Subgroups — C₃×#5111 #10592208

Order 576.

286 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×#67	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#830	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×(C₂×D₃)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#815	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#435	—	3	1	no	maximal
9	#5111	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×#67	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
13	C₂×C₃²×C₈	—	2	2	no	—
14	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
16	C₃×#217	C₄	1	1	yes	characteristic
17	C₃×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
18	C₃×Dic₃⋊C₄	C₄	1	1	yes	characteristic
19	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
23	C₂×C₃×M₁₆	—	3	1	no	—
24	C₂²×C₃×C₈	—	2	1	no	—
25	C₂²×C₃×C₈	D₃	1	1	yes	characteristic, complemented
26	C₂²×C₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×C₄×D₄	—	3	1	no	—
28	C₄×#67	C₂×C₃	1	1	yes	characteristic
29	C₂×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic, complemented
30	C₃×C₄⋊C₈	—	3	1	no	—
31	C₃×C₂²⋊C₈	—	3	1	no	—
32	C₃×C₂²⋊C₈	—	3	1	no	—
33	C₃×#97	—	3	1	no	—
34	#830	C₂×C₃	1	1	yes	characteristic, complemented
35	(C₂×D₃)⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	#815	C₂×C₃	1	1	yes	characteristic, complemented
37	Dic₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂³×C₃²	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₄	—	2	1	no	—
41	C₂×C₃²×C₄	—	2	1	no	—
42	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
44	C₃×#67	—	2	2	no	—
45	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×D₃	—	2	1	no	—
49	C₃×C₄×D₃	—	2	1	no	—
50	C₃²×C₈	—	2	1	no	—
51	C₃²×C₈	D₄	1	2	yes	complemented
52	C₃×C₃⋊C₈	—	2	1	no	—
53	C₃×C₃⋊C₈	—	2	1	no	—
54	#435	—	3	1	no	Sylow
55	C₂×C₃×D₄	—	3	1	no	—
56	C₂²×C₃×C₄	—	2	1	no	—
57	C₂²×C₃×C₄	—	3	1	no	—
58	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
60	C₂×#67	C₃×C₄	1	1	yes	characteristic
61	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
62	C₃×M₁₆	—	6	2	no	—
63	C₂×C₃×C₈	—	2	4	no	—
64	C₂×C₃×C₈	—	2	2	no	—
65	C₂×C₃×C₈	—	2	2	no	—
66	C₂×C₃×C₈	—	2	1	no	—
67	C₂×C₃×C₈	—	2	1	no	—
68	C₂×C₃×C₈	—	3	1	no	—
69	C₂×C₃×C₈	—	3	1	no	—
70	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
71	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×C₄⋊C₄	—	3	1	no	—
75	C₃×C₂²⋊C₄	—	3	1	no	—
76	C₃×C₂²⋊C₄	—	3	1	no	—
77	C₃×C₄²	—	3	1	no	—
78	#217	C₃×C₄	1	1	yes	characteristic
79	(C₂×D₃)⋊C₄	C₃×C₄	1	1	yes	characteristic
80	Dic₃⋊C₄	C₃×C₄	1	1	yes	characteristic
81	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
84	ΓL(1,𝔽₂₅)	—	2	2	no	—
85	C₂²×C₃²	—	2	1	no	—
86	C₂²×C₃²	—	2	1	no	—
87	C₂²×C₃²	M₁₆	1	2	yes	complemented
88	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×D₃	—	2	1	no	—
90	C₂×C₃×D₃	—	2	1	no	—
91	C₂×C₃×D₃	—	2	1	no	—
92	C₃²×C₄	—	2	1	no	—
93	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₃²×C₄	#41	1	1	yes	characteristic
95	C₃×Dic₃	—	2	1	no	—
96	C₃×Dic₃	—	2	1	no	—
97	C₃×Dic₃	—	2	1	no	—
98	C₂×M₁₆	—	3	1	no	—
99	C₂²×C₈	C₃×D₃	1	1	yes	characteristic, complemented
100	C₄×D₄	—	3	1	no	—
101	C₄⋊C₈	—	3	1	no	—
102	C₂²⋊C₈	—	3	1	no	—
103	C₂²⋊C₈	—	3	1	no	—
104	#97	—	3	1	no	—
105	C₂³×C₃	—	2	1	no	—
106	C₂³×C₃	—	3	1	no	—
107	C₂³×C₃	C₄×D₃	1	1	yes	characteristic
108	C₂³×C₃	C₂×C₃×C₄	1	1	yes	characteristic
109	C₂²×D₃	C₂×C₃×C₄	1	1	yes	characteristic
110	C₃×D₄	—	6	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	1	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	2	1	no	—
119	C₂×C₃×C₄	—	3	1	no	—
120	C₂×C₃×C₄	—	3	1	no	—
121	C₂×C₃×C₄	—	3	1	no	—
122	C₂×C₃×C₄	—	6	1	no	—
123	C₂×C₃×C₄	—	6	1	no	—
124	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
127	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
128	#67	—	2	2	no	—
129	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
132	C₄×D₃	—	2	1	no	—
133	C₄×D₃	—	2	1	no	—
134	C₃×C₈	—	2	2	no	—
135	C₃×C₈	—	2	1	no	—
136	C₃×C₈	—	2	1	no	—
137	C₃×C₈	—	2	2	no	—
138	C₃×C₈	—	6	1	no	—
139	C₃×C₈	—	6	1	no	—
140	C₃×C₈	#67	1	2	yes	complemented
141	C₃×C₈	C₃×D₄	1	2	yes	complemented
142	C₃⋊C₈	—	2	1	no	—
143	C₃⋊C₈	—	2	1	no	—
144	C₂×C₃²	—	2	2	no	—
145	C₂×C₃²	C₄×D₄	1	1	yes	characteristic
146	C₂×C₃²	C₂×M₁₆	1	1	yes	characteristic
147	C₂×C₃²	#131	1	1	yes	characteristic
148	C₃×D₃	—	4	1	no	—
149	C₂×D₄	—	3	1	no	—
150	C₂²×C₄	—	3	1	no	—
151	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
152	M₁₆	—	6	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	3	1	no	—
155	C₂×C₈	—	3	1	no	—
156	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
157	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
158	C₄⋊C₄	—	3	1	no	—
159	C₂²⋊C₄	—	3	1	no	—
160	C₂²⋊C₄	—	3	1	no	—
161	C₄²	—	3	1	no	—
162	C₂²×C₃	—	2	2	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	—	2	1	no	—
165	C₂²×C₃	—	2	1	no	—
166	C₂²×C₃	—	2	1	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	2	2	no	—
169	C₂²×C₃	—	2	1	no	—
170	C₂²×C₃	—	6	1	no	—
171	C₂²×C₃	—	6	1	no	—
172	C₂²×C₃	—	6	1	no	—
173	C₂²×C₃	ΓL(1,𝔽₂₅)	1	2	yes	complemented
174	C₂²×C₃	C₃×M₁₆	1	2	yes	complemented
175	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
176	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
177	C₂×D₃	—	2	1	no	—
178	C₂×D₃	—	2	1	no	—
179	C₂×D₃	—	2	1	no	—
180	C₃×C₄	—	2	2	no	—
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	—	2	1	no	—
185	C₃×C₄	—	6	1	no	—
186	C₃×C₄	—	6	1	no	—
187	C₃×C₄	—	6	1	no	—
188	C₃×C₄	#235	1	1	yes	characteristic
189	C₃×C₄	C₂×#67	1	1	yes	characteristic
190	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
191	C₃×C₄	C₃×#41	1	1	yes	characteristic
192	Dic₃	—	2	1	no	—
193	Dic₃	—	2	1	no	—
194	Dic₃	—	2	1	no	—
195	C₃²	#435	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	3	1	no	—
197	C₂³	C₃×C₄×D₃	1	1	yes	characteristic
198	D₄	—	6	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	3	1	no	—
202	C₂×C₄	—	3	1	no	—
203	C₂×C₄	—	3	1	no	—
204	C₂×C₄	—	6	1	no	—
205	C₂×C₄	—	6	1	no	—
206	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	—	6	1	no	—
210	C₈	—	6	1	no	—
211	C₈	C₃×#67	1	2	yes	complemented
212	C₂×C₃	—	2	4	no	—
213	C₂×C₃	—	2	2	no	—
214	C₂×C₃	—	2	2	no	—
215	C₂×C₃	—	2	1	no	—
216	C₂×C₃	—	2	1	no	—
217	C₂×C₃	—	2	1	no	—
218	C₂×C₃	—	12	1	no	—
219	C₂×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
220	C₂×C₃	#901	1	1	yes	characteristic
221	C₂×C₃	C₄×#67	1	1	yes	characteristic
222	C₂×C₃	C₃×C₄×D₄	1	1	yes	characteristic
223	C₂×C₃	C₂×C₃×M₁₆	1	1	yes	characteristic
224	C₂×C₃	C₃×#131	1	1	yes	characteristic
225	D₃	—	4	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	6	1	no	—
229	C₂²	—	6	1	no	—
230	C₂²	—	6	1	no	—
231	C₂²	C₃×ΓL(1,𝔽₂₅)	1	2	yes	complemented
232	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	6	1	no	—
235	C₄	—	6	1	no	—
236	C₄	—	6	1	no	—
237	C₄	C₃×#235	1	1	yes	characteristic
238	C₄	C₂×C₃×#67	1	1	yes	characteristic
239	C₃	—	2	1	no	—
240	C₃	#5111	1	1	yes	characteristic, direct product factor
241	C₃	C₃×#435	1	1	yes	characteristic, complemented
242	C₂	—	2	2	no	—
243	C₂	—	12	1	no	—
244	C₂	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
245	C₂	C₃×#901	1	1	yes	characteristic
246	C₂	C₃×C₄×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃²×C₈	C₂
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂×C₃×#67	C₄
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₂²×C₃×C₈	D₃
hypocenter	C₃	C₃×#435
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×C₈	C₂	semidirect
3. C₂×C₃×ΓL(1,𝔽₂₅)	C₂	semidirect
4. C₃×#830	C₂	semidirect
5. C₃×(C₂×D₃)⋊C₈	C₂	semidirect
6. C₃×#815	C₂	semidirect
7. C₃×Dic₃⋊C₈	C₂	semidirect
9. #5111	C₃	direct (N × H)
25. C₂²×C₃×C₈	D₃	semidirect
26. C₂²×C₃×C₈	C₂×C₃	semidirect
29. C₂×ΓL(1,𝔽₂₅)	C₂×C₃	semidirect
34. #830	C₂×C₃	semidirect
35. (C₂×D₃)⋊C₈	C₂×C₃	semidirect
36. #815	C₂×C₃	semidirect
37. Dic₃⋊C₈	C₂×C₃	semidirect
51. C₃²×C₈	D₄	semidirect
87. C₂²×C₃²	M₁₆	semidirect
99. C₂²×C₈	C₃×D₃	semidirect
140. C₃×C₈	#67	semidirect
141. C₃×C₈	C₃×D₄	semidirect
173. C₂²×C₃	ΓL(1,𝔽₂₅)	semidirect
174. C₂²×C₃	C₃×M₁₆	semidirect
195. C₃²	#435	semidirect
211. C₈	C₃×#67	semidirect
231. C₂²	C₃×ΓL(1,𝔽₂₅)	semidirect
240. C₃	#5111	direct (N × H)
241. C₃	C₃×#435	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₈	2
C₃×C₄×#67	2
C₂×C₃×ΓL(1,𝔽₂₅)	2
C₃×#830	2
C₃×(C₂×D₃)⋊C₈	2
C₃×#815	2
C₃×Dic₃⋊C₈	2
C₃×#435	3
#5111	3

Subgroups — C3×#5111 #10592208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5111 #10592208