Subgroups — C₃×#5682 #10595779

Order 576.

470 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#946	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#938	C₂	1	1	yes	maximal, characteristic
5	C₃×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#886	C₂	1	2	yes	maximal, complemented
7	C₃×#885	C₂	1	2	yes	maximal, complemented
8	C₃×#879	C₂	1	2	yes	maximal, complemented
9	C₃×#878	C₂	1	2	yes	maximal, complemented
10	C₃×#872	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#542	—	3	1	no	maximal
13	#5682	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×#67	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
18	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×Dic₆	C₂²	1	2	yes	—
20	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃²×C₄²	C₂²	1	1	yes	characteristic
23	C₃×#217	C₂²	1	2	yes	—
24	C₃×#217	C₂²	1	2	yes	—
25	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
26	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
30	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
31	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
32	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
33	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
34	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₂²	1	2	yes	—
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	3	1	no	—
38	C₃×#126	—	3	2	no	—
39	C₃×#124	—	2	1	no	—
40	C₃×#124	—	3	2	no	—
41	C₃×#124	D₃	1	1	yes	characteristic, complemented
42	C₃×#124	C₂×C₃	1	1	yes	characteristic, complemented
43	C₃×#123	—	3	2	no	—
44	C₃×C₂²⋊Q₈	—	3	1	no	—
45	C₃×C₂²⋊Q₈	—	3	2	no	—
46	C₃×#121	—	3	1	no	—
47	C₃×C₄×Q₈	—	3	1	no	—
48	C₃×C₄×D₄	—	3	1	no	—
49	C₃×#117	—	3	1	no	—
50	#946	C₂×C₃	1	1	yes	characteristic, complemented
51	Q₈×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
52	#938	C₂×C₃	1	1	yes	characteristic
53	D₄×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
54	#886	C₂×C₃	1	2	yes	complemented
55	#885	C₂×C₃	1	2	yes	complemented
56	#879	C₂×C₃	1	2	yes	complemented
57	#878	C₂×C₃	1	2	yes	complemented
58	#872	C₂×C₃	1	1	yes	characteristic, complemented
59	#869	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂³×C₃²	C₂³	1	2	yes	—
61	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
62	C₃²×Q₈	—	2	2	no	—
63	C₃²×D₄	—	2	2	no	—
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×C₃²×C₄	C₂³	1	1	yes	characteristic
68	C₃×#67	—	4	2	no	—
69	C₂×C₃×Dic₃	—	4	2	no	—
70	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
71	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
72	C₂×C₃×Dic₃	C₂³	1	2	yes	—
73	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
74	C₂×C₃×Dic₃	C₂³	1	2	yes	—
75	C₃×C₄×D₃	—	2	2	no	—
76	C₃×Dic₆	—	4	2	no	—
77	#542	—	3	1	no	Sylow
78	C₂×C₃×Q₈	—	2	1	no	—
79	C₂×C₃×Q₈	—	3	2	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	—	2	1	no	—
83	C₂×C₃×D₄	—	3	2	no	—
84	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
85	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
86	C₂²×C₃×C₄	—	3	2	no	—
87	C₂²×C₃×C₄	—	3	1	no	—
88	C₂×#67	C₂²×C₃	1	2	yes	—
89	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
90	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
91	C₂×Dic₆	C₂²×C₃	1	2	yes	—
92	C₃×C₄⋊C₄	—	3	1	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	—	3	1	no	—
97	C₃×C₄⋊C₄	—	3	2	no	—
98	C₃×C₄⋊C₄	—	3	2	no	—
99	C₃×C₂²⋊C₄	—	2	2	no	—
100	C₃×C₂²⋊C₄	—	2	2	no	—
101	C₃×C₂²⋊C₄	—	3	2	no	—
102	C₃×C₂²⋊C₄	—	3	2	no	—
103	C₃×C₂²⋊C₄	—	3	1	no	—
104	C₃×C₂²⋊C₄	—	3	1	no	—
105	C₃×C₂²⋊C₄	—	3	2	no	—
106	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
107	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
108	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
109	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
110	C₃×C₄²	—	2	1	no	—
111	C₃×C₄²	—	3	1	no	—
112	C₃×C₄²	—	3	2	no	—
113	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
114	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
115	#217	C₂²×C₃	1	2	yes	—
116	#217	C₂²×C₃	1	2	yes	—
117	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
118	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
119	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
120	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
121	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
122	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
123	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
124	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
125	Dic₃⋊C₄	C₂²×C₃	1	1	yes	characteristic
126	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
127	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
128	C₄×Dic₃	C₂²×C₃	1	2	yes	—
129	C₂²×C₃²	—	2	2	no	—
130	C₂²×C₃²	—	2	2	no	—
131	C₂²×C₃²	—	2	2	no	—
132	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
133	C₂×C₃×D₃	—	2	1	no	—
134	C₂×C₃×D₃	—	2	1	no	—
135	C₂×C₃×D₃	—	2	1	no	—
136	C₃²×C₄	—	2	2	no	—
137	C₃²×C₄	—	2	1	no	—
138	C₃²×C₄	—	2	1	no	—
139	C₃²×C₄	#41	1	2	yes	—
140	C₃×Dic₃	—	2	1	no	—
141	C₃×Dic₃	—	2	2	no	—
142	C₃×Dic₃	—	2	2	no	—
143	C₃×Dic₃	—	2	1	no	—
144	C₃×Dic₃	—	2	1	no	—
145	C₄⋊Q₈	—	3	1	no	—
146	#126	—	3	2	no	—
147	#124	—	3	2	no	—
148	#124	C₃×D₃	1	1	yes	characteristic, complemented
149	#123	—	3	2	no	—
150	C₂²⋊Q₈	—	3	1	no	—
151	C₂²⋊Q₈	—	3	2	no	—
152	#121	—	3	1	no	—
153	C₄×Q₈	—	3	1	no	—
154	C₄×D₄	—	3	1	no	—
155	#117	—	3	1	no	—
156	C₂³×C₃	—	2	2	no	—
157	C₂³×C₃	—	3	1	no	—
158	C₂³×C₃	C₂²×D₃	1	2	yes	—
159	C₂³×C₃	C₂³×C₃	1	2	yes	—
160	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
161	C₃×Q₈	—	2	2	no	—
162	C₃×Q₈	—	2	2	no	—
163	C₃×Q₈	—	4	2	no	—
164	C₃×Q₈	—	12	2	no	—
165	C₃×D₄	—	2	2	no	—
166	C₃×D₄	—	2	2	no	—
167	C₃×D₄	—	4	2	no	—
168	C₃×D₄	—	12	2	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	3	2	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	3	2	no	—
178	C₂×C₃×C₄	—	6	2	no	—
179	C₂×C₃×C₄	—	12	2	no	—
180	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
181	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
185	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
188	#67	—	4	2	no	—
189	C₂×Dic₃	—	4	2	no	—
190	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
191	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
192	C₂×Dic₃	C₂³×C₃	1	2	yes	—
193	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
194	C₂×Dic₃	C₂³×C₃	1	2	yes	—
195	C₄×D₃	—	2	2	no	—
196	Dic₆	—	4	2	no	—
197	C₂×C₃²	—	4	2	no	—
198	C₂×C₃²	C₂×#41	1	1	yes	characteristic
199	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
201	C₃×D₃	—	4	1	no	—
202	C₂×Q₈	—	3	2	no	—
203	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
204	C₂×D₄	—	3	2	no	—
205	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
206	C₂²×C₄	—	3	2	no	—
207	C₂²×C₄	—	3	1	no	—
208	C₄⋊C₄	—	3	1	no	—
209	C₄⋊C₄	—	3	1	no	—
210	C₄⋊C₄	—	3	1	no	—
211	C₄⋊C₄	—	3	1	no	—
212	C₄⋊C₄	—	3	2	no	—
213	C₄⋊C₄	—	3	2	no	—
214	C₄⋊C₄	—	3	2	no	—
215	C₂²⋊C₄	—	3	2	no	—
216	C₂²⋊C₄	—	3	2	no	—
217	C₂²⋊C₄	—	3	1	no	—
218	C₂²⋊C₄	—	3	1	no	—
219	C₂²⋊C₄	—	3	2	no	—
220	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
221	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
222	C₄²	—	3	1	no	—
223	C₄²	—	3	2	no	—
224	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	2	2	no	—
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	4	2	no	—
233	C₂²×C₃	—	4	2	no	—
234	C₂²×C₃	—	4	2	no	—
235	C₂²×C₃	—	6	1	no	—
236	C₂²×C₃	—	6	1	no	—
237	C₂²×C₃	—	6	1	no	—
238	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
239	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
240	C₂×D₃	—	2	1	no	—
241	C₂×D₃	—	2	1	no	—
242	C₂×D₃	—	2	1	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	1	no	—
248	C₃×C₄	—	2	1	no	—
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	—	4	2	no	—
251	C₃×C₄	—	4	1	no	—
252	C₃×C₄	—	4	1	no	—
253	C₃×C₄	—	6	1	no	—
254	C₃×C₄	—	6	1	no	—
255	C₃×C₄	—	6	2	no	—
256	C₃×C₄	—	6	2	no	—
257	C₃×C₄	—	6	1	no	—
258	C₃×C₄	#237	1	2	yes	—
259	C₃×C₄	C₃×#41	1	2	yes	—
260	Dic₃	—	2	1	no	—
261	Dic₃	—	2	2	no	—
262	Dic₃	—	2	2	no	—
263	Dic₃	—	2	1	no	—
264	Dic₃	—	2	1	no	—
265	C₃²	#542	1	1	yes	characteristic, complemented, Sylow
266	C₂³	—	3	1	no	—
267	C₂³	C₂²×C₃×D₃	1	2	yes	—
268	Q₈	—	2	2	no	—
269	Q₈	—	12	2	no	—
270	D₄	—	2	2	no	—
271	D₄	—	12	2	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	3	1	no	—
274	C₂×C₄	—	3	2	no	—
275	C₂×C₄	—	3	1	no	—
276	C₂×C₄	—	3	2	no	—
277	C₂×C₄	—	6	2	no	—
278	C₂×C₄	—	12	2	no	—
279	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
280	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
282	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	1	no	—
285	C₂×C₃	—	2	1	no	—
286	C₂×C₃	—	4	2	no	—
287	C₂×C₃	—	4	2	no	—
288	C₂×C₃	—	8	2	no	—
289	C₂×C₃	—	12	1	no	—
290	C₂×C₃	C₂×#237	1	1	yes	characteristic
291	C₂×C₃	#1009	1	1	yes	characteristic
292	C₂×C₃	#1010	1	1	yes	characteristic
293	C₂×C₃	C₂×C₃×#41	1	1	yes	characteristic
294	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
295	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
296	D₃	—	4	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	6	1	no	—
301	C₂²	—	6	1	no	—
302	C₂²	—	6	1	no	—
303	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	6	1	no	—
308	C₄	—	6	1	no	—
309	C₄	—	6	2	no	—
310	C₄	—	6	2	no	—
311	C₄	—	6	1	no	—
312	C₄	C₃×#237	1	2	yes	—
313	C₃	—	2	1	no	—
314	C₃	#5682	1	1	yes	characteristic, direct product factor
315	C₃	C₃×#542	1	1	yes	characteristic, complemented
316	C₂	—	4	2	no	—
317	C₂	—	12	1	no	—
318	C₂	C₂×C₃×#237	1	1	yes	characteristic
319	C₂	C₃×#1009	1	1	yes	characteristic
320	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×#124	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₃×#938	C₂
hypercenter	C₃×#124	D₃
hypocenter	C₃	C₃×#542
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. C₃²×#124	C₂	semidirect
2. C₃×#946	C₂	semidirect
3. C₃×Q₈×Dic₃	C₂	semidirect
5. C₃×D₄×Dic₃	C₂	semidirect
6. C₃×#886	C₂	semidirect
7. C₃×#885	C₂	semidirect
8. C₃×#879	C₂	semidirect
9. C₃×#878	C₂	semidirect
10. C₃×#872	C₂	semidirect
11. C₃×#869	C₂	semidirect
13. #5682	C₃	direct (N × H)
41. C₃×#124	D₃	semidirect
42. C₃×#124	C₂×C₃	semidirect
50. #946	C₂×C₃	semidirect
51. Q₈×Dic₃	C₂×C₃	semidirect
53. D₄×Dic₃	C₂×C₃	semidirect
54. #886	C₂×C₃	semidirect
55. #885	C₂×C₃	semidirect
56. #879	C₂×C₃	semidirect
57. #878	C₂×C₃	semidirect
58. #872	C₂×C₃	semidirect
59. #869	C₂×C₃	semidirect
148. #124	C₃×D₃	semidirect
265. C₃²	#542	semidirect
314. C₃	#5682	direct (N × H)
315. C₃	C₃×#542	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#124	2
C₃×#946	2
C₃×Q₈×Dic₃	2
C₃×#938	2
C₃×D₄×Dic₃	2
C₃×#886	2
C₃×#885	2
C₃×#879	2
C₃×#878	2
C₃×#872	2
C₃×#869	2
C₃×#542	3
#5682	3

Subgroups — C3×#5682 #10595779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5682 #10595779