Subgroups — C₃×#12667 #10631274

Order 768.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#364	C₂	1	1	yes	maximal, characteristic
2	C₃×#2110	C₂	1	2	yes	maximal
3	C₃×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₃×#1378	C₂	1	2	yes	maximal
6	#12667	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₃×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₃×#108	C₄	1	1	yes	characteristic
11	C₃×#427	C₄	1	1	yes	characteristic
12	C₃×#405	C₂²	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₃×#364	C₄	1	2	yes	complemented
15	C₃×#364	C₄	1	2	yes	complemented
16	C₃×#347	—	2	4	no	—
17	C₃×#346	C₄	1	2	yes	complemented
18	C₃×C₄×C₁₆	C₄	1	2	yes	complemented
19	C₃×#339	C₄	1	2	yes	—
20	C₂×#364	C₂×C₃	1	1	yes	characteristic
21	#2110	C₂×C₃	1	2	yes	—
22	C₄×M₃₂	C₂×C₃	1	1	yes	characteristic
23	#1772	C₂×C₃	1	1	yes	characteristic
24	#1378	C₂×C₃	1	2	yes	—
25	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
28	C₃×#117	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃×M₃₂	—	2	2	no	—
31	C₃×M₃₂	—	2	1	no	—
32	C₃×M₃₂	—	2	1	no	—
33	C₃×M₃₂	C₂×C₄	1	4	yes	—
34	C₃×M₃₂	D₄	1	2	yes	—
35	C₃×M₃₂	Q₈	1	2	yes	—
36	C₂×C₃×C₁₆	—	2	2	no	—
37	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
39	C₃×#108	—	2	1	no	—
40	C₃×#108	—	2	1	no	—
41	C₃×#107	—	2	1	no	—
42	C₃×#106	—	2	1	no	—
43	C₃×#97	—	2	2	no	—
44	C₃×C₄×C₈	—	2	2	no	—
45	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₂×M₃₂	C₂²×C₃	1	2	yes	—
50	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
51	C₂×M₃₂	C₂²×C₃	1	1	yes	characteristic
52	C₂×#108	C₃×C₄	1	1	yes	characteristic
53	#427	C₃×C₄	1	1	yes	characteristic
54	#405	C₂²×C₃	1	2	yes	—
55	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
56	#364	C₃×C₄	1	2	yes	complemented
57	#364	C₃×C₄	1	2	yes	complemented
58	#347	—	2	4	no	—
59	#346	C₃×C₄	1	2	yes	complemented
60	C₄×C₁₆	C₃×C₄	1	2	yes	complemented
61	#339	C₃×C₄	1	2	yes	—
62	C₂²×C₃×C₄	—	2	1	no	—
63	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₃×M₁₆	—	2	2	no	—
65	C₃×M₁₆	—	2	2	no	—
66	C₂×C₃×C₈	—	2	2	no	—
67	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
68	C₂×C₃×C₈	C₄²	1	1	yes	characteristic
69	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
70	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
71	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
72	C₂×C₃×C₈	M₁₆	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₃×C₈	#41	1	1	yes	characteristic
78	C₂×C₃×C₈	#41	1	1	yes	characteristic
79	C₃×C₄⋊C₄	—	2	2	no	—
80	C₃×C₂²⋊C₄	—	2	2	no	—
81	C₃×C₄²	—	2	1	no	—
82	C₃×C₄²	—	2	2	no	—
83	C₃×C₄²	C₄²	1	1	yes	characteristic
84	C₃×C₄²	C₄²	1	1	yes	characteristic
85	C₃×C₁₆	—	2	2	no	—
86	C₃×C₁₆	—	2	4	no	—
87	C₃×C₁₆	C₄⋊C₄	1	4	yes	—
88	C₂×M₁₆	C₂×C₃×C₄	1	2	yes	—
89	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
91	#117	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
93	M₃₂	—	2	2	no	—
94	M₃₂	—	2	1	no	—
95	M₃₂	—	2	1	no	—
96	M₃₂	C₂×C₃×C₄	1	4	yes	—
97	M₃₂	C₃×D₄	1	2	yes	—
98	M₃₂	C₃×Q₈	1	2	yes	—
99	C₂×C₁₆	—	2	2	no	—
100	C₂×C₁₆	C₂×C₃×C₄	1	2	yes	—
101	C₂×C₁₆	C₂×C₃×C₄	1	2	yes	—
102	#108	—	2	1	no	—
103	#108	—	2	1	no	—
104	#107	—	2	1	no	—
105	#106	—	2	1	no	—
106	#97	—	2	2	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
109	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
110	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
111	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
112	C₂³×C₃	#131	1	1	yes	characteristic
113	C₂×C₃×C₄	—	2	1	no	—
114	C₂×C₃×C₄	—	2	1	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	—	4	2	no	—
117	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
119	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
120	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₃×C₄	#131	1	1	yes	characteristic
123	C₃×C₈	—	4	2	no	—
124	C₃×C₈	#97	1	1	yes	characteristic
125	C₃×C₈	#97	1	1	yes	characteristic
126	C₃×C₈	#97	1	1	yes	characteristic
127	C₃×C₈	#97	1	1	yes	characteristic
128	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₃×C₈	#117	1	1	yes	characteristic
130	C₃×C₈	C₄×D₄	1	1	yes	characteristic
131	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
132	C₂²×C₄	—	2	1	no	—
133	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	2	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	C₃×C₄²	1	1	yes	characteristic
138	C₂×C₈	C₃×C₄²	1	1	yes	characteristic
139	C₂×C₈	C₃×M₁₆	1	1	yes	characteristic
140	C₂×C₈	C₃×M₁₆	1	1	yes	characteristic
141	C₂×C₈	C₃×M₁₆	1	1	yes	characteristic
142	C₂×C₈	C₃×M₁₆	1	1	yes	characteristic
143	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
145	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
147	C₂×C₈	C₃×#41	1	1	yes	characteristic
148	C₂×C₈	C₃×#41	1	1	yes	characteristic
149	C₄⋊C₄	—	2	2	no	—
150	C₂²⋊C₄	—	2	2	no	—
151	C₄²	—	2	1	no	—
152	C₄²	—	2	2	no	—
153	C₄²	C₃×C₄²	1	1	yes	characteristic
154	C₄²	C₃×C₄²	1	1	yes	characteristic
155	C₁₆	—	2	2	no	—
156	C₁₆	—	2	4	no	—
157	C₁₆	C₃×C₄⋊C₄	1	4	yes	—
158	C₂²×C₃	—	4	1	no	—
159	C₂²×C₃	#405	1	1	yes	characteristic
160	C₂²×C₃	#436	1	1	yes	characteristic
161	C₂²×C₃	#446	1	1	yes	characteristic
162	C₃×C₄	—	2	2	no	—
163	C₃×C₄	—	4	1	no	—
164	C₃×C₄	—	8	2	no	—
165	C₃×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
166	C₃×C₄	C₂×#97	1	1	yes	characteristic
167	C₃×C₄	#436	1	1	yes	characteristic
168	C₃×C₄	#446	1	1	yes	characteristic
169	C₂³	C₃×#131	1	1	yes	characteristic
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
175	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
176	C₂×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
177	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
178	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
179	C₂×C₄	C₃×#131	1	1	yes	characteristic
180	C₈	—	4	2	no	—
181	C₈	C₃×#97	1	1	yes	characteristic
182	C₈	C₃×#97	1	1	yes	characteristic
183	C₈	C₃×#97	1	1	yes	characteristic
184	C₈	C₃×#97	1	1	yes	characteristic
185	C₈	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
186	C₈	C₃×#117	1	1	yes	characteristic
187	C₈	C₃×C₄×D₄	1	1	yes	characteristic
188	C₈	C₃×C₄×Q₈	1	1	yes	characteristic
189	C₂×C₃	—	2	1	no	—
190	C₂×C₃	—	2	1	no	—
191	C₂×C₃	—	2	1	no	—
192	C₂×C₃	#1770	1	1	yes	characteristic
193	C₂²	—	4	1	no	—
194	C₂²	C₃×#405	1	1	yes	characteristic
195	C₂²	C₃×#436	1	1	yes	characteristic
196	C₂²	C₃×#446	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	4	1	no	—
199	C₄	—	8	2	no	—
200	C₄	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
201	C₄	C₂×C₃×#97	1	1	yes	characteristic
202	C₄	C₃×#436	1	1	yes	characteristic
203	C₄	C₃×#446	1	1	yes	characteristic
204	C₃	#12667	1	1	yes	characteristic, direct product factor, Sylow
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	C₃×#1770	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	#117
commutator	C₂×C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂×C₃	#1770
prime radical	C₂³×C₃	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×M₃₂ ◁ G

Complements

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

N	H ≅ G/N	type
6. #12667	C₃	direct (N × H)
14. C₃×#364	C₄	semidirect
15. C₃×#364	C₄	semidirect
17. C₃×#346	C₄	semidirect
18. C₃×C₄×C₁₆	C₄	semidirect
56. #364	C₃×C₄	semidirect
57. #364	C₃×C₄	semidirect
59. #346	C₃×C₄	semidirect
60. C₄×C₁₆	C₃×C₄	semidirect
204. C₃	#12667	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#364	2
C₃×#2110	2
C₃×C₄×M₃₂	2
C₃×#1772	2
C₃×#1378	2
#12667	3

Subgroups — C3×#12667 #10631274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12667 #10631274