Subgroups — C₃×#68077 #11777344

Order 1152.

417 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×D₁₂	C₂	1	2	yes	maximal, complemented
3	C₃²×#322	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4459	C₂	1	2	yes	maximal, complemented
5	C₃×#4457	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#1580	—	3	1	no	maximal
7	#68077	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₂₄	C₄	1	1	yes	characteristic
9	C₂×C₃×D₃×C₈	—	2	2	no	—
10	C₃×C₄×D₁₂	C₂²	1	2	yes	—
11	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#821	C₄	1	2	yes	—
15	C₃×(C₂×D₃)⋊C₈	—	2	2	no	—
16	C₃×#818	C₄	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₃×C₄×D₈	—	3	1	no	—
19	C₃×C₈×D₄	—	3	2	no	—
20	C₄×D₂₄	C₂×C₃	1	1	yes	characteristic
21	C₈×D₁₂	C₂×C₃	1	2	yes	complemented
22	C₃×#334	—	3	1	no	—
23	C₃×D₄⋊C₈	—	3	2	no	—
24	C₃×#322	—	2	1	no	—
25	C₃×#322	D₃	1	1	yes	characteristic, complemented
26	C₃×#322	C₂×C₃	1	1	yes	characteristic, complemented
27	#4459	C₂×C₃	1	2	yes	complemented
28	#4457	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	1	no	—
32	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃²×C₈	D₄	1	2	yes	—
35	C₃²×C₄²	C₂³	1	1	yes	characteristic
36	C₃×(C₂×D₃)⋊C₄	—	2	2	no	—
37	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₃⋊C₈	—	2	2	no	—
39	C₃×D₂₄	C₈	1	4	yes	complemented
40	C₃×D₃×C₈	—	4	4	no	—
41	#1580	—	3	1	no	Sylow
42	C₂×C₃×D₈	—	3	1	no	—
43	C₂²×C₃×C₈	—	6	2	no	—
44	C₃×C₄×D₄	—	3	2	no	—
45	C₂×D₂₄	C₃×C₄	1	1	yes	characteristic
46	C₂×D₃×C₈	—	2	2	no	—
47	C₄×D₁₂	C₂²×C₃	1	2	yes	—
48	C₃×#107	—	3	1	no	—
49	C₃×C₄⋊C₈	—	3	2	no	—
50	C₃×#102	—	3	2	no	—
51	C₃×C₂²⋊C₈	—	6	2	no	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃×C₄×C₈	—	2	1	no	—
55	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
57	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
58	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
60	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
61	#821	C₃×C₄	1	2	yes	—
62	(C₂×D₃)⋊C₈	—	2	2	no	—
63	#818	C₃×C₄	1	1	yes	characteristic
64	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	2	yes	—
65	C₂²×C₃×D₃	—	2	2	no	—
66	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃²×C₄	#41	1	1	yes	characteristic
69	C₂×C₃×Dic₃	—	2	2	no	—
70	C₃×D₁₂	—	2	2	no	—
71	C₃×D₁₂	C₂×C₈	1	4	yes	complemented
72	C₃×C₄×D₃	—	4	2	no	—
73	C₃×C₄×D₃	—	4	2	no	—
74	C₃²×C₈	—	2	1	no	—
75	C₃²×C₈	—	2	2	no	—
76	C₃²×C₈	—	4	1	no	—
77	C₃²×C₈	C₂×C₈	1	2	yes	complemented
78	C₃×C₃⋊C₈	—	4	2	no	—
79	C₄×D₈	—	3	1	no	—
80	C₈×D₄	—	3	2	no	—
81	#334	—	3	1	no	—
82	D₄⋊C₈	—	3	2	no	—
83	#322	C₃×D₃	1	1	yes	characteristic, complemented
84	C₂×C₃×D₄	—	3	2	no	—
85	C₂²×C₃×C₄	—	6	2	no	—
86	C₂×D₁₂	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₄×D₃	—	2	2	no	—
88	C₃×D₈	—	3	4	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	1	no	—
91	C₂×C₃×C₈	—	2	1	no	—
92	C₂×C₃×C₈	—	2	1	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	2	no	—
95	C₂×C₃×C₈	—	6	2	no	—
96	C₂×C₃×C₈	—	12	4	no	—
97	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
98	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₃×C₈	D₁₂	1	2	yes	—
100	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
101	C₂×C₃×C₈	C₂×C₃×C₄	1	1	yes	characteristic
102	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
103	C₃×C₄⋊C₄	—	3	2	no	—
104	C₃×C₂²⋊C₄	—	6	2	no	—
105	C₃×C₄²	—	2	1	no	—
106	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
107	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
108	(C₂×D₃)⋊C₄	—	2	2	no	—
109	C₄⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
110	C₂×C₃⋊C₈	—	2	2	no	—
111	D₂₄	C₃×C₈	1	4	yes	complemented
112	D₃×C₈	—	4	4	no	—
113	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
114	C₂×C₃×D₃	—	2	4	no	—
115	C₂×C₃×D₃	—	4	2	no	—
116	C₂×C₃×D₃	—	4	2	no	—
117	C₃²×C₄	—	2	1	no	—
118	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
119	C₃²×C₄	#131	1	1	yes	characteristic
120	C₃²×C₄	AGL(1,ℤ/8)	1	2	yes	—
121	C₃×Dic₃	—	4	2	no	—
122	C₂×D₈	—	3	1	no	—
123	C₂²×C₈	—	6	2	no	—
124	C₄×D₄	—	3	2	no	—
125	#107	—	3	1	no	—
126	C₄⋊C₈	—	3	2	no	—
127	#102	—	3	2	no	—
128	C₂²⋊C₈	—	6	2	no	—
129	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
130	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
131	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
132	C₂³×C₃	—	6	2	no	—
133	C₂²×D₃	—	2	2	no	—
134	C₃×D₄	—	3	4	no	—
135	C₃×D₄	—	6	2	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	6	2	no	—
140	C₂×C₃×C₄	—	12	2	no	—
141	C₂×C₃×C₄	—	12	2	no	—
142	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
144	C₂×C₃×C₄	#235	1	1	yes	characteristic
145	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
148	C₂×Dic₃	—	2	2	no	—
149	D₁₂	—	2	2	no	—
150	D₁₂	C₂×C₃×C₈	1	4	yes	complemented
151	C₄×D₃	—	4	2	no	—
152	C₄×D₃	—	4	2	no	—
153	C₃×C₈	—	2	2	no	—
154	C₃×C₈	—	2	1	no	—
155	C₃×C₈	—	2	2	no	—
156	C₃×C₈	—	2	1	no	—
157	C₃×C₈	—	2	2	no	—
158	C₃×C₈	—	2	2	no	—
159	C₃×C₈	—	4	2	no	—
160	C₃×C₈	—	4	2	no	—
161	C₃×C₈	—	4	1	no	—
162	C₃×C₈	—	4	1	no	—
163	C₃×C₈	—	12	2	no	—
164	C₃×C₈	D₃×C₈	1	2	yes	complemented
165	C₃×C₈	C₂×C₃×C₈	1	2	yes	complemented
166	C₃⋊C₈	—	4	2	no	—
167	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
168	C₂×C₃²	#442	1	1	yes	characteristic
169	C₂×C₃²	#444	1	1	yes	characteristic
170	C₃×D₃	—	4	4	no	—
171	C₂×D₄	—	3	2	no	—
172	C₂²×C₄	—	6	2	no	—
173	D₈	—	3	4	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	6	2	no	—
176	C₂×C₈	—	12	4	no	—
177	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
178	C₂×C₈	C₃×C₄×D₃	1	1	yes	characteristic
179	C₂×C₈	C₃×D₁₂	1	2	yes	—
180	C₄⋊C₄	—	3	2	no	—
181	C₂²⋊C₄	—	6	2	no	—
182	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
183	C₂²×C₃	—	2	1	no	—
184	C₂²×C₃	—	6	4	no	—
185	C₂²×C₃	—	12	2	no	—
186	C₂²×C₃	—	12	2	no	—
187	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
188	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
189	C₂×D₃	—	2	4	no	—
190	C₂×D₃	—	4	2	no	—
191	C₂×D₃	—	4	2	no	—
192	C₃×C₄	—	2	2	no	—
193	C₃×C₄	—	2	1	no	—
194	C₃×C₄	—	2	1	no	—
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	12	2	no	—
199	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
200	C₃×C₄	#901	1	1	yes	characteristic
201	C₃×C₄	#908	1	2	yes	—
202	C₃×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
203	C₃×C₄	C₃×#131	1	1	yes	characteristic
204	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
205	Dic₃	—	4	2	no	—
206	C₃²	#1580	1	1	yes	characteristic, complemented, Sylow
207	C₂³	—	6	2	no	—
208	D₄	—	3	4	no	—
209	D₄	—	6	2	no	—
210	C₂×C₄	—	6	2	no	—
211	C₂×C₄	—	12	2	no	—
212	C₂×C₄	—	12	2	no	—
213	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
215	C₂×C₄	C₃×#235	1	1	yes	characteristic
216	C₈	—	2	1	no	—
217	C₈	—	2	2	no	—
218	C₈	—	4	1	no	—
219	C₈	—	12	2	no	—
220	C₈	C₃×D₃×C₈	1	2	yes	complemented
221	C₂×C₃	—	2	1	no	—
222	C₂×C₃	—	2	1	no	—
223	C₂×C₃	—	2	1	no	—
224	C₂×C₃	—	12	4	no	—
225	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
226	C₂×C₃	#4711	1	1	yes	characteristic
227	C₂×C₃	#4717	1	1	yes	characteristic
228	C₂×C₃	C₃×C₈×D₄	1	1	yes	characteristic
229	C₂×C₃	C₃×#442	1	1	yes	characteristic
230	C₂×C₃	C₃×#444	1	1	yes	characteristic
231	D₃	—	4	4	no	—
232	C₂²	—	6	4	no	—
233	C₂²	—	12	2	no	—
234	C₂²	—	12	2	no	—
235	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	12	2	no	—
238	C₄	C₂×C₃×D₃×C₈	1	1	yes	characteristic
239	C₄	C₃×#901	1	1	yes	characteristic
240	C₄	C₃×#908	1	2	yes	—
241	C₃	—	2	1	no	—
242	C₃	#68077	1	1	yes	characteristic, direct product factor
243	C₃	C₃×#1580	1	1	yes	characteristic, complemented
244	C₂	—	12	4	no	—
245	C₂	C₃×C₈×D₁₂	1	1	yes	characteristic
246	C₂	C₃×#4711	1	1	yes	characteristic
247	C₂	C₃×#4717	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₃×C₈
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#322	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×D₂₄	C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#322	D₃
hypocenter	C₃	C₃×#1580
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×#322

Complements

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

N	H ≅ G/N	type
2. C₃×C₈×D₁₂	C₂	semidirect
3. C₃²×#322	C₂	semidirect
4. C₃×#4459	C₂	semidirect
5. C₃×#4457	C₂	semidirect
7. #68077	C₃	direct (N × H)
21. C₈×D₁₂	C₂×C₃	semidirect
25. C₃×#322	D₃	semidirect
26. C₃×#322	C₂×C₃	semidirect
27. #4459	C₂×C₃	semidirect
28. #4457	C₂×C₃	semidirect
39. C₃×D₂₄	C₈	semidirect
71. C₃×D₁₂	C₂×C₈	semidirect
77. C₃²×C₈	C₂×C₈	semidirect
83. #322	C₃×D₃	semidirect
111. D₂₄	C₃×C₈	semidirect
150. D₁₂	C₂×C₃×C₈	semidirect
164. C₃×C₈	D₃×C₈	semidirect
165. C₃×C₈	C₂×C₃×C₈	semidirect
206. C₃²	#1580	semidirect
220. C₈	C₃×D₃×C₈	semidirect
242. C₃	#68077	direct (N × H)
243. C₃	C₃×#1580	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×D₂₄	2
C₃×C₈×D₁₂	2
C₃²×#322	2
C₃×#4459	2
C₃×#4457	2
C₃×#1580	3
#68077	3

Subgroups — C3×#68077 #11777344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#68077 #11777344