Subgroups — C₃×#12665 #10631272

Order 768.

326 conjugacy classes of subgroups.

The subgroup lattice has 188 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×#2108	C₂	1	2	yes	maximal
3	C₂×C₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₃×#1772	C₂	1	1	yes	maximal, characteristic
5	C₃×#1379	C₂	1	2	yes	maximal
6	#12665	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₁₆	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	4	yes	complemented
15	C₃×#346	—	2	2	no	—
16	C₃×#346	C₄	1	4	yes	complemented
17	C₃×C₄×C₁₆	—	2	2	no	—
18	C₃×#339	C₄	1	2	yes	—
19	C₂×#364	C₂×C₃	1	1	yes	characteristic
20	#2108	C₂×C₃	1	2	yes	—
21	C₂×#346	C₂×C₃	1	1	yes	characteristic
22	#1772	C₂×C₃	1	1	yes	characteristic
23	#1379	C₂×C₃	1	2	yes	—
24	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
27	C₃×#117	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
29	C₃×M₃₂	—	2	2	no	—
30	C₃×M₃₂	C₂×C₄	1	4	yes	—
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₂×C₃×C₁₆	—	2	1	no	—
33	C₂×C₃×C₁₆	—	2	1	no	—
34	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₁₆	D₄	1	2	yes	—
37	C₂×C₃×C₁₆	Q₈	1	2	yes	—
38	C₃×#108	C₈	1	4	yes	—
39	C₃×#107	C₈	1	2	yes	—
40	C₃×#106	C₈	1	2	yes	—
41	C₃×#97	—	2	2	no	—
42	C₃×C₄×C₈	—	2	2	no	—
43	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
44	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
45	C₂×M₃₂	C₂²×C₃	1	2	yes	—
46	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₁₆	C₂²×C₃	1	1	yes	characteristic
48	C₂×#108	C₃×C₄	1	1	yes	characteristic
49	#427	C₃×C₄	1	1	yes	characteristic
50	#405	C₂²×C₃	1	2	yes	—
51	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
52	#364	C₃×C₄	1	4	yes	complemented
53	#346	—	2	2	no	—
54	#346	C₃×C₄	1	4	yes	complemented
55	C₄×C₁₆	—	2	2	no	—
56	#339	C₃×C₄	1	2	yes	—
57	C₂²×C₃×C₄	—	2	1	no	—
58	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃×M₁₆	—	2	2	no	—
60	C₃×M₁₆	C₂×C₈	1	4	yes	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	C₄²	1	2	yes	—
63	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
64	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₈	#41	1	1	yes	characteristic
71	C₂×C₃×C₈	#41	1	1	yes	characteristic
72	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
73	C₃×C₂²⋊C₄	—	2	2	no	—
74	C₃×C₄²	—	2	2	no	—
75	C₃×C₄²	—	2	1	no	—
76	C₃×C₄²	C₄²	1	2	yes	—
77	C₃×C₁₆	—	2	2	no	—
78	C₃×C₁₆	—	2	4	no	—
79	C₃×C₁₆	C₄⋊C₄	1	4	yes	—
80	C₂×M₁₆	C₂×C₃×C₄	1	2	yes	—
81	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
83	#117	C₂×C₃×C₄	1	2	yes	—
84	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
85	M₃₂	—	2	2	no	—
86	M₃₂	C₂×C₃×C₄	1	4	yes	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	1	no	—
90	C₂×C₁₆	C₂×C₃×C₄	1	2	yes	—
91	C₂×C₁₆	C₂×C₃×C₄	1	2	yes	—
92	C₂×C₁₆	C₃×D₄	1	2	yes	—
93	C₂×C₁₆	C₃×Q₈	1	2	yes	—
94	#108	C₃×C₈	1	4	yes	—
95	#107	C₃×C₈	1	2	yes	—
96	#106	C₃×C₈	1	2	yes	—
97	#97	—	2	2	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
100	C₄×C₈	C₂×C₃×C₄	1	2	yes	—
101	C₂³×C₃	#131	1	1	yes	characteristic
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	4	no	—
104	C₂×C₃×C₄	—	4	2	no	—
105	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄	C₄×Q₈	1	1	yes	characteristic
108	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₃×C₄	#131	1	1	yes	characteristic
111	C₃×C₈	—	2	4	no	—
112	C₃×C₈	C₄×C₈	1	2	yes	—
113	C₃×C₈	C₄×C₈	1	2	yes	—
114	C₃×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₃×C₈	#117	1	1	yes	characteristic
116	C₃×C₈	C₄×D₄	1	1	yes	characteristic
117	C₃×C₈	C₄×Q₈	1	1	yes	characteristic
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
120	M₁₆	—	2	2	no	—
121	M₁₆	C₂×C₃×C₈	1	4	yes	—
122	C₂×C₈	—	2	2	no	—
123	C₂×C₈	C₃×C₄²	1	2	yes	—
124	C₂×C₈	C₂×C₃×C₈	1	1	yes	characteristic
125	C₂×C₈	C₂×C₃×C₈	1	1	yes	characteristic
126	C₂×C₈	C₂×C₃×C₈	1	2	yes	—
127	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
128	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
129	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
131	C₂×C₈	C₃×#41	1	1	yes	characteristic
132	C₂×C₈	C₃×#41	1	1	yes	characteristic
133	C₄⋊C₄	C₂×C₃×C₈	1	4	yes	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	1	no	—
137	C₄²	C₃×C₄²	1	2	yes	—
138	C₁₆	—	2	2	no	—
139	C₁₆	—	2	4	no	—
140	C₁₆	C₃×C₄⋊C₄	1	4	yes	—
141	C₂²×C₃	—	2	2	no	—
142	C₂²×C₃	#405	1	1	yes	characteristic
143	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
144	C₂²×C₃	C₈×Q₈	1	1	yes	characteristic
145	C₃×C₄	—	4	2	no	—
146	C₃×C₄	—	4	4	no	—
147	C₃×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
148	C₃×C₄	C₂×C₄×C₈	1	1	yes	characteristic
149	C₃×C₄	C₈×D₄	1	1	yes	characteristic
150	C₃×C₄	C₈×Q₈	1	1	yes	characteristic
151	C₂³	C₃×#131	1	1	yes	characteristic
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	2	no	—
155	C₂×C₄	C₂×C₃×C₄²	1	1	yes	characteristic
156	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
157	C₂×C₄	C₃×C₄×Q₈	1	1	yes	characteristic
158	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
159	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
160	C₂×C₄	C₃×#131	1	1	yes	characteristic
161	C₈	—	2	4	no	—
162	C₈	C₃×C₄×C₈	1	2	yes	—
163	C₈	C₃×C₄×C₈	1	2	yes	—
164	C₈	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
165	C₈	C₃×#117	1	1	yes	characteristic
166	C₈	C₃×C₄×D₄	1	1	yes	characteristic
167	C₈	C₃×C₄×Q₈	1	1	yes	characteristic
168	C₂×C₃	—	2	1	no	—
169	C₂×C₃	—	2	1	no	—
170	C₂×C₃	C₈×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₃	#2171	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	C₃×#405	1	1	yes	characteristic
174	C₂²	C₃×C₈×D₄	1	1	yes	characteristic
175	C₂²	C₃×C₈×Q₈	1	1	yes	characteristic
176	C₄	—	4	2	no	—
177	C₄	—	4	4	no	—
178	C₄	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
179	C₄	C₂×C₃×C₄×C₈	1	1	yes	characteristic
180	C₄	C₃×C₈×D₄	1	1	yes	characteristic
181	C₄	C₃×C₈×Q₈	1	1	yes	characteristic
182	C₃	#12665	1	1	yes	characteristic, direct product factor, Sylow
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	C₃×C₈×C₄⋊C₄	1	1	yes	characteristic
186	C₂	C₃×#2171	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₄
commutator	C₄	C₂×C₃×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂²×C₃	C₈×D₄
prime radical	C₂³×C₃	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
6. #12665	C₃	direct (N × H)
14. C₃×#364	C₄	semidirect
16. C₃×#346	C₄	semidirect
52. #364	C₃×C₄	semidirect
54. #346	C₃×C₄	semidirect
182. C₃	#12665	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×#2108	2
C₂×C₃×#346	2
C₃×#1772	2
C₃×#1379	2
#12665	3

Subgroups — C3×#12665 #10631272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12665 #10631272