Subgroups — C₃×#68075 #11773695

Order 1152.

461 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×ΓL(1,𝔽₂₅)	C₂	1	2	yes	maximal
2	C₃×C₄×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
3	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4454	C₂	1	2	yes	maximal, complemented
5	C₃×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1455	—	3	1	no	maximal
7	#68075	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	2	yes	—
9	C₂×C₃×ΓL(1,𝔽₂₅)	C₄	1	4	yes	—
10	C₃×C₄²×D₃	C₂²	1	1	yes	characteristic
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	2	yes	—
13	C₃×#815	C₄	1	2	yes	—
14	C₃×#815	C₄	1	4	yes	—
15	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×M₁₆	—	3	2	no	—
18	C₃×C₄×M₁₆	—	3	1	no	—
19	C₄×ΓL(1,𝔽₂₅)	C₂×C₃	1	2	yes	—
20	C₄×ΓL(1,𝔽₂₅)	C₂×C₃	1	1	yes	characteristic
21	C₃×#322	—	2	1	no	—
22	C₃×#322	—	3	2	no	—
23	C₃×#322	D₃	1	1	yes	characteristic, complemented
24	C₃×#322	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×C₈²	—	3	1	no	—
26	#4454	C₂×C₃	1	2	yes	complemented
27	C₈×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×D₃	C₂×C₄	1	2	yes	—
30	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
31	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
32	C₃²×C₄²	C₂³	1	1	yes	characteristic
33	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄×Dic₃	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
37	C₃×ΓL(1,𝔽₂₅)	—	2	4	no	—
38	C₃×ΓL(1,𝔽₂₅)	—	2	8	no	—
39	#1455	—	3	1	no	Sylow
40	C₂×C₃×M₁₆	—	3	4	no	—
41	C₂×C₃×M₁₆	—	3	2	no	—
42	C₂×C₃×C₄²	—	3	1	no	—
43	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	2	yes	—
44	C₂×ΓL(1,𝔽₂₅)	C₃×C₄	1	4	yes	—
45	C₄²×D₃	C₂²×C₃	1	1	yes	characteristic
46	C₃×#97	—	3	4	no	—
47	C₃×#97	—	3	2	no	—
48	C₃×C₄×C₈	—	2	1	no	—
49	C₃×C₄×C₈	—	2	2	no	—
50	C₃×C₄×C₈	—	3	2	no	—
51	C₃×C₄×C₈	—	3	1	no	—
52	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
54	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
56	#815	C₃×C₄	1	2	yes	—
57	#815	C₃×C₄	1	4	yes	—
58	C₄×C₃⋊C₈	C₂²×C₃	1	2	yes	—
59	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
60	C₂²×C₃×D₃	C₄²	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	C₂²×C₄	1	2	yes	—
63	C₂×C₃×Dic₃	C₄²	1	1	yes	characteristic
64	C₂×C₃×Dic₃	C₄²	1	2	yes	—
65	C₃×C₄×D₃	—	2	2	no	—
66	C₃×C₄×D₃	—	2	4	no	—
67	C₃²×C₈	—	2	4	no	—
68	C₃²×C₈	M₁₆	1	4	yes	complemented
69	C₃×C₃⋊C₈	—	2	2	no	—
70	C₃×C₃⋊C₈	M₁₆	1	8	yes	complemented
71	C₄×M₁₆	—	3	2	no	—
72	C₄×M₁₆	—	3	1	no	—
73	#322	—	3	2	no	—
74	#322	C₃×D₃	1	1	yes	characteristic, complemented
75	C₈²	—	3	1	no	—
76	C₂²×C₃×C₄	—	3	1	no	—
77	C₂²×C₃×C₄	—	3	2	no	—
78	C₂×C₄×D₃	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂×C₄×D₃	C₂×C₃×C₄	1	2	yes	—
80	C₃×M₁₆	—	6	8	no	—
81	C₃×M₁₆	—	6	4	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	4	no	—
84	C₂×C₃×C₈	—	3	4	no	—
85	C₂×C₃×C₈	—	3	2	no	—
86	C₂×C₃×C₈	C₄×D₃	1	4	yes	—
87	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
88	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
89	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
90	C₃×C₄²	—	2	1	no	—
91	C₃×C₄²	—	3	1	no	—
92	C₃×C₄²	—	3	2	no	—
93	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
94	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
95	C₄×Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
96	C₄×Dic₃	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₃⋊C₈	C₂×C₃×C₄	1	2	yes	—
98	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
99	ΓL(1,𝔽₂₅)	—	2	4	no	—
100	ΓL(1,𝔽₂₅)	—	2	8	no	—
101	C₂²×C₃²	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₃×D₃	—	2	1	no	—
103	C₂×C₃×D₃	—	2	2	no	—
104	C₃²×C₄	C₂×M₁₆	1	2	yes	—
105	C₃²×C₄	C₂×M₁₆	1	4	yes	—
106	C₃×Dic₃	—	2	1	no	—
107	C₃×Dic₃	—	2	2	no	—
108	C₂×M₁₆	—	3	4	no	—
109	C₂×M₁₆	—	3	2	no	—
110	C₂×C₄²	—	3	1	no	—
111	#97	—	3	4	no	—
112	#97	—	3	2	no	—
113	C₄×C₈	—	3	2	no	—
114	C₄×C₈	—	3	1	no	—
115	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
116	C₄×C₈	C₂×C₃×D₃	1	2	yes	—
117	C₂³×C₃	—	3	1	no	—
118	C₂²×D₃	C₃×C₄²	1	1	yes	characteristic
119	C₂×C₃×C₄	—	2	1	no	—
120	C₂×C₃×C₄	—	2	2	no	—
121	C₂×C₃×C₄	—	3	1	no	—
122	C₂×C₃×C₄	—	3	2	no	—
123	C₂×C₃×C₄	—	6	2	no	—
124	C₂×C₃×C₄	—	6	4	no	—
125	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
126	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
127	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂×C₃×C₄	C₂²×C₃×C₄	1	2	yes	—
129	C₂×Dic₃	C₃×C₄²	1	1	yes	characteristic
130	C₂×Dic₃	C₃×C₄²	1	2	yes	—
131	C₄×D₃	—	2	2	no	—
132	C₄×D₃	—	2	4	no	—
133	C₃×C₈	—	2	4	no	—
134	C₃×C₈	—	2	4	no	—
135	C₃×C₈	—	2	4	no	—
136	C₃×C₈	—	3	8	no	—
137	C₃×C₈	—	4	4	no	—
138	C₃×C₈	—	6	2	no	—
139	C₃×C₈	ΓL(1,𝔽₂₅)	1	4	yes	complemented
140	C₃×C₈	C₃×M₁₆	1	4	yes	complemented
141	C₃⋊C₈	—	2	2	no	—
142	C₃⋊C₈	C₃×M₁₆	1	8	yes	complemented
143	C₂×C₃²	C₄×M₁₆	1	1	yes	characteristic
144	C₂×C₃²	C₄×M₁₆	1	2	yes	—
145	C₃×D₃	—	4	1	no	—
146	C₂²×C₄	—	3	1	no	—
147	C₂²×C₄	—	3	2	no	—
148	M₁₆	—	6	8	no	—
149	M₁₆	—	6	4	no	—
150	C₂×C₈	—	3	4	no	—
151	C₂×C₈	—	3	2	no	—
152	C₂×C₈	C₃×C₄×D₃	1	4	yes	—
153	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
154	C₄²	—	3	1	no	—
155	C₄²	—	3	2	no	—
156	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
157	C₂²×C₃	—	2	1	no	—
158	C₂²×C₃	—	6	1	no	—
159	C₂²×C₃	—	6	2	no	—
160	C₂²×C₃	C₄²×D₃	1	1	yes	characteristic
161	C₂²×C₃	C₂×C₃×C₄²	1	1	yes	characteristic
162	C₂×D₃	—	2	1	no	—
163	C₂×D₃	—	2	2	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	2	4	no	—
166	C₃×C₄	—	6	1	no	—
167	C₃×C₄	—	6	2	no	—
168	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
169	C₃×C₄	D₃×M₁₆	1	4	yes	—
170	C₃×C₄	C₂×C₃×M₁₆	1	2	yes	—
171	C₃×C₄	C₂×C₃×M₁₆	1	4	yes	—
172	Dic₃	—	2	1	no	—
173	Dic₃	—	2	2	no	—
174	C₃²	#1455	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	—	6	2	no	—
179	C₂×C₄	—	6	4	no	—
180	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₃×C₄×D₃	1	2	yes	—
182	C₈	—	2	4	no	—
183	C₈	—	3	8	no	—
184	C₈	—	6	2	no	—
185	C₈	C₃×ΓL(1,𝔽₂₅)	1	4	yes	complemented
186	C₂×C₃	—	2	1	no	—
187	C₂×C₃	—	2	2	no	—
188	C₂×C₃	—	12	1	no	—
189	C₂×C₃	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
190	C₂×C₃	#4707	1	2	yes	—
191	C₂×C₃	C₃×C₄×M₁₆	1	1	yes	characteristic
192	C₂×C₃	C₃×C₄×M₁₆	1	2	yes	—
193	D₃	—	4	1	no	—
194	C₂²	—	6	1	no	—
195	C₂²	—	6	2	no	—
196	C₂²	C₃×C₄²×D₃	1	1	yes	characteristic
197	C₄	—	6	1	no	—
198	C₄	—	6	2	no	—
199	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	2	yes	—
200	C₄	C₃×D₃×M₁₆	1	4	yes	—
201	C₃	—	2	1	no	—
202	C₃	#68075	1	1	yes	characteristic, direct product factor
203	C₃	C₃×#1455	1	1	yes	characteristic, complemented
204	C₂	—	12	1	no	—
205	C₂	C₃×C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
206	C₂	C₃×#4707	1	2	yes	—

	subgroup	G/H
center	C₃×C₄²	C₂²×D₃
commutator	C₂²×C₃	C₂×C₃×C₄²
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#322	C₂
socle	C₂²×C₃²	C₂×C₄²
prime radical	C₂²×C₃×D₃	C₄²
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#322	D₃
hypocenter	C₃	C₃×#1455
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₄² ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
3. C₃²×#322	C₂	semidirect
4. C₃×#4454	C₂	semidirect
5. C₃×C₈×C₃⋊C₈	C₂	semidirect
7. #68075	C₃	direct (N × H)
23. C₃×#322	D₃	semidirect
24. C₃×#322	C₂×C₃	semidirect
26. #4454	C₂×C₃	semidirect
27. C₈×C₃⋊C₈	C₂×C₃	semidirect
68. C₃²×C₈	M₁₆	semidirect
70. C₃×C₃⋊C₈	M₁₆	semidirect
74. #322	C₃×D₃	semidirect
139. C₃×C₈	ΓL(1,𝔽₂₅)	semidirect
140. C₃×C₈	C₃×M₁₆	semidirect
142. C₃⋊C₈	C₃×M₁₆	semidirect
174. C₃²	#1455	semidirect
185. C₈	C₃×ΓL(1,𝔽₂₅)	semidirect
202. C₃	#68075	direct (N × H)
203. C₃	C₃×#1455	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×ΓL(1,𝔽₂₅)	2
C₃²×#322	2
C₃×#4454	2
C₃×C₈×C₃⋊C₈	2
C₃×#1455	3
#68075	3

Subgroups — C3×#68075 #11773695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68075 #11773695