Subgroups — C₃×#8132 #10626739

Order 768.

478 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1744	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#97	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#1273	C₂	1	4	yes	maximal
5	#8132	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#97	C₄	1	2	yes	—
7	C₂×C₃×#97	C₄	1	2	yes	—
8	C₂×C₃×C₄×C₈	C₄	1	4	yes	—
9	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₃×C₄³	C₂²	1	1	yes	characteristic
13	C₃×#336	C₄	1	4	yes	—
14	#1744	C₂×C₃	1	1	yes	characteristic
15	C₄×#97	C₂×C₃	1	1	yes	characteristic
16	C₄²×C₈	C₂×C₃	1	1	yes	characteristic
17	#1273	C₂×C₃	1	4	yes	—
18	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
19	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
26	C₃×#97	—	2	4	no	—
27	C₃×#97	C₈	1	8	yes	complemented
28	C₃×C₄×C₈	—	2	8	no	—
29	C₃×C₄×C₈	—	2	4	no	—
30	C₃×C₄×C₈	—	2	4	no	—
31	C₃×C₄×C₈	—	2	4	no	—
32	C₂×#97	C₃×C₄	1	2	yes	—
33	C₂×#97	C₃×C₄	1	2	yes	—
34	C₂×C₄×C₈	C₃×C₄	1	4	yes	—
35	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
36	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
37	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
38	C₄³	C₂²×C₃	1	1	yes	characteristic
39	#336	C₃×C₄	1	4	yes	—
40	C₂²×C₃×C₄	C₄²	1	2	yes	—
41	C₂²×C₃×C₄	C₄²	1	2	yes	—
42	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₃×C₈	—	2	4	no	—
46	C₂×C₃×C₈	—	2	4	no	—
47	C₂×C₃×C₈	—	2	4	no	—
48	C₂×C₃×C₈	—	2	4	no	—
49	C₂×C₃×C₈	—	2	4	no	—
50	C₂×C₃×C₈	—	2	8	no	—
51	C₂×C₃×C₈	C₂×C₈	1	8	yes	—
52	C₂×C₃×C₈	M₁₆	1	8	yes	—
53	C₃×C₄²	—	2	4	no	—
54	C₃×C₄²	—	2	4	no	—
55	C₃×C₄²	C₂×C₈	1	4	yes	—
56	C₃×C₄²	M₁₆	1	4	yes	—
57	C₃×C₄²	#41	1	4	yes	—
58	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
59	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
60	C₂²×C₈	C₂×C₃×C₄	1	4	yes	—
61	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
62	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
63	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
66	#97	—	2	4	no	—
67	#97	C₃×C₈	1	8	yes	complemented
68	C₄×C₈	—	2	8	no	—
69	C₄×C₈	—	2	4	no	—
70	C₄×C₈	—	2	4	no	—
71	C₄×C₈	—	2	4	no	—
72	C₂³×C₃	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₃×C₄	—	2	2	no	—
74	C₂×C₃×C₄	—	2	2	no	—
75	C₂×C₃×C₄	—	2	2	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	C₄×C₈	1	4	yes	—
78	C₂×C₃×C₄	#97	1	4	yes	—
79	C₂×C₃×C₄	#117	1	2	yes	—
80	C₂×C₃×C₄	#117	1	2	yes	—
81	C₂×C₃×C₄	#117	1	2	yes	—
82	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
83	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
84	C₂×C₃×C₄	C₂×M₁₆	1	4	yes	—
85	C₂×C₃×C₄	#131	1	4	yes	—
86	C₃×C₈	—	2	8	no	—
87	C₃×C₈	—	2	8	no	—
88	C₃×C₈	—	4	4	no	—
89	C₂²×C₄	C₃×C₄²	1	2	yes	—
90	C₂²×C₄	C₃×C₄²	1	2	yes	—
91	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	8	no	—
100	C₂×C₈	C₂×C₃×C₈	1	8	yes	—
101	C₂×C₈	C₃×M₁₆	1	8	yes	—
102	C₄²	—	2	4	no	—
103	C₄²	—	2	4	no	—
104	C₄²	C₂×C₃×C₈	1	4	yes	—
105	C₄²	C₃×M₁₆	1	4	yes	—
106	C₄²	C₃×#41	1	4	yes	—
107	C₂²×C₃	#376	1	1	yes	characteristic
108	C₂²×C₃	C₂×C₄×C₈	1	1	yes	characteristic
109	C₂²×C₃	C₂×#97	1	1	yes	characteristic
110	C₂²×C₃	C₄×M₁₆	1	2	yes	—
111	C₂²×C₃	#405	1	2	yes	—
112	C₃×C₄	—	2	4	no	—
113	C₃×C₄	—	2	4	no	—
114	C₃×C₄	#431	1	4	yes	—
115	C₃×C₄	#432	1	4	yes	—
116	C₃×C₄	#433	1	4	yes	—
117	C₂³	C₂×C₃×C₄²	1	1	yes	characteristic
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	C₃×C₄×C₈	1	4	yes	—
123	C₂×C₄	C₃×#97	1	4	yes	—
124	C₂×C₄	C₃×#117	1	2	yes	—
125	C₂×C₄	C₃×#117	1	2	yes	—
126	C₂×C₄	C₃×#117	1	2	yes	—
127	C₂×C₄	C₂²×C₃×C₈	1	2	yes	—
128	C₂×C₄	C₂×C₃×M₁₆	1	2	yes	—
129	C₂×C₄	C₂×C₃×M₁₆	1	4	yes	—
130	C₂×C₄	C₃×#131	1	4	yes	—
131	C₈	—	2	8	no	—
132	C₈	—	2	8	no	—
133	C₈	—	4	4	no	—
134	C₂×C₃	C₈×M₁₆	1	2	yes	—
135	C₂×C₃	#1451	1	2	yes	—
136	C₂×C₃	#1744	1	1	yes	characteristic
137	C₂×C₃	#1745	1	1	yes	characteristic
138	C₂×C₃	#1746	1	1	yes	characteristic
139	C₂²	C₃×#376	1	1	yes	characteristic
140	C₂²	C₂×C₃×C₄×C₈	1	1	yes	characteristic
141	C₂²	C₂×C₃×#97	1	1	yes	characteristic
142	C₂²	C₃×C₄×M₁₆	1	2	yes	—
143	C₂²	C₃×#405	1	2	yes	—
144	C₄	—	2	4	no	—
145	C₄	—	2	4	no	—
146	C₄	C₃×#431	1	4	yes	—
147	C₄	C₃×#432	1	4	yes	—
148	C₄	C₃×#433	1	4	yes	—
149	C₃	#8132	1	1	yes	characteristic, direct product factor, Sylow
150	C₂	C₃×C₈×M₁₆	1	2	yes	—
151	C₂	C₃×#1451	1	2	yes	—
152	C₂	C₃×#1744	1	1	yes	characteristic
153	C₂	C₃×#1745	1	1	yes	characteristic
154	C₂	C₃×#1746	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
5. #8132	C₃	direct (N × H)
27. C₃×#97	C₈	semidirect
67. #97	C₃×C₈	semidirect
149. C₃	#8132	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#1744	2
C₃×C₄×#97	2
C₃×C₄²×C₈	2
C₃×#1273	2
#8132	3

Subgroups — C3×#8132 #10626739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#8132 #10626739