Subgroups — C₃×#5689 #10595786

Order 576.

440 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#897	C₂	1	2	yes	maximal
3	C₃×#896	C₂	1	2	yes	maximal
4	C₃×#892	C₂	1	1	yes	maximal, characteristic
5	C₃×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₃×#890	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#889	C₂	1	2	yes	maximal, complemented
8	C₃×#888	C₂	1	2	yes	maximal, complemented
9	C₃×#888	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#872	C₂	1	1	yes	maximal, characteristic
11	C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#544	—	3	1	no	maximal
13	#5689	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
15	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×Dic₆	C₂²	1	2	yes	—
17	C₂×C₃×Dic₆	C₂²	1	2	yes	—
18	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
20	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
21	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₃²×C₄²	C₂²	1	1	yes	characteristic
23	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₃×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
26	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
29	C₃×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
30	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
32	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
33	C₃×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
34	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
35	C₃×C₄×Dic₃	C₂²	1	2	yes	—
36	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	3	2	no	—
38	C₃×C₄⋊Q₈	—	3	1	no	—
39	C₃×C₄⋊Q₈	—	3	1	no	—
40	C₃×#125	—	2	1	no	—
41	C₃×#125	—	3	2	no	—
42	C₃×#125	—	3	1	no	—
43	C₃×#125	D₃	1	1	yes	characteristic, complemented
44	C₃×#125	C₂×C₃	1	1	yes	characteristic, complemented
45	C₃×C₂²⋊Q₈	—	3	2	no	—
46	C₃×C₂²⋊Q₈	—	3	2	no	—
47	C₃×#117	—	3	1	no	—
48	C₃×#117	—	3	1	no	—
49	C₂×C₃×C₄⋊C₄	—	3	1	no	—
50	#897	C₂×C₃	1	2	yes	—
51	#896	C₂×C₃	1	2	yes	—
52	#892	C₂×C₃	1	1	yes	characteristic
53	D₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
54	#890	C₂×C₃	1	1	yes	characteristic, complemented
55	#889	C₂×C₃	1	2	yes	complemented
56	#888	C₂×C₃	1	1	yes	characteristic, complemented
57	#888	C₂×C₃	1	2	yes	complemented
58	#872	C₂×C₃	1	1	yes	characteristic
59	#869	C₂×C₃	1	1	yes	characteristic, complemented
60	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂³	1	2	yes	—
62	C₂×C₃²×C₄	C₂³	1	2	yes	—
63	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
64	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
65	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
66	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
67	C₂×C₃×Dic₃	C₂³	1	2	yes	—
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	C₃×C₄×D₃	—	4	2	no	—
72	C₃×C₄×D₃	Q₈	1	4	yes	complemented
73	C₃×Dic₆	—	4	2	no	—
74	C₃×Dic₆	—	4	2	no	—
75	#544	—	3	1	no	Sylow
76	C₂×C₃×Q₈	—	3	2	no	—
77	C₂×C₃×Q₈	—	3	2	no	—
78	C₂²×C₃×C₄	—	3	1	no	—
79	C₂²×C₃×C₄	—	3	2	no	—
80	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
81	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
82	C₂×Dic₆	C₂²×C₃	1	2	yes	—
83	C₂×Dic₆	C₂²×C₃	1	2	yes	—
84	C₃×C₄⋊C₄	—	2	1	no	—
85	C₃×C₄⋊C₄	—	2	1	no	—
86	C₃×C₄⋊C₄	—	2	2	no	—
87	C₃×C₄⋊C₄	—	2	2	no	—
88	C₃×C₄⋊C₄	—	3	2	no	—
89	C₃×C₄⋊C₄	—	3	1	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	1	no	—
95	C₃×C₄⋊C₄	—	3	2	no	—
96	C₃×C₄⋊C₄	—	3	2	no	—
97	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
98	C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
99	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
100	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
101	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
102	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
103	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
104	C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
105	C₃×C₂²⋊C₄	—	3	1	no	—
106	C₃×C₂²⋊C₄	—	3	2	no	—
107	C₃×C₂²⋊C₄	—	3	1	no	—
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	—	3	2	no	—
110	C₃×C₄²	—	3	1	no	—
111	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
112	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
113	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
114	(C₂×D₃)⋊C₄	C₂²×C₃	1	1	yes	characteristic
115	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
116	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
117	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
118	C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
119	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
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	2	yes	—
127	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
128	C₂×C₃×D₃	—	2	1	no	—
129	C₂×C₃×D₃	—	2	1	no	—
130	C₂×C₃×D₃	C₂×Q₈	1	2	yes	—
131	C₃²×C₄	—	2	1	no	—
132	C₃²×C₄	—	2	2	no	—
133	C₃²×C₄	—	2	2	no	—
134	C₃²×C₄	—	2	1	no	—
135	C₃²×C₄	C₂×Q₈	1	2	yes	—
136	C₃×Dic₃	—	2	1	no	—
137	C₃×Dic₃	—	2	2	no	—
138	C₃×Dic₃	—	2	2	no	—
139	C₃×Dic₃	—	2	1	no	—
140	C₃×Dic₃	C₂×Q₈	1	2	yes	—
141	C₄⋊Q₈	—	3	2	no	—
142	C₄⋊Q₈	—	3	1	no	—
143	C₄⋊Q₈	—	3	1	no	—
144	#125	—	3	2	no	—
145	#125	—	3	1	no	—
146	#125	C₃×D₃	1	1	yes	characteristic, complemented
147	C₂²⋊Q₈	—	3	2	no	—
148	C₂²⋊Q₈	—	3	2	no	—
149	#117	—	3	1	no	—
150	#117	—	3	1	no	—
151	C₂×C₄⋊C₄	—	3	1	no	—
152	C₂³×C₃	—	3	1	no	—
153	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
154	C₃×Q₈	—	12	2	no	—
155	C₃×Q₈	—	12	2	no	—
156	C₂×C₃×C₄	—	2	1	no	—
157	C₂×C₃×C₄	—	2	1	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	2	2	no	—
161	C₂×C₃×C₄	—	3	1	no	—
162	C₂×C₃×C₄	—	3	4	no	—
163	C₂×C₃×C₄	—	3	1	no	—
164	C₂×C₃×C₄	—	3	1	no	—
165	C₂×C₃×C₄	—	3	2	no	—
166	C₂×C₃×C₄	—	3	2	no	—
167	C₂×C₃×C₄	—	12	2	no	—
168	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
169	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
172	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
174	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
175	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
176	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
178	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
179	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
180	C₂×Dic₃	C₂³×C₃	1	2	yes	—
181	C₂×Dic₃	C₂³×C₃	1	2	yes	—
182	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
183	C₄×D₃	—	4	2	no	—
184	C₄×D₃	C₃×Q₈	1	4	yes	complemented
185	Dic₆	—	4	2	no	—
186	Dic₆	—	4	2	no	—
187	C₂×C₃²	C₂²×Q₈	1	1	yes	characteristic
188	C₂×C₃²	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₃²	2₋¹⁺⁴	1	1	yes	characteristic
190	C₃×D₃	—	2	2	no	—
191	C₂×Q₈	—	3	2	no	—
192	C₂×Q₈	—	3	2	no	—
193	C₂²×C₄	—	3	1	no	—
194	C₂²×C₄	—	3	2	no	—
195	C₄⋊C₄	—	3	2	no	—
196	C₄⋊C₄	—	3	1	no	—
197	C₄⋊C₄	—	3	1	no	—
198	C₄⋊C₄	—	3	1	no	—
199	C₄⋊C₄	—	3	1	no	—
200	C₄⋊C₄	—	3	2	no	—
201	C₄⋊C₄	—	3	2	no	—
202	C₄⋊C₄	—	3	2	no	—
203	C₄⋊C₄	—	3	2	no	—
204	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
205	C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
206	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
207	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
208	C₂²⋊C₄	—	3	1	no	—
209	C₂²⋊C₄	—	3	2	no	—
210	C₂²⋊C₄	—	3	1	no	—
211	C₄²	—	3	2	no	—
212	C₄²	—	3	1	no	—
213	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
214	C₂²×C₃	—	2	1	no	—
215	C₂²×C₃	—	3	2	no	—
216	C₂²×C₃	—	6	1	no	—
217	C₂²×C₃	—	6	1	no	—
218	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
219	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
220	C₂×D₃	—	2	1	no	—
221	C₂×D₃	—	2	1	no	—
222	C₂×D₃	C₂×C₃×Q₈	1	2	yes	—
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	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	—	3	2	no	—
233	C₃×C₄	—	4	2	no	—
234	C₃×C₄	—	4	1	no	—
235	C₃×C₄	—	4	1	no	—
236	C₃×C₄	—	4	2	no	—
237	C₃×C₄	—	6	2	no	—
238	C₃×C₄	—	6	1	no	—
239	C₃×C₄	—	6	2	no	—
240	C₃×C₄	—	6	1	no	—
241	C₃×C₄	D₃×Q₈	1	2	yes	—
242	C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
243	Dic₃	—	2	1	no	—
244	Dic₃	—	2	2	no	—
245	Dic₃	—	2	2	no	—
246	Dic₃	—	2	1	no	—
247	Dic₃	C₂×C₃×Q₈	1	2	yes	—
248	C₃²	#544	1	1	yes	characteristic, complemented, Sylow
249	C₂³	—	3	1	no	—
250	Q₈	—	12	2	no	—
251	Q₈	—	12	2	no	—
252	C₂×C₄	—	3	1	no	—
253	C₂×C₄	—	3	4	no	—
254	C₂×C₄	—	3	1	no	—
255	C₂×C₄	—	3	1	no	—
256	C₂×C₄	—	3	2	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	12	2	no	—
259	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
260	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
263	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	1	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	—	6	2	no	—
268	C₂×C₃	C₂×D₃×Q₈	1	1	yes	characteristic
269	C₂×C₃	#1009	1	1	yes	characteristic
270	C₂×C₃	#1010	1	1	yes	characteristic
271	C₂×C₃	C₂²×C₃×Q₈	1	1	yes	characteristic
272	C₂×C₃	C₃×H(5,𝔽₂)	1	1	yes	characteristic
273	C₂×C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic
274	D₃	—	2	2	no	—
275	C₂²	—	3	2	no	—
276	C₂²	—	6	1	no	—
277	C₂²	—	6	1	no	—
278	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	1	no	—
283	C₄	—	3	2	no	—
284	C₄	—	6	2	no	—
285	C₄	—	6	1	no	—
286	C₄	—	6	2	no	—
287	C₄	—	6	1	no	—
288	C₄	C₃×D₃×Q₈	1	2	yes	—
289	C₃	—	2	1	no	—
290	C₃	#5689	1	1	yes	characteristic, direct product factor
291	C₃	C₃×#544	1	1	yes	characteristic, complemented
292	C₂	—	6	2	no	—
293	C₂	C₂×C₃×D₃×Q₈	1	1	yes	characteristic
294	C₂	C₃×#1009	1	1	yes	characteristic
295	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×#125	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂²×C₃×D₃	C₂³
hypercenter	C₃×#125	D₃
hypocenter	C₃	C₃×#544
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#125

Complements

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

N	H ≅ G/N	type
1. C₃²×#125	C₂	semidirect
6. C₃×#890	C₂	semidirect
7. C₃×#889	C₂	semidirect
8. C₃×#888	C₂	semidirect
9. C₃×#888	C₂	semidirect
11. C₃×#869	C₂	semidirect
13. #5689	C₃	direct (N × H)
43. C₃×#125	D₃	semidirect
44. C₃×#125	C₂×C₃	semidirect
54. #890	C₂×C₃	semidirect
55. #889	C₂×C₃	semidirect
56. #888	C₂×C₃	semidirect
57. #888	C₂×C₃	semidirect
59. #869	C₂×C₃	semidirect
72. C₃×C₄×D₃	Q₈	semidirect
146. #125	C₃×D₃	semidirect
184. C₄×D₃	C₃×Q₈	semidirect
248. C₃²	#544	semidirect
290. C₃	#5689	direct (N × H)
291. C₃	C₃×#544	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#125	2
C₃×#897	2
C₃×#896	2
C₃×#892	2
C₃×D₃×C₄⋊C₄	2
C₃×#890	2
C₃×#889	2
C₃×#888	2
C₃×#872	2
C₃×#869	2
C₃×#544	3
#5689	3

Subgroups — C3×#5689 #10595786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5689 #10595786