Subgroups — C₃×#5724 #10595821

Order 576.

574 conjugacy classes of subgroups.

The subgroup lattice has 275 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#239	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#947	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#944	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#896	C₂	1	4	yes	maximal, complemented
7	C₃×#894	C₂	1	4	yes	maximal, complemented
8	C₃×#875	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#872	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#536	—	3	1	no	maximal
11	#5724	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
14	C₃×#239	—	2	4	no	—
15	C₃×D₃×Q₈	—	2	4	no	—
16	C₂×C₃×D₁₂	C₂²	1	2	yes	—
17	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
19	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
20	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
21	C₂×C₃×Dic₆	C₂²	1	2	yes	—
22	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
23	C₃²×C₄⋊C₄	C₂²	1	4	yes	—
24	C₃²×C₄²	C₂²	1	1	yes	characteristic
25	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
27	C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
28	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
29	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₃×#41	—	3	1	no	—
32	C₂²×C₃×Q₈	—	3	1	no	—
33	C₂×#239	C₂×C₃	1	1	yes	characteristic
34	C₂×D₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×C₄⋊Q₈	—	2	1	no	—
36	C₃×C₄⋊Q₈	—	3	1	no	—
37	C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
38	C₃×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
39	C₃×#124	—	3	1	no	—
40	C₃×#124	—	3	1	no	—
41	C₃×#123	—	3	4	no	—
42	C₃×C₂²⋊Q₈	—	3	4	no	—
43	C₃×#117	—	3	1	no	—
44	#947	C₂×C₃	1	1	yes	characteristic, complemented
45	#944	C₂×C₃	1	1	yes	characteristic, complemented
46	#896	C₂×C₃	1	4	yes	complemented
47	#894	C₂×C₃	1	4	yes	complemented
48	#875	C₂×C₃	1	1	yes	characteristic, complemented
49	#872	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂²×C₃×D₃	C₂³	1	2	yes	—
51	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
52	C₃²×Q₈	—	2	2	no	—
53	C₃²×Q₈	—	2	2	no	—
54	C₂×C₃²×C₄	C₂³	1	2	yes	—
55	C₂×C₃²×C₄	C₂³	1	2	yes	—
56	C₂×C₃²×C₄	C₂³	1	2	yes	—
57	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
58	C₂×C₃×Dic₃	C₂³	1	2	yes	—
59	C₂×C₃×Dic₃	C₂³	1	2	yes	—
60	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
61	C₃×D₁₂	—	2	4	no	—
62	C₃×D₁₂	—	2	2	no	—
63	C₃×C₄×D₃	—	2	4	no	—
64	C₃×C₄×D₃	—	2	4	no	—
65	C₃×C₄×D₃	D₄	1	4	yes	—
66	C₃×Dic₆	—	2	2	no	—
67	C₃×Dic₆	—	2	4	no	—
68	#536	—	3	1	no	Sylow
69	C₃×#41	—	6	4	no	—
70	C₂×C₃×Q₈	—	2	1	no	—
71	C₂×C₃×Q₈	—	2	1	no	—
72	C₂×C₃×Q₈	—	3	2	no	—
73	C₂×C₃×Q₈	—	3	1	no	—
74	C₂×C₃×Q₈	—	6	4	no	—
75	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
76	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
77	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₂×C₃×D₄	—	3	2	no	—
80	C₂×C₃×D₄	—	3	1	no	—
81	C₂²×C₃×C₄	—	3	1	no	—
82	C₂²×C₃×C₄	—	3	2	no	—
83	C₂²×C₃×C₄	—	3	2	no	—
84	#239	—	2	4	no	—
85	D₃×Q₈	—	2	4	no	—
86	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
87	C₂×D₁₂	C₂²×C₃	1	2	yes	—
88	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
89	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
90	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
91	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
92	C₂×Dic₆	C₂²×C₃	1	2	yes	—
93	C₃×C₄⋊C₄	—	2	4	no	—
94	C₃×C₄⋊C₄	—	3	2	no	—
95	C₃×C₄⋊C₄	—	3	4	no	—
96	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
97	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
98	C₃×C₂²⋊C₄	—	3	2	no	—
99	C₃×C₂²⋊C₄	—	3	4	no	—
100	C₃×C₂²⋊C₄	—	3	4	no	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	3	1	no	—
103	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
104	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
105	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
106	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
107	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
108	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
109	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
110	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
111	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
112	C₂×C₃×D₃	—	2	2	no	—
113	C₂×C₃×D₃	—	2	2	no	—
114	C₂×C₃×D₃	—	2	4	no	—
115	C₂×C₃×D₃	C₂×D₄	1	2	yes	—
116	C₃²×C₄	—	2	2	no	—
117	C₃²×C₄	—	2	2	no	—
118	C₃²×C₄	—	2	2	no	—
119	C₃²×C₄	C₂×D₄	1	2	yes	—
120	C₃×Dic₃	—	2	2	no	—
121	C₃×Dic₃	—	2	2	no	—
122	C₃×Dic₃	C₂×D₄	1	2	yes	—
123	C₂×#41	—	3	1	no	—
124	C₂²×Q₈	—	3	1	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic, complemented
127	#124	—	3	1	no	—
128	#124	—	3	1	no	—
129	#123	—	3	4	no	—
130	C₂²⋊Q₈	—	3	4	no	—
131	#117	—	3	1	no	—
132	C₂³×C₃	—	3	1	no	—
133	C₂³×C₃	—	3	2	no	—
134	C₂²×D₃	C₂³×C₃	1	2	yes	—
135	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
136	C₃×Q₈	—	2	2	no	—
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	—	2	2	no	—
139	C₃×Q₈	—	2	2	no	—
140	C₃×Q₈	—	4	2	no	—
141	C₃×Q₈	—	4	2	no	—
142	C₃×Q₈	—	6	4	no	—
143	C₃×Q₈	—	6	2	no	—
144	C₃×D₄	—	6	4	no	—
145	C₃×D₄	—	6	2	no	—
146	C₂×C₃×C₄	—	2	2	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	3	1	no	—
151	C₂×C₃×C₄	—	3	4	no	—
152	C₂×C₃×C₄	—	3	2	no	—
153	C₂×C₃×C₄	—	3	2	no	—
154	C₂×C₃×C₄	—	6	4	no	—
155	C₂×C₃×C₄	—	6	4	no	—
156	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
157	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
158	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
159	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
161	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
162	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
163	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
164	C₂×Dic₃	C₂³×C₃	1	2	yes	—
165	C₂×Dic₃	C₂³×C₃	1	2	yes	—
166	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
167	D₁₂	—	2	4	no	—
168	D₁₂	—	2	2	no	—
169	C₄×D₃	—	2	4	no	—
170	C₄×D₃	—	2	4	no	—
171	C₄×D₃	C₃×D₄	1	4	yes	—
172	Dic₆	—	2	2	no	—
173	Dic₆	—	2	4	no	—
174	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
176	C₃×D₃	—	2	2	no	—
177	C₃×D₃	—	4	2	no	—
178	#41	—	6	4	no	—
179	C₂×Q₈	—	3	2	no	—
180	C₂×Q₈	—	3	1	no	—
181	C₂×Q₈	—	6	4	no	—
182	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
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	2	no	—
188	C₂²×C₄	—	3	2	no	—
189	C₄⋊C₄	—	3	2	no	—
190	C₄⋊C₄	—	3	4	no	—
191	C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
192	C₂²⋊C₄	—	3	2	no	—
193	C₂²⋊C₄	—	3	4	no	—
194	C₂²⋊C₄	—	3	4	no	—
195	C₄²	—	3	1	no	—
196	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
197	C₂²×C₃	—	2	1	no	—
198	C₂²×C₃	—	3	2	no	—
199	C₂²×C₃	—	6	2	no	—
200	C₂²×C₃	—	6	2	no	—
201	C₂²×C₃	—	6	4	no	—
202	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
203	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
204	C₂×D₃	—	2	2	no	—
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	—	2	4	no	—
207	C₂×D₃	C₂×C₃×D₄	1	2	yes	—
208	C₃×C₄	—	2	4	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	3	2	no	—
217	C₃×C₄	—	4	2	no	—
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	6	2	no	—
220	C₃×C₄	—	6	2	no	—
221	C₃×C₄	D₃×D₄	1	2	yes	—
222	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	2	no	—
225	Dic₃	C₂×C₃×D₄	1	2	yes	—
226	C₃²	#536	1	1	yes	characteristic, complemented, Sylow
227	C₂³	—	3	1	no	—
228	C₂³	—	3	2	no	—
229	Q₈	—	2	2	no	—
230	Q₈	—	2	2	no	—
231	Q₈	—	6	4	no	—
232	Q₈	—	6	2	no	—
233	D₄	—	6	4	no	—
234	D₄	—	6	2	no	—
235	C₂×C₄	—	3	1	no	—
236	C₂×C₄	—	3	4	no	—
237	C₂×C₄	—	3	2	no	—
238	C₂×C₄	—	3	2	no	—
239	C₂×C₄	—	6	4	no	—
240	C₂×C₄	—	6	4	no	—
241	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
242	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
243	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	—	6	2	no	—
248	C₂×C₃	—	12	2	no	—
249	C₂×C₃	C₂×D₃×D₄	1	1	yes	characteristic
250	C₂×C₃	#1007	1	2	yes	—
251	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
252	C₂×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
253	D₃	—	2	2	no	—
254	D₃	—	4	2	no	—
255	C₂²	—	3	2	no	—
256	C₂²	—	6	2	no	—
257	C₂²	—	6	2	no	—
258	C₂²	—	6	4	no	—
259	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	3	2	no	—
264	C₄	—	6	2	no	—
265	C₄	—	6	2	no	—
266	C₄	C₃×D₃×D₄	1	2	yes	—
267	C₃	—	2	1	no	—
268	C₃	#5724	1	1	yes	characteristic, direct product factor
269	C₃	C₃×#536	1	1	yes	characteristic, complemented
270	C₂	—	6	2	no	—
271	C₂	—	12	2	no	—
272	C₂	C₂×C₃×D₃×D₄	1	1	yes	characteristic
273	C₂	C₃×#1007	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×C₄⋊Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×#239	C₂
hypercenter	C₃×C₄⋊Q₈	D₃
hypocenter	C₃	C₃×#536
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×D₃×Q₈	C₂	semidirect
3. C₃²×C₄⋊Q₈	C₂	semidirect
4. C₃×#947	C₂	semidirect
5. C₃×#944	C₂	semidirect
6. C₃×#896	C₂	semidirect
7. C₃×#894	C₂	semidirect
8. C₃×#875	C₂	semidirect
9. C₃×#872	C₂	semidirect
11. #5724	C₃	direct (N × H)
34. C₂×D₃×Q₈	C₂×C₃	semidirect
37. C₃×C₄⋊Q₈	D₃	semidirect
38. C₃×C₄⋊Q₈	C₂×C₃	semidirect
44. #947	C₂×C₃	semidirect
45. #944	C₂×C₃	semidirect
46. #896	C₂×C₃	semidirect
47. #894	C₂×C₃	semidirect
48. #875	C₂×C₃	semidirect
49. #872	C₂×C₃	semidirect
126. C₄⋊Q₈	C₃×D₃	semidirect
226. C₃²	#536	semidirect
268. C₃	#5724	direct (N × H)
269. C₃	C₃×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#239	2
C₂×C₃×D₃×Q₈	2
C₃²×C₄⋊Q₈	2
C₃×#947	2
C₃×#944	2
C₃×#896	2
C₃×#894	2
C₃×#875	2
C₃×#872	2
C₃×#536	3
#5724	3

Subgroups — C3×#5724 #10595821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5724 #10595821