Subgroups — C₃×#70516 #11786090

Order 1152.

439 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#423	C₂	1	1	yes	maximal, characteristic
2	C₃×#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#4558	C₂	1	4	yes	maximal
5	C₃×#1837	—	3	1	no	maximal
6	#70516	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃×#208	C₂²	1	2	yes	—
10	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
11	C₃²×C₄⋊C₈	—	2	2	no	—
12	C₃×C₄⋊(C₃⋊C₈)	—	2	2	no	—
13	C₃×C₄×C₃⋊C₈	C₄	1	2	yes	—
14	C₃×C₄×C₃⋊C₈	C₄	1	2	yes	—
15	C₃×#423	—	2	1	no	—
16	C₃×#423	—	3	1	no	—
17	C₃×#423	D₃	1	1	yes	characteristic
18	C₃×#423	C₂×C₃	1	1	yes	characteristic
19	C₂×C₃×C₄×C₈	—	3	1	no	—
20	#4924	C₂×C₃	1	1	yes	characteristic
21	C₂×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃×#341	—	3	4	no	—
23	#4558	C₂×C₃	1	4	yes	—
24	C₂²×C₃²×C₄	Q₈	1	2	yes	—
25	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₃²×M₁₆	—	2	4	no	—
27	C₂×C₃²×C₈	—	2	2	no	—
28	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃²×C₄²	D₄	1	2	yes	—
31	C₃×#208	—	2	4	no	—
32	C₂×C₃×C₃⋊C₈	—	2	2	no	—
33	C₂×C₃×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃×C₃⋊C₈	C₂×C₄	1	4	yes	—
36	#1837	—	3	1	no	Sylow
37	C₂×C₃×M₁₆	—	2	2	no	—
38	C₂×C₃×M₁₆	—	3	2	no	—
39	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
40	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
41	C₂²×C₃×C₈	—	3	2	no	—
42	C₂×C₃×C₄²	—	2	1	no	—
43	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
45	C₂×#208	C₂²×C₃	1	2	yes	—
46	C₂²×C₃⋊C₈	C₂²×C₃	1	2	yes	—
47	C₃×C₄⋊C₈	—	2	4	no	—
48	C₃×C₄⋊C₈	—	2	2	no	—
49	C₃×C₄⋊C₈	—	2	2	no	—
50	C₃×C₄⋊C₈	—	6	2	no	—
51	C₃×C₄×C₈	—	3	2	no	—
52	C₃×C₄×C₈	—	3	2	no	—
53	C₄⋊(C₃⋊C₈)	—	2	2	no	—
54	C₄×C₃⋊C₈	C₃×C₄	1	2	yes	—
55	C₄×C₃⋊C₈	C₃×C₄	1	2	yes	—
56	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₃²×C₄	—	2	4	no	—
58	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
59	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
60	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃²×C₄	#41	1	2	yes	—
63	C₂×C₃²×C₄	#41	1	2	yes	—
64	C₃²×C₈	—	4	2	no	—
65	C₃×C₃⋊C₈	—	2	4	no	—
66	C₃×C₃⋊C₈	—	4	2	no	—
67	#423	—	3	1	no	—
68	#423	C₃×D₃	1	1	yes	characteristic
69	C₂×C₄×C₈	—	3	1	no	—
70	#341	—	3	4	no	—
71	C₂²×C₃×C₄	—	2	2	no	—
72	C₂²×C₃×C₄	—	2	1	no	—
73	C₂²×C₃×C₄	Dic₆	1	2	yes	—
74	C₂²×C₃×C₄	C₃×Q₈	1	2	yes	—
75	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
77	C₃×M₁₆	—	2	4	no	—
78	C₃×M₁₆	—	2	4	no	—
79	C₃×M₁₆	—	4	4	no	—
80	C₃×M₁₆	—	6	4	no	—
81	C₂×C₃×C₈	—	2	4	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₂×C₃×C₈	—	3	4	no	—
85	C₂×C₃×C₈	—	6	2	no	—
86	C₂×C₃×C₈	—	6	2	no	—
87	C₂×C₃×C₈	—	6	2	no	—
88	C₃×C₄²	—	2	1	no	—
89	C₃×C₄²	—	2	1	no	—
90	C₃×C₄²	—	2	2	no	—
91	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
92	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
93	C₃×C₄²	#67	1	2	yes	—
94	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
95	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
96	C₃×C₄²	C₃×D₄	1	2	yes	—
97	#208	—	2	4	no	—
98	C₂×C₃⋊C₈	—	2	2	no	—
99	C₂×C₃⋊C₈	—	2	2	no	—
100	C₂×C₃⋊C₈	—	2	2	no	—
101	C₂×C₃⋊C₈	C₂×C₃×C₄	1	4	yes	—
102	C₂²×C₃²	—	2	2	no	—
103	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃²	#125	1	2	yes	—
105	C₃²×C₄	—	2	2	no	—
106	C₃²×C₄	#108	1	4	yes	—
107	C₃²×C₄	#117	1	2	yes	—
108	C₃²×C₄	#124	1	1	yes	characteristic
109	C₃²×C₄	#124	1	1	yes	characteristic
110	C₂×M₁₆	—	3	2	no	—
111	C₂×M₁₆	C₂×C₃×D₃	1	2	yes	—
112	C₂²×C₈	—	3	2	no	—
113	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	6	2	no	—
116	C₄×C₈	—	3	2	no	—
117	C₄×C₈	—	3	2	no	—
118	C₂³×C₃	—	2	1	no	—
119	C₂³×C₃	C₂×Dic₆	1	1	yes	characteristic
120	C₂³×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
121	C₂×C₃×C₄	—	2	4	no	—
122	C₂×C₃×C₄	—	2	4	no	—
123	C₂×C₃×C₄	—	2	2	no	—
124	C₂×C₃×C₄	—	2	2	no	—
125	C₂×C₃×C₄	—	2	1	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	2	2	no	—
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	4	4	no	—
130	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
131	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
132	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
133	C₂×C₃×C₄	C₃×C₄⋊C₄	1	2	yes	—
134	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂×C₃×C₄	#237	1	2	yes	—
136	C₂×C₃×C₄	#239	1	2	yes	—
137	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×C₃×C₄	C₃×#41	1	2	yes	—
141	C₂×C₃×C₄	C₃×#41	1	2	yes	—
142	C₃×C₈	—	4	4	no	—
143	C₃×C₈	—	4	2	no	—
144	C₃×C₈	—	4	2	no	—
145	C₃×C₈	—	6	4	no	—
146	C₃×C₈	—	12	2	no	—
147	C₃⋊C₈	—	2	4	no	—
148	C₃⋊C₈	—	4	2	no	—
149	C₂×C₃²	—	2	2	no	—
150	C₂×C₃²	#382	1	1	yes	characteristic
151	C₂×C₃²	C₂×#108	1	2	yes	—
152	C₂²×C₄	C₃×Dic₆	1	2	yes	—
153	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
154	M₁₆	—	2	4	no	—
155	M₁₆	—	6	4	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	3	4	no	—
158	C₂×C₈	—	6	2	no	—
159	C₂×C₈	—	6	2	no	—
160	C₂×C₈	—	6	2	no	—
161	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
162	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
163	C₄²	C₃×#67	1	2	yes	—
164	C₂²×C₃	—	2	2	no	—
165	C₂²×C₃	—	2	2	no	—
166	C₂²×C₃	—	2	2	no	—
167	C₂²×C₃	—	2	1	no	—
168	C₂²×C₃	—	4	2	no	—
169	C₂²×C₃	#890	1	2	yes	—
170	C₂²×C₃	C₂×Dic₃⋊C₄	1	1	yes	characteristic
171	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×C₃	C₃×#125	1	2	yes	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	2	2	no	—
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	1	no	—
178	C₃×C₄	—	2	4	no	—
179	C₃×C₄	—	4	2	no	—
180	C₃×C₄	#822	1	4	yes	—
181	C₃×C₄	C₃×#108	1	4	yes	—
182	C₃×C₄	#892	1	2	yes	—
183	C₃×C₄	#936	1	1	yes	characteristic
184	C₃×C₄	#947	1	1	yes	characteristic
185	C₃×C₄	C₃×#117	1	2	yes	—
186	C₃×C₄	C₃×#124	1	1	yes	characteristic
187	C₃×C₄	C₃×#124	1	1	yes	characteristic
188	C₃²	#1837	1	1	yes	characteristic, complemented, Sylow
189	C₂³	C₂×C₃×Dic₆	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
192	C₂×C₄	C₃×Dic₃⋊C₄	1	2	yes	—
193	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
194	C₂×C₄	C₃×#237	1	2	yes	—
195	C₂×C₄	C₃×#239	1	2	yes	—
196	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
197	C₈	—	4	2	no	—
198	C₈	—	6	4	no	—
199	C₈	—	12	2	no	—
200	C₂×C₃	—	2	2	no	—
201	C₂×C₃	—	2	2	no	—
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	—	4	2	no	—
205	C₂×C₃	#4977	1	1	yes	characteristic
206	C₂×C₃	C₂×#822	1	2	yes	—
207	C₂×C₃	C₃×#382	1	1	yes	characteristic
208	C₂×C₃	C₂×C₃×#108	1	2	yes	—
209	C₂²	—	2	2	no	—
210	C₂²	C₃×#890	1	2	yes	—
211	C₂²	C₂×C₃×Dic₃⋊C₄	1	1	yes	characteristic
212	C₄	—	2	2	no	—
213	C₄	C₃×#822	1	4	yes	—
214	C₄	C₃×#892	1	2	yes	—
215	C₄	C₃×#936	1	1	yes	characteristic
216	C₄	C₃×#947	1	1	yes	characteristic
217	C₃	—	2	1	no	—
218	C₃	#70516	1	1	yes	characteristic, direct product factor
219	C₃	C₃×#1837	1	1	yes	characteristic, complemented
220	C₂	—	2	2	no	—
221	C₂	C₃×#4977	1	1	yes	characteristic
222	C₂	C₂×C₃×#822	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#423	C₂
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#423	D₃
hypocenter	C₃	C₃×#1837
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#423

Complements

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

N	H ≅ G/N	type
6. #70516	C₃	direct (N × H)
188. C₃²	#1837	semidirect
218. C₃	#70516	direct (N × H)
219. C₃	C₃×#1837	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#423	2
C₃×#4924	2
C₂×C₃×C₄×C₃⋊C₈	2
C₃×#4558	2
C₃×#1837	3
#70516	3

Subgroups — C3×#70516 #11786090

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70516 #11786090