Subgroups — C₂×#69743 #10948166

Order 768.

798 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#812	C₂	1	2	yes	maximal
3	#69743	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	4	yes	maximal, complemented
5	C₂×#2150	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₈	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₄²	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×C₈	C₄	1	4	yes	—
10	C₂×C₃×C₄×C₈	C₄	1	2	yes	—
11	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	4	yes	—
13	C₂×#812	—	2	8	no	—
14	C₂×#812	C₂²	1	8	yes	—
15	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
16	C₄⋊(C₃⋊C₁₆)	C₂²	1	16	yes	complemented
17	C₂²×M₃₂	—	3	2	no	—
18	C₂²×C₄×C₈	D₃	1	1	yes	characteristic
19	#2150	—	3	8	no	—
20	C₂×C₄⋊C₁₆	—	3	4	no	—
21	C₂³×C₃×C₄	C₈	1	2	yes	—
22	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₈	—	2	4	no	—
24	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₈	D₄	1	4	yes	—
29	C₂²×C₃×C₈	Q₈	1	4	yes	—
30	C₂²×C₃×C₈	C₂³	1	2	yes	—
31	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₈	1	1	yes	characteristic
33	C₂×C₃×C₄²	C₈	1	4	yes	—
34	C₂×C₃×C₄²	C₈	1	1	yes	characteristic
35	C₂×C₃×C₄²	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
37	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
38	C₃×C₄×C₈	C₂×C₄	1	8	yes	—
39	C₃×C₄×C₈	C₂³	1	4	yes	—
40	#812	—	2	16	no	—
41	C₂×C₃⋊C₁₆	—	2	8	no	—
42	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
43	C₂³×C₈	Dic₃	1	1	yes	characteristic
44	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₄²	Dic₃	1	1	yes	characteristic
46	C₂×M₃₂	—	3	8	no	—
47	C₂×M₃₂	—	6	8	no	—
48	C₂²×C₁₆	—	3	4	no	—
49	C₂×C₄×C₈	Dic₃	1	2	yes	—
50	C₂×C₄×C₈	Dic₃	1	4	yes	—
51	C₂×C₄×C₈	C₂×D₃	1	4	yes	—
52	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
53	C₄⋊C₁₆	—	3	16	no	—
54	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₃×C₄	—	2	8	no	—
56	C₂²×C₃×C₄	C₄⋊C₄	1	4	yes	—
57	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
58	C₂²×C₃×C₄	C₂×C₈	1	8	yes	—
59	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
60	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
62	C₂²×C₃×C₄	M₁₆	1	4	yes	—
63	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
64	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₈	—	2	8	no	—
66	C₂×C₃×C₈	—	2	4	no	—
67	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	—
68	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
69	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
70	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
74	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
75	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
76	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
77	C₃×C₄²	C₂×C₈	1	4	yes	—
78	C₃×C₄²	C₂×C₈	1	4	yes	—
79	C₃×C₄²	C₂×C₈	1	4	yes	—
80	C₃×C₄²	C₂²×C₄	1	4	yes	—
81	C₃⋊C₁₆	—	2	8	no	—
82	C₂³×C₄	C₃⋊C₈	1	2	yes	—
83	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
84	C₂²×C₈	—	2	4	no	—
85	C₂²×C₈	Dic₆	1	4	yes	—
86	C₂²×C₈	D₁₂	1	4	yes	—
87	C₂²×C₈	C₂×Dic₃	1	4	yes	—
88	C₂²×C₈	C₂×Dic₃	1	2	yes	—
89	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×Dic₃	1	2	yes	—
91	C₂²×C₈	C₂²×D₃	1	2	yes	—
92	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₄²	C₃⋊C₈	1	1	yes	characteristic
94	C₂×C₄²	C₃⋊C₈	1	4	yes	—
95	C₂×C₄²	C₃⋊C₈	1	1	yes	characteristic
96	C₂×C₄²	C₂×Dic₃	1	4	yes	—
97	C₂×C₄²	C₂×Dic₃	1	2	yes	—
98	M₃₂	—	6	16	no	—
99	C₂×C₁₆	—	3	8	no	—
100	C₂×C₁₆	—	6	8	no	—
101	C₄×C₈	C₂×Dic₃	1	8	yes	—
102	C₄×C₈	C₂×Dic₃	1	4	yes	—
103	C₄×C₈	C₂²×D₃	1	4	yes	—
104	C₂³×C₃	—	2	4	no	—
105	C₂³×C₃	C₄⋊C₈	1	4	yes	—
106	C₂³×C₃	C₂²×C₈	1	2	yes	—
107	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	4	no	—
109	C₂×C₃×C₄	—	2	16	no	—
110	C₂×C₃×C₄	C₄⋊C₈	1	8	yes	—
111	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
112	C₂×C₃×C₄	M₃₂	1	8	yes	complemented
113	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
114	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
116	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
117	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
119	C₂×C₃×C₄	C₂×M₁₆	1	4	yes	—
120	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
121	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
122	C₃×C₈	—	2	4	no	—
123	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
124	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
125	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
126	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
127	C₂⁴	C₂×C₃⋊C₈	1	1	yes	characteristic
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
130	C₂²×C₄	C₂×C₃⋊C₈	1	8	yes	—
131	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
132	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
134	C₂²×C₄	#208	1	4	yes	—
135	C₂²×C₄	C₄⋊Dic₃	1	4	yes	—
136	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
138	C₂×C₈	—	2	8	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₄⋊Dic₃	1	4	yes	—
141	C₂×C₈	C₄⋊Dic₃	1	8	yes	—
142	C₂×C₈	C₂×Dic₆	1	4	yes	—
143	C₂×C₈	C₂×Dic₆	1	2	yes	—
144	C₂×C₈	C₂×D₁₂	1	2	yes	—
145	C₂×C₈	C₂×D₁₂	1	4	yes	—
146	C₂×C₈	C₂²×Dic₃	1	4	yes	—
147	C₂×C₈	C₂²×Dic₃	1	2	yes	—
148	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
149	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
150	C₄²	C₂×C₃⋊C₈	1	4	yes	—
151	C₄²	C₂×C₃⋊C₈	1	4	yes	—
152	C₄²	C₂×C₃⋊C₈	1	4	yes	—
153	C₄²	C₂²×Dic₃	1	4	yes	—
154	C₁₆	—	6	8	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	—	2	8	no	—
157	C₂²×C₃	C₂×C₄⋊C₈	1	4	yes	—
158	C₂²×C₃	C₂×C₄⋊C₈	1	2	yes	—
159	C₂²×C₃	C₂×M₃₂	1	4	yes	—
160	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
161	C₃×C₄	—	2	4	no	—
162	C₃×C₄	C₂×C₄⋊C₈	1	2	yes	—
163	C₃×C₄	C₂×C₄⋊C₈	1	4	yes	—
164	C₃×C₄	C₂×M₃₂	1	8	yes	complemented
165	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
166	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
167	C₂³	—	2	4	no	—
168	C₂³	C₄⋊(C₃⋊C₈)	1	4	yes	—
169	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
170	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	16	no	—
173	C₂×C₄	C₄⋊(C₃⋊C₈)	1	8	yes	—
174	C₂×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
175	C₂×C₄	#812	1	8	yes	complemented
176	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
177	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
178	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
179	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
180	C₂×C₄	C₂×#208	1	4	yes	—
181	C₂×C₄	C₂×#208	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
184	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	C₂×C₄⋊Dic₃	1	4	yes	—
187	C₈	C₂×C₄⋊Dic₃	1	2	yes	—
188	C₈	C₂²×Dic₆	1	1	yes	characteristic
189	C₈	C₂²×D₁₂	1	1	yes	characteristic
190	C₂×C₃	—	2	4	no	—
191	C₂×C₃	#2150	1	4	yes	complemented
192	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
193	C₂×C₃	C₂²×M₃₂	1	2	yes	—
194	C₂²	—	2	4	no	—
195	C₂²	—	2	8	no	—
196	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
197	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
198	C₂²	C₂×#812	1	4	yes	—
199	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
200	C₄	—	2	4	no	—
201	C₄	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
202	C₄	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
203	C₄	C₂×#812	1	8	yes	complemented
204	C₄	C₂²×#208	1	1	yes	characteristic
205	C₄	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
206	C₃	C₂×#2150	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	2	4	no	—
208	C₂	#69743	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
210	C₂	C₂²×#812	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂²×C₃×C₄×C₈	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂⁴×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂²×C₄×C₈	D₃
hypocenter	C₃	C₂×#2150
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #69743	C₂	direct (N × H)
4. C₂×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
16. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
112. C₂×C₃×C₄	M₃₂	semidirect
164. C₃×C₄	C₂×M₃₂	semidirect
175. C₂×C₄	#812	semidirect
191. C₂×C₃	#2150	semidirect
203. C₄	C₂×#812	semidirect
206. C₃	C₂×#2150	semidirect
208. C₂	#69743	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₄×C₈	2
C₂²×#812	2
#69743	2
C₂×C₄⋊(C₃⋊C₁₆)	2
C₂×#2150	3

Subgroups — C2×#69743 #10948166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69743 #10948166