Subgroups — C₃×#5664 #10595761

Order 576.

511 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#940	C₂	1	1	yes	maximal, characteristic
3	C₃×#935	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×#67	C₂	1	2	yes	maximal, complemented
5	C₃×#894	C₂	1	2	yes	maximal, complemented
6	C₃×#887	C₂	1	2	yes	maximal, complemented
7	C₃×#885	C₂	1	2	yes	maximal, complemented
8	C₃×#885	C₂	1	2	yes	maximal, complemented
9	C₃×#67⋊C₄	C₂	1	2	yes	maximal, complemented
10	C₃×#559	—	3	1	no	maximal
11	#5664	C₃	1	1	yes	maximal, characteristic, direct product factor
12	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
14	C₂×C₃×#67	C₂²	1	2	yes	—
15	C₂×C₃×#67	C₂²	1	2	yes	—
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	C₂×C₃×Dic₆	C₂²	1	2	yes	—
20	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×#217	C₂²	1	1	yes	characteristic
24	C₃×#217	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×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄×Dic₃	C₂²	1	2	yes	—
31	C₃×C₄×Dic₃	C₂²	1	2	yes	—
32	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
33	C₃×#127	—	3	1	no	—
34	C₃×#124	—	3	2	no	—
35	C₃×#124	—	3	2	no	—
36	C₃×#123	—	2	1	no	—
37	C₃×#123	—	3	1	no	—
38	C₃×#123	—	3	2	no	—
39	C₃×#123	D₃	1	1	yes	characteristic, complemented
40	C₃×#123	C₂×C₃	1	1	yes	characteristic, complemented
41	C₃×C₄×Q₈	—	3	2	no	—
42	C₃×C₄×D₄	—	3	2	no	—
43	C₃×C₄×D₄	—	3	2	no	—
44	#940	C₂×C₃	1	1	yes	characteristic
45	#935	C₂×C₃	1	1	yes	characteristic, complemented
46	C₄×#67	C₂×C₃	1	2	yes	complemented
47	#894	C₂×C₃	1	2	yes	complemented
48	#887	C₂×C₃	1	2	yes	complemented
49	#885	C₂×C₃	1	2	yes	complemented
50	#885	C₂×C₃	1	2	yes	complemented
51	#67⋊C₄	C₂×C₃	1	2	yes	complemented
52	C₂³×C₃²	C₂³	1	1	yes	characteristic
53	C₂³×C₃²	C₂³	1	1	yes	characteristic
54	C₂²×C₃×D₃	C₂³	1	2	yes	—
55	C₃²×D₄	—	4	1	no	—
56	C₂×C₃²×C₄	—	4	1	no	—
57	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
58	C₂×C₃²×C₄	C₂³	1	2	yes	—
59	C₂×C₃²×C₄	C₂³	1	2	yes	—
60	C₃×#67	—	2	4	no	—
61	C₃×#67	—	2	4	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	C₃×Dic₆	—	2	4	no	—
69	#559	—	3	1	no	Sylow
70	C₂×C₃×Q₈	—	3	2	no	—
71	C₂×C₃×D₄	—	2	1	no	—
72	C₂×C₃×D₄	—	3	2	no	—
73	C₂×C₃×D₄	—	3	2	no	—
74	C₂×C₃×D₄	—	3	1	no	—
75	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
76	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₃×C₄	—	2	1	no	—
78	C₂²×C₃×C₄	—	3	1	no	—
79	C₂²×C₃×C₄	—	3	2	no	—
80	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
82	C₂×#67	C₂²×C₃	1	2	yes	—
83	C₂×#67	C₂²×C₃	1	2	yes	—
84	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
85	C₂×D₁₂	C₂²×C₃	1	1	yes	characteristic
86	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
87	C₂×Dic₆	C₂²×C₃	1	2	yes	—
88	C₃×C₄⋊C₄	—	2	2	no	—
89	C₃×C₄⋊C₄	—	3	2	no	—
90	C₃×C₄⋊C₄	—	3	2	no	—
91	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
92	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
93	C₃×C₂²⋊C₄	—	2	2	no	—
94	C₃×C₂²⋊C₄	—	2	1	no	—
95	C₃×C₂²⋊C₄	—	3	2	no	—
96	C₃×C₂²⋊C₄	—	3	1	no	—
97	C₃×C₂²⋊C₄	—	3	2	no	—
98	C₃×C₂²⋊C₄	—	3	2	no	—
99	C₃×C₂²⋊C₄	—	3	2	no	—
100	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
101	C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
102	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
103	C₃×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
104	C₃×C₄²	—	3	2	no	—
105	C₃×C₄²	—	3	2	no	—
106	C₃×C₄²	—	3	1	no	—
107	#217	C₂²×C₃	1	1	yes	characteristic
108	#217	C₂²×C₃	1	2	yes	—
109	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
110	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
111	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
112	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
113	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
114	C₄×Dic₃	C₂²×C₃	1	2	yes	—
115	C₄×Dic₃	C₂²×C₃	1	2	yes	—
116	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
117	C₂²×C₃²	—	2	1	no	—
118	C₂²×C₃²	—	2	2	no	—
119	C₂²×C₃²	—	2	2	no	—
120	C₂²×C₃²	—	2	1	no	—
121	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
122	C₂×C₃×D₃	—	2	2	no	—
123	C₂×C₃×D₃	—	2	2	no	—
124	C₂×C₃×D₃	—	2	2	no	—
125	C₃²×C₄	—	2	2	no	—
126	C₃²×C₄	—	2	2	no	—
127	C₃²×C₄	—	2	1	no	—
128	C₃×Dic₃	—	2	2	no	—
129	C₃×Dic₃	—	2	2	no	—
130	C₃×Dic₃	#41	1	4	yes	—
131	#127	—	3	1	no	—
132	#124	—	3	2	no	—
133	#124	—	3	2	no	—
134	#123	—	3	1	no	—
135	#123	—	3	2	no	—
136	#123	C₃×D₃	1	1	yes	characteristic, complemented
137	C₄×Q₈	—	3	2	no	—
138	C₄×D₄	—	3	2	no	—
139	C₄×D₄	—	3	2	no	—
140	C₂³×C₃	—	2	1	no	—
141	C₂³×C₃	—	2	1	no	—
142	C₂³×C₃	—	3	2	no	—
143	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
144	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
145	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
146	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
147	C₂²×D₃	C₂³×C₃	1	2	yes	—
148	C₃×Q₈	—	6	4	no	—
149	C₃×D₄	—	4	1	no	—
150	C₃×D₄	—	4	1	no	—
151	C₃×D₄	—	4	2	no	—
152	C₃×D₄	—	6	4	no	—
153	C₃×D₄	—	6	4	no	—
154	C₃×D₄	—	12	1	no	—
155	C₂×C₃×C₄	—	2	2	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	3	2	no	—
159	C₂×C₃×C₄	—	3	2	no	—
160	C₂×C₃×C₄	—	3	2	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	1	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	—	12	2	no	—
165	C₂×C₃×C₄	—	12	1	no	—
166	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
167	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
168	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
171	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
172	#67	—	2	4	no	—
173	#67	—	2	4	no	—
174	C₂×Dic₃	—	4	1	no	—
175	C₂×Dic₃	C₂³×C₃	1	2	yes	—
176	C₂×Dic₃	C₂³×C₃	1	2	yes	—
177	C₂×Dic₃	C₂³×C₃	1	2	yes	—
178	D₁₂	—	4	1	no	—
179	C₄×D₃	—	4	2	no	—
180	Dic₆	—	2	4	no	—
181	C₂×C₃²	—	4	1	no	—
182	C₂×C₃²	—	4	1	no	—
183	C₂×C₃²	C₂×#41	1	2	yes	—
184	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
185	C₃×D₃	—	4	2	no	—
186	C₂×Q₈	—	3	2	no	—
187	C₂×D₄	—	3	2	no	—
188	C₂×D₄	—	3	2	no	—
189	C₂×D₄	—	3	1	no	—
190	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
191	C₂²×C₄	—	3	1	no	—
192	C₂²×C₄	—	3	2	no	—
193	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
194	C₄⋊C₄	—	3	2	no	—
195	C₄⋊C₄	—	3	2	no	—
196	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
197	C₂²⋊C₄	—	3	2	no	—
198	C₂²⋊C₄	—	3	1	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	C₂²⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
204	C₄²	—	3	2	no	—
205	C₄²	—	3	2	no	—
206	C₄²	—	3	1	no	—
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	—	4	1	no	—
219	C₂²×C₃	—	4	2	no	—
220	C₂²×C₃	—	6	2	no	—
221	C₂²×C₃	—	6	2	no	—
222	C₂²×C₃	—	6	2	no	—
223	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
224	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
225	C₂×D₃	—	2	2	no	—
226	C₂×D₃	—	2	2	no	—
227	C₂×D₃	—	2	2	no	—
228	C₃×C₄	—	2	2	no	—
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	2	no	—
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	2	2	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	3	4	no	—
235	C₃×C₄	—	4	2	no	—
236	C₃×C₄	—	4	1	no	—
237	C₃×C₄	—	4	2	no	—
238	C₃×C₄	—	6	2	no	—
239	C₃×C₄	—	6	2	no	—
240	Dic₃	—	2	2	no	—
241	Dic₃	—	2	2	no	—
242	Dic₃	C₃×#41	1	4	yes	—
243	C₃²	#559	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	3	2	no	—
245	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
246	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
247	Q₈	—	6	4	no	—
248	D₄	—	4	1	no	—
249	D₄	—	6	4	no	—
250	D₄	—	6	4	no	—
251	D₄	—	12	1	no	—
252	C₂×C₄	—	3	2	no	—
253	C₂×C₄	—	3	2	no	—
254	C₂×C₄	—	3	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	12	2	no	—
257	C₂×C₄	—	12	1	no	—
258	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
259	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
260	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
261	C₂×C₃	—	2	1	no	—
262	C₂×C₃	—	2	2	no	—
263	C₂×C₃	—	4	2	no	—
264	C₂×C₃	—	4	1	no	—
265	C₂×C₃	—	4	1	no	—
266	C₂×C₃	—	4	1	no	—
267	C₂×C₃	—	4	1	no	—
268	C₂×C₃	—	8	1	no	—
269	C₂×C₃	—	12	2	no	—
270	C₂×C₃	#1004	1	1	yes	characteristic
271	C₂×C₃	D₃×#41	1	2	yes	—
272	C₂×C₃	C₂×C₃×#41	1	2	yes	—
273	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
274	D₃	—	4	2	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	6	2	no	—
280	C₂²	—	6	2	no	—
281	C₂²	—	6	2	no	—
282	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	3	4	no	—
287	C₄	—	6	2	no	—
288	C₄	—	6	2	no	—
289	C₃	—	2	1	no	—
290	C₃	#5664	1	1	yes	characteristic, direct product factor
291	C₃	C₃×#559	1	1	yes	characteristic, complemented
292	C₂	—	4	1	no	—
293	C₂	—	4	1	no	—
294	C₂	—	12	2	no	—
295	C₂	C₃×#1004	1	1	yes	characteristic
296	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×#123	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₃×#940	C₂
hypercenter	C₃×#123	D₃
hypocenter	C₃	C₃×#559
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#123

Complements

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

N	H ≅ G/N	type
1. C₃²×#123	C₂	semidirect
3. C₃×#935	C₂	semidirect
4. C₃×C₄×#67	C₂	semidirect
5. C₃×#894	C₂	semidirect
6. C₃×#887	C₂	semidirect
7. C₃×#885	C₂	semidirect
8. C₃×#885	C₂	semidirect
9. C₃×#67⋊C₄	C₂	semidirect
11. #5664	C₃	direct (N × H)
39. C₃×#123	D₃	semidirect
40. C₃×#123	C₂×C₃	semidirect
45. #935	C₂×C₃	semidirect
46. C₄×#67	C₂×C₃	semidirect
47. #894	C₂×C₃	semidirect
48. #887	C₂×C₃	semidirect
49. #885	C₂×C₃	semidirect
50. #885	C₂×C₃	semidirect
51. #67⋊C₄	C₂×C₃	semidirect
136. #123	C₃×D₃	semidirect
243. C₃²	#559	semidirect
290. C₃	#5664	direct (N × H)
291. C₃	C₃×#559	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#123	2
C₃×#940	2
C₃×#935	2
C₃×C₄×#67	2
C₃×#894	2
C₃×#887	2
C₃×#885	2
C₃×#67⋊C₄	2
C₃×#559	3
#5664	3

Subgroups — C3×#5664 #10595761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5664 #10595761