Subgroups — #10948771

Order 768.

662 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#4549	C₂	1	4	yes	maximal, complemented
3	D₄×C₃⋊C₁₆	C₂	1	8	yes	maximal, complemented
4	#69744	C₂	1	2	yes	maximal, complemented
5	#33486	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄×D₄	C₄	1	1	yes	characteristic
8	C₃×C₈×D₄	C₄	1	4	yes	—
9	C₃×C₈×D₄	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₃×C₂²⋊C₈	C₄	1	2	yes	—
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃⋊C₁₆	C₂²	1	8	yes	complemented
14	#4549	C₂²	1	4	yes	—
15	#4549	C₂²	1	8	yes	complemented
16	C₄⋊(C₃⋊C₁₆)	C₂²	1	4	yes	complemented
17	C₄×C₃⋊C₁₆	C₂²	1	4	yes	complemented
18	C₂×C₈×D₄	D₃	1	1	yes	characteristic
19	D₄×C₁₆	—	3	8	no	—
20	#2162	—	3	2	no	—
21	C₂×C₂²⋊C₁₆	—	3	4	no	—
22	C₂²×C₃×D₄	C₈	1	1	yes	characteristic
23	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₈	—	4	4	no	—
25	C₂²×C₃×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂³	1	4	yes	—
30	C₃×C₄×D₄	C₈	1	4	yes	—
31	C₃×C₄×D₄	C₂×C₄	1	4	yes	—
32	C₂×C₃×C₄⋊C₄	C₈	1	1	yes	characteristic
33	C₂×C₃×C₂²⋊C₄	C₈	1	2	yes	—
34	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
37	C₃×C₄×C₈	C₂×C₄	1	2	yes	—
38	C₃×C₄×C₈	C₂³	1	2	yes	—
39	C₂×C₃⋊C₁₆	—	4	8	no	—
40	C₂×C₃⋊C₁₆	C₂³	1	8	yes	—
41	C₂³×C₈	C₂×D₃	1	2	yes	—
42	C₂×C₄×D₄	Dic₃	1	1	yes	characteristic
43	C₂²×C₁₆	—	3	8	no	—
44	C₈×D₄	Dic₃	1	4	yes	—
45	C₈×D₄	C₂×D₃	1	4	yes	—
46	C₂×C₄⋊C₈	Dic₃	1	1	yes	characteristic
47	C₂×C₂²⋊C₈	Dic₃	1	2	yes	—
48	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
49	C₄⋊C₁₆	—	3	4	no	—
50	C₂²⋊C₁₆	—	3	4	no	—
51	C₂²⋊C₁₆	—	3	8	no	—
52	C₄×C₁₆	—	3	4	no	—
53	C₂⁴×C₃	C₂×C₈	1	2	yes	—
54	C₂×C₃×D₄	C₁₆	1	8	yes	complemented
55	C₂×C₃×D₄	C₂×C₈	1	4	yes	—
56	C₂²×C₃×C₄	—	4	2	no	—
57	C₂²×C₃×C₄	—	4	2	no	—
58	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₃×C₄	C₂×C₈	1	2	yes	—
60	C₂²×C₃×C₄	C₂×C₈	1	4	yes	—
61	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₃×C₄	C₂²×C₄	1	4	yes	—
63	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₈	—	2	8	no	—
65	C₂×C₃×C₈	—	2	2	no	—
66	C₂×C₃×C₈	—	4	4	no	—
67	C₂×C₃×C₈	—	4	2	no	—
68	C₂×C₃×C₈	—	4	1	no	—
69	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
71	C₂×C₃×C₈	C₂²×C₄	1	4	yes	—
72	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
73	C₃×C₄⋊C₄	C₂×C₈	1	4	yes	—
74	C₃×C₂²⋊C₄	C₂×C₈	1	8	yes	—
75	C₃×C₄²	C₂×C₈	1	2	yes	—
76	C₃×C₄²	C₂²×C₄	1	2	yes	—
77	C₃⋊C₁₆	—	2	8	no	—
78	C₂²×D₄	C₃⋊C₈	1	1	yes	characteristic
79	C₂³×C₄	C₂×Dic₃	1	2	yes	—
80	C₂²×C₈	—	4	4	no	—
81	C₂²×C₈	C₂×Dic₃	1	2	yes	—
82	C₂²×C₈	C₂×Dic₃	1	4	yes	—
83	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
84	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
85	C₂²×C₈	C₂²×D₃	1	4	yes	—
86	C₄×D₄	C₃⋊C₈	1	4	yes	—
87	C₄×D₄	C₂×Dic₃	1	4	yes	—
88	C₂×C₄⋊C₄	C₃⋊C₈	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	C₃⋊C₈	1	2	yes	—
90	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
91	C₂×C₁₆	—	3	8	no	—
92	C₂×C₁₆	—	12	8	no	—
93	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
94	C₂²⋊C₈	C₂×Dic₃	1	8	yes	—
95	C₄×C₈	C₂×Dic₃	1	2	yes	—
96	C₄×C₈	C₂²×D₃	1	2	yes	—
97	C₂³×C₃	—	4	2	no	—
98	C₂³×C₃	—	4	2	no	—
99	C₂³×C₃	C₂×C₁₆	1	8	yes	complemented
100	C₂³×C₃	C₂²×C₈	1	4	yes	—
101	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
102	C₃×D₄	C₂×C₁₆	1	16	yes	complemented
103	C₂×C₃×C₄	—	2	1	no	—
104	C₂×C₃×C₄	—	2	4	no	—
105	C₂×C₃×C₄	—	2	1	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	—	4	4	no	—
108	C₂×C₃×C₄	—	4	2	no	—
109	C₂×C₃×C₄	—	4	1	no	—
110	C₂×C₃×C₄	C₂×C₁₆	1	4	yes	complemented
111	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
113	C₂×C₃×C₄	C₂²×C₈	1	4	yes	—
114	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
115	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
116	C₃×C₈	—	2	1	no	—
117	C₃×C₈	—	2	4	no	—
118	C₃×C₈	—	2	2	no	—
119	C₃×C₈	H(5,𝔽₂)	1	2	yes	—
120	C₂⁴	C₂×C₃⋊C₈	1	2	yes	—
121	C₂×D₄	C₃⋊C₁₆	1	8	yes	complemented
122	C₂×D₄	C₂×C₃⋊C₈	1	4	yes	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₃⋊C₈	1	2	yes	—
127	C₂²×C₄	C₂×C₃⋊C₈	1	4	yes	—
128	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
129	C₂²×C₄	C₂²×Dic₃	1	4	yes	—
130	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
131	C₂×C₈	—	2	8	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	4	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	1	no	—
136	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₃	1	2	yes	—
138	C₂×C₈	C₂²×Dic₃	1	4	yes	—
139	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
140	C₄⋊C₄	C₂×C₃⋊C₈	1	4	yes	—
141	C₂²⋊C₄	C₂×C₃⋊C₈	1	8	yes	—
142	C₄²	C₂×C₃⋊C₈	1	2	yes	—
143	C₄²	C₂²×Dic₃	1	2	yes	—
144	C₁₆	—	6	8	no	—
145	C₂²×C₃	—	2	1	no	—
146	C₂²×C₃	—	2	4	no	—
147	C₂²×C₃	—	2	1	no	—
148	C₂²×C₃	—	2	4	no	—
149	C₂²×C₃	—	4	8	no	—
150	C₂²×C₃	C₂²×C₁₆	1	2	yes	—
151	C₂²×C₃	C₂²×C₁₆	1	8	yes	—
152	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	2	4	no	—
156	C₃×C₄	C₂²×C₁₆	1	4	yes	—
157	C₃×C₄	#518	1	1	yes	characteristic
158	C₃×C₄	#568	1	1	yes	characteristic
159	C₂³	—	4	2	no	—
160	C₂³	—	4	2	no	—
161	C₂³	C₂×C₃⋊C₁₆	1	8	yes	complemented
162	C₂³	C₂²×C₃⋊C₈	1	4	yes	—
163	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
164	D₄	C₂×C₃⋊C₁₆	1	16	yes	complemented
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	C₂×C₃⋊C₁₆	1	4	yes	complemented
173	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
174	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
175	C₂×C₄	C₂²×C₃⋊C₈	1	4	yes	—
176	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
177	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
178	C₈	—	2	1	no	—
179	C₈	—	2	4	no	—
180	C₈	—	2	2	no	—
181	C₈	#1004	1	2	yes	—
182	C₂×C₃	—	2	2	no	—
183	C₂×C₃	—	2	8	no	—
184	C₂×C₃	#2972	1	1	yes	characteristic
185	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
186	C₂×C₃	#3407	1	1	yes	characteristic
187	C₂²	—	2	1	no	—
188	C₂²	—	2	4	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	4	8	no	—
192	C₂²	C₂²×C₃⋊C₁₆	1	2	yes	—
193	C₂²	C₂²×C₃⋊C₁₆	1	8	yes	—
194	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	4	no	—
198	C₄	C₂²×C₃⋊C₁₆	1	4	yes	—
199	C₄	#5798	1	1	yes	characteristic
200	C₄	#5819	1	1	yes	characteristic
201	C₃	#33486	1	1	yes	characteristic, complemented, Sylow
202	C₂	—	2	2	no	—
203	C₂	—	2	8	no	—
204	C₂	#82301	1	1	yes	characteristic
205	C₂	C₂³×C₃⋊C₁₆	1	1	yes	characteristic
206	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator	C₂×C₃	C₂³×C₁₆
Fitting	C₂×C₃×C₈×D₄	C₂
socle	C₂²×C₃	C₂³×C₈
prime radical	C₂²×C₃×D₄	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	C₂×C₈×D₄	D₃
hypocenter	C₃	#33486
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₂×#4549	C₂	semidirect
3. D₄×C₃⋊C₁₆	C₂	semidirect
4. #69744	C₂	semidirect
13. C₂²×C₃⋊C₁₆	C₂²	semidirect
15. #4549	C₂²	semidirect
16. C₄⋊(C₃⋊C₁₆)	C₂²	semidirect
17. C₄×C₃⋊C₁₆	C₂²	semidirect
54. C₂×C₃×D₄	C₁₆	semidirect
99. C₂³×C₃	C₂×C₁₆	semidirect
102. C₃×D₄	C₂×C₁₆	semidirect
110. C₂×C₃×C₄	C₂×C₁₆	semidirect
121. C₂×D₄	C₃⋊C₁₆	semidirect
161. C₂³	C₂×C₃⋊C₁₆	semidirect
164. D₄	C₂×C₃⋊C₁₆	semidirect
172. C₂×C₄	C₂×C₃⋊C₁₆	semidirect
201. C₃	#33486	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×D₄	2
C₂×#4549	2
D₄×C₃⋊C₁₆	2
#69744	2
#33486	3