Subgroups — C₂×#69882 #10948170

Order 768.

734 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	C₂×C₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#812	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#69882	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₃⋊C₁₆)	C₂	1	4	yes	maximal, complemented
6	C₂×#2151	—	3	1	no	maximal, Sylow
7	C₂²×C₃×M₁₆	C₄	1	1	yes	characteristic
8	C₂³×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×#117	C₄	1	1	yes	characteristic
10	C₃×#405	C₄	1	4	yes	—
11	C₃×#405	C₂²	1	4	yes	—
12	C₂×C₃×#97	C₄	1	2	yes	—
13	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
14	C₂×#812	—	2	4	no	—
15	C₂×#812	C₂²	1	4	yes	—
16	C₂²×C₃⋊C₁₆	—	2	4	no	—
17	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
18	C₂²×C₃⋊C₁₆	C₂²	1	4	yes	—
19	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
20	C₄⋊(C₃⋊C₁₆)	C₂²	1	16	yes	complemented
21	C₂²×M₃₂	—	3	1	no	—
22	C₂³×C₁₆	—	3	1	no	—
23	C₂×#405	D₃	1	1	yes	characteristic
24	#2151	—	3	8	no	—
25	C₂×C₄⋊C₁₆	—	3	4	no	—
26	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×M₁₆	—	2	4	no	—
28	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₈	D₄	1	4	yes	—
33	C₂²×C₃×C₈	Q₈	1	4	yes	—
34	C₂²×C₃×C₈	C₂³	1	2	yes	—
35	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
36	C₃×#117	C₈	1	4	yes	—
37	C₃×#117	C₂×C₄	1	4	yes	—
38	C₂×C₃×C₄⋊C₄	C₈	1	2	yes	—
39	C₂×C₃×C₂²⋊C₄	C₈	1	2	yes	—
40	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
41	C₃×#97	C₂×C₄	1	8	yes	—
42	C₃×C₄×C₈	C₂×C₄	1	4	yes	—
43	C₃×C₄×C₈	C₂³	1	4	yes	—
44	#812	—	2	8	no	—
45	C₂×C₃⋊C₁₆	—	2	4	no	—
46	C₂×C₃⋊C₁₆	—	2	8	no	—
47	C₂×C₃⋊C₁₆	—	2	4	no	—
48	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
49	C₂×C₃⋊C₁₆	C₂³	1	4	yes	—
50	C₂²×M₁₆	Dic₃	1	1	yes	characteristic
51	C₂³×C₈	C₂×D₃	1	1	yes	characteristic
52	C₂×#117	Dic₃	1	1	yes	characteristic
53	C₂×M₃₂	—	3	4	no	—
54	C₂×M₃₂	—	6	4	no	—
55	C₂²×C₁₆	—	3	4	no	—
56	C₂²×C₁₆	—	3	2	no	—
57	C₂²×C₁₆	—	3	2	no	—
58	C₂²×C₁₆	—	6	4	no	—
59	#405	Dic₃	1	4	yes	—
60	#405	C₂×D₃	1	4	yes	—
61	C₂×#97	Dic₃	1	2	yes	—
62	C₂×C₄×C₈	C₂×D₃	1	2	yes	—
63	C₄⋊C₁₆	—	3	16	no	—
64	C₂⁴×C₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₄⋊C₄	1	4	yes	—
66	C₂²×C₃×C₄	C₂×C₈	1	4	yes	—
67	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
68	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
69	C₂²×C₃×C₄	M₁₆	1	4	yes	—
70	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
72	C₃×M₁₆	—	2	8	no	—
73	C₂×C₃×C₈	—	2	4	no	—
74	C₂×C₃×C₈	C₄⋊C₄	1	4	yes	—
75	C₂×C₃×C₈	C₄⋊C₄	1	8	yes	—
76	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₈	C₂×D₄	1	4	yes	—
81	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
82	C₂×C₃×C₈	C₂×Q₈	1	2	yes	—
83	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
84	C₃×C₄⋊C₄	C₂×C₈	1	8	yes	—
85	C₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
86	C₃×C₄²	C₂×C₈	1	4	yes	—
87	C₃×C₄²	C₂²×C₄	1	4	yes	—
88	C₃⋊C₁₆	—	2	4	no	—
89	C₃⋊C₁₆	—	2	4	no	—
90	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
91	C₂×M₁₆	—	2	4	no	—
92	C₂×M₁₆	C₂×Dic₃	1	4	yes	—
93	C₂²×C₈	Dic₆	1	4	yes	—
94	C₂²×C₈	D₁₂	1	4	yes	—
95	C₂²×C₈	C₂×Dic₃	1	2	yes	—
96	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
97	C₂²×C₈	C₂×Dic₃	1	2	yes	—
98	C₂²×C₈	C₂²×D₃	1	2	yes	—
99	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
100	#117	C₃⋊C₈	1	4	yes	—
101	#117	C₂×Dic₃	1	4	yes	—
102	C₂×C₄⋊C₄	C₃⋊C₈	1	2	yes	—
103	C₂×C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
104	C₂×C₄²	C₂×Dic₃	1	2	yes	—
105	M₃₂	—	6	8	no	—
106	C₂×C₁₆	—	3	4	no	—
107	C₂×C₁₆	—	3	4	no	—
108	C₂×C₁₆	—	6	4	no	—
109	C₂×C₁₆	—	6	8	no	—
110	C₂×C₁₆	—	6	4	no	—
111	#97	C₂×Dic₃	1	8	yes	—
112	C₄×C₈	C₂×Dic₃	1	4	yes	—
113	C₄×C₈	C₂²×D₃	1	4	yes	—
114	C₂³×C₃	—	2	4	no	—
115	C₂³×C₃	C₄⋊C₈	1	4	yes	—
116	C₂³×C₃	C₂²×C₈	1	2	yes	—
117	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₃×C₄	—	2	8	no	—
119	C₂×C₃×C₄	C₄⋊C₈	1	8	yes	—
120	C₂×C₃×C₄	C₄⋊C₈	1	4	yes	—
121	C₂×C₃×C₄	C₂×C₄⋊C₄	1	4	yes	—
122	C₂×C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₃×C₄	C₂²×C₈	1	8	yes	—
124	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
126	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
127	C₂×C₃×C₄	C₂×M₁₆	1	4	yes	—
128	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
129	C₃×C₈	—	2	4	no	—
130	C₃×C₈	C₂×C₄⋊C₄	1	2	yes	—
131	C₃×C₈	C₂×C₄⋊C₄	1	4	yes	—
132	C₃×C₈	C₂²×D₄	1	1	yes	characteristic
133	C₃×C₈	C₂²×Q₈	1	1	yes	characteristic
134	C₂⁴	C₂×C₃⋊C₈	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₃⋊C₈	1	4	yes	—
136	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
138	C₂²×C₄	#208	1	4	yes	—
139	C₂²×C₄	C₄⋊Dic₃	1	4	yes	—
140	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
141	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
142	M₁₆	—	2	8	no	—
143	C₂×C₈	—	2	4	no	—
144	C₂×C₈	C₄⋊Dic₃	1	4	yes	—
145	C₂×C₈	C₄⋊Dic₃	1	8	yes	—
146	C₂×C₈	C₂×Dic₆	1	4	yes	—
147	C₂×C₈	C₂×Dic₆	1	2	yes	—
148	C₂×C₈	C₂×D₁₂	1	2	yes	—
149	C₂×C₈	C₂×D₁₂	1	4	yes	—
150	C₂×C₈	C₂²×Dic₃	1	4	yes	—
151	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
152	C₂×C₈	C₂²×Dic₃	1	2	yes	—
153	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
154	C₄⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
155	C₂²⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
156	C₄²	C₂×C₃⋊C₈	1	4	yes	—
157	C₄²	C₂²×Dic₃	1	4	yes	—
158	C₁₆	—	6	4	no	—
159	C₁₆	—	6	4	no	—
160	C₂²×C₃	—	2	4	no	—
161	C₂²×C₃	—	2	8	no	—
162	C₂²×C₃	C₂×C₄⋊C₈	1	4	yes	—
163	C₂²×C₃	C₂×C₄⋊C₈	1	2	yes	—
164	C₂²×C₃	#504	1	4	yes	—
165	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
166	C₃×C₄	—	2	8	no	—
167	C₃×C₄	C₂×C₄⋊C₈	1	2	yes	—
168	C₃×C₄	C₂×C₄⋊C₈	1	4	yes	—
169	C₃×C₄	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
171	C₂³	—	2	4	no	—
172	C₂³	C₄⋊(C₃⋊C₈)	1	4	yes	—
173	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
174	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	C₄⋊(C₃⋊C₈)	1	8	yes	—
177	C₂×C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
178	C₂×C₄	C₂²×C₃⋊C₈	1	8	yes	—
179	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
180	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
181	C₂×C₄	C₂×#208	1	2	yes	—
182	C₂×C₄	C₂×#208	1	4	yes	—
183	C₂×C₄	C₂×C₄⋊Dic₃	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₃	1	2	yes	—
185	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
186	C₈	—	2	4	no	—
187	C₈	C₂×C₄⋊Dic₃	1	4	yes	—
188	C₈	C₂×C₄⋊Dic₃	1	2	yes	—
189	C₈	C₂²×Dic₆	1	1	yes	characteristic
190	C₈	C₂²×D₁₂	1	1	yes	characteristic
191	C₂×C₃	—	2	4	no	—
192	C₂×C₃	#2151	1	4	yes	complemented
193	C₂×C₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
194	C₂×C₃	C₂×#504	1	2	yes	—
195	C₂²	—	2	4	no	—
196	C₂²	—	2	8	no	—
197	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
198	C₂²	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
199	C₂²	#5140	1	4	yes	—
200	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
201	C₄	—	2	8	no	—
202	C₄	C₂×C₄⋊(C₃⋊C₈)	1	2	yes	—
203	C₄	C₂×C₄⋊(C₃⋊C₈)	1	4	yes	—
204	C₄	C₂²×#208	1	1	yes	characteristic
205	C₄	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
206	C₃	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
207	C₂	—	2	4	no	—
208	C₂	#69882	1	4	yes	direct product factor
209	C₂	C₂²×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
210	C₂	C₂×#5140	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₃
Fitting	C₂×C₃×#405	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂⁴×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×#405	D₃
hypocenter	C₃	C₂×#2151
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
4. #69882	C₂	direct (N × H)
5. C₂×C₄⋊(C₃⋊C₁₆)	C₂	semidirect
20. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
192. C₂×C₃	#2151	semidirect
206. C₃	C₂×#2151	semidirect
208. C₂	#69882	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×#405	2
C₂²×#812	2
C₂³×C₃⋊C₁₆	2
#69882	2
C₂×C₄⋊(C₃⋊C₁₆)	2
C₂×#2151	3

Subgroups — C2×#69882 #10948170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#69882 #10948170