Subgroups — C₃×#5645 #10595742

Order 576.

473 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#946	C₂	1	2	yes	maximal, complemented
3	C₃×#930	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#898	C₂	1	2	yes	maximal, complemented
6	C₃×#894	C₂	1	2	yes	maximal, complemented
7	C₃×#890	C₂	1	2	yes	maximal, complemented
8	C₃×#886	C₂	1	2	yes	maximal, complemented
9	C₃×#67⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₃×#539	—	3	1	no	maximal
11	#5645	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×#67	C₂²	1	2	yes	—
15	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
19	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
28	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
31	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₃×C₄×Dic₃	C₂²	1	2	yes	—
33	C₃×#126	—	3	2	no	—
34	C₃×#125	—	3	2	no	—
35	C₃×#123	—	3	2	no	—
36	C₃×#123	—	3	2	no	—
37	C₃×C₂²⋊Q₈	—	2	1	no	—
38	C₃×C₂²⋊Q₈	—	3	2	no	—
39	C₃×C₂²⋊Q₈	D₃	1	1	yes	characteristic, complemented
40	C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×#121	—	3	1	no	—
42	C₃×C₄×D₄	—	3	2	no	—
43	C₂×C₃×C₄⋊C₄	—	3	1	no	—
44	#946	C₂×C₃	1	2	yes	complemented
45	#930	C₂×C₃	1	1	yes	characteristic
46	C₂×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
47	#898	C₂×C₃	1	2	yes	complemented
48	#894	C₂×C₃	1	2	yes	complemented
49	#890	C₂×C₃	1	2	yes	complemented
50	#886	C₂×C₃	1	2	yes	complemented
51	#67⋊C₄	C₂×C₃	1	2	yes	complemented
52	C₂³×C₃²	C₂³	1	1	yes	characteristic
53	C₂²×C₃×D₃	C₂³	1	2	yes	—
54	C₃²×Q₈	—	4	1	no	—
55	C₂×C₃²×C₄	—	4	1	no	—
56	C₂×C₃²×C₄	C₂³	1	2	yes	—
57	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
59	C₂×C₃²×C₄	C₂³	1	2	yes	—
60	C₃×#67	—	2	4	no	—
61	C₂×C₃×Dic₃	—	2	2	no	—
62	C₂×C₃×Dic₃	—	4	1	no	—
63	C₂×C₃×Dic₃	C₂³	1	2	yes	—
64	C₂×C₃×Dic₃	C₂³	1	2	yes	—
65	C₂×C₃×Dic₃	C₂³	1	2	yes	—
66	C₃×D₁₂	—	4	1	no	—
67	C₃×C₄×D₃	—	4	2	no	—
68	#539	—	3	1	no	Sylow
69	C₂×C₃×Q₈	—	2	1	no	—
70	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
72	C₂×C₃×D₄	—	3	2	no	—
73	C₂×C₃×D₄	—	3	1	no	—
74	C₂²×C₃×C₄	—	2	1	no	—
75	C₂²×C₃×C₄	—	3	1	no	—
76	C₂²×C₃×C₄	—	3	1	no	—
77	C₂²×C₃×C₄	—	3	2	no	—
78	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
79	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
80	C₂×#67	C₂²×C₃	1	2	yes	—
81	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
82	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
83	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	—	2	1	no	—
87	C₃×C₄⋊C₄	—	3	2	no	—
88	C₃×C₄⋊C₄	—	3	2	no	—
89	C₃×C₄⋊C₄	—	3	2	no	—
90	C₃×C₄⋊C₄	—	3	1	no	—
91	C₃×C₄⋊C₄	—	3	2	no	—
92	C₃×C₄⋊C₄	—	3	2	no	—
93	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
94	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
95	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
96	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
97	C₃×C₂²⋊C₄	—	2	2	no	—
98	C₃×C₂²⋊C₄	—	3	2	no	—
99	C₃×C₂²⋊C₄	—	3	2	no	—
100	C₃×C₂²⋊C₄	—	3	2	no	—
101	C₃×C₂²⋊C₄	—	3	2	no	—
102	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
103	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
104	C₃×C₄²	—	3	2	no	—
105	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
106	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
107	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
108	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
109	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
110	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
111	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
112	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
113	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
114	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
115	C₄×Dic₃	C₂²×C₃	1	2	yes	—
116	C₂²×C₃²	—	2	1	no	—
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	#41	1	2	yes	—
119	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
120	C₂×C₃×D₃	—	2	2	no	—
121	C₂×C₃×D₃	—	2	2	no	—
122	C₂×C₃×D₃	—	2	2	no	—
123	C₃²×C₄	—	2	1	no	—
124	C₃²×C₄	—	2	2	no	—
125	C₃²×C₄	—	2	1	no	—
126	C₃²×C₄	—	2	2	no	—
127	C₃×Dic₃	—	2	2	no	—
128	C₃×Dic₃	—	2	2	no	—
129	C₃×Dic₃	—	2	2	no	—
130	#126	—	3	2	no	—
131	#125	—	3	2	no	—
132	#123	—	3	2	no	—
133	#123	—	3	2	no	—
134	C₂²⋊Q₈	—	3	2	no	—
135	C₂²⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
136	#121	—	3	1	no	—
137	C₄×D₄	—	3	2	no	—
138	C₂×C₄⋊C₄	—	3	1	no	—
139	C₂³×C₃	—	2	1	no	—
140	C₂³×C₃	—	3	2	no	—
141	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
142	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
143	C₂²×D₃	C₂³×C₃	1	2	yes	—
144	C₃×Q₈	—	4	1	no	—
145	C₃×Q₈	—	4	1	no	—
146	C₃×Q₈	—	4	2	no	—
147	C₃×D₄	—	6	4	no	—
148	C₃×D₄	—	12	1	no	—
149	C₂×C₃×C₄	—	2	1	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	2	2	no	—
152	C₂×C₃×C₄	—	2	1	no	—
153	C₂×C₃×C₄	—	3	2	no	—
154	C₂×C₃×C₄	—	3	2	no	—
155	C₂×C₃×C₄	—	3	2	no	—
156	C₂×C₃×C₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	6	2	no	—
160	C₂×C₃×C₄	—	12	2	no	—
161	C₂×C₃×C₄	—	12	1	no	—
162	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
163	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
165	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
167	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
170	#67	—	2	4	no	—
171	C₂×Dic₃	—	2	2	no	—
172	C₂×Dic₃	—	4	1	no	—
173	C₂×Dic₃	C₂³×C₃	1	2	yes	—
174	C₂×Dic₃	C₂³×C₃	1	2	yes	—
175	C₂×Dic₃	C₂³×C₃	1	2	yes	—
176	D₁₂	—	4	1	no	—
177	C₄×D₃	—	4	2	no	—
178	C₂×C₃²	—	2	2	no	—
179	C₂×C₃²	C₂×#41	1	1	yes	characteristic
180	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
182	C₃×D₃	—	4	2	no	—
183	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
184	C₂×D₄	—	3	2	no	—
185	C₂×D₄	—	3	1	no	—
186	C₂²×C₄	—	3	1	no	—
187	C₂²×C₄	—	3	1	no	—
188	C₂²×C₄	—	3	2	no	—
189	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
190	C₄⋊C₄	—	3	2	no	—
191	C₄⋊C₄	—	3	2	no	—
192	C₄⋊C₄	—	3	2	no	—
193	C₄⋊C₄	—	3	1	no	—
194	C₄⋊C₄	—	3	2	no	—
195	C₄⋊C₄	—	3	2	no	—
196	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
197	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
198	C₂²⋊C₄	—	3	2	no	—
199	C₂²⋊C₄	—	3	2	no	—
200	C₂²⋊C₄	—	3	2	no	—
201	C₂²⋊C₄	—	3	2	no	—
202	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
203	C₄²	—	3	2	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	2	1	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	4	1	no	—
211	C₂²×C₃	—	4	1	no	—
212	C₂²×C₃	—	6	2	no	—
213	C₂²×C₃	—	6	2	no	—
214	C₂²×C₃	—	6	2	no	—
215	C₂²×C₃	#239	1	2	yes	—
216	C₂²×C₃	C₃×#41	1	2	yes	—
217	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
218	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
219	C₂×D₃	—	2	2	no	—
220	C₂×D₃	—	2	2	no	—
221	C₂×D₃	—	2	2	no	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	4	1	no	—
232	C₃×C₄	—	4	2	no	—
233	C₃×C₄	—	4	2	no	—
234	C₃×C₄	—	6	2	no	—
235	C₃×C₄	—	6	2	no	—
236	C₃×C₄	—	6	2	no	—
237	Dic₃	—	2	2	no	—
238	Dic₃	—	2	2	no	—
239	Dic₃	—	2	2	no	—
240	C₃²	#539	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	3	2	no	—
242	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
243	Q₈	—	4	1	no	—
244	D₄	—	6	4	no	—
245	D₄	—	12	1	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	2	no	—
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	6	2	no	—
251	C₂×C₄	—	12	2	no	—
252	C₂×C₄	—	12	1	no	—
253	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
254	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
255	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
256	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
257	C₂×C₃	—	2	2	no	—
258	C₂×C₃	—	2	2	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	2	1	no	—
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	4	2	no	—
263	C₂×C₃	—	12	2	no	—
264	C₂×C₃	#1004	1	1	yes	characteristic
265	C₂×C₃	C₂×#239	1	1	yes	characteristic
266	C₂×C₃	#1010	1	1	yes	characteristic
267	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
268	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
270	D₃	—	4	2	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	6	2	no	—
274	C₂²	—	6	2	no	—
275	C₂²	—	6	2	no	—
276	C₂²	C₃×#239	1	2	yes	—
277	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	6	2	no	—
283	C₄	—	6	2	no	—
284	C₄	—	6	2	no	—
285	C₃	—	2	1	no	—
286	C₃	#5645	1	1	yes	characteristic, direct product factor
287	C₃	C₃×#539	1	1	yes	characteristic, complemented
288	C₂	—	2	2	no	—
289	C₂	—	12	2	no	—
290	C₂	C₃×#1004	1	1	yes	characteristic
291	C₂	C₂×C₃×#239	1	1	yes	characteristic
292	C₂	C₃×#1010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×C₂²⋊Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₃×#930	C₂
hypercenter	C₃×C₂²⋊Q₈	D₃
hypocenter	C₃	C₃×#539
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₃²×C₂²⋊Q₈	C₂	semidirect
2. C₃×#946	C₂	semidirect
4. C₂×C₃×Dic₃⋊C₄	C₂	semidirect
5. C₃×#898	C₂	semidirect
6. C₃×#894	C₂	semidirect
7. C₃×#890	C₂	semidirect
8. C₃×#886	C₂	semidirect
9. C₃×#67⋊C₄	C₂	semidirect
11. #5645	C₃	direct (N × H)
39. C₃×C₂²⋊Q₈	D₃	semidirect
40. C₃×C₂²⋊Q₈	C₂×C₃	semidirect
44. #946	C₂×C₃	semidirect
46. C₂×Dic₃⋊C₄	C₂×C₃	semidirect
47. #898	C₂×C₃	semidirect
48. #894	C₂×C₃	semidirect
49. #890	C₂×C₃	semidirect
50. #886	C₂×C₃	semidirect
51. #67⋊C₄	C₂×C₃	semidirect
135. C₂²⋊Q₈	C₃×D₃	semidirect
240. C₃²	#539	semidirect
286. C₃	#5645	direct (N × H)
287. C₃	C₃×#539	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂²⋊Q₈	2
C₃×#946	2
C₃×#930	2
C₂×C₃×Dic₃⋊C₄	2
C₃×#898	2
C₃×#894	2
C₃×#890	2
C₃×#886	2
C₃×#67⋊C₄	2
C₃×#539	3
#5645	3

Subgroups — C3×#5645 #10595742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5645 #10595742