Subgroups — C₃×#5568 #10595665

Order 576.

459 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#947	C₂	1	1	yes	maximal, characteristic
3	C₃×#944	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#898	C₂	1	2	yes	maximal, complemented
5	C₃×#896	C₂	1	2	yes	maximal, complemented
6	C₃×#893	C₂	1	1	yes	maximal, characteristic
7	C₃×#887	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#876	C₂	1	2	yes	maximal, complemented
9	C₃×#875	C₂	1	1	yes	maximal, characteristic
10	C₃×#872	C₂	1	2	yes	maximal, complemented
11	C₃×C₄×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#556	—	3	1	no	maximal
13	#5568	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
17	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
19	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃²×C₄²	C₂²	1	1	yes	characteristic
22	C₃²×C₄²	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×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₃×(C₂×D₃)⋊C₄	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×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₂²	1	2	yes	—
35	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	3	1	no	—
38	C₃×#126	—	3	2	no	—
39	C₃×#126	—	3	2	no	—
40	C₃×#124	—	3	1	no	—
41	C₃×#124	—	3	1	no	—
42	C₃×C₂²⋊Q₈	—	3	2	no	—
43	C₃×C₄×Q₈	—	2	1	no	—
44	C₃×C₄×Q₈	—	3	1	no	—
45	C₃×C₄×Q₈	—	3	1	no	—
46	C₃×C₄×Q₈	D₃	1	1	yes	characteristic, complemented
47	C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
48	C₃×C₄×D₄	—	3	1	no	—
49	C₃×#117	—	3	2	no	—
50	#947	C₂×C₃	1	1	yes	characteristic
51	#944	C₂×C₃	1	1	yes	characteristic, complemented
52	#898	C₂×C₃	1	2	yes	complemented
53	#896	C₂×C₃	1	2	yes	complemented
54	#893	C₂×C₃	1	1	yes	characteristic
55	#887	C₂×C₃	1	1	yes	characteristic, complemented
56	#876	C₂×C₃	1	2	yes	complemented
57	#875	C₂×C₃	1	1	yes	characteristic
58	#872	C₂×C₃	1	2	yes	complemented
59	C₄×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂²×C₃×D₃	C₂³	1	2	yes	—
61	C₃²×Q₈	—	2	2	no	—
62	C₂×C₃²×C₄	C₂³	1	2	yes	—
63	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂³	1	2	yes	—
65	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
68	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
69	C₂×C₃×Dic₃	C₂³	1	2	yes	—
70	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
71	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
72	C₃×D₁₂	—	2	2	no	—
73	C₃×C₄×D₃	—	2	4	no	—
74	C₃×Dic₆	—	2	2	no	—
75	C₃×Dic₆	—	2	2	no	—
76	#556	—	3	1	no	Sylow
77	C₂×C₃×Q₈	—	2	1	no	—
78	C₂×C₃×Q₈	—	3	1	no	—
79	C₂×C₃×Q₈	—	3	1	no	—
80	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
81	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×D₄	—	3	1	no	—
83	C₂²×C₃×C₄	—	3	2	no	—
84	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
86	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
87	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	—	2	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	—	3	1	no	—
94	C₃×C₄⋊C₄	—	3	2	no	—
95	C₃×C₄⋊C₄	—	3	1	no	—
96	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
97	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
98	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
99	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
100	C₃×C₂²⋊C₄	—	3	2	no	—
101	C₃×C₂²⋊C₄	—	3	2	no	—
102	C₃×C₂²⋊C₄	—	3	2	no	—
103	C₃×C₂²⋊C₄	—	3	2	no	—
104	C₃×C₂²⋊C₄	—	3	2	no	—
105	C₃×C₄²	—	2	2	no	—
106	C₃×C₄²	—	2	1	no	—
107	C₃×C₄²	—	3	2	no	—
108	C₃×C₄²	—	3	1	no	—
109	C₃×C₄²	—	3	1	no	—
110	C₃×C₄²	C₂×D₃	1	2	yes	—
111	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
112	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
113	C₃×C₄²	C₂²×C₃	1	2	yes	—
114	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
115	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
116	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
117	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
118	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
119	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
120	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
121	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
122	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
123	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
124	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
125	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
126	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
127	C₄×Dic₃	C₂²×C₃	1	2	yes	—
128	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
129	C₂×C₃×D₃	—	2	2	no	—
130	C₂×C₃×D₃	—	2	2	no	—
131	C₂×C₃×D₃	—	2	2	no	—
132	C₃²×C₄	—	2	2	no	—
133	C₃²×C₄	—	2	1	no	—
134	C₃²×C₄	—	2	1	no	—
135	C₃²×C₄	—	2	2	no	—
136	C₃²×C₄	#41	1	2	yes	—
137	C₃×Dic₃	—	2	2	no	—
138	C₃×Dic₃	—	2	1	no	—
139	C₃×Dic₃	—	2	1	no	—
140	C₃×Dic₃	—	2	1	no	—
141	C₃×Dic₃	#41	1	2	yes	—
142	C₄⋊Q₈	—	3	1	no	—
143	#126	—	3	2	no	—
144	#126	—	3	2	no	—
145	#124	—	3	1	no	—
146	#124	—	3	1	no	—
147	C₂²⋊Q₈	—	3	2	no	—
148	C₄×Q₈	—	3	1	no	—
149	C₄×Q₈	—	3	1	no	—
150	C₄×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
151	C₄×D₄	—	3	1	no	—
152	#117	—	3	2	no	—
153	C₂³×C₃	—	3	2	no	—
154	C₂²×D₃	C₂³×C₃	1	2	yes	—
155	C₃×Q₈	—	2	4	no	—
156	C₃×Q₈	—	2	2	no	—
157	C₃×Q₈	—	2	2	no	—
158	C₃×Q₈	—	6	2	no	—
159	C₃×Q₈	—	6	2	no	—
160	C₃×D₄	—	6	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	3	1	no	—
167	C₂×C₃×C₄	—	3	1	no	—
168	C₂×C₃×C₄	—	3	1	no	—
169	C₂×C₃×C₄	—	3	2	no	—
170	C₂×C₃×C₄	—	3	1	no	—
171	C₂×C₃×C₄	—	6	4	no	—
172	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
173	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
175	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
178	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
180	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
182	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
183	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
184	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
185	C₂×Dic₃	C₂³×C₃	1	2	yes	—
186	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
187	D₁₂	—	2	2	no	—
188	C₄×D₃	—	2	4	no	—
189	Dic₆	—	2	2	no	—
190	Dic₆	—	2	2	no	—
191	C₂×C₃²	C₂×#41	1	1	yes	characteristic
192	C₂×C₃²	C₂×#41	1	1	yes	characteristic
193	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
194	C₃×D₃	—	4	2	no	—
195	C₂×Q₈	—	3	1	no	—
196	C₂×Q₈	—	3	1	no	—
197	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
198	C₂×D₄	—	3	1	no	—
199	C₂²×C₄	—	3	2	no	—
200	C₄⋊C₄	—	3	1	no	—
201	C₄⋊C₄	—	3	2	no	—
202	C₄⋊C₄	—	3	2	no	—
203	C₄⋊C₄	—	3	1	no	—
204	C₄⋊C₄	—	3	1	no	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
207	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
208	C₂²⋊C₄	—	3	2	no	—
209	C₂²⋊C₄	—	3	2	no	—
210	C₂²⋊C₄	—	3	2	no	—
211	C₂²⋊C₄	—	3	2	no	—
212	C₂²⋊C₄	—	3	2	no	—
213	C₄²	—	3	2	no	—
214	C₄²	—	3	1	no	—
215	C₄²	—	3	1	no	—
216	C₄²	C₂×C₃×D₃	1	2	yes	—
217	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	6	2	no	—
220	C₂²×C₃	—	6	2	no	—
221	C₂²×C₃	—	6	2	no	—
222	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
223	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
224	C₂×D₃	—	2	2	no	—
225	C₂×D₃	—	2	2	no	—
226	C₂×D₃	—	2	2	no	—
227	C₃×C₄	—	2	4	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	2	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	3	2	no	—
239	C₃×C₄	—	4	2	no	—
240	C₃×C₄	—	4	1	no	—
241	C₃×C₄	—	6	2	no	—
242	C₃×C₄	—	6	1	no	—
243	C₃×C₄	—	6	1	no	—
244	C₃×C₄	—	6	1	no	—
245	C₃×C₄	#235	1	2	yes	—
246	C₃×C₄	C₃×#41	1	2	yes	—
247	Dic₃	—	2	2	no	—
248	Dic₃	—	2	1	no	—
249	Dic₃	—	2	1	no	—
250	Dic₃	—	2	1	no	—
251	Dic₃	C₃×#41	1	2	yes	—
252	C₃²	#556	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	3	2	no	—
254	Q₈	—	2	2	no	—
255	Q₈	—	6	2	no	—
256	Q₈	—	6	2	no	—
257	D₄	—	6	2	no	—
258	C₂×C₄	—	3	1	no	—
259	C₂×C₄	—	3	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	2	no	—
262	C₂×C₄	—	3	1	no	—
263	C₂×C₄	—	6	4	no	—
264	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
265	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
267	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	2	1	no	—
272	C₂×C₃	—	12	2	no	—
273	C₂×C₃	C₂×#235	1	1	yes	characteristic
274	C₂×C₃	#1007	1	1	yes	characteristic
275	C₂×C₃	D₃×#41	1	1	yes	characteristic
276	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
277	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
278	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
279	D₃	—	4	2	no	—
280	C₂²	—	6	2	no	—
281	C₂²	—	6	2	no	—
282	C₂²	—	6	2	no	—
283	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	1	no	—
288	C₄	—	3	2	no	—
289	C₄	—	6	2	no	—
290	C₄	—	6	1	no	—
291	C₄	—	6	1	no	—
292	C₄	—	6	1	no	—
293	C₄	C₃×#235	1	2	yes	—
294	C₃	—	2	1	no	—
295	C₃	#5568	1	1	yes	characteristic, direct product factor
296	C₃	C₃×#556	1	1	yes	characteristic, complemented
297	C₂	—	12	2	no	—
298	C₂	C₂×C₃×#235	1	1	yes	characteristic
299	C₂	C₃×#1007	1	1	yes	characteristic
300	C₂	C₃×D₃×#41	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×C₄×Q₈	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×D₁₂	C₂²
hypercenter	C₃×C₄×Q₈	D₃
hypocenter	C₃	C₃×#556
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
1. C₃²×C₄×Q₈	C₂	semidirect
3. C₃×#944	C₂	semidirect
4. C₃×#898	C₂	semidirect
5. C₃×#896	C₂	semidirect
7. C₃×#887	C₂	semidirect
8. C₃×#876	C₂	semidirect
10. C₃×#872	C₂	semidirect
11. C₃×C₄×Dic₆	C₂	semidirect
13. #5568	C₃	direct (N × H)
46. C₃×C₄×Q₈	D₃	semidirect
47. C₃×C₄×Q₈	C₂×C₃	semidirect
51. #944	C₂×C₃	semidirect
52. #898	C₂×C₃	semidirect
53. #896	C₂×C₃	semidirect
55. #887	C₂×C₃	semidirect
56. #876	C₂×C₃	semidirect
58. #872	C₂×C₃	semidirect
59. C₄×Dic₆	C₂×C₃	semidirect
150. C₄×Q₈	C₃×D₃	semidirect
252. C₃²	#556	semidirect
295. C₃	#5568	direct (N × H)
296. C₃	C₃×#556	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄×Q₈	2
C₃×#947	2
C₃×#944	2
C₃×#898	2
C₃×#896	2
C₃×#893	2
C₃×#887	2
C₃×#876	2
C₃×#875	2
C₃×#872	2
C₃×C₄×Dic₆	2
C₃×#556	3
#5568	3

Subgroups — C3×#5568 #10595665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5568 #10595665