Subgroups — C₂×#69280 #10948667

Order 768.

582 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₃⋊C₁₆	C₂	1	2	yes	maximal
4	C₂×Dic₃×C₁₆	C₂	1	2	yes	maximal
5	C₂×C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#69280	C₂	1	8	yes	maximal, direct product factor
7	C₂×Q₈×C₁₆	—	3	1	no	maximal, Sylow
8	C₂×C₄×Dic₆	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₃⋊C₈	C₄	1	2	yes	—
12	C₂×C₈×Dic₃	C₂²	1	2	yes	—
13	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
16	C₈×Dic₆	C₄	1	4	yes	—
17	C₈×Dic₆	C₂²	1	4	yes	—
18	C₃×C₄⋊C₁₆	C₂²	1	4	yes	—
19	Dic₃⋊C₁₆	C₂²	1	8	yes	—
20	Dic₃×C₁₆	C₂²	1	8	yes	—
21	C₄×C₃⋊C₁₆	C₂²	1	4	yes	—
22	C₂×C₈×Q₈	—	3	1	no	—
23	Q₈×C₁₆	—	3	8	no	—
24	C₂×C₄⋊C₁₆	—	3	2	no	—
25	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic
26	C₂×C₄×C₁₆	—	3	2	no	—
27	C₂×C₄×C₁₆	—	3	1	no	—
28	C₂²×Dic₆	C₈	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊Dic₃	C₈	1	1	yes	characteristic
33	C₂×Dic₃⋊C₄	C₈	1	2	yes	—
34	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
35	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
36	C₄×Dic₆	C₈	1	4	yes	—
37	C₄×Dic₆	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₁₆	—	2	4	no	—
39	C₂×C₃×C₁₆	C₂³	1	4	yes	—
40	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
41	C₃×C₄×C₈	C₂³	1	2	yes	—
42	Dic₃⋊C₈	C₂×C₄	1	8	yes	—
43	C₈×Dic₃	C₂×C₄	1	4	yes	—
44	C₈×Dic₃	C₂³	1	4	yes	—
45	C₂×C₃⋊C₁₆	—	2	2	no	—
46	C₂×C₃⋊C₁₆	Q₈	1	4	yes	complemented
47	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
48	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
49	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
50	C₂×C₄×Q₈	—	3	1	no	—
51	C₂²×C₁₆	—	3	1	no	—
52	C₂²×C₁₆	—	3	1	no	—
53	C₂²×C₁₆	C₂×D₃	1	2	yes	—
54	C₈×Q₈	—	3	4	no	—
55	C₈×Q₈	—	3	4	no	—
56	C₂×C₄⋊C₈	—	3	2	no	—
57	C₂×C₄⋊C₈	—	3	1	no	—
58	C₂×C₄×C₈	—	3	2	no	—
59	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
60	C₄⋊C₁₆	—	3	8	no	—
61	C₄⋊C₁₆	C₂×D₃	1	4	yes	—
62	C₄×C₁₆	—	3	8	no	—
63	C₄×C₁₆	—	3	4	no	—
64	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₃	C₂×C₈	1	2	yes	—
68	C₂²×Dic₃	C₂×C₈	1	2	yes	—
69	C₂×Dic₆	C₁₆	1	8	yes	complemented
70	C₂×Dic₆	C₂×C₈	1	4	yes	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
75	C₂×C₃×C₈	#41	1	2	yes	—
76	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
77	C₃×C₄²	C₂×C₈	1	2	yes	—
78	C₃×C₄²	C₂²×C₄	1	2	yes	—
79	C₄⋊Dic₃	C₂×C₈	1	4	yes	—
80	Dic₃⋊C₄	C₂×C₈	1	8	yes	—
81	C₄×Dic₃	C₂×C₈	1	4	yes	—
82	C₄×Dic₃	C₂²×C₄	1	4	yes	—
83	C₂×C₃⋊C₈	—	2	4	no	—
84	C₂×C₃⋊C₈	C₂²×C₄	1	4	yes	—
85	C₃×C₁₆	—	2	4	no	—
86	C₃⋊C₁₆	—	2	2	no	—
87	C₃⋊C₁₆	C₂×Q₈	1	4	yes	complemented
88	C₂²×Q₈	—	3	1	no	—
89	C₂²×C₈	—	3	2	no	—
90	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
92	C₄×Q₈	—	3	4	no	—
93	C₄×Q₈	—	3	4	no	—
94	C₂×C₄⋊C₄	—	3	2	no	—
95	C₂×C₄⋊C₄	—	3	1	no	—
96	C₂×C₄²	—	3	2	no	—
97	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
98	C₂×C₁₆	—	2	4	no	—
99	C₂×C₁₆	—	3	4	no	—
100	C₂×C₁₆	—	3	2	no	—
101	C₂×C₁₆	—	3	2	no	—
102	C₂×C₁₆	—	6	2	no	—
103	C₂×C₁₆	C₂²×D₃	1	4	yes	—
104	C₄⋊C₈	—	3	8	no	—
105	C₄⋊C₈	—	3	4	no	—
106	C₄×C₈	—	3	4	no	—
107	C₄×C₈	—	3	4	no	—
108	C₄×C₈	C₄×D₃	1	2	yes	—
109	C₄×C₈	C₂²×D₃	1	2	yes	—
110	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	C₂×C₁₆	1	4	yes	—
113	C₂×C₃×C₄	C₄×Q₈	1	2	yes	—
114	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
115	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
117	C₂×C₃×C₄	#131	1	2	yes	—
118	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
119	C₂×Dic₃	—	2	4	no	—
120	C₂×Dic₃	C₂×C₁₆	1	8	yes	—
121	C₂×Dic₃	C₂²×C₈	1	4	yes	—
122	C₂×Dic₃	C₂²×C₈	1	4	yes	—
123	Dic₆	C₂×C₁₆	1	16	yes	complemented
124	C₃×C₈	—	2	2	no	—
125	C₃×C₈	C₄×Q₈	1	2	yes	—
126	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
127	C₃×C₈	C₂×#41	1	1	yes	characteristic
128	C₃⋊C₈	—	2	4	no	—
129	C₂×Q₈	—	3	8	no	—
130	C₂×Q₈	—	3	4	no	—
131	C₂²×C₄	—	3	2	no	—
132	C₂²×C₄	—	3	2	no	—
133	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
134	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	3	4	no	—
138	C₂×C₈	—	6	4	no	—
139	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₈	#237	1	2	yes	—
142	C₂×C₈	D₃×Q₈	1	2	yes	—
143	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
144	C₄⋊C₄	—	3	8	no	—
145	C₄⋊C₄	—	3	4	no	—
146	C₄²	—	3	4	no	—
147	C₄²	—	3	4	no	—
148	C₄²	D₃×C₈	1	2	yes	—
149	C₄²	C₂×C₄×D₃	1	2	yes	—
150	C₁₆	—	2	4	no	—
151	C₁₆	—	3	4	no	—
152	C₁₆	—	6	2	no	—
153	C₂²×C₃	C₈×Q₈	1	2	yes	—
154	C₂²×C₃	C₂²×C₁₆	1	2	yes	—
155	C₂²×C₃	#504	1	2	yes	—
156	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
157	C₃×C₄	—	2	2	no	—
158	C₃×C₄	C₈×Q₈	1	2	yes	—
159	C₃×C₄	C₂²×C₁₆	1	4	yes	—
160	C₃×C₄	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₃×C₄	C₂×#131	1	1	yes	characteristic
162	Dic₃	—	2	4	no	—
163	Dic₃	C₂²×C₁₆	1	8	yes	—
164	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
165	Q₈	—	3	16	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	3	8	no	—
168	C₂×C₄	—	3	4	no	—
169	C₂×C₄	—	3	4	no	—
170	C₂×C₄	—	6	4	no	—
171	C₂×C₄	D₃×C₁₆	1	4	yes	—
172	C₂×C₄	#887	1	2	yes	—
173	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
174	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
175	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
176	C₂×C₄	#907	1	2	yes	—
177	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
178	C₈	—	2	2	no	—
179	C₈	—	6	4	no	—
180	C₈	#887	1	2	yes	—
181	C₈	C₂×#237	1	1	yes	characteristic
182	C₈	C₂×D₃×Q₈	1	1	yes	characteristic
183	C₂×C₃	Q₈×C₁₆	1	4	yes	complemented
184	C₂×C₃	C₂×C₈×Q₈	1	1	yes	characteristic
185	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
186	C₂×C₃	C₂×#504	1	1	yes	characteristic
187	C₂²	#4830	1	2	yes	—
188	C₂²	C₂×D₃×C₁₆	1	2	yes	—
189	C₂²	#4907	1	2	yes	—
190	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	3	8	no	—
193	C₄	—	6	4	no	—
194	C₄	#4830	1	2	yes	—
195	C₄	C₂×D₃×C₁₆	1	4	yes	—
196	C₄	C₂×#887	1	1	yes	characteristic
197	C₄	C₂×#907	1	1	yes	characteristic
198	C₃	C₂×Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
199	C₂	#69280	1	4	yes	direct product factor
200	C₂	C₂×#4830	1	1	yes	characteristic
201	C₂	C₂²×D₃×C₁₆	1	1	yes	characteristic
202	C₂	C₂×#4907	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₃
commutator	C₂×C₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₃	C₂²×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	C₂×Q₈×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
6. #69280	C₂	direct (N × H)
46. C₂×C₃⋊C₁₆	Q₈	semidirect
69. C₂×Dic₆	C₁₆	semidirect
87. C₃⋊C₁₆	C₂×Q₈	semidirect
123. Dic₆	C₂×C₁₆	semidirect
183. C₂×C₃	Q₈×C₁₆	semidirect
198. C₃	C₂×Q₈×C₁₆	semidirect
199. C₂	#69280	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×C₈×Dic₆	2
C₂×C₃×C₄⋊C₁₆	2
C₂×Dic₃⋊C₁₆	2
C₂×Dic₃×C₁₆	2
C₂×C₄×C₃⋊C₁₆	2
#69280	2
C₂×Q₈×C₁₆	3

Subgroups — C2×#69280 #10948667

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69280 #10948667