Finite Group Databasebeta

← #152820

Subgroups — #152820

Order 512.

212 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

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

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1425C211yesmaximal, characteristic, complemented
2C2×#1417C211yesmaximal, characteristic
3#13615C211yesmaximal, characteristic
4#13241C211yesmaximal, characteristic
5#12670C211yesmaximal, characteristic, complemented
6#7473C211yesmaximal, characteristic, complemented
7#7465C211yesmaximal, characteristic, complemented
8C2×#508C2211yescharacteristic
9C22×C32C2211yescharacteristic
10C22×C32C2211yescharacteristic
11C2×#368C2211yescharacteristic
12C2×#368C2211yescharacteristic
13#2158C2211yescharacteristic
14C2×#359—21no—
15C2×C4×C16C2211yescharacteristic
16#1974—21no—
17#1772—21no—
18#1425—22no—
19#1417—22no—
20#1416C411yescharacteristic, complemented
21#1416C411yescharacteristic, complemented
22#1415C411yescharacteristic, complemented
23#1415C411yescharacteristic, complemented
24#1382—21no—
25C2×#135—21no—
26#508—21no—
27#508—21no—
28#508—21no—
29#508—21no—
30C2×Q32C2×C411yescharacteristic
31C2×SD32—21no—
32C2×D16C2×C411yescharacteristic
33C22×C16D411yescharacteristic
34C22×C16C2311yescharacteristic
35C2×#108—21no—
36C2×#108—21no—
37#427—21no—
38C2×#104—41no—
39#408—41no—
40#405—41no—
41C2×C4×C8D411yescharacteristic
42C2×C32—21no—
43C2×C32—21no—
44C2×C32—21no—
45C2×C32—21no—
46C2×C32—21no—
47C2×C32C2×C411yescharacteristic
48C2×C32C2×C411yescharacteristic
49#368—21no—
50#368—21no—
51#368—21no—
52#368D412yes—
53#367—21no—
54#366C2×C411yescharacteristic
55#366C2×C411yescharacteristic
56#359—42no—
57C4×C16—21no—
58C4×C16—21no—
59#339—41no—
60C2×#41—41no—
61#135—41no—
62#135—41no—
63#135—41no—
64#135—41no—
65C2×Q16—21no—
66C2×SD16—41no—
67C2×D8—21no—
68C2×M16—41no—
69C2×M16—41no—
70C22×C8D811yescharacteristic
71C22×C8C2×D411yescharacteristic
72#117—41no—
73C2×C42D811yescharacteristic
74Q32—21no—
75Q32—21no—
76SD32—41no—
77SD32—41no—
78D16—21no—
79D16—21no—
80C2×C16—21no—
81C2×C16—21no—
82C2×C16—21no—
83C2×C16C22×C411yescharacteristic
84C2×C16C2×D411yescharacteristic
85C2×C16#4111yescharacteristic
86C2×C16#4111yescharacteristic
87C2×C16#4111yescharacteristic
88C2×C16#4111yescharacteristic
89#108—22no—
90#108—41no—
91#108—41no—
92#108—41no—
93#107—21no—
94#107—21no—
95#106—41no—
96#104—82no—
97C22⋊C8—81no—
98C22⋊C8—81no—
99#97—81no—
100C4×C8—21no—
101C4×C8—21no—
102C4×C8—81no—
103C32—21no—
104C32—21no—
105C32—41no—
106#41—81no—
107#41—81no—
108#41—81no—
109#41—81no—
110C2×Q8—41no—
111C2×D4—41no—
112C2×D4—81no—
113C22×C4—21no—
114C22×C4—81no—
115C22×C4C2×D811yescharacteristic
116Q16—41no—
117Q16—41no—
118SD16—81no—
119SD16—81no—
120D8—41no—
121D8—41no—
122M16—42no—
123M16—81no—
124M16—81no—
125M16—81no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—81no—
129C2×C8—81no—
130C2×C8D1611yescharacteristic
131C2×C8D1611yescharacteristic
132C2×C8C4×D411yescharacteristic
133C2×C8#12111yescharacteristic
134C2×C8#12311yescharacteristic
135C2×C8#13511yescharacteristic
136C2×C8AGL(1,ℤ/8)11yescharacteristic
137C2×C8#13711yescharacteristic
138C4⋊C4—41no—
139C4⋊C4—41no—
140C22⋊C4—81no—
141C42—21no—
142C42—21no—
143C42—81no—
144C16—41no—
145C16#11711yescharacteristic
146C16C4×D411yescharacteristic
147C16#12411yescharacteristic
148C16#12611yescharacteristic
149C23—81no—
150C23#50811yescharacteristic
151Q8—81no—
152Q8—81no—
153D4—81no—
154D4—81no—
155D4—161no—
156D4—161no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—161no—
164C2×C4—161no—
165C2×C4C4×D811yescharacteristic
166C2×C4#45911yescharacteristic
167C2×C4#48011yescharacteristic
168C2×C4C2×D1611yescharacteristic
169C2×C4#50911yescharacteristic
170C2×C4#51011yescharacteristic
171C8—21no—
172C8—21no—
173C8—82no—
174C8—161no—
175C8#38811yescharacteristic
176C8#44011yescharacteristic
177C8#46411yescharacteristic
178C8#48211yescharacteristic
179C22—21no—
180C22—21no—
181C22—161no—
182C22—161no—
183C22—161no—
184C22C4×D1611yescharacteristic
185C22#221111yescharacteristic
186C22#223411yescharacteristic
187C4—42no—
188C4—161no—
189C4—161no—
190C4—161no—
191C4#192511yescharacteristic
192C4#217511yescharacteristic
193C4#220611yescharacteristic
194C4#223211yescharacteristic
195C2—21no—
196C2—21no—
197C2—321no—
198C2#1313011yescharacteristic
199C2#1365511yescharacteristic
200C2#1365611yescharacteristic
subgroupG/H
centerC2×C4C2×D16
commutatorC2×C16C22×C4
Frattini, cosocleC22×C16C23
socleC22C4×D16
prime radicalC2×#508C22
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G
LECSG ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1425C2semidirect
5. #12670C2semidirect
6. #7473C2semidirect
7. #7465C2semidirect
20. #1416C4semidirect
21. #1416C4semidirect
22. #1415C4semidirect
23. #1415C4semidirect

Maximal subgroups

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

subgroupindex
C2×#14252
C2×#14172
#136152
#132412
#126702
#74732
#74652