Subgroups — C₃×#13264 #10631871

Order 768.

386 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₃×#1374	C₂	1	4	yes	maximal
5	#13264	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
9	C₃×C₈×Q₈	—	2	2	no	—
10	C₃×C₈×Q₈	—	2	2	no	—
11	C₂×C₃×C₄⋊C₈	C₄	1	2	yes	—
12	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×C₈	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	C₂×C₈×Q₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
19	C₂×#346	C₂×C₃	1	1	yes	characteristic
20	#1374	C₂×C₃	1	4	yes	—
21	C₂²×C₃×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃×C₄×Q₈	—	2	2	no	—
26	C₃×C₄×Q₈	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₃×C₄⋊C₄	C₈	1	2	yes	—
29	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
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	—	2	4	no	—
35	C₂×C₃×C₁₆	Q₈	1	4	yes	complemented
36	C₃×C₄⋊C₈	—	2	4	no	—
37	C₃×C₄⋊C₈	—	2	2	no	—
38	C₃×C₄×C₈	—	2	2	no	—
39	C₃×C₄×C₈	—	2	2	no	—
40	C₃×C₄×C₈	D₄	1	2	yes	—
41	C₃×C₄×C₈	D₄	1	2	yes	—
42	C₂×C₄×Q₈	C₃×C₄	1	1	yes	characteristic
43	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
44	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
45	C₈×Q₈	—	2	2	no	—
46	C₈×Q₈	—	2	2	no	—
47	C₂×C₄⋊C₈	C₃×C₄	1	2	yes	—
48	C₂×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
50	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₄⋊C₁₆	—	2	2	no	—
52	#346	—	2	2	no	—
53	C₂×C₃×Q₈	—	2	2	no	—
54	C₂×C₃×Q₈	—	2	4	no	—
55	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
56	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
59	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₈	—	2	2	no	—
61	C₂×C₃×C₈	—	2	2	no	—
62	C₂×C₃×C₈	—	2	4	no	—
63	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₈	#41	1	2	yes	—
68	C₃×C₄⋊C₄	—	2	4	no	—
69	C₃×C₄⋊C₄	—	2	2	no	—
70	C₃×C₄²	—	2	2	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	C₂²⋊C₄	1	2	yes	—
73	C₃×C₄²	M₁₆	1	2	yes	—
74	C₃×C₁₆	—	2	4	no	—
75	C₃×C₁₆	—	4	2	no	—
76	C₂²×Q₈	C₃×C₈	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₃×C₄	1	2	yes	—
78	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
79	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
80	C₄×Q₈	—	2	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₂×C₄⋊C₄	C₃×C₈	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₃×C₈	1	2	yes	—
84	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
85	C₂×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂×C₁₆	—	2	2	no	—
87	C₂×C₁₆	—	2	2	no	—
88	C₂×C₁₆	—	2	2	no	—
89	C₂×C₁₆	—	2	4	no	—
90	C₂×C₁₆	C₃×Q₈	1	4	yes	complemented
91	C₄⋊C₈	—	2	4	no	—
92	C₄⋊C₈	—	2	2	no	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	—	2	2	no	—
95	C₄×C₈	C₃×D₄	1	2	yes	—
96	C₄×C₈	C₃×D₄	1	2	yes	—
97	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
98	C₃×Q₈	—	2	8	no	—
99	C₂×C₃×C₄	—	2	2	no	—
100	C₂×C₃×C₄	—	2	2	no	—
101	C₂×C₃×C₄	—	2	2	no	—
102	C₂×C₃×C₄	—	2	2	no	—
103	C₂×C₃×C₄	—	2	2	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
106	C₂×C₃×C₄	C₂²⋊C₈	1	2	yes	—
107	C₂×C₃×C₄	M₃₂	1	4	yes	—
108	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
110	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₃×C₄	#131	1	2	yes	—
112	C₃×C₈	—	2	2	no	—
113	C₃×C₈	—	4	2	no	—
114	C₃×C₈	C₂²⋊Q₈	1	2	yes	—
115	C₃×C₈	#124	1	1	yes	characteristic
116	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×Q₈	—	2	2	no	—
118	C₂×Q₈	—	2	4	no	—
119	C₂²×C₄	C₂×C₃×C₈	1	2	yes	—
120	C₂²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
121	C₂²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
122	C₂²×C₄	C₂×C₃×C₈	1	2	yes	—
123	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	4	no	—
127	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
129	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
130	C₂×C₈	C₂×C₃×Q₈	1	2	yes	—
131	C₂×C₈	C₃×#41	1	2	yes	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₃×C₂²⋊C₄	1	2	yes	—
137	C₄²	C₃×M₁₆	1	2	yes	—
138	C₁₆	—	2	4	no	—
139	C₁₆	—	4	2	no	—
140	C₂²×C₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₃	C₈×Q₈	1	2	yes	—
142	C₂²×C₃	C₂×M₃₂	1	2	yes	—
143	C₂²×C₃	#504	1	2	yes	—
144	C₃×C₄	—	2	2	no	—
145	C₃×C₄	—	2	4	no	—
146	C₃×C₄	—	2	2	no	—
147	C₃×C₄	—	4	2	no	—
148	C₃×C₄	#391	1	1	yes	characteristic
149	C₃×C₄	#408	1	1	yes	characteristic
150	C₃×C₄	#446	1	2	yes	—
151	C₂³	C₂²×C₃×C₈	1	1	yes	characteristic
152	Q₈	—	2	8	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	C₃×C₂²⋊C₈	1	2	yes	—
160	C₂×C₄	C₃×C₂²⋊C₈	1	2	yes	—
161	C₂×C₄	C₃×M₃₂	1	4	yes	—
162	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₄	C₃×C₄×Q₈	1	2	yes	—
164	C₂×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
165	C₂×C₄	C₃×#131	1	2	yes	—
166	C₈	—	2	2	no	—
167	C₈	—	4	2	no	—
168	C₈	C₃×C₂²⋊Q₈	1	2	yes	—
169	C₈	C₃×#124	1	1	yes	characteristic
170	C₈	C₃×C₄⋊Q₈	1	1	yes	characteristic
171	C₂×C₃	#1984	1	1	yes	characteristic
172	C₂×C₃	#2112	1	1	yes	characteristic
173	C₂×C₃	#2113	1	1	yes	characteristic
174	C₂×C₃	#2183	1	4	yes	—
175	C₂²	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
176	C₂²	C₃×C₈×Q₈	1	2	yes	—
177	C₂²	C₂×C₃×M₃₂	1	2	yes	—
178	C₂²	C₃×#504	1	2	yes	—
179	C₄	—	2	2	no	—
180	C₄	—	2	4	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	2	no	—
183	C₄	C₃×#391	1	1	yes	characteristic
184	C₄	C₃×#408	1	1	yes	characteristic
185	C₄	C₃×#446	1	2	yes	—
186	C₃	#13264	1	1	yes	characteristic, direct product factor, Sylow
187	C₂	C₃×#1984	1	1	yes	characteristic
188	C₂	C₃×#2112	1	1	yes	characteristic
189	C₂	C₃×#2113	1	1	yes	characteristic
190	C₂	C₃×#2183	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
5. #13264	C₃	direct (N × H)
35. C₂×C₃×C₁₆	Q₈	semidirect
90. C₂×C₁₆	C₃×Q₈	semidirect
186. C₃	#13264	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×C₈×Q₈	2
C₂×C₃×C₄⋊C₁₆	2
C₂×C₃×#346	2
C₃×#1374	2
#13264	3

Subgroups — C3×#13264 #10631871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#13264 #10631871