Finite Group Databasebeta

← #125495

Subgroups — #125495

Order 512.

252 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 64 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
1#33270C211yesmaximal, characteristic
2#12659C211yesmaximal, characteristic
3#8800C211yesmaximal, characteristic
4#7127C211yesmaximal, characteristic
5#7126C211yesmaximal, characteristic
6#7111C211yesmaximal, characteristic
7#7111C211yesmaximal, characteristic
8#2863—21no—
9C2×#364—21no—
10C2×#347—21no—
11#1971C411yescharacteristic
12#1971C2211yescharacteristic
13#1969C411yescharacteristic
14#1944C2211yescharacteristic
15#1943C411yescharacteristic
16#1941C411yescharacteristic
17#1935—21no—
18#1935—21no—
19#1933—21no—
20C4×#97C2211yescharacteristic
21#1377C2211yescharacteristic
22#1377C2211yescharacteristic
23#1281—21no—
24#1276—21no—
25#1275C2211yescharacteristic
26#1275C2211yescharacteristic
27C22×Q16C2×C411yescharacteristic
28C2×M32—21no—
29C2×M32—21no—
30#431—21no—
31#431—21no—
32C2×#107C2×C411yescharacteristic
33C2×#106C2×C411yescharacteristic
34C2×#106C2×C411yescharacteristic
35C2×#103—21no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#97C2×C411yescharacteristic
39C2×#97D411yescharacteristic
40C2×#97D411yescharacteristic
41C2×#97C2311yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#391—21no—
45#391—21no—
46#391—21no—
47#389—21no—
48#389—21no—
49#389—21no—
50#387—41no—
51#387—41no—
52#387—41no—
53#383—41no—
54C43D411yescharacteristic
55#364—24no—
56#347—42no—
57#340—21no—
58#340—21no—
59#340—21no—
60C22×Q8—21no—
61C2×Q16—22no—
62C2×Q16—41no—
63C2×Q16—41no—
64C22×C8C22⋊C411yescharacteristic
65C22×C8C22⋊C411yescharacteristic
66C22×C8C22×C411yescharacteristic
67C22×C8C2×D411yescharacteristic
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—41no—
72C2×C4⋊C4—41no—
73C2×C4⋊C4—41no—
74C2×C42—21no—
75C2×C42—21no—
76C2×C42C22⋊C411yescharacteristic
77C2×C42C22⋊C411yescharacteristic
78C2×C42C2×D411yescharacteristic
79M32—22no—
80M32—22no—
81M32—41no—
82M32—41no—
83C2×C16—41no—
84C2×C16—41no—
85#107—41no—
86#106—22no—
87#106—41no—
88C4⋊C8—41no—
89C4⋊C8—41no—
90#103—42no—
91#103—42no—
92#103—41no—
93#103—41no—
94C22⋊C8—41no—
95C22⋊C8—41no—
96#97—21no—
97#97—21no—
98#97—41no—
99#97—41no—
100#97—41no—
101C4×C8—22no—
102C4×C8—22no—
103C4×C8—41no—
104C4×C8—41no—
105#95—41no—
106#95—41no—
107#95—41no—
108#95—41no—
109#95—41no—
110#95—41no—
111#95—41no—
112#95—41no—
113#95—41no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—41no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4C2×C22⋊C411yescharacteristic
127C22×C4AGL(1,ℤ/8)11yescharacteristic
128C22×C4AGL(1,ℤ/8)11yescharacteristic
129Q16—82no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—81no—
140C2×C8—81no—
141C2×C8#10412yes—
142C2×C8#13112yes—
143C2×C8AGL(1,ℤ/8)11yescharacteristic
144C2×C8#13711yescharacteristic
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—82no—
152C4⋊C4—82no—
153C4⋊C4—82no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—21no—
159C42—21no—
160C42—42no—
161C42—81no—
162C16—42no—
163C16—42no—
164C23#41811yescharacteristic
165Q8—81no—
166Q8—81no—
167Q8—81no—
168Q8—81no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4#40811yescharacteristic
192C2×C4#41011yescharacteristic
193C2×C4#41311yescharacteristic
194C2×C4#41711yescharacteristic
195C2×C4C2×#10411yescharacteristic
196C2×C4#42111yescharacteristic
197C8—22no—
198C8—21no—
199C8—21no—
200C8—22no—
201C8—41no—
202C8—161no—
203C8—161no—
204C22—21no—
205C22—21no—
206C22#148411yescharacteristic
207C22#151411yescharacteristic
208C22#155711yescharacteristic
209C4—42no—
210C4—41no—
211C4—41no—
212C4—81no—
213C4—161no—
214C4—161no—
215C4—161no—
216C4—161no—
217C4#147311yescharacteristic
218C4#149511yescharacteristic
219C4#150911yescharacteristic
220C4#153211yescharacteristic
221C2—21no—
222C2—21no—
223C2#771211yescharacteristic
224C2#1183611yescharacteristic
225C2#1183811yescharacteristic
subgroupG/H
center, socleC22#1514
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC23#418
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332702
#126592
#88002
#71272
#71262
#71112