Subgroups — C₃×#13267 #10631874

Order 768.

334 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2960	C₂	1	1	yes	maximal, characteristic
2	C₃×#2150	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₃×#1379	C₂	1	4	yes	maximal
5	#13267	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₃×#533	C₄	1	1	yes	characteristic
7	C₂×C₃×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×#446	—	2	2	no	—
10	C₃×C₈×Q₈	—	2	2	no	—
11	C₃×#424	C₄	1	2	yes	—
12	C₃×#423	C₄	1	1	yes	characteristic
13	C₃×#405	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊C₁₆	—	2	2	no	—
16	C₃×#346	—	2	2	no	—
17	#2960	C₂×C₃	1	1	yes	characteristic
18	#2150	C₂×C₃	1	1	yes	characteristic
19	C₂×#346	C₂×C₃	1	1	yes	characteristic
20	#1379	C₂×C₃	1	4	yes	—
21	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
24	C₃×C₄⋊Q₈	C₈	1	2	yes	—
25	C₃×#125	C₈	1	2	yes	—
26	C₃×C₂²⋊Q₈	—	2	2	no	—
27	C₃×C₄×Q₈	—	2	2	no	—
28	C₃×#117	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
30	C₃×M₃₂	—	2	4	no	—
31	C₂×C₃×C₁₆	—	2	2	no	—
32	C₂×C₃×C₁₆	—	2	2	no	—
33	C₂×C₃×C₁₆	—	2	2	no	—
34	C₂×C₃×C₁₆	Q₈	1	4	yes	complemented
35	C₃×C₄⋊C₈	—	2	4	no	—
36	C₃×C₄⋊C₈	—	2	2	no	—
37	C₃×#97	—	2	2	no	—
38	C₃×C₄×C₈	—	2	2	no	—
39	C₃×C₄×C₈	D₄	1	2	yes	—
40	C₃×C₄×C₈	D₄	1	2	yes	—
41	#533	C₃×C₄	1	1	yes	characteristic
42	C₂×M₃₂	C₂²×C₃	1	2	yes	—
43	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
44	#446	—	2	2	no	—
45	C₈×Q₈	—	2	2	no	—
46	#424	C₃×C₄	1	2	yes	—
47	#423	C₃×C₄	1	1	yes	characteristic
48	#405	C₂²×C₃	1	2	yes	—
49	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₄⋊C₁₆	—	2	2	no	—
51	#346	—	2	2	no	—
52	C₂×C₃×Q₈	—	2	2	no	—
53	C₂²×C₃×C₄	M₁₆	1	2	yes	—
54	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
55	C₃×M₁₆	—	2	4	no	—
56	C₂×C₃×C₈	—	2	2	no	—
57	C₂×C₃×C₈	—	2	2	no	—
58	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
62	C₂×C₃×C₈	#41	1	2	yes	—
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
66	C₃×C₂²⋊C₄	—	2	2	no	—
67	C₃×C₄²	—	2	2	no	—
68	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
69	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
70	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
71	C₃×C₁₆	—	2	4	no	—
72	C₃×C₁₆	—	4	2	no	—
73	C₂×M₁₆	C₂×C₃×C₄	1	2	yes	—
74	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
76	C₄⋊Q₈	C₃×C₈	1	2	yes	—
77	#125	C₃×C₈	1	2	yes	—
78	C₂²⋊Q₈	—	2	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	#117	C₂×C₃×C₄	1	2	yes	—
81	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
82	M₃₂	—	2	4	no	—
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	C₃×Q₈	1	4	yes	complemented
87	C₄⋊C₈	—	2	4	no	—
88	C₄⋊C₈	—	2	2	no	—
89	#97	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	C₃×D₄	1	2	yes	—
92	C₄×C₈	C₃×D₄	1	2	yes	—
93	C₂³×C₃	C₂×M₁₆	1	1	yes	characteristic
94	C₃×Q₈	—	4	4	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	—	4	2	no	—
97	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
98	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
99	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
101	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₃×C₄	#131	1	2	yes	—
103	C₃×C₈	—	2	2	no	—
104	C₃×C₈	—	4	2	no	—
105	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
106	C₃×C₈	#124	1	1	yes	characteristic
107	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
108	C₂×Q₈	—	2	2	no	—
109	C₂²×C₄	C₃×M₁₆	1	2	yes	—
110	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
111	M₁₆	—	2	4	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
115	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
116	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
117	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
118	C₂×C₈	C₃×#41	1	2	yes	—
119	C₄⋊C₄	—	2	4	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	C₂×C₃×C₈	1	4	yes	—
122	C₂²⋊C₄	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
125	C₄²	C₂×C₃×C₈	1	1	yes	characteristic
126	C₄²	C₂×C₃×C₈	1	1	yes	characteristic
127	C₁₆	—	2	4	no	—
128	C₁₆	—	4	2	no	—
129	C₂²×C₃	—	2	2	no	—
130	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂²×C₃	#446	1	2	yes	—
132	C₃×C₄	—	2	2	no	—
133	C₃×C₄	—	2	2	no	—
134	C₃×C₄	—	4	4	no	—
135	C₃×C₄	#391	1	1	yes	characteristic
136	C₃×C₄	#408	1	1	yes	characteristic
137	C₃×C₄	C₈×Q₈	1	2	yes	—
138	C₂³	C₂×C₃×M₁₆	1	1	yes	characteristic
139	Q₈	—	4	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	C₃×C₂²⋊C₈	1	2	yes	—
143	C₂×C₄	C₃×C₂²⋊C₈	1	2	yes	—
144	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
146	C₂×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
147	C₂×C₄	C₃×#131	1	2	yes	—
148	C₈	—	2	2	no	—
149	C₈	—	4	2	no	—
150	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
151	C₈	C₃×#124	1	1	yes	characteristic
152	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
153	C₂×C₃	—	2	2	no	—
154	C₂×C₃	#1984	1	1	yes	characteristic
155	C₂×C₃	#2117	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²	C₃×#446	1	2	yes	—
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₃×#391	1	1	yes	characteristic
163	C₄	C₃×#408	1	1	yes	characteristic
164	C₄	C₃×C₈×Q₈	1	2	yes	—
165	C₃	#13267	1	1	yes	characteristic, direct product factor, Sylow
166	C₂	—	2	2	no	—
167	C₂	C₃×#1984	1	1	yes	characteristic
168	C₂	C₃×#2117	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₂²⋊C₈
prime radical	C₂³×C₃	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
5. #13267	C₃	direct (N × H)
34. C₂×C₃×C₁₆	Q₈	semidirect
86. C₂×C₁₆	C₃×Q₈	semidirect
165. C₃	#13267	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×#2960	2
C₃×#2150	2
C₂×C₃×#346	2
C₃×#1379	2
#13267	3

Subgroups — C3×#13267 #10631874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13267 #10631874