Subgroups — C₃×#4839 #10591936

Order 576.

247 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×#872	C₂	1	1	yes	maximal, characteristic
4	C₃²×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#424	—	3	1	no	maximal
9	#4839	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄²	C₂²	1	1	yes	characteristic
14	C₃×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
15	C₃×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
16	C₃×Dic₃⋊C₄	C₄	1	1	yes	characteristic
17	C₃×Dic₃⋊C₄	C₄	1	1	yes	characteristic
18	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
22	C₃×D₃×C₈	—	2	2	no	—
23	C₂×C₃×M₁₆	—	3	1	no	—
24	C₂²×C₃×C₈	—	3	1	no	—
25	C₃×#117	—	3	1	no	—
26	C₂×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
27	C₂×D₃×C₈	C₂×C₃	1	1	yes	characteristic
28	#872	C₂×C₃	1	1	yes	characteristic
29	C₃×C₄⋊C₈	—	2	1	no	—
30	C₃×C₄⋊C₈	—	3	1	no	—
31	C₃×C₄⋊C₈	—	3	1	no	—
32	C₃×C₄⋊C₈	—	3	1	no	—
33	C₃×C₄⋊C₈	D₃	1	1	yes	characteristic, complemented
34	C₃×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
35	Dic₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
36	Dic₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
37	C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂²×C₃×D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
42	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×D₃	D₄	1	2	yes	—
46	C₃×C₄×D₃	Q₈	1	2	yes	—
47	C₃²×C₈	—	2	1	no	—
48	C₃²×C₈	—	2	1	no	—
49	C₃×C₃⋊C₈	—	2	1	no	—
50	C₃×C₃⋊C₈	—	2	1	no	—
51	#424	—	3	1	no	Sylow
52	C₂²×C₃×C₄	—	3	1	no	—
53	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
54	C₃×M₁₆	—	6	2	no	—
55	C₂×C₃×C₈	—	2	1	no	—
56	C₂×C₃×C₈	—	2	1	no	—
57	C₂×C₃×C₈	—	3	1	no	—
58	C₂×C₃×C₈	—	3	1	no	—
59	C₂×C₃×C₈	—	6	2	no	—
60	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₃×C₄⋊C₄	—	3	1	no	—
65	C₃×C₄⋊C₄	—	3	1	no	—
66	C₃×C₂²⋊C₄	—	3	1	no	—
67	C₃×C₂²⋊C₄	—	3	1	no	—
68	C₃×C₄²	—	2	1	no	—
69	C₃×C₄²	—	3	1	no	—
70	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
72	(C₂×D₃)⋊C₄	C₃×C₄	1	1	yes	characteristic
73	(C₂×D₃)⋊C₄	C₃×C₄	1	1	yes	characteristic
74	Dic₃⋊C₄	C₃×C₄	1	1	yes	characteristic
75	Dic₃⋊C₄	C₃×C₄	1	1	yes	characteristic
76	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
78	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
79	ΓL(1,𝔽₂₅)	—	2	2	no	—
80	D₃×C₈	—	2	2	no	—
81	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×D₃	—	2	1	no	—
83	C₂×C₃×D₃	—	2	1	no	—
84	C₂×C₃×D₃	C₄⋊C₄	1	2	yes	—
85	C₃²×C₄	—	2	1	no	—
86	C₃²×C₄	—	2	1	no	—
87	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
88	C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
89	C₃×Dic₃	—	2	1	no	—
90	C₃×Dic₃	—	2	1	no	—
91	C₃×Dic₃	C₄⋊C₄	1	2	yes	—
92	C₂×M₁₆	—	3	1	no	—
93	C₂²×C₈	—	3	1	no	—
94	#117	—	3	1	no	—
95	C₄⋊C₈	—	3	1	no	—
96	C₄⋊C₈	—	3	1	no	—
97	C₄⋊C₈	—	3	1	no	—
98	C₄⋊C₈	C₃×D₃	1	1	yes	characteristic, complemented
99	C₂³×C₃	—	3	1	no	—
100	C₂²×D₃	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₃×C₄	—	2	1	no	—
102	C₂×C₃×C₄	—	2	1	no	—
103	C₂×C₃×C₄	—	2	1	no	—
104	C₂×C₃×C₄	—	3	1	no	—
105	C₂×C₃×C₄	—	3	2	no	—
106	C₂×C₃×C₄	—	3	2	no	—
107	C₂×C₃×C₄	—	3	1	no	—
108	C₂×C₃×C₄	—	3	1	no	—
109	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
110	C₂×C₃×C₄	C₄×D₃	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
113	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
115	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
116	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
118	C₄×D₃	C₃×D₄	1	2	yes	—
119	C₄×D₃	C₃×Q₈	1	2	yes	—
120	C₃×C₈	—	2	1	no	—
121	C₃×C₈	—	2	1	no	—
122	C₃×C₈	—	2	1	no	—
123	C₃×C₈	—	2	1	no	—
124	C₃×C₈	—	4	1	no	—
125	C₃×C₈	—	4	1	no	—
126	C₃×C₈	—	6	1	no	—
127	C₃×C₈	—	6	1	no	—
128	C₃⋊C₈	—	2	1	no	—
129	C₃⋊C₈	—	2	1	no	—
130	C₂×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃²	#131	1	1	yes	characteristic
132	C₂×C₃²	#131	1	1	yes	characteristic
133	C₃×D₃	—	2	2	no	—
134	C₂²×C₄	—	3	1	no	—
135	M₁₆	—	6	2	no	—
136	C₂×C₈	—	3	1	no	—
137	C₂×C₈	—	3	1	no	—
138	C₂×C₈	—	6	2	no	—
139	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
141	C₄⋊C₄	—	3	1	no	—
142	C₄⋊C₄	—	3	1	no	—
143	C₂²⋊C₄	—	3	1	no	—
144	C₂²⋊C₄	—	3	1	no	—
145	C₄²	—	3	1	no	—
146	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
147	C₂²×C₃	—	2	1	no	—
148	C₂²×C₃	—	3	2	no	—
149	C₂²×C₃	—	6	1	no	—
150	C₂²×C₃	—	6	1	no	—
151	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
152	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
153	C₂×D₃	—	2	1	no	—
154	C₂×D₃	—	2	1	no	—
155	C₂×D₃	C₃×C₄⋊C₄	1	2	yes	—
156	C₃×C₄	—	2	2	no	—
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	—	2	1	no	—
159	C₃×C₄	—	2	1	no	—
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	2	1	no	—
162	C₃×C₄	—	2	1	no	—
163	C₃×C₄	—	3	2	no	—
164	C₃×C₄	—	4	1	no	—
165	C₃×C₄	—	6	1	no	—
166	C₃×C₄	—	6	1	no	—
167	C₃×C₄	D₃×D₄	1	1	yes	characteristic
168	C₃×C₄	D₃×Q₈	1	1	yes	characteristic
169	C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
170	C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
171	Dic₃	—	2	1	no	—
172	Dic₃	—	2	1	no	—
173	Dic₃	C₃×C₄⋊C₄	1	2	yes	—
174	C₃²	#424	1	1	yes	characteristic, complemented, Sylow
175	C₂³	—	3	1	no	—
176	C₂×C₄	—	3	1	no	—
177	C₂×C₄	—	3	2	no	—
178	C₂×C₄	—	3	2	no	—
179	C₂×C₄	—	3	1	no	—
180	C₂×C₄	—	3	1	no	—
181	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
182	C₂×C₄	C₃×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	6	1	no	—
187	C₈	—	6	1	no	—
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	—	6	2	no	—
192	C₂×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₃	#901	1	1	yes	characteristic
194	C₂×C₃	#907	1	1	yes	characteristic
195	C₂×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
196	C₂×C₃	C₃×#131	1	1	yes	characteristic
197	C₂×C₃	C₃×#131	1	1	yes	characteristic
198	D₃	—	2	2	no	—
199	C₂²	—	3	2	no	—
200	C₂²	—	6	1	no	—
201	C₂²	—	6	1	no	—
202	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	3	2	no	—
206	C₄	—	6	1	no	—
207	C₄	—	6	1	no	—
208	C₄	C₃×D₃×D₄	1	1	yes	characteristic
209	C₄	C₃×D₃×Q₈	1	1	yes	characteristic
210	C₃	—	2	1	no	—
211	C₃	#4839	1	1	yes	characteristic, direct product factor
212	C₃	C₃×#424	1	1	yes	characteristic, complemented
213	C₂	—	6	2	no	—
214	C₂	C₃×D₃×C₄⋊C₄	1	1	yes	characteristic
215	C₂	C₃×#901	1	1	yes	characteristic
216	C₂	C₃×#907	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	C₃²×C₄⋊C₈	C₂
socle	C₂²×C₃²	C₂²×C₄
prime radical	C₂²×C₃×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³×C₃
hypercenter	C₃×C₄⋊C₈	D₃
hypocenter	C₃	C₃×#424
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃²×C₄⋊C₈	C₂	semidirect
5. C₃×Dic₃⋊C₈	C₂	semidirect
6. C₃×Dic₃⋊C₈	C₂	semidirect
7. C₃×C₄⋊(C₃⋊C₈)	C₂	semidirect
9. #4839	C₃	direct (N × H)
33. C₃×C₄⋊C₈	D₃	semidirect
34. C₃×C₄⋊C₈	C₂×C₃	semidirect
35. Dic₃⋊C₈	C₂×C₃	semidirect
36. Dic₃⋊C₈	C₂×C₃	semidirect
37. C₄⋊(C₃⋊C₈)	C₂×C₃	semidirect
98. C₄⋊C₈	C₃×D₃	semidirect
174. C₃²	#424	semidirect
211. C₃	#4839	direct (N × H)
212. C₃	C₃×#424	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×ΓL(1,𝔽₂₅)	2
C₂×C₃×D₃×C₈	2
C₃×#872	2
C₃²×C₄⋊C₈	2
C₃×Dic₃⋊C₈	2
C₃×C₄⋊(C₃⋊C₈)	2
C₃×#424	3
#4839	3

Subgroups — C3×#4839 #10591936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#4839 #10591936