Finite Group Databasebeta

← #125354

Subgroups — #125354

Order 512.

302 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 94 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#33267C211yesmaximal, characteristic
2C4×C4⋊C16C211yesmaximal, characteristic
3#8274C211yesmaximal, characteristic
4#7121C212yesmaximal
5#7113C212yesmaximal
6#2863—21no—
7C2×C4⋊C16—21no—
8C2×C4×C16—21no—
9#1969C411yescharacteristic
10#1969C411yescharacteristic
11#1969C2211yescharacteristic
12#1942C411yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1933—21no—
16#1933—21no—
17#1933—21no—
18#1767—21no—
19C42×C8C2211yescharacteristic
20C2×#335—21no—
21#1374C2212yes—
22#1277C2212yes—
23C22×Q16C2×C411yescharacteristic
24C22×C16—22no—
25C2×#107C2×C411yescharacteristic
26C2×#107C2×C411yescharacteristic
27C2×#107C2×C411yescharacteristic
28C2×C4⋊C8—22no—
29C2×#103—21no—
30C2×#103—21no—
31C2×#103—21no—
32C2×#97—41no—
33C2×C4×C8C2×C411yescharacteristic
34C2×C4×C8C2×C411yescharacteristic
35C2×C4×C8D411yescharacteristic
36C2×C4×C8D411yescharacteristic
37C2×C4×C8D411yescharacteristic
38C2×C4×C8C2311yescharacteristic
39#391—21no—
40#391—21no—
41#391—21no—
42#389—21no—
43#389—21no—
44#389—21no—
45#387—41no—
46#387—41no—
47#387—41no—
48#383—41no—
49C43D411yescharacteristic
50C4⋊C16—24no—
51C4×C16—42no—
52#340—21no—
53#340—21no—
54#340—21no—
55#336—41no—
56#335—24no—
57C22×Q8—21no—
58C2×Q16—22no—
59C2×Q16—41no—
60C2×Q16—41no—
61C22×C8—42no—
62C22×C8C22⋊C411yescharacteristic
63C22×C8C22⋊C411yescharacteristic
64C22×C8C22×C411yescharacteristic
65C22×C8C2×D411yescharacteristic
66C2×C4⋊C4—21no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—41no—
70C2×C4⋊C4—41no—
71C2×C4⋊C4—41no—
72C2×C42C22⋊C411yescharacteristic
73C2×C42C22⋊C411yescharacteristic
74C2×C42D811yescharacteristic
75C2×C42D811yescharacteristic
76C2×C42SD1611yescharacteristic
77C2×C42SD1611yescharacteristic
78C2×C42C2×D411yescharacteristic
79C2×C16—24no—
80C2×C16—42no—
81C2×C16—42no—
82#107—22no—
83#107—41no—
84#107—41no—
85C4⋊C8—24no—
86C4⋊C8—42no—
87#103—42no—
88#103—42no—
89#103—41no—
90#103—41no—
91#97—82no—
92C4×C8—22no—
93C4×C8—22no—
94C4×C8—22no—
95C4×C8—21no—
96C4×C8—21no—
97C4×C8—41no—
98C4×C8—41no—
99#95—41no—
100#95—41no—
101#95—41no—
102#95—41no—
103#95—41no—
104#95—41no—
105#95—41no—
106#95—41no—
107#95—41no—
108C2×Q8—41no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8—41no—
113C2×Q8—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4#10211yescharacteristic
119C22×C4#10211yescharacteristic
120C22×C4#10211yescharacteristic
121C22×C4#10211yescharacteristic
122C22×C4C2×C22⋊C411yescharacteristic
123C22×C4C2×D811yescharacteristic
124C22×C4C2×SD1611yescharacteristic
125Q16—82no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—44no—
135C2×C8—82no—
136C2×C8—82no—
137C2×C8#10412yes—
138C2×C8#10412yes—
139C2×C8#13112yes—
140C2×C8AGL(1,ℤ/8)11yescharacteristic
141C2×C8#13711yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C4⋊C4—82no—
149C4⋊C4—82no—
150C4⋊C4—82no—
151C42—22no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—41no—
159C42—41no—
160C42SD3212yes—
161C42Q3212yes—
162C16—44no—
163C23C2×#10211yescharacteristic
164Q8—81no—
165Q8—81no—
166Q8—81no—
167Q8—81no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
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—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4#35812yes—
195C2×C4#35812yes—
196C2×C4#35912yes—
197C2×C4#40811yescharacteristic
198C2×C4#41011yescharacteristic
199C2×C4#41311yescharacteristic
200C2×C4#41711yescharacteristic
201C2×C4C2×#10411yescharacteristic
202C2×C4C2×#10411yescharacteristic
203C2×C4C2×SD3211yescharacteristic
204C2×C4C2×Q3211yescharacteristic
205C8—22no—
206C8—22no—
207C8—21no—
208C8—21no—
209C8—22no—
210C8—84no—
211C22#149211yescharacteristic
212C22#150211yescharacteristic
213C22#154711yescharacteristic
214C22C2×#35811yescharacteristic
215C22C2×#35911yescharacteristic
216C22#214511yescharacteristic
217C22#214811yescharacteristic
218C4—22no—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—161no—
226C4—161no—
227C4—161no—
228C4—161no—
229C4#147311yescharacteristic
230C4#148311yescharacteristic
231C4#150911yescharacteristic
232C4#152411yescharacteristic
233C2#768611yescharacteristic
234C2#1167211yescharacteristic
235C2#1168211yescharacteristic
236C2#1174311yescharacteristic
237C2#1174911yescharacteristic
238C2#1197811yescharacteristic
239C2#1198311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#332672
C4×C4⋊C162
#82742
#71212
#71132