Subgroups — C₃×#4689 #10591786

Order 576.

282 conjugacy classes of subgroups.

The subgroup lattice has 183 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×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#803	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#405	—	3	1	no	maximal
9	#4689	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	2	yes	—
15	C₃×Dic₃⋊C₄	C₄	1	2	yes	—
16	C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×ΓL(1,𝔽₂₅)	C₄	1	4	yes	complemented
20	C₃×D₃×C₈	C₄	1	4	yes	complemented
21	C₂×C₃×M₁₆	—	3	1	no	—
22	C₂²×C₃×C₈	—	3	1	no	—
23	C₃×#117	—	3	1	no	—
24	C₂×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
25	C₂×D₃×C₈	C₂×C₃	1	1	yes	characteristic
26	#872	C₂×C₃	1	1	yes	characteristic
27	C₃×#97	—	3	1	no	—
28	C₃×#97	—	3	1	no	—
29	C₃×C₄×C₈	—	2	1	no	—
30	C₃×C₄×C₈	—	3	1	no	—
31	C₃×C₄×C₈	D₃	1	1	yes	characteristic, complemented
32	C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic, complemented
33	#815	C₂×C₃	1	1	yes	characteristic, complemented
34	C₈×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
35	#803	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂²×C₃×D₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
38	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
39	C₂×C₃×Dic₃	C₂×C₄	1	2	yes	—
40	C₂×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×D₃	C₂×C₄	1	2	yes	—
42	C₃×C₄×D₃	C₂×C₄	1	2	yes	—
43	C₃²×C₈	C₂×C₄	1	2	yes	—
44	C₃²×C₈	C₂×C₄	1	2	yes	—
45	C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
46	C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
47	#405	—	3	1	no	Sylow
48	C₂²×C₃×C₄	—	3	1	no	—
49	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
50	C₃×M₁₆	—	3	4	no	—
51	C₂×C₃×C₈	—	2	1	no	—
52	C₂×C₃×C₈	—	2	1	no	—
53	C₂×C₃×C₈	—	3	1	no	—
54	C₂×C₃×C₈	—	3	1	no	—
55	C₂×C₃×C₈	—	3	4	no	—
56	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×C₄⋊C₄	—	3	2	no	—
61	C₃×C₂²⋊C₄	—	3	2	no	—
62	C₃×C₄²	—	2	1	no	—
63	C₃×C₄²	—	3	1	no	—
64	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
65	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
66	(C₂×D₃)⋊C₄	C₃×C₄	1	2	yes	—
67	Dic₃⋊C₄	C₃×C₄	1	2	yes	—
68	C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	ΓL(1,𝔽₂₅)	C₃×C₄	1	4	yes	complemented
72	D₃×C₈	C₃×C₄	1	4	yes	complemented
73	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×D₃	—	2	2	no	—
75	C₂×C₃×D₃	C₄²	1	2	yes	—
76	C₃²×C₄	—	2	2	no	—
77	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
78	C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
79	C₃×Dic₃	—	2	2	no	—
80	C₃×Dic₃	C₄²	1	2	yes	—
81	C₂×M₁₆	—	3	1	no	—
82	C₂²×C₈	—	3	1	no	—
83	#117	—	3	1	no	—
84	#97	—	3	1	no	—
85	#97	—	3	1	no	—
86	C₄×C₈	—	3	1	no	—
87	C₄×C₈	C₃×D₃	1	1	yes	characteristic, complemented
88	C₂³×C₃	—	3	1	no	—
89	C₂²×D₃	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄	—	2	2	no	—
91	C₂×C₃×C₄	—	2	1	no	—
92	C₂×C₃×C₄	—	3	1	no	—
93	C₂×C₃×C₄	—	3	2	no	—
94	C₂×C₃×C₄	—	3	2	no	—
95	C₂×C₃×C₄	—	3	2	no	—
96	C₂×C₃×C₄	C₄×D₃	1	2	yes	—
97	C₂×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
98	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
100	C₂×Dic₃	C₂×C₃×C₄	1	2	yes	—
101	C₂×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
102	C₄×D₃	C₂×C₃×C₄	1	2	yes	—
103	C₄×D₃	C₂×C₃×C₄	1	2	yes	—
104	C₃×C₈	—	2	2	no	—
105	C₃×C₈	—	2	2	no	—
106	C₃×C₈	—	3	2	no	—
107	C₃×C₈	—	3	2	no	—
108	C₃×C₈	C₄×D₃	1	2	yes	—
109	C₃×C₈	C₄×D₃	1	2	yes	—
110	C₃×C₈	C₂×C₃×C₄	1	2	yes	—
111	C₃×C₈	C₂×C₃×C₄	1	2	yes	—
112	C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
113	C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
114	C₂×C₃²	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₃²	#131	1	2	yes	—
116	C₃×D₃	—	2	2	no	—
117	C₂²×C₄	—	3	1	no	—
118	M₁₆	—	3	4	no	—
119	C₂×C₈	—	3	1	no	—
120	C₂×C₈	—	3	1	no	—
121	C₂×C₈	—	3	4	no	—
122	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
123	C₂×C₈	C₂×C₃×D₃	1	1	yes	characteristic
124	C₄⋊C₄	—	3	2	no	—
125	C₂²⋊C₄	—	3	2	no	—
126	C₄²	—	3	1	no	—
127	C₄²	C₂×C₃×D₃	1	1	yes	characteristic
128	C₂²×C₃	—	2	1	no	—
129	C₂²×C₃	—	3	2	no	—
130	C₂²×C₃	—	6	2	no	—
131	C₂²×C₃	C₂×C₄×D₃	1	1	yes	characteristic
132	C₂²×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
133	C₂×D₃	—	2	2	no	—
134	C₂×D₃	C₃×C₄²	1	2	yes	—
135	C₃×C₄	—	2	4	no	—
136	C₃×C₄	—	2	2	no	—
137	C₃×C₄	—	2	2	no	—
138	C₃×C₄	—	2	1	no	—
139	C₃×C₄	—	2	1	no	—
140	C₃×C₄	—	3	2	no	—
141	C₃×C₄	—	6	2	no	—
142	C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
143	C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
144	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
146	Dic₃	—	2	2	no	—
147	Dic₃	C₃×C₄²	1	2	yes	—
148	C₃²	#405	1	1	yes	characteristic, complemented, Sylow
149	C₂³	—	3	1	no	—
150	C₂×C₄	—	3	1	no	—
151	C₂×C₄	—	3	2	no	—
152	C₂×C₄	—	3	2	no	—
153	C₂×C₄	—	3	2	no	—
154	C₂×C₄	C₃×C₄×D₃	1	2	yes	—
155	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
156	C₈	—	3	2	no	—
157	C₈	—	3	2	no	—
158	C₈	C₃×C₄×D₃	1	2	yes	—
159	C₈	C₃×C₄×D₃	1	2	yes	—
160	C₂×C₃	—	2	2	no	—
161	C₂×C₃	—	2	1	no	—
162	C₂×C₃	—	6	2	no	—
163	C₂×C₃	C₄²×D₃	1	1	yes	characteristic
164	C₂×C₃	#901	1	2	yes	—
165	C₂×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
166	C₂×C₃	C₃×#131	1	2	yes	—
167	D₃	—	2	2	no	—
168	C₂²	—	3	2	no	—
169	C₂²	—	6	2	no	—
170	C₂²	C₂×C₃×C₄×D₃	1	1	yes	characteristic
171	C₄	—	2	2	no	—
172	C₄	—	3	2	no	—
173	C₄	—	6	2	no	—
174	C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
175	C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
176	C₃	—	2	1	no	—
177	C₃	#4689	1	1	yes	characteristic, direct product factor
178	C₃	C₃×#405	1	1	yes	characteristic, complemented
179	C₂	—	6	2	no	—
180	C₂	C₃×C₄²×D₃	1	1	yes	characteristic
181	C₂	C₃×#901	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₃×C₄²
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×#405
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₈ ◁ 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×#815	C₂	semidirect
6. C₃×C₈×Dic₃	C₂	semidirect
7. C₃×#803	C₂	semidirect
9. #4689	C₃	direct (N × H)
19. C₃×ΓL(1,𝔽₂₅)	C₄	semidirect
20. C₃×D₃×C₈	C₄	semidirect
31. C₃×C₄×C₈	D₃	semidirect
32. C₃×C₄×C₈	C₂×C₃	semidirect
33. #815	C₂×C₃	semidirect
34. C₈×Dic₃	C₂×C₃	semidirect
35. #803	C₂×C₃	semidirect
71. ΓL(1,𝔽₂₅)	C₃×C₄	semidirect
72. D₃×C₈	C₃×C₄	semidirect
87. C₄×C₈	C₃×D₃	semidirect
148. C₃²	#405	semidirect
177. C₃	#4689	direct (N × H)
178. C₃	C₃×#405	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×#815	2
C₃×C₈×Dic₃	2
C₃×#803	2
C₃×#405	3
#4689	3

Subgroups — C3×#4689 #10591786

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#4689 #10591786