Subgroups — C₃×#68525 #11777952

Order 1152.

379 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#4838	C₂	1	2	yes	maximal, complemented
2	C₃×C₄×D₂₄	C₂	1	1	yes	maximal, characteristic
3	C₃²×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4483	C₂	1	2	yes	maximal, complemented
5	C₃×#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1594	—	3	1	no	maximal
7	#68525	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₂₄	C₄	1	1	yes	characteristic
9	C₂×C₃×ΓL(1,𝔽₂₅)	—	2	2	no	—
10	C₃×C₄×D₁₂	C₂²	1	2	yes	—
11	C₃²×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃×#821	C₄	1	2	yes	—
14	C₃×(C₂×D₃)⋊C₈	—	2	2	no	—
15	C₃×#818	C₄	1	1	yes	characteristic
16	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×D₈	—	3	1	no	—
19	C₃×#436	—	3	2	no	—
20	#4838	C₂×C₃	1	2	yes	complemented
21	C₄×D₂₄	C₂×C₃	1	1	yes	characteristic
22	C₃×#334	—	2	1	no	—
23	C₃×#334	D₃	1	1	yes	characteristic, complemented
24	C₃×#334	C₂×C₃	1	1	yes	characteristic, complemented
25	C₃×D₄⋊C₈	—	3	2	no	—
26	C₃×#322	—	3	1	no	—
27	#4483	C₂×C₃	1	2	yes	complemented
28	#4455	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×C₃×D₁₂	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄×D₃	—	2	2	no	—
31	C₂×C₃²×C₈	—	2	2	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₃²×C₄²	C₂³	1	1	yes	characteristic
35	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
36	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₃⋊C₈	—	2	1	no	—
38	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
39	C₃×D₂₄	—	2	2	no	—
40	C₃×ΓL(1,𝔽₂₅)	—	4	4	no	—
41	#1594	—	3	1	no	Sylow
42	C₂×C₃×D₈	—	3	1	no	—
43	C₂×C₃×M₁₆	—	6	2	no	—
44	C₃×C₄×D₄	—	3	2	no	—
45	C₂×D₂₄	C₃×C₄	1	1	yes	characteristic
46	C₂×ΓL(1,𝔽₂₅)	—	2	2	no	—
47	C₄×D₁₂	C₂²×C₃	1	2	yes	—
48	C₃×#107	—	3	1	no	—
49	C₃×C₄⋊C₈	—	2	2	no	—
50	C₃×C₄⋊C₈	C₂×D₃	1	2	yes	—
51	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
52	C₃×#102	—	3	2	no	—
53	C₃×C₂²⋊C₈	—	6	2	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	—	3	1	no	—
56	C₃×C₄×C₈	—	3	1	no	—
57	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
59	#821	C₃×C₄	1	2	yes	—
60	(C₂×D₃)⋊C₈	—	2	2	no	—
61	#818	C₃×C₄	1	1	yes	characteristic
62	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
63	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂²×C₃×D₃	—	2	2	no	—
65	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	#41	1	1	yes	characteristic
68	C₂×C₃×Dic₃	—	2	2	no	—
69	C₃×D₁₂	—	2	2	no	—
70	C₃×D₁₂	—	2	2	no	—
71	C₃×C₄×D₃	—	4	2	no	—
72	C₃×C₄×D₃	—	4	2	no	—
73	C₃²×C₈	—	2	1	no	—
74	C₃²×C₈	—	4	2	no	—
75	C₃²×C₈	M₁₆	1	2	yes	complemented
76	C₃×C₃⋊C₈	—	2	2	no	—
77	C₃×C₃⋊C₈	—	4	1	no	—
78	C₄×D₈	—	3	1	no	—
79	#436	—	3	2	no	—
80	#334	C₃×D₃	1	1	yes	characteristic, complemented
81	D₄⋊C₈	—	3	2	no	—
82	#322	—	3	1	no	—
83	C₂×C₃×D₄	—	3	2	no	—
84	C₂²×C₃×C₄	—	6	2	no	—
85	C₂×D₁₂	C₂×C₃×C₄	1	2	yes	—
86	C₂×C₄×D₃	—	2	2	no	—
87	C₃×D₈	—	6	2	no	—
88	C₃×M₁₆	—	12	4	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	3	2	no	—
94	C₂×C₃×C₈	—	4	2	no	—
95	C₂×C₃×C₈	—	6	1	no	—
96	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
97	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
99	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
100	C₃×C₄⋊C₄	—	3	2	no	—
101	C₃×C₂²⋊C₄	—	6	2	no	—
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
104	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
105	(C₂×D₃)⋊C₄	—	2	2	no	—
106	C₄⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
107	C₂×C₃⋊C₈	—	2	1	no	—
108	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
109	D₂₄	—	2	2	no	—
110	ΓL(1,𝔽₂₅)	—	4	4	no	—
111	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
112	C₂×C₃×D₃	—	4	2	no	—
113	C₂×C₃×D₃	—	4	2	no	—
114	C₂×C₃×D₃	—	4	2	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
117	C₃²×C₄	#131	1	1	yes	characteristic
118	C₃²×C₄	AGL(1,ℤ/8)	1	2	yes	—
119	C₃×Dic₃	—	4	2	no	—
120	C₂×D₈	—	3	1	no	—
121	C₂×M₁₆	—	6	2	no	—
122	C₄×D₄	—	3	2	no	—
123	#107	—	3	1	no	—
124	C₄⋊C₈	C₂×C₃×D₃	1	2	yes	—
125	#102	—	3	2	no	—
126	C₂²⋊C₈	—	6	2	no	—
127	C₄×C₈	—	3	1	no	—
128	C₄×C₈	—	3	1	no	—
129	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
130	C₂³×C₃	—	6	2	no	—
131	C₂²×D₃	—	2	2	no	—
132	C₃×D₄	—	6	2	no	—
133	C₃×D₄	—	6	2	no	—
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	6	2	no	—
138	C₂×C₃×C₄	—	12	2	no	—
139	C₂×C₃×C₄	—	12	2	no	—
140	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
142	C₂×C₃×C₄	#239	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
146	C₂×Dic₃	—	2	2	no	—
147	D₁₂	—	2	2	no	—
148	D₁₂	—	2	2	no	—
149	C₄×D₃	—	4	2	no	—
150	C₄×D₃	—	4	2	no	—
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	—	2	1	no	—
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	—	4	1	no	—
155	C₃×C₈	—	4	2	no	—
156	C₃×C₈	—	4	2	no	—
157	C₃×C₈	—	6	2	no	—
158	C₃×C₈	—	8	2	no	—
159	C₃×C₈	—	12	1	no	—
160	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
161	C₃×C₈	C₃×M₁₆	1	2	yes	complemented
162	C₃⋊C₈	—	2	2	no	—
163	C₃⋊C₈	—	4	1	no	—
164	C₂×C₃²	#436	1	1	yes	characteristic
165	C₂×C₃²	#442	1	1	yes	characteristic
166	C₂×C₃²	#444	1	1	yes	characteristic
167	C₃×D₃	—	8	2	no	—
168	C₂×D₄	—	3	2	no	—
169	C₂²×C₄	—	6	2	no	—
170	D₈	—	6	2	no	—
171	M₁₆	—	12	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	3	2	no	—
174	C₂×C₈	—	6	1	no	—
175	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
176	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
177	C₄⋊C₄	—	3	2	no	—
178	C₂²⋊C₄	—	6	2	no	—
179	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	12	2	no	—
182	C₂²×C₃	—	12	2	no	—
183	C₂²×C₃	—	12	2	no	—
184	C₂²×C₃	#893	1	1	yes	characteristic
185	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
186	C₂×D₃	—	4	2	no	—
187	C₂×D₃	—	4	2	no	—
188	C₂×D₃	—	4	2	no	—
189	C₃×C₄	—	2	1	no	—
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	2	1	no	—
192	C₃×C₄	—	2	1	no	—
193	C₃×C₄	—	2	2	no	—
194	C₃×C₄	—	4	1	no	—
195	C₃×C₄	—	12	2	no	—
196	C₃×C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
197	C₃×C₄	#907	1	1	yes	characteristic
198	C₃×C₄	#914	1	2	yes	—
199	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
200	C₃×C₄	C₃×#131	1	1	yes	characteristic
201	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
202	Dic₃	—	4	2	no	—
203	C₃²	#1594	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	6	2	no	—
205	D₄	—	6	2	no	—
206	D₄	—	6	2	no	—
207	C₂×C₄	—	6	2	no	—
208	C₂×C₄	—	12	2	no	—
209	C₂×C₄	—	12	2	no	—
210	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
211	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
212	C₂×C₄	C₃×#239	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	4	2	no	—
215	C₈	—	6	2	no	—
216	C₈	—	12	1	no	—
217	C₈	C₃×ΓL(1,𝔽₂₅)	1	2	yes	complemented
218	C₂×C₃	—	2	1	no	—
219	C₂×C₃	—	2	1	no	—
220	C₂×C₃	—	2	1	no	—
221	C₂×C₃	—	24	2	no	—
222	C₂×C₃	#4838	1	1	yes	characteristic
223	C₂×C₃	#4869	1	1	yes	characteristic
224	C₂×C₃	#4894	1	1	yes	characteristic
225	C₂×C₃	C₃×#436	1	1	yes	characteristic
226	C₂×C₃	C₃×#442	1	1	yes	characteristic
227	C₂×C₃	C₃×#444	1	1	yes	characteristic
228	D₃	—	8	2	no	—
229	C₂²	—	12	2	no	—
230	C₂²	—	12	2	no	—
231	C₂²	—	12	2	no	—
232	C₂²	C₃×#893	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	12	2	no	—
235	C₄	C₂×C₃×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
236	C₄	C₃×#907	1	1	yes	characteristic
237	C₄	C₃×#914	1	2	yes	—
238	C₃	—	2	1	no	—
239	C₃	#68525	1	1	yes	characteristic, direct product factor
240	C₃	C₃×#1594	1	1	yes	characteristic, complemented
241	C₂	—	24	2	no	—
242	C₂	C₃×#4838	1	1	yes	characteristic
243	C₂	C₃×#4869	1	1	yes	characteristic
244	C₂	C₃×#4894	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#334	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₂₄	C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#334	D₃
hypocenter	C₃	C₃×#1594
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×#334

Complements

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

N	H ≅ G/N	type
1. C₃×#4838	C₂	semidirect
3. C₃²×#334	C₂	semidirect
4. C₃×#4483	C₂	semidirect
5. C₃×#4455	C₂	semidirect
7. #68525	C₃	direct (N × H)
20. #4838	C₂×C₃	semidirect
23. C₃×#334	D₃	semidirect
24. C₃×#334	C₂×C₃	semidirect
27. #4483	C₂×C₃	semidirect
28. #4455	C₂×C₃	semidirect
75. C₃²×C₈	M₁₆	semidirect
80. #334	C₃×D₃	semidirect
160. C₃×C₈	ΓL(1,𝔽₂₅)	semidirect
161. C₃×C₈	C₃×M₁₆	semidirect
203. C₃²	#1594	semidirect
217. C₈	C₃×ΓL(1,𝔽₂₅)	semidirect
239. C₃	#68525	direct (N × H)
240. C₃	C₃×#1594	semidirect

Maximal subgroups

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

subgroup	index
C₃×#4838	2
C₃×C₄×D₂₄	2
C₃²×#334	2
C₃×#4483	2
C₃×#4455	2
C₃×#1594	3
#68525	3

Subgroups — C3×#68525 #11777952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68525 #11777952