Subgroups — C₃×#5668 #10595765

Order 576.

658 conjugacy classes of subgroups.

The subgroup lattice has 293 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, complemented
2	C₂×C₃×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#944	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#940	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#884	C₂	1	4	yes	maximal, complemented
7	C₃×#67⋊C₄	C₂	1	4	yes	maximal, complemented
8	C₃×#875	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×C₄²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#532	—	3	1	no	maximal
11	#5668	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×#67	C₂²	1	4	yes	—
15	C₂×C₃×#67	C₂²	1	2	yes	—
16	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
17	C₃×#239	—	2	4	no	—
18	C₃×#237	—	2	4	no	—
19	C₂×C₃×D₁₂	C₂²	1	2	yes	—
20	C₂×C₃×D₁₂	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
22	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
23	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
24	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
25	C₃²×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₃²×C₄²	C₂²	1	1	yes	characteristic
27	C₃×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
28	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
29	C₃×C₄×Dic₃	C₂²	1	2	yes	—
30	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₃×#41	—	3	1	no	—
32	C₂×C₃×#41	—	3	1	no	—
33	C₂×#239	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#237	C₂×C₃	1	1	yes	characteristic, complemented
35	C₃×C₄⋊Q₈	—	3	1	no	—
36	C₃×#127	—	3	1	no	—
37	C₃×#124	—	2	1	no	—
38	C₃×#124	—	3	1	no	—
39	C₃×#124	D₃	1	1	yes	characteristic, complemented
40	C₃×#124	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×#121	—	3	4	no	—
42	C₃×C₄×D₄	—	3	4	no	—
43	C₂×C₃×C₄²	—	3	1	no	—
44	#944	C₂×C₃	1	1	yes	characteristic, complemented
45	#940	C₂×C₃	1	1	yes	characteristic, complemented
46	#884	C₂×C₃	1	4	yes	complemented
47	#67⋊C₄	C₂×C₃	1	4	yes	complemented
48	#875	C₂×C₃	1	1	yes	characteristic, complemented
49	C₄²×D₃	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂³×C₃²	C₂³	1	2	yes	—
51	C₂²×C₃×D₃	C₂³	1	2	yes	—
52	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
53	C₃²×Q₈	—	2	2	no	—
54	C₃²×D₄	—	2	2	no	—
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	C₃×#67	—	2	4	no	—
59	C₃×#67	—	2	8	no	—
60	C₂×C₃×Dic₃	—	2	4	no	—
61	C₂×C₃×Dic₃	C₂³	1	2	yes	—
62	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
63	C₂×C₃×Dic₃	C₂³	1	2	yes	—
64	C₃×D₁₂	—	2	4	no	—
65	C₃×D₁₂	—	2	2	no	—
66	C₃×C₄×D₃	—	2	4	no	—
67	C₃×C₄×D₃	—	2	4	no	—
68	C₃×C₄×D₃	D₄	1	4	yes	complemented
69	C₃×Dic₆	—	2	2	no	—
70	#532	—	3	1	no	Sylow
71	C₃×#41	—	6	4	no	—
72	C₃×#41	—	6	4	no	—
73	C₂×C₃×Q₈	—	2	1	no	—
74	C₂×C₃×Q₈	—	3	1	no	—
75	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
76	C₂×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×D₄	—	2	1	no	—
78	C₂×C₃×D₄	—	3	4	no	—
79	C₂×C₃×D₄	—	3	2	no	—
80	C₂×C₃×D₄	—	3	2	no	—
81	C₂×C₃×D₄	—	3	1	no	—
82	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
83	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	—	3	1	no	—
85	C₂²×C₃×C₄	—	3	2	no	—
86	C₂²×C₃×C₄	—	3	2	no	—
87	C₂²×C₃×C₄	—	3	2	no	—
88	C₂×#67	C₂²×C₃	1	4	yes	—
89	C₂×#67	C₂²×C₃	1	2	yes	—
90	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
91	#239	—	2	4	no	—
92	#237	—	2	4	no	—
93	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
94	C₂×D₁₂	C₂²×C₃	1	2	yes	—
95	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
96	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
97	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
98	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
99	C₃×C₄⋊C₄	—	3	4	no	—
100	C₃×C₂²⋊C₄	—	2	4	no	—
101	C₃×C₂²⋊C₄	—	3	4	no	—
102	C₃×C₂²⋊C₄	C₂×D₃	1	4	yes	—
103	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
104	C₃×C₄²	—	2	1	no	—
105	C₃×C₄²	—	3	2	no	—
106	C₃×C₄²	—	3	1	no	—
107	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
108	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
109	(C₂×D₃)⋊C₄	C₂²×C₃	1	4	yes	—
110	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
111	C₄×Dic₃	C₂²×C₃	1	2	yes	—
112	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
113	C₂²×C₃²	—	2	2	no	—
114	C₂²×C₃²	—	2	4	no	—
115	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
116	C₂×C₃×D₃	—	2	2	no	—
117	C₂×C₃×D₃	—	2	2	no	—
118	C₂×C₃×D₃	—	2	4	no	—
119	C₂×C₃×D₃	C₂×D₄	1	2	yes	—
120	C₃²×C₄	—	2	2	no	—
121	C₃²×C₄	—	2	2	no	—
122	C₃²×C₄	C₂×D₄	1	2	yes	—
123	C₃×Dic₃	—	2	2	no	—
124	C₃×Dic₃	C₂×D₄	1	2	yes	—
125	C₃×Dic₃	#41	1	4	yes	complemented
126	C₂×#41	—	3	1	no	—
127	C₂×#41	—	3	1	no	—
128	C₄⋊Q₈	—	3	1	no	—
129	#127	—	3	1	no	—
130	#124	—	3	1	no	—
131	#124	C₃×D₃	1	1	yes	characteristic, complemented
132	#121	—	3	4	no	—
133	C₄×D₄	—	3	4	no	—
134	C₂×C₄²	—	3	1	no	—
135	C₂³×C₃	—	2	2	no	—
136	C₂³×C₃	—	3	1	no	—
137	C₂³×C₃	—	3	2	no	—
138	C₂³×C₃	C₂²×D₃	1	2	yes	—
139	C₂³×C₃	C₂³×C₃	1	2	yes	—
140	C₂²×D₃	C₂³×C₃	1	2	yes	—
141	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
142	C₃×Q₈	—	2	2	no	—
143	C₃×Q₈	—	2	2	no	—
144	C₃×Q₈	—	4	2	no	—
145	C₃×Q₈	—	6	2	no	—
146	C₃×D₄	—	2	2	no	—
147	C₃×D₄	—	2	2	no	—
148	C₃×D₄	—	4	2	no	—
149	C₃×D₄	—	6	4	no	—
150	C₃×D₄	—	6	8	no	—
151	C₃×D₄	—	6	4	no	—
152	C₃×D₄	—	6	2	no	—
153	C₂×C₃×C₄	—	2	1	no	—
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	3	2	no	—
157	C₂×C₃×C₄	—	3	1	no	—
158	C₂×C₃×C₄	—	3	4	no	—
159	C₂×C₃×C₄	—	3	2	no	—
160	C₂×C₃×C₄	—	6	4	no	—
161	C₂×C₃×C₄	—	6	4	no	—
162	C₂×C₃×C₄	—	6	4	no	—
163	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
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	2	yes	—
168	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
169	#67	—	2	4	no	—
170	#67	—	2	8	no	—
171	C₂×Dic₃	—	2	4	no	—
172	C₂×Dic₃	C₂³×C₃	1	2	yes	—
173	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
174	C₂×Dic₃	C₂³×C₃	1	2	yes	—
175	D₁₂	—	2	4	no	—
176	D₁₂	—	2	2	no	—
177	C₄×D₃	—	2	4	no	—
178	C₄×D₃	—	2	4	no	—
179	C₄×D₃	C₃×D₄	1	4	yes	complemented
180	Dic₆	—	2	2	no	—
181	C₂×C₃²	—	4	2	no	—
182	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
183	C₂×C₃²	C₂×#41	1	2	yes	—
184	C₃×D₃	—	2	2	no	—
185	C₃×D₃	—	4	2	no	—
186	#41	—	6	4	no	—
187	#41	—	6	4	no	—
188	C₂×Q₈	—	3	1	no	—
189	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
190	C₂×D₄	—	3	4	no	—
191	C₂×D₄	—	3	2	no	—
192	C₂×D₄	—	3	2	no	—
193	C₂×D₄	—	3	1	no	—
194	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
195	C₂²×C₄	—	3	1	no	—
196	C₂²×C₄	—	3	2	no	—
197	C₂²×C₄	—	3	2	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₄⋊C₄	—	3	4	no	—
200	C₂²⋊C₄	—	3	4	no	—
201	C₂²⋊C₄	C₂×C₃×D₃	1	4	yes	—
202	C₄²	—	3	2	no	—
203	C₄²	—	3	1	no	—
204	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	4	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	4	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	3	2	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	4	no	—
213	C₂²×C₃	—	6	2	no	—
214	C₂²×C₃	—	6	2	no	—
215	C₂²×C₃	—	6	4	no	—
216	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
217	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
218	C₂×D₃	—	2	2	no	—
219	C₂×D₃	—	2	2	no	—
220	C₂×D₃	—	2	4	no	—
221	C₂×D₃	C₂×C₃×D₄	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	3	2	no	—
228	C₃×C₄	—	3	4	no	—
229	C₃×C₄	—	4	2	no	—
230	C₃×C₄	—	4	2	no	—
231	C₃×C₄	—	6	2	no	—
232	C₃×C₄	D₃×D₄	1	2	yes	—
233	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
234	Dic₃	—	2	2	no	—
235	Dic₃	C₂×C₃×D₄	1	2	yes	—
236	Dic₃	C₃×#41	1	4	yes	complemented
237	C₃²	#532	1	1	yes	characteristic, complemented, Sylow
238	C₂³	—	3	1	no	—
239	C₂³	—	3	2	no	—
240	C₂³	C₂²×C₃×D₃	1	2	yes	—
241	Q₈	—	2	2	no	—
242	Q₈	—	6	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	6	4	no	—
245	D₄	—	6	8	no	—
246	D₄	—	6	4	no	—
247	D₄	—	6	2	no	—
248	C₂×C₄	—	3	2	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	4	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	6	4	no	—
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	—	6	4	no	—
255	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
256	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
257	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	—	4	2	no	—
262	C₂×C₃	—	6	2	no	—
263	C₂×C₃	—	8	2	no	—
264	C₂×C₃	—	12	2	no	—
265	C₂×C₃	C₂×D₃×D₄	1	1	yes	characteristic
266	C₂×C₃	D₃×#41	1	2	yes	—
267	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
268	C₂×C₃	C₂×C₃×#41	1	2	yes	—
269	D₃	—	2	2	no	—
270	D₃	—	4	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	3	2	no	—
274	C₂²	—	6	2	no	—
275	C₂²	—	6	2	no	—
276	C₂²	—	6	4	no	—
277	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	3	2	no	—
281	C₄	—	3	4	no	—
282	C₄	—	6	2	no	—
283	C₄	C₃×D₃×D₄	1	2	yes	—
284	C₃	—	2	1	no	—
285	C₃	#5668	1	1	yes	characteristic, direct product factor
286	C₃	C₃×#532	1	1	yes	characteristic, complemented
287	C₂	—	4	2	no	—
288	C₂	—	6	2	no	—
289	C₂	—	12	2	no	—
290	C₂	C₂×C₃×D₃×D₄	1	1	yes	characteristic
291	C₂	C₃×D₃×#41	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×#124	C₂
socle	C₂²×C₃²	C₂⁴
hypercenter	C₃×#124	D₃
hypocenter	C₃	C₃×#532
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#239	C₂	semidirect
2. C₂×C₃×#237	C₂	semidirect
3. C₃²×#124	C₂	semidirect
4. C₃×#944	C₂	semidirect
5. C₃×#940	C₂	semidirect
6. C₃×#884	C₂	semidirect
7. C₃×#67⋊C₄	C₂	semidirect
8. C₃×#875	C₂	semidirect
9. C₃×C₄²×D₃	C₂	semidirect
11. #5668	C₃	direct (N × H)
33. C₂×#239	C₂×C₃	semidirect
34. C₂×#237	C₂×C₃	semidirect
39. C₃×#124	D₃	semidirect
40. C₃×#124	C₂×C₃	semidirect
44. #944	C₂×C₃	semidirect
45. #940	C₂×C₃	semidirect
46. #884	C₂×C₃	semidirect
47. #67⋊C₄	C₂×C₃	semidirect
48. #875	C₂×C₃	semidirect
49. C₄²×D₃	C₂×C₃	semidirect
68. C₃×C₄×D₃	D₄	semidirect
125. C₃×Dic₃	#41	semidirect
131. #124	C₃×D₃	semidirect
179. C₄×D₃	C₃×D₄	semidirect
236. Dic₃	C₃×#41	semidirect
237. C₃²	#532	semidirect
285. C₃	#5668	direct (N × H)
286. C₃	C₃×#532	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×#237	2
C₃²×#124	2
C₃×#944	2
C₃×#940	2
C₃×#884	2
C₃×#67⋊C₄	2
C₃×#875	2
C₃×C₄²×D₃	2
C₃×#532	3
#5668	3

Subgroups — C3×#5668 #10595765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5668 #10595765