Subgroups — C₃×#5675 #10595772

Order 576.

578 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#237	C₂	1	1	yes	maximal, characteristic
3	C₃²×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#944	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#936	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#883	C₂	1	4	yes	maximal, complemented
7	C₃×#878	C₂	1	4	yes	maximal, complemented
8	C₃×#872	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#869	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#536	—	3	1	no	maximal
11	#5675	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	2	yes	—
15	C₂²×C₃×Dic₃	C₂²	1	2	yes	—
16	C₃×D₃×Q₈	—	2	4	no	—
17	C₃×#237	—	2	4	no	—
18	C₂×C₃×C₄×D₃	C₂²	1	2	yes	—
19	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₆	C₂²	1	2	yes	—
21	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
22	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
23	C₃²×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₃²×C₄²	C₂²	1	1	yes	characteristic
25	C₃×#217	C₂²	1	4	yes	—
26	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
28	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₃×Dic₃⋊C₄	C₂²	1	4	yes	—
30	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₃×#41	—	3	1	no	—
32	C₂²×C₃×Q₈	—	3	1	no	—
33	C₂×D₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
34	C₂×#237	C₂×C₃	1	1	yes	characteristic
35	C₃×C₄⋊Q₈	—	3	1	no	—
36	C₃×C₄⋊Q₈	—	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×#123	—	3	4	no	—
42	C₃×C₂²⋊Q₈	—	3	4	no	—
43	C₃×#117	—	3	1	no	—
44	#944	C₂×C₃	1	1	yes	characteristic, complemented
45	#936	C₂×C₃	1	1	yes	characteristic, complemented
46	#883	C₂×C₃	1	4	yes	complemented
47	#878	C₂×C₃	1	4	yes	complemented
48	#872	C₂×C₃	1	1	yes	characteristic, complemented
49	#869	C₂×C₃	1	1	yes	characteristic, complemented
50	C₂³×C₃²	C₂³	1	2	yes	—
51	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
52	C₃²×Q₈	—	2	2	no	—
53	C₃²×D₄	—	2	2	no	—
54	C₂×C₃²×C₄	C₂³	1	2	yes	—
55	C₂×C₃²×C₄	C₂³	1	2	yes	—
56	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
57	C₃×#67	—	2	4	no	—
58	C₂×C₃×Dic₃	—	2	4	no	—
59	C₂×C₃×Dic₃	C₂³	1	2	yes	—
60	C₂×C₃×Dic₃	C₂³	1	2	yes	—
61	C₂×C₃×Dic₃	C₂³	1	2	yes	—
62	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
63	C₃×C₄×D₃	—	2	4	no	—
64	C₃×C₄×D₃	D₄	1	4	yes	—
65	C₃×Dic₆	—	2	2	no	—
66	C₃×Dic₆	—	2	2	no	—
67	C₃×Dic₆	—	2	4	no	—
68	#536	—	3	1	no	Sylow
69	C₃×#41	—	6	4	no	—
70	C₂×C₃×Q₈	—	2	1	no	—
71	C₂×C₃×Q₈	—	3	2	no	—
72	C₂×C₃×Q₈	—	3	1	no	—
73	C₂×C₃×Q₈	—	3	1	no	—
74	C₂×C₃×Q₈	—	6	4	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	2	no	—
79	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
80	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	—	3	1	no	—
82	C₂²×C₃×C₄	—	3	2	no	—
83	C₂²×C₃×C₄	—	3	2	no	—
84	C₂×#67	C₂²×C₃	1	2	yes	—
85	C₂²×Dic₃	C₂²×C₃	1	2	yes	—
86	D₃×Q₈	—	2	4	no	—
87	#237	—	2	4	no	—
88	C₂×C₄×D₃	C₂²×C₃	1	2	yes	—
89	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
90	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
91	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
92	C₂×Dic₆	C₂²×C₃	1	2	yes	—
93	C₃×C₄⋊C₄	—	3	2	no	—
94	C₃×C₄⋊C₄	—	3	4	no	—
95	C₃×C₄⋊C₄	—	3	4	no	—
96	C₃×C₂²⋊C₄	—	2	4	no	—
97	C₃×C₂²⋊C₄	—	3	4	no	—
98	C₃×C₂²⋊C₄	—	3	2	no	—
99	C₃×C₂²⋊C₄	C₂×D₃	1	4	yes	—
100	C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
101	C₃×C₄²	—	2	1	no	—
102	C₃×C₄²	—	3	1	no	—
103	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
104	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
105	#217	C₂²×C₃	1	4	yes	—
106	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
107	C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
108	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
109	Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
110	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
111	C₂²×C₃²	—	2	2	no	—
112	C₂²×C₃²	—	2	4	no	—
113	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
114	C₂×C₃×D₃	—	2	2	no	—
115	C₂×C₃×D₃	C₂×D₄	1	2	yes	—
116	C₃²×C₄	—	2	2	no	—
117	C₃²×C₄	—	2	2	no	—
118	C₃²×C₄	C₂×D₄	1	2	yes	—
119	C₃×Dic₃	—	2	2	no	—
120	C₃×Dic₃	—	2	2	no	—
121	C₃×Dic₃	—	2	2	no	—
122	C₃×Dic₃	C₂×D₄	1	2	yes	—
123	C₂×#41	—	3	1	no	—
124	C₂²×Q₈	—	3	1	no	—
125	C₄⋊Q₈	—	3	1	no	—
126	C₄⋊Q₈	—	3	1	no	—
127	#124	—	3	1	no	—
128	#124	C₃×D₃	1	1	yes	characteristic, complemented
129	#123	—	3	4	no	—
130	C₂²⋊Q₈	—	3	4	no	—
131	#117	—	3	1	no	—
132	C₂³×C₃	—	2	2	no	—
133	C₂³×C₃	—	3	1	no	—
134	C₂³×C₃	C₂²×D₃	1	2	yes	—
135	C₂³×C₃	C₂³×C₃	1	2	yes	—
136	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
137	C₃×Q₈	—	2	2	no	—
138	C₃×Q₈	—	2	2	no	—
139	C₃×Q₈	—	4	2	no	—
140	C₃×Q₈	—	6	4	no	—
141	C₃×Q₈	—	6	2	no	—
142	C₃×Q₈	—	6	2	no	—
143	C₃×D₄	—	2	2	no	—
144	C₃×D₄	—	2	2	no	—
145	C₃×D₄	—	4	2	no	—
146	C₃×D₄	—	6	4	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	2	2	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	3	1	no	—
151	C₂×C₃×C₄	—	3	4	no	—
152	C₂×C₃×C₄	—	3	2	no	—
153	C₂×C₃×C₄	—	3	2	no	—
154	C₂×C₃×C₄	—	3	2	no	—
155	C₂×C₃×C₄	—	6	4	no	—
156	C₂×C₃×C₄	—	6	4	no	—
157	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
158	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
159	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
161	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
162	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
163	#67	—	2	4	no	—
164	C₂×Dic₃	—	2	4	no	—
165	C₂×Dic₃	C₂³×C₃	1	2	yes	—
166	C₂×Dic₃	C₂³×C₃	1	2	yes	—
167	C₂×Dic₃	C₂³×C₃	1	2	yes	—
168	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
169	C₄×D₃	—	2	4	no	—
170	C₄×D₃	C₃×D₄	1	4	yes	—
171	Dic₆	—	2	2	no	—
172	Dic₆	—	2	2	no	—
173	Dic₆	—	2	4	no	—
174	C₂×C₃²	—	4	2	no	—
175	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
177	C₃×D₃	—	2	2	no	—
178	#41	—	6	4	no	—
179	C₂×Q₈	—	3	2	no	—
180	C₂×Q₈	—	3	1	no	—
181	C₂×Q₈	—	3	1	no	—
182	C₂×Q₈	—	6	4	no	—
183	C₂×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
184	C₂×D₄	—	3	2	no	—
185	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
186	C₂²×C₄	—	3	1	no	—
187	C₂²×C₄	—	3	2	no	—
188	C₂²×C₄	—	3	2	no	—
189	C₄⋊C₄	—	3	2	no	—
190	C₄⋊C₄	—	3	4	no	—
191	C₄⋊C₄	—	3	4	no	—
192	C₂²⋊C₄	—	3	4	no	—
193	C₂²⋊C₄	—	3	2	no	—
194	C₂²⋊C₄	C₂×C₃×D₃	1	4	yes	—
195	C₄²	—	3	1	no	—
196	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	4	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	—	2	1	no	—
202	C₂²×C₃	—	3	2	no	—
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	—	4	4	no	—
205	C₂²×C₃	—	6	2	no	—
206	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
207	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
208	C₂×D₃	—	2	2	no	—
209	C₂×D₃	C₂×C₃×D₄	1	2	yes	—
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	2	no	—
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	—	2	2	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	3	2	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	—	4	2	no	—
218	C₃×C₄	—	6	2	no	—
219	C₃×C₄	—	6	2	no	—
220	C₃×C₄	—	6	2	no	—
221	C₃×C₄	D₃×D₄	1	2	yes	—
222	C₃×C₄	C₂×C₃×D₄	1	2	yes	—
223	Dic₃	—	2	2	no	—
224	Dic₃	—	2	2	no	—
225	Dic₃	—	2	2	no	—
226	Dic₃	C₂×C₃×D₄	1	2	yes	—
227	C₃²	#536	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	3	1	no	—
229	C₂³	C₂²×C₃×D₃	1	2	yes	—
230	Q₈	—	2	2	no	—
231	Q₈	—	6	4	no	—
232	Q₈	—	6	2	no	—
233	Q₈	—	6	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	6	4	no	—
236	C₂×C₄	—	3	1	no	—
237	C₂×C₄	—	3	4	no	—
238	C₂×C₄	—	3	2	no	—
239	C₂×C₄	—	3	2	no	—
240	C₂×C₄	—	3	2	no	—
241	C₂×C₄	—	6	4	no	—
242	C₂×C₄	—	6	4	no	—
243	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
244	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
245	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	4	2	no	—
249	C₂×C₃	—	4	2	no	—
250	C₂×C₃	—	6	2	no	—
251	C₂×C₃	—	8	2	no	—
252	C₂×C₃	C₂×D₃×D₄	1	1	yes	characteristic
253	C₂×C₃	#1010	1	2	yes	—
254	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
255	C₂×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
256	D₃	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	3	2	no	—
260	C₂²	—	6	2	no	—
261	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	3	2	no	—
265	C₄	—	6	2	no	—
266	C₄	—	6	2	no	—
267	C₄	—	6	2	no	—
268	C₄	C₃×D₃×D₄	1	2	yes	—
269	C₃	—	2	1	no	—
270	C₃	#5675	1	1	yes	characteristic, direct product factor
271	C₃	C₃×#536	1	1	yes	characteristic, complemented
272	C₂	—	4	2	no	—
273	C₂	—	6	2	no	—
274	C₂	C₂×C₃×D₃×D₄	1	1	yes	characteristic
275	C₂	C₃×#1010	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
prime radical	C₂×C₃×#237	C₂
hypercenter	C₃×#124	D₃
hypocenter	C₃	C₃×#536
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×D₃×Q₈	C₂	semidirect
3. C₃²×#124	C₂	semidirect
4. C₃×#944	C₂	semidirect
5. C₃×#936	C₂	semidirect
6. C₃×#883	C₂	semidirect
7. C₃×#878	C₂	semidirect
8. C₃×#872	C₂	semidirect
9. C₃×#869	C₂	semidirect
11. #5675	C₃	direct (N × H)
33. C₂×D₃×Q₈	C₂×C₃	semidirect
39. C₃×#124	D₃	semidirect
40. C₃×#124	C₂×C₃	semidirect
44. #944	C₂×C₃	semidirect
45. #936	C₂×C₃	semidirect
46. #883	C₂×C₃	semidirect
47. #878	C₂×C₃	semidirect
48. #872	C₂×C₃	semidirect
49. #869	C₂×C₃	semidirect
128. #124	C₃×D₃	semidirect
227. C₃²	#536	semidirect
270. C₃	#5675	direct (N × H)
271. C₃	C₃×#536	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃×Q₈	2
C₂×C₃×#237	2
C₃²×#124	2
C₃×#944	2
C₃×#936	2
C₃×#883	2
C₃×#878	2
C₃×#872	2
C₃×#869	2
C₃×#536	3
#5675	3

Subgroups — C3×#5675 #10595772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5675 #10595772