830	C₂×C₃	semidirect
36. C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
37. #803	C₂×C₃	semidirect
53. C₃×C₃⋊C₈	D₄	semidirect
93. C₂²×C₃²	M₁₆	semidirect
151. C₃⋊C₈	C₃×D₄	semidirect
187. C₂²×C₃	#208	semidirect
188. C₂²×C₃	C₃×M₁₆	semidirect
209. C₃²	#435	semidirect
246. C₂²	C₃×#208	semidirect
255. C₃	#5011	direct (N × H)
256. C₃	C₃×#435	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×D₄	2
C₂×C₃×#208	2
C₂²×C₃×C₃⋊C₈	2
C₃×#830	2
C₃×C₄⋊(C₃⋊C₈)	2
C₃×#803	2
C₃×#435	3
#5011	3

Subgroups — C3×#5011 #10592108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5011 #10592108