Subgroups — C₃×#69765 #11786229

Order 1152.

641 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×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#818	C₂	1	4	yes	maximal
3	C₃×#4935	C₂	1	2	yes	maximal
4	C₃×#1845	—	3	1	no	maximal
5	#69765	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂²×C₃²×C₈	C₂²	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊Dic₃	—	2	4	no	—
9	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
10	C₃²×C₄×C₈	C₄	1	4	yes	complemented
11	C₃×#818	—	2	8	no	—
12	C₂×C₃×#107	—	3	4	no	—
13	C₂×C₃×C₄×C₈	—	2	1	no	—
14	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
15	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
16	C₃×#384	—	3	2	no	—
17	C₂×#818	C₂×C₃	1	4	yes	—
18	#4935	C₂×C₃	1	2	yes	—
19	C₂²×C₃²×C₄	D₄	1	2	yes	—
20	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
21	C₂²×C₃×Dic₃	—	2	4	no	—
22	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
23	C₂×C₃²×C₈	D₄	1	4	yes	—
24	C₂×C₃²×C₈	Q₈	1	4	yes	—
25	C₃²×C₄²	C₂×C₄	1	2	yes	—
26	C₃²×C₄²	Q₈	1	2	yes	—
27	C₃×C₄⋊Dic₃	—	2	8	no	—
28	C₃×C₄⋊Dic₃	—	4	8	no	—
29	#1845	—	3	1	no	Sylow
30	C₂²×C₃×C₈	—	2	2	no	—
31	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
32	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
33	C₂×C₃×C₄⋊C₄	—	3	4	no	—
34	C₂×C₃×C₄⋊C₄	—	6	4	no	—
35	C₂×C₃×C₄²	—	2	1	no	—
36	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
37	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₄⋊Dic₃	—	2	4	no	—
39	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
40	C₃×#107	—	6	8	no	—
41	C₃×C₄×C₈	—	2	4	no	—
42	C₃×C₄×C₈	Dic₃	1	4	yes	complemented
43	C₃×C₄×C₈	C₃×C₄	1	4	yes	complemented
44	#818	—	2	8	no	—
45	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃²×C₄	C₄⋊C₄	1	4	yes	—
47	C₂×C₃²×C₄	D₈	1	4	yes	—
48	C₂×C₃²×C₄	Q₁₆	1	4	yes	—
49	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
51	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
52	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₃×Dic₃	—	4	4	no	—
54	C₂×C₃×Dic₃	—	4	8	no	—
55	C₃²×C₈	C₄⋊C₄	1	8	yes	complemented
56	C₂×#107	—	3	4	no	—
57	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
58	#384	—	3	2	no	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	1	no	—
61	C₂²×C₃×C₄	—	6	4	no	—
62	C₂²×C₃×C₄	D₁₂	1	2	yes	—
63	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
64	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
66	C₂²×Dic₃	—	2	4	no	—
67	C₂×C₃×C₈	—	2	4	no	—
68	C₂×C₃×C₈	—	2	4	no	—
69	C₂×C₃×C₈	—	2	4	no	—
70	C₂×C₃×C₈	Dic₆	1	4	yes	—
71	C₂×C₃×C₈	D₁₂	1	4	yes	—
72	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
73	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
74	C₂×C₃×C₈	C₃×D₄	1	4	yes	—
75	C₂×C₃×C₈	C₃×Q₈	1	4	yes	—
76	C₃×C₄⋊C₄	—	6	8	no	—
77	C₃×C₄⋊C₄	—	12	8	no	—
78	C₃×C₄²	—	2	2	no	—
79	C₃×C₄²	—	2	2	no	—
80	C₃×C₄²	Dic₆	1	2	yes	—
81	C₃×C₄²	C₂×Dic₃	1	2	yes	—
82	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
83	C₃×C₄²	C₃×Q₈	1	2	yes	—
84	C₄⋊Dic₃	—	2	8	no	—
85	C₄⋊Dic₃	—	4	8	no	—
86	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₃²	#127	1	1	yes	characteristic
88	C₂²×C₃²	C₄⋊Q₈	1	1	yes	characteristic
89	C₂²×C₃²	C₂×D₈	1	2	yes	—
90	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
91	C₃²×C₄	#107	1	8	yes	complemented
92	C₃²×C₄	C₂×C₄⋊C₄	1	2	yes	—
93	C₃²×C₄	C₄⋊Q₈	1	2	yes	—
94	C₃×Dic₃	—	8	4	no	—
95	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
96	C₂×C₄⋊C₄	—	3	4	no	—
97	C₂×C₄⋊C₄	—	6	4	no	—
98	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
99	#107	—	6	8	no	—
100	C₄×C₈	C₃×Dic₃	1	4	yes	complemented
101	C₂³×C₃	—	2	1	no	—
102	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
103	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	2	1	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	2	no	—
110	C₂×C₃×C₄	—	2	1	no	—
111	C₂×C₃×C₄	—	12	4	no	—
112	C₂×C₃×C₄	—	12	8	no	—
113	C₂×C₃×C₄	D₂₄	1	4	yes	—
114	C₂×C₃×C₄	Dic₁₂	1	4	yes	—
115	C₂×C₃×C₄	C₄⋊Dic₃	1	4	yes	—
116	C₂×C₃×C₄	C₃×C₄⋊C₄	1	4	yes	—
117	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
118	C₂×C₃×C₄	C₃×Q₁₆	1	4	yes	—
119	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×Dic₆	1	2	yes	—
121	C₂×C₃×C₄	C₂×D₁₂	1	2	yes	—
122	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
123	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
125	C₂×C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
126	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
127	C₂×Dic₃	—	4	4	no	—
128	C₂×Dic₃	—	4	8	no	—
129	C₃×C₈	—	2	8	no	—
130	C₃×C₈	C₄⋊Dic₃	1	8	yes	complemented
131	C₃×C₈	C₃×C₄⋊C₄	1	8	yes	complemented
132	C₂×C₃²	#384	1	1	yes	characteristic
133	C₂×C₃²	C₂×#107	1	2	yes	—
134	C₂×C₃²	#493	1	1	yes	characteristic
135	C₂×C₃²	#494	1	1	yes	characteristic
136	C₂×C₃²	#500	1	2	yes	—
137	C₂²×C₄	—	6	4	no	—
138	C₂²×C₄	C₃×D₁₂	1	2	yes	—
139	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
140	C₂×C₈	C₃×Dic₆	1	4	yes	—
141	C₂×C₈	C₃×D₁₂	1	4	yes	—
142	C₂×C₈	C₂×C₃×Dic₃	1	4	yes	—
143	C₄⋊C₄	—	6	8	no	—
144	C₄⋊C₄	—	12	8	no	—
145	C₄²	C₃×Dic₆	1	2	yes	—
146	C₄²	C₂×C₃×Dic₃	1	2	yes	—
147	C₂²×C₃	—	2	2	no	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	2	1	no	—
150	C₂²×C₃	—	2	1	no	—
151	C₂²×C₃	—	2	1	no	—
152	C₂²×C₃	#869	1	1	yes	characteristic
153	C₂²×C₃	#874	1	1	yes	characteristic
154	C₂²×C₃	C₂×D₂₄	1	2	yes	—
155	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
156	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
157	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₃	C₃×#127	1	1	yes	characteristic
159	C₂²×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
160	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
161	C₂²×C₃	C₂×C₃×Q₁₆	1	2	yes	—
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	2	8	no	—
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	—	24	4	no	—
166	C₃×C₄	#818	1	8	yes	complemented
167	C₃×C₄	C₃×#107	1	8	yes	complemented
168	C₃×C₄	#869	1	2	yes	—
169	C₃×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
170	C₃×C₄	C₂×C₃×C₄⋊C₄	1	2	yes	—
171	C₃×C₄	C₃×C₄⋊Q₈	1	2	yes	—
172	Dic₃	—	8	4	no	—
173	C₃²	#1845	1	1	yes	characteristic, complemented, Sylow
174	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
175	C₂×C₄	—	12	8	no	—
176	C₂×C₄	—	12	4	no	—
177	C₂×C₄	C₃×D₂₄	1	4	yes	—
178	C₂×C₄	C₃×Dic₁₂	1	4	yes	—
179	C₂×C₄	C₃×C₄⋊Dic₃	1	4	yes	—
180	C₂×C₄	C₂×C₃×Dic₆	1	2	yes	—
181	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
182	C₂×C₄	C₂×C₃×D₁₂	1	2	yes	—
183	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
184	C₈	C₃×C₄⋊Dic₃	1	8	yes	complemented
185	C₂×C₃	—	2	1	no	—
186	C₂×C₃	—	2	2	no	—
187	C₂×C₃	—	2	1	no	—
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	—	2	2	no	—
190	C₂×C₃	#4682	1	2	yes	—
191	C₂×C₃	#4695	1	1	yes	characteristic
192	C₂×C₃	#4699	1	1	yes	characteristic
193	C₂×C₃	#4935	1	1	yes	characteristic
194	C₂×C₃	C₂×#818	1	2	yes	—
195	C₂×C₃	C₃×#384	1	1	yes	characteristic
196	C₂×C₃	C₂×C₃×#107	1	2	yes	—
197	C₂×C₃	C₃×#493	1	1	yes	characteristic
198	C₂×C₃	C₃×#494	1	1	yes	characteristic
199	C₂×C₃	C₃×#500	1	2	yes	—
200	C₂²	C₃×#869	1	1	yes	characteristic
201	C₂²	C₃×#874	1	1	yes	characteristic
202	C₂²	C₂×C₃×D₂₄	1	2	yes	—
203	C₂²	C₂×C₃×Dic₁₂	1	2	yes	—
204	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
205	C₄	—	24	4	no	—
206	C₄	C₃×#818	1	8	yes	complemented
207	C₄	C₃×#869	1	2	yes	—
208	C₄	C₂×C₃×C₄⋊Dic₃	1	2	yes	—
209	C₃	—	2	1	no	—
210	C₃	#69765	1	1	yes	characteristic, direct product factor
211	C₃	C₃×#1845	1	1	yes	characteristic, complemented
212	C₂	C₃×#4682	1	2	yes	—
213	C₂	C₃×#4695	1	1	yes	characteristic
214	C₂	C₃×#4699	1	1	yes	characteristic
215	C₂	C₃×#4935	1	1	yes	characteristic
216	C₂	C₂×C₃×#818	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₄×C₈	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₃
hypocenter	C₃	C₃×#1845
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
5. #69765	C₃	direct (N × H)
10. C₃²×C₄×C₈	C₄	semidirect
42. C₃×C₄×C₈	Dic₃	semidirect
43. C₃×C₄×C₈	C₃×C₄	semidirect
55. C₃²×C₈	C₄⋊C₄	semidirect
91. C₃²×C₄	#107	semidirect
100. C₄×C₈	C₃×Dic₃	semidirect
130. C₃×C₈	C₄⋊Dic₃	semidirect
131. C₃×C₈	C₃×C₄⋊C₄	semidirect
166. C₃×C₄	#818	semidirect
167. C₃×C₄	C₃×#107	semidirect
173. C₃²	#1845	semidirect
184. C₈	C₃×C₄⋊Dic₃	semidirect
206. C₄	C₃×#818	semidirect
210. C₃	#69765	direct (N × H)
211. C₃	C₃×#1845	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₄×C₈	2
C₂×C₃×#818	2
C₃×#4935	2
C₃×#1845	3
#69765	3

Subgroups — C3×#69765 #11786229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69765 #11786229