Finite Group Databasebeta

← #122607

Subgroups — #122607

Order 512.

297 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 70 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#29879C211yesmaximal, characteristic
2#11459C211yesmaximal, characteristic
3#7986C211yesmaximal, characteristic
4#7083C212yesmaximal, complemented
5#7081C212yesmaximal, complemented
6C2×#499C411yescharacteristic
7C2×#432—21no—
8C2×#431C2211yescharacteristic
9#2851—21no—
10#2162—21no—
11#2088—21no—
12#2087C2211yescharacteristic
13#2086C2211yescharacteristic
14#2077—21no—
15#2076C411yescharacteristic
16#2075C411yescharacteristic
17#1991—21no—
18#1835C411yescharacteristic
19#1322C2212yes—
20#1314—21no—
21#1290—21no—
22#1290C2212yes—
23C2×#125—21no—
24C22×C42D411yescharacteristic
25#499—41no—
26#499—41no—
27#432—22no—
28#432—22no—
29#432—41no—
30#432—41no—
31#431—41no—
32#431—41no—
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×#106—21no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8D411yescharacteristic
38C2×C4⋊C8C2311yescharacteristic
39C2×C22⋊C8—41no—
40C2×C22⋊C8C2×C411yescharacteristic
41C2×C22⋊C8D411yescharacteristic
42C2×#97—22no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45#389—21no—
46#389—21no—
47#387—41no—
48#387—41no—
49#385—41no—
50#385—41no—
51#382—21no—
52#380—21no—
53#380—21no—
54C4⋊C16—41no—
55C22⋊C16—22no—
56C4×C16—41no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4D811yescharacteristic
66C23×C4SD1611yescharacteristic
67C22×C8—42no—
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22×C411yescharacteristic
71C22×C8C2×D411yescharacteristic
72#125—41no—
73#125—41no—
74#125—41no—
75#125—41no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4—41no—
82C2×C4⋊C4—41no—
83C2×C4⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C42—21no—
87C2×C42—21no—
88C2×C42—21no—
89C2×C42—21no—
90C2×C42C22⋊C411yescharacteristic
91C2×C42D811yescharacteristic
92C2×C42SD1611yescharacteristic
93C2×C42C2×D411yescharacteristic
94C2×C16—42no—
95#107—41no—
96#107—41no—
97#106—42no—
98C4⋊C8—41no—
99C4⋊C8—41no—
100C4⋊C8—42no—
101C4⋊C8—42no—
102C22⋊C8—41no—
103C22⋊C8—41no—
104C22⋊C8—42no—
105C22⋊C8—42no—
106#97—24no—
107#97—42no—
108C4×C8—21no—
109C4×C8—21no—
110C4×C8—21no—
111C4×C8—21no—
112#95—41no—
113#95—41no—
114#95—41no—
115#95—41no—
116#95—41no—
117#95—41no—
118C24#10211yescharacteristic
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—81no—
134C22×C4#10211yescharacteristic
135C22×C4#10211yescharacteristic
136C22×C4#10211yescharacteristic
137C22×C4C2×C22⋊C411yescharacteristic
138C22×C4C2×D811yescharacteristic
139C22×C4C2×SD1611yescharacteristic
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—44no—
149C2×C8—84no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—82no—
159C4⋊C4—82no—
160C4⋊C4—81no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C22⋊C4—82no—
165C22⋊C4—82no—
166C42—21no—
167C42—21no—
168C42—21no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42#9911yescharacteristic
173C42#9911yescharacteristic
174C42#10411yescharacteristic
175C42#10411yescharacteristic
176C16—82no—
177C23—21no—
178C23—21no—
179C23—41no—
180C23—41no—
181C23#35912yes—
182C23C2×#10211yescharacteristic
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4#32711yescharacteristic
215C2×C4#32711yescharacteristic
216C2×C4#32711yescharacteristic
217C2×C4#32711yescharacteristic
218C2×C4C2×#9911yescharacteristic
219C2×C4C2×#10411yescharacteristic
220C8—41no—
221C8—41no—
222C8—41no—
223C8—41no—
224C8—84no—
225C22—21no—
226C22—21no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22C2×#32711yescharacteristic
233C22C2×#35911yescharacteristic
234C22#214811yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—161no—
244C4—161no—
245C4—161no—
246C4—161no—
247C4#212311yescharacteristic
248C4#213511yescharacteristic
249C2—21no—
250C2—21no—
251C2—42no—
252C2#1168511yescharacteristic
253C2#1170011yescharacteristic
254C2#1170911yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7083C2semidirect
5. #7081C2semidirect

Maximal subgroups

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

subgroupindex
#298792
#114592
#79862
#70832
#70812