Finite Group Databasebeta

Subgroups — #72321925

Order 1024.

285 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512773	C₂	1	1	yes	maximal, characteristic
2	#153582	C₂	1	1	yes	maximal, characteristic, complemented
3	#151056	C₂	1	1	yes	maximal, characteristic, complemented
4	#94613	C₂	1	4	yes	maximal, complemented
5	#33981	C₄	1	1	yes	characteristic
6	C₂×#2178	—	2	1	no	—
7	#13656	—	2	1	no	—
8	#13656	C₂²	1	2	yes	—
9	#13655	—	2	1	no	—
10	#13655	C₄	1	2	yes	—
11	#13636	C₄	1	1	yes	characteristic
12	#13620	—	2	1	no	—
13	C₄×M₆₄	C₂²	1	1	yes	characteristic
14	#12024	—	2	1	no	—
15	#7544	—	2	1	no	—
16	#7513	—	2	1	no	—
17	#7511	C₂²	1	2	yes	complemented
18	#7463	—	2	1	no	—
19	#7463	C₂²	1	2	yes	complemented
20	C₂×#508	—	2	1	no	—
21	C₂×#443	—	4	1	no	—
22	#2264	—	2	2	no	—
23	#2263	—	2	2	no	—
24	#2262	—	2	1	no	—
25	#2262	C₂×C₄	1	2	yes	—
26	C₂×M₆₄	C₂×C₄	1	1	yes	characteristic
27	C₂×M₆₄	D₄	1	1	yes	characteristic
28	#2178	—	2	2	no	—
29	#2178	—	2	2	no	—
30	#2178	—	4	1	no	—
31	#2178	—	4	1	no	—
32	C₂×#368	—	2	1	no	—
33	C₂×#359	—	4	1	no	—
34	C₂×C₄×C₁₆	D₄	1	1	yes	characteristic
35	#1568	—	4	1	no	—
36	M₁₂₈	—	2	2	no	—
37	#1425	—	2	1	no	—
38	#1425	C₂×C₄	1	2	yes	—
39	#1417	—	2	2	no	—
40	#1417	—	2	2	no	—
41	#1397	D₄	1	1	yes	characteristic
42	#1397	C₂³	1	1	yes	characteristic
43	C₄×C₃₂	C₂×C₄	1	1	yes	characteristic
44	C₄×C₃₂	D₄	1	1	yes	characteristic
45	#1372	—	2	2	no	—
46	#1371	—	4	1	no	—
47	C₂×#135	—	4	1	no	—
48	C₂×#131	—	8	1	no	—
49	#508	—	2	2	no	—
50	#508	—	2	2	no	—
51	#508	—	4	1	no	—
52	#508	—	4	1	no	—
53	C₂×Q₃₂	—	2	1	no	—
54	C₂×SD₃₂	—	4	1	no	—
55	C₂×D₁₆	—	2	1	no	—
56	C₂²×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₆	SD₁₆	1	1	yes	characteristic
58	#443	—	4	2	no	—
59	#443	—	4	2	no	—
60	#443	—	8	1	no	—
61	#443	—	8	1	no	—
62	#432	—	8	1	no	—
63	C₂×#108	—	4	1	no	—
64	C₂×#104	—	8	1	no	—
65	C₂×C₄×C₈	D₈	1	1	yes	characteristic
66	Q₆₄	—	2	1	no	—
67	Q₆₄	M₁₆	1	2	yes	—
68	SD₆₄	—	2	2	no	—
69	SD₆₄	—	2	2	no	—
70	D₃₂	—	2	1	no	—
71	D₃₂	M₁₆	1	2	yes	—
72	M₆₄	—	2	2	no	—
73	M₆₄	—	4	1	no	—
74	C₂×C₃₂	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃₂	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃₂	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃₂	C₂×D₄	1	1	yes	characteristic
78	#368	—	2	2	no	—
79	#368	—	4	1	no	—
80	#359	—	4	2	no	—
81	#359	—	4	2	no	—
82	C₄×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
83	C₄×C₁₆	D₈	1	1	yes	characteristic
84	C₄×C₁₆	SD₁₆	1	1	yes	characteristic
85	C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
86	#335	—	4	2	no	—
87	#334	—	8	1	no	—
88	C₆₄	—	4	2	no	—
89	C₂×#41	—	8	1	no	—
90	#135	—	4	2	no	—
91	#135	—	4	2	no	—
92	#135	—	8	1	no	—
93	#135	—	8	1	no	—
94	C₂×Q₁₆	—	4	1	no	—
95	C₂×SD₁₆	—	8	1	no	—
96	C₂×D₈	—	4	1	no	—
97	#131	—	8	2	no	—
98	#131	—	8	2	no	—
99	#131	—	16	1	no	—
100	#131	—	16	1	no	—
101	C₂×M₁₆	—	8	1	no	—
102	C₂×M₁₆	—	16	1	no	—
103	C₂²×C₈	—	16	1	no	—
104	C₂²×C₈	#102	1	1	yes	characteristic
105	C₂²×C₈	SD₃₂	1	1	yes	characteristic
106	C₂×C₄²	D₁₆	1	1	yes	characteristic
107	Q₃₂	—	2	2	no	—
108	Q₃₂	—	4	1	no	—
109	SD₃₂	—	4	2	no	—
110	SD₃₂	—	4	2	no	—
111	D₁₆	—	2	2	no	—
112	D₁₆	—	4	1	no	—
113	C₂×C₁₆	—	2	1	no	—
114	C₂×C₁₆	—	2	1	no	—
115	C₂×C₁₆	—	4	1	no	—
116	C₂×C₁₆	#102	1	1	yes	characteristic
117	C₂×C₁₆	#102	1	1	yes	characteristic
118	C₂×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
120	#108	—	4	2	no	—
121	#108	—	8	1	no	—
122	C₄⋊C₈	—	8	2	no	—
123	#104	—	8	2	no	—
124	#104	—	8	2	no	—
125	C₂²⋊C₈	—	16	1	no	—
126	#97	—	16	1	no	—
127	C₄×C₈	#102	1	1	yes	characteristic
128	C₄×C₈	D₁₆	1	1	yes	characteristic
129	C₄×C₈	SD₃₂	1	1	yes	characteristic
130	C₄×C₈	C₂×D₈	1	1	yes	characteristic
131	C₃₂	—	2	1	no	—
132	C₃₂	—	2	1	no	—
133	C₃₂	—	2	1	no	—
134	C₃₂	C₂×M₁₆	1	2	yes	—
135	#41	—	8	2	no	—
136	#41	—	8	2	no	—
137	#41	—	16	1	no	—
138	#41	—	16	1	no	—
139	C₂×Q₈	—	8	1	no	—
140	C₂×D₄	—	8	1	no	—
141	C₂×D₄	—	16	1	no	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	16	1	no	—
144	C₂²×C₄	#357	1	1	yes	characteristic
145	Q₁₆	—	4	2	no	—
146	Q₁₆	—	8	1	no	—
147	SD₁₆	—	8	2	no	—
148	SD₁₆	—	8	2	no	—
149	D₈	—	4	2	no	—
150	D₈	—	8	1	no	—
151	M₁₆	—	8	2	no	—
152	M₁₆	—	16	2	no	—
153	M₁₆	—	16	1	no	—
154	M₁₆	—	16	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	16	2	no	—
160	C₂×C₈	—	16	1	no	—
161	C₂×C₈	—	16	2	no	—
162	C₂×C₈	—	16	2	no	—
163	C₂×C₈	#357	1	1	yes	characteristic
164	C₂×C₈	#357	1	1	yes	characteristic
165	C₂×C₈	C₂×#102	1	1	yes	characteristic
166	C₂×C₈	C₂×SD₃₂	1	1	yes	characteristic
167	C₄²	—	2	1	no	—
168	C₄²	#357	1	1	yes	characteristic
169	C₄²	C₂×D₁₆	1	1	yes	characteristic
170	C₁₆	—	2	1	no	—
171	C₁₆	—	2	1	no	—
172	C₁₆	—	2	1	no	—
173	C₁₆	#407	1	1	yes	characteristic
174	C₁₆	#421	1	1	yes	characteristic
175	C₂³	—	16	1	no	—
176	C₂³	#1418	1	1	yes	characteristic
177	Q₈	—	8	2	no	—
178	Q₈	—	16	1	no	—
179	D₄	—	8	2	no	—
180	D₄	—	16	1	no	—
181	D₄	—	16	2	no	—
182	D₄	—	16	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	8	1	no	—
188	C₂×C₄	—	16	2	no	—
189	C₂×C₄	—	16	1	no	—
190	C₂×C₄	—	16	2	no	—
191	C₂×C₄	#1418	1	1	yes	characteristic
192	C₂×C₄	C₂×#357	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	16	2	no	—
196	C₈	—	32	2	no	—
197	C₈	#1419	1	1	yes	characteristic
198	C₈	#1419	1	1	yes	characteristic
199	C₈	#1486	1	1	yes	characteristic
200	C₈	#2143	1	1	yes	characteristic
201	C₂²	—	2	1	no	—
202	C₂²	—	4	1	no	—
203	C₂²	—	16	2	no	—
204	C₂²	—	32	2	no	—
205	C₂²	C₂×#1418	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	1	no	—
209	C₄	—	16	2	no	—
210	C₄	#11657	1	1	yes	characteristic
211	C₄	C₂×#1419	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	4	1	no	—
214	C₂	—	32	2	no	—
215	C₂	#150138	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#150138
commutator	C₂×C₃₂	C₂²×C₄
Frattini, cosocle	#1397	C₂³
prime radical	#33981	C₄
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ C₄×M₆₄ ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #153582	C₂	semidirect
3. #151056	C₂	semidirect
4. #94613	C₂	semidirect
17. #7511	C₂²	semidirect
19. #7463	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#512773	2
#153582	2
#151056	2
#94613	2