Subgroups — C₃×#69762 #11786031

Order 1152.

577 conjugacy classes of subgroups.

The subgroup lattice has 236 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	C₃×#4935	C₂	1	2	yes	maximal
3	C₃×#4551	C₂	1	4	yes	maximal
4	C₃×#1834	—	3	1	no	maximal
5	#69762	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	2	no	—
9	C₂×C₃×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₃×C₄⋊Dic₃	C₂²	1	4	yes	—
11	C₃²×C₄×C₈	C₄	1	4	yes	complemented
12	C₂×C₃×C₄×C₈	—	2	1	no	—
13	C₂×C₃×C₄×C₈	D₃	1	1	yes	characteristic
14	C₂×C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
15	C₃×#384	—	3	2	no	—
16	#4935	C₂×C₃	1	2	yes	—
17	C₃×#340	—	3	4	no	—
18	#4551	C₂×C₃	1	4	yes	—
19	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
20	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
21	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₂²×C₃×Dic₃	—	2	4	no	—
23	C₂×C₃²×C₈	—	2	4	no	—
24	C₂×C₃²×C₈	C₂×C₄	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	4	no	—
29	C₃×C₄⋊Dic₃	—	4	4	no	—
30	#1834	—	3	1	no	Sylow
31	C₂²×C₃×C₈	—	2	2	no	—
32	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
33	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
34	C₂×C₃×C₄⋊C₄	—	3	4	no	—
35	C₂×C₃×C₄⋊C₄	—	6	2	no	—
36	C₂×C₃×C₄⋊C₄	—	6	2	no	—
37	C₂×C₃×C₄²	—	2	1	no	—
38	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
39	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
40	C₂×C₄⋊Dic₃	—	2	2	no	—
41	C₂×C₄⋊Dic₃	—	2	2	no	—
42	C₂×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
43	C₃×C₄×C₈	—	2	4	no	—
44	C₃×C₄×C₈	Dic₃	1	4	yes	complemented
45	C₃×C₄×C₈	C₃×C₄	1	4	yes	complemented
46	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
48	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
49	C₂×C₃²×C₄	D₈	1	2	yes	—
50	C₂×C₃²×C₄	SD₁₆	1	4	yes	—
51	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
52	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₃²×C₄	#41	1	4	yes	—
55	C₂×C₃×Dic₃	—	4	4	no	—
56	C₂×C₃×Dic₃	—	4	4	no	—
57	C₂×C₃×Dic₃	—	4	4	no	—
58	C₃²×C₈	—	2	4	no	—
59	C₂×C₄×C₈	C₃×D₃	1	1	yes	characteristic
60	#384	—	3	2	no	—
61	#340	—	3	4	no	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	—	2	1	no	—
64	C₂²×C₃×C₄	—	2	1	no	—
65	C₂²×C₃×C₄	—	6	4	no	—
66	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
67	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
70	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
72	C₂²×Dic₃	—	2	4	no	—
73	C₂×C₃×C₈	—	2	8	no	—
74	C₂×C₃×C₈	—	2	4	no	—
75	C₂×C₃×C₈	—	2	4	no	—
76	C₂×C₃×C₈	—	2	4	no	—
77	C₂×C₃×C₈	C₂×Dic₃	1	4	yes	—
78	C₂×C₃×C₈	C₂×C₃×C₄	1	4	yes	—
79	C₃×C₄⋊C₄	—	6	8	no	—
80	C₃×C₄⋊C₄	—	12	4	no	—
81	C₃×C₄⋊C₄	—	12	4	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	—	2	2	no	—
84	C₃×C₄²	Dic₆	1	2	yes	—
85	C₃×C₄²	C₂×Dic₃	1	2	yes	—
86	C₃×C₄²	C₂×C₃×C₄	1	2	yes	—
87	C₃×C₄²	C₃×Q₈	1	2	yes	—
88	C₄⋊Dic₃	—	2	8	no	—
89	C₄⋊Dic₃	—	4	4	no	—
90	C₄⋊Dic₃	—	4	4	no	—
91	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₃²	#124	1	2	yes	—
93	C₂²×C₃²	C₂×D₈	1	1	yes	characteristic
94	C₂²×C₃²	C₂×SD₁₆	1	2	yes	—
95	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
96	C₃²×C₄	#106	1	4	yes	—
97	C₃²×C₄	#107	1	4	yes	—
98	C₃²×C₄	#117	1	2	yes	—
99	C₃²×C₄	#125	1	2	yes	—
100	C₃×Dic₃	—	8	4	no	—
101	C₂²×C₈	C₂×C₃×D₃	1	2	yes	—
102	C₂×C₄⋊C₄	—	3	4	no	—
103	C₂×C₄⋊C₄	—	6	2	no	—
104	C₂×C₄⋊C₄	—	6	2	no	—
105	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
106	C₄×C₈	C₃×Dic₃	1	4	yes	complemented
107	C₂³×C₃	—	2	1	no	—
108	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
109	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	2	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	1	no	—
116	C₂×C₃×C₄	—	2	4	no	—
117	C₂×C₃×C₄	—	2	1	no	—
118	C₂×C₃×C₄	—	12	4	no	—
119	C₂×C₃×C₄	—	12	4	no	—
120	C₂×C₃×C₄	—	12	4	no	—
121	C₂×C₃×C₄	#204	1	4	yes	—
122	C₂×C₃×C₄	D₂₄	1	2	yes	—
123	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
124	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
125	C₂×C₃×C₄	C₄⋊Dic₃	1	2	yes	—
126	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
127	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
128	C₂×C₃×C₄	C₃×D₈	1	2	yes	—
129	C₂×C₃×C₄	C₃×SD₁₆	1	4	yes	—
130	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
131	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
132	C₂×C₃×C₄	#235	1	4	yes	—
133	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
135	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
136	C₂×C₃×C₄	C₃×#41	1	4	yes	—
137	C₂×Dic₃	—	4	4	no	—
138	C₂×Dic₃	—	4	4	no	—
139	C₂×Dic₃	—	4	4	no	—
140	C₃×C₈	—	2	8	no	—
141	C₃×C₈	—	2	4	no	—
142	C₃×C₈	—	2	4	no	—
143	C₂×C₃²	#382	1	1	yes	characteristic
144	C₂×C₃²	C₂×#106	1	1	yes	characteristic
145	C₂×C₃²	C₂×#107	1	1	yes	characteristic
146	C₂×C₃²	#486	1	2	yes	—
147	C₂×C₃²	#487	1	2	yes	—
148	C₂²×C₄	—	6	4	no	—
149	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
150	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂×C₃×Dic₃	1	4	yes	—
154	C₄⋊C₄	—	6	8	no	—
155	C₄⋊C₄	—	12	4	no	—
156	C₄⋊C₄	—	12	4	no	—
157	C₄²	C₃×Dic₆	1	2	yes	—
158	C₄²	C₂×C₃×Dic₃	1	2	yes	—
159	C₂²×C₃	—	2	2	no	—
160	C₂²×C₃	—	2	1	no	—
161	C₂²×C₃	—	2	1	no	—
162	C₂²×C₃	—	2	1	no	—
163	C₂²×C₃	—	2	2	no	—
164	C₂²×C₃	#875	1	2	yes	—
165	C₂²×C₃	C₂×#204	1	2	yes	—
166	C₂²×C₃	C₂×D₂₄	1	1	yes	characteristic
167	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
168	C₂²×C₃	C₂×C₄⋊Dic₃	1	1	yes	characteristic
169	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₃	C₃×#124	1	2	yes	—
171	C₂²×C₃	C₂×C₃×D₈	1	1	yes	characteristic
172	C₂²×C₃	C₂×C₃×SD₁₆	1	2	yes	—
173	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	4	no	—
176	C₃×C₄	—	2	4	no	—
177	C₃×C₄	—	2	2	no	—
178	C₃×C₄	—	24	4	no	—
179	C₃×C₄	#817	1	4	yes	—
180	C₃×C₄	#818	1	4	yes	—
181	C₃×C₄	C₃×#106	1	4	yes	—
182	C₃×C₄	C₃×#107	1	4	yes	—
183	C₃×C₄	#870	1	2	yes	—
184	C₃×C₄	#926	1	2	yes	—
185	C₃×C₄	C₃×#117	1	2	yes	—
186	C₃×C₄	C₃×#125	1	2	yes	—
187	Dic₃	—	8	4	no	—
188	C₃²	#1834	1	1	yes	characteristic, complemented, Sylow
189	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
190	C₂×C₄	—	12	4	no	—
191	C₂×C₄	—	12	4	no	—
192	C₂×C₄	—	12	4	no	—
193	C₂×C₄	C₃×#204	1	4	yes	—
194	C₂×C₄	C₃×D₂₄	1	2	yes	—
195	C₂×C₄	C₃×Dic₁₂	1	2	yes	—
196	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
197	C₂×C₄	C₃×C₄⋊Dic₃	1	2	yes	—
198	C₂×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
199	C₂×C₄	C₃×#235	1	4	yes	—
200	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
201	C₈	—	2	4	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	—	2	2	no	—
205	C₂×C₃	—	2	1	no	—
206	C₂×C₃	—	2	1	no	—
207	C₂×C₃	#4681	1	2	yes	—
208	C₂×C₃	#4694	1	2	yes	—
209	C₂×C₃	#4936	1	1	yes	characteristic
210	C₂×C₃	C₂×#817	1	1	yes	characteristic
211	C₂×C₃	C₂×#818	1	1	yes	characteristic
212	C₂×C₃	C₃×#382	1	1	yes	characteristic
213	C₂×C₃	C₂×C₃×#106	1	1	yes	characteristic
214	C₂×C₃	C₂×C₃×#107	1	1	yes	characteristic
215	C₂×C₃	C₃×#486	1	2	yes	—
216	C₂×C₃	C₃×#487	1	2	yes	—
217	C₂²	C₃×#875	1	2	yes	—
218	C₂²	C₂×C₃×#204	1	2	yes	—
219	C₂²	C₂×C₃×D₂₄	1	1	yes	characteristic
220	C₂²	C₂×C₃×Dic₁₂	1	1	yes	characteristic
221	C₂²	C₂×C₃×C₄⋊Dic₃	1	1	yes	characteristic
222	C₄	—	24	4	no	—
223	C₄	C₃×#817	1	4	yes	—
224	C₄	C₃×#818	1	4	yes	—
225	C₄	C₃×#870	1	2	yes	—
226	C₄	C₃×#926	1	2	yes	—
227	C₃	—	2	1	no	—
228	C₃	#69762	1	1	yes	characteristic, direct product factor
229	C₃	C₃×#1834	1	1	yes	characteristic, complemented
230	C₂	C₃×#4681	1	2	yes	—
231	C₂	C₃×#4694	1	2	yes	—
232	C₂	C₃×#4936	1	1	yes	characteristic
233	C₂	C₂×C₃×#817	1	1	yes	characteristic
234	C₂	C₂×C₃×#818	1	1	yes	characteristic

	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×#1834
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. #69762	C₃	direct (N × H)
11. C₃²×C₄×C₈	C₄	semidirect
44. C₃×C₄×C₈	Dic₃	semidirect
45. C₃×C₄×C₈	C₃×C₄	semidirect
106. C₄×C₈	C₃×Dic₃	semidirect
188. C₃²	#1834	semidirect
228. C₃	#69762	direct (N × H)
229. C₃	C₃×#1834	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
C₃×#4935	2
C₃×#4551	2
C₃×#1834	3
#69762	3

Subgroups — C3×#69762 #11786031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69762 #11786031