Subgroups — #72260432

Order 1024.

533 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	#512789	C₂	1	1	yes	maximal, characteristic
2	#483475	C₂	1	1	yes	maximal, characteristic
3	#483470	C₂	1	1	yes	maximal, characteristic
4	#151068	C₂	1	3	yes	maximal, complemented
5	#151067	C₂	1	3	yes	maximal, complemented
6	#151055	C₂	1	3	yes	maximal, complemented
7	#151038	C₂	1	3	yes	maximal, complemented
8	C₁₆×#41	C₂²	1	1	yes	characteristic
9	#17837	—	2	1	no	—
10	#17828	—	2	1	no	—
11	Q₈×C₃₂	C₄	1	1	yes	characteristic
12	Q₈×C₃₂	C₄	1	1	yes	characteristic
13	#13646	C₄	1	3	yes	—
14	#13646	C₄	1	3	yes	—
15	#13640	C₂²	1	3	yes	—
16	C₄×M₆₄	C₂²	1	3	yes	—
17	#12036	—	2	3	no	—
18	#12033	—	2	3	no	—
19	#12023	C₂²	1	3	yes	—
20	#12023	C₂²	1	3	yes	—
21	#12008	C₂²	1	3	yes	—
22	C₂×#1372	C₂²	1	3	yes	—
23	#7463	C₂²	1	3	yes	—
24	#7463	C₂²	1	6	yes	—
25	#7463	C₂²	1	3	yes	—
26	#7463	C₂²	1	3	yes	—
27	#7463	C₂²	1	1	yes	characteristic
28	#2991	—	4	1	no	—
29	C₈×#41	D₄	1	1	yes	characteristic
30	#2906	—	4	1	no	—
31	C₂×M₆₄	C₂×C₄	1	3	yes	—
32	C₂×M₆₄	C₂×C₄	1	3	yes	—
33	Q₈×C₁₆	C₂×C₄	1	1	yes	characteristic
34	Q₈×C₁₆	Q₈	1	1	yes	characteristic
35	D₄×C₁₆	C₂×C₄	1	3	yes	—
36	D₄×C₁₆	Q₈	1	3	yes	—
37	#2162	D₄	1	3	yes	—
38	C₂×C₄×C₁₆	C₂³	1	3	yes	—
39	#1574	—	4	3	no	—
40	#1571	—	4	3	no	—
41	#1567	—	2	3	no	—
42	#1567	—	2	3	no	—
43	#1564	—	2	3	no	—
44	C₂×#335	—	2	3	no	—
45	C₄⋊C₃₂	C₂×C₄	1	3	yes	—
46	C₄⋊C₃₂	C₂×C₄	1	3	yes	—
47	C₂²⋊C₃₂	C₂×C₄	1	6	yes	—
48	#1397	C₂³	1	3	yes	—
49	#1397	C₂³	1	1	yes	characteristic
50	C₄×C₃₂	C₂×C₄	1	3	yes	—
51	C₄×C₃₂	C₂×C₄	1	3	yes	—
52	#1372	—	2	3	no	—
53	#1372	—	2	3	no	—
54	#1372	—	2	3	no	—
55	#1372	C₂³	1	1	yes	characteristic
56	#1372	C₂³	1	3	yes	—
57	#1372	C₂³	1	1	yes	characteristic
58	#1372	C₂³	1	3	yes	—
59	#1371	—	2	3	no	—
60	C₂×#131	—	8	1	no	—
61	C₄×#41	D₈	1	1	yes	characteristic
62	C₂²×C₁₆	C₂²×C₄	1	3	yes	—
63	C₂²×C₁₆	C₂×Q₈	1	3	yes	—
64	C₈×Q₈	—	8	1	no	—
65	C₈×Q₈	C₄⋊C₄	1	1	yes	characteristic
66	C₈×Q₈	Q₁₆	1	1	yes	characteristic
67	#436	—	8	3	no	—
68	C₈×D₄	C₄⋊C₄	1	3	yes	—
69	C₈×D₄	Q₁₆	1	3	yes	—
70	#431	—	4	3	no	—
71	#431	D₈	1	3	yes	—
72	#424	—	8	3	no	—
73	#423	—	4	3	no	—
74	#423	—	4	3	no	—
75	C₂×C₄⋊C₈	—	4	3	no	—
76	C₂×C₄×C₈	C₂×D₄	1	3	yes	—
77	M₆₄	—	2	6	no	—
78	M₆₄	—	4	3	no	—
79	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃₂	C₂²×C₄	1	3	yes	—
81	C₂×C₃₂	C₂²×C₄	1	3	yes	—
82	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
83	C₄⋊C₁₆	C₄⋊C₄	1	3	yes	—
84	C₄⋊C₁₆	C₄⋊C₄	1	3	yes	—
85	C₂²⋊C₁₆	C₄⋊C₄	1	6	yes	—
86	C₄×C₁₆	C₂²×C₄	1	3	yes	—
87	C₄×C₁₆	C₂×D₄	1	3	yes	—
88	C₄×C₁₆	C₂×Q₈	1	3	yes	—
89	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
90	#335	—	2	1	no	—
91	#335	—	2	3	no	—
92	#335	—	2	1	no	—
93	#335	—	2	3	no	—
94	#335	—	4	3	no	—
95	#335	—	4	3	no	—
96	#335	—	4	3	no	—
97	#334	—	4	3	no	—
98	C₂×#41	SD₃₂	1	1	yes	characteristic
99	#131	—	16	4	no	—
100	C₂×M₁₆	—	8	3	no	—
101	C₂×M₁₆	—	8	3	no	—
102	C₂²×C₈	—	8	3	no	—
103	C₂²×C₈	C₂×C₄⋊C₄	1	3	yes	—
104	C₂²×C₈	C₂×Q₁₆	1	3	yes	—
105	C₄×Q₈	#107	1	1	yes	characteristic
106	C₄×Q₈	SD₃₂	1	1	yes	characteristic
107	C₄×D₄	#107	1	3	yes	—
108	C₄×D₄	SD₃₂	1	3	yes	—
109	#117	SD₃₂	1	3	yes	—
110	C₂×C₄²	C₂×D₈	1	3	yes	—
111	C₂×C₁₆	—	2	3	no	—
112	C₂×C₁₆	—	2	3	no	—
113	C₂×C₁₆	—	4	3	no	—
114	C₂×C₁₆	C₂×C₄⋊C₄	1	3	yes	—
115	C₂×C₁₆	C₂×C₄⋊C₄	1	3	yes	—
116	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₁₆	C₂²×Q₈	1	1	yes	characteristic
118	C₄⋊C₈	—	4	1	no	—
119	C₄⋊C₈	—	4	3	no	—
120	C₄⋊C₈	—	4	1	no	—
121	C₄⋊C₈	—	4	3	no	—
122	C₄⋊C₈	—	8	3	no	—
123	C₄⋊C₈	—	8	3	no	—
124	C₄⋊C₈	—	8	3	no	—
125	C₄⋊C₈	#107	1	3	yes	—
126	C₄⋊C₈	#107	1	3	yes	—
127	C₂²⋊C₈	—	8	3	no	—
128	C₂²⋊C₈	#107	1	6	yes	—
129	C₄×C₈	—	8	3	no	—
130	C₄×C₈	C₂×C₄⋊C₄	1	3	yes	—
131	C₄×C₈	C₂×D₈	1	3	yes	—
132	C₄×C₈	C₂×Q₁₆	1	3	yes	—
133	C₄×C₈	C₂²×D₄	1	1	yes	characteristic
134	C₃₂	—	2	3	no	—
135	C₃₂	—	2	3	no	—
136	C₃₂	—	2	1	no	—
137	C₃₂	#131	1	2	yes	complemented
138	#41	—	2	4	no	—
139	C₂×Q₈	#367	1	1	yes	characteristic
140	C₂×D₄	#367	1	3	yes	—
141	C₂²×C₄	C₂×#107	1	3	yes	—
142	C₂²×C₄	C₂×SD₃₂	1	3	yes	—
143	C₂²×C₄	C₂×SD₃₂	1	3	yes	—
144	M₁₆	—	16	6	no	—
145	M₁₆	—	32	3	no	—
146	C₂×C₈	—	2	3	no	—
147	C₂×C₈	—	2	3	no	—
148	C₂×C₈	—	4	3	no	—
149	C₂×C₈	—	8	1	no	—
150	C₂×C₈	—	8	3	no	—
151	C₂×C₈	—	8	1	no	—
152	C₂×C₈	—	8	3	no	—
153	C₂×C₈	—	16	6	no	—
154	C₂×C₈	C₂×#107	1	3	yes	—
155	C₂×C₈	C₂×#107	1	3	yes	—
156	C₂×C₈	C₂²×C₄⋊C₄	1	1	yes	characteristic
157	C₂×C₈	C₂²×Q₁₆	1	1	yes	characteristic
158	C₄⋊C₄	#367	1	3	yes	—
159	C₄⋊C₄	#367	1	3	yes	—
160	C₂²⋊C₄	#367	1	6	yes	—
161	C₄²	C₂×#107	1	3	yes	—
162	C₄²	C₂×SD₃₂	1	3	yes	—
163	C₄²	C₂×SD₃₂	1	3	yes	—
164	C₄²	C₂²×D₈	1	1	yes	characteristic
165	C₁₆	—	2	3	no	—
166	C₁₆	—	2	1	no	—
167	C₁₆	—	2	3	no	—
168	C₁₆	C₂×#131	1	1	yes	characteristic
169	C₁₆	#568	1	1	yes	characteristic
170	C₂³	C₂×#367	1	3	yes	—
171	Q₈	#1426	1	4	yes	—
172	D₄	—	2	6	no	—
173	C₂×C₄	—	2	6	no	—
174	C₂×C₄	—	2	3	no	—
175	C₂×C₄	—	2	3	no	—
176	C₂×C₄	—	4	3	no	—
177	C₂×C₄	C₂×#367	1	3	yes	—
178	C₂×C₄	C₂×#367	1	3	yes	—
179	C₂×C₄	C₂×#367	1	3	yes	—
180	C₂×C₄	C₂²×#107	1	1	yes	characteristic
181	C₂×C₄	C₂²×SD₃₂	1	1	yes	characteristic
182	C₂×C₄	C₂²×SD₃₂	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	3	no	—
185	C₈	—	2	3	no	—
186	C₈	—	16	1	no	—
187	C₈	—	16	1	no	—
188	C₈	—	16	3	no	—
189	C₈	—	16	3	no	—
190	C₈	#2906	1	1	yes	characteristic
191	C₈	#2916	1	1	yes	characteristic
192	C₂²	—	2	3	no	—
193	C₂²	—	2	3	no	—
194	C₂²	—	2	3	no	—
195	C₂²	C₂²×#367	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	2	3	no	—
200	C₄	C₂×#1426	1	6	yes	—
201	C₄	#17828	1	1	yes	characteristic
202	C₄	#33414	1	1	yes	characteristic
203	C₂	—	4	3	no	—
204	C₂	#483470	1	1	yes	characteristic
205	C₂	C₂²×#1426	1	1	yes	characteristic
206	C₂	#512724	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×#367
commutator	C₂×C₁₆	C₂³×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁴
prime radical	C₂×#41	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ C₁₆×#41 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #151068	C₂	semidirect
5. #151067	C₂	semidirect
6. #151055	C₂	semidirect
7. #151038	C₂	semidirect
137. C₃₂	#131	semidirect

Maximal subgroups

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

subgroup	index
#512789	2
#483475	2
#483470	2
#151068	2
#151067	2
#151055	2
#151038	2