Finite Group Databasebeta

← #138125

Subgroups — #138125

Order 512.

339 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 96 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#33419C211yesmaximal, characteristic
2#33214C211yesmaximal, characteristic
3#11935C212yesmaximal, complemented
4#8601C211yesmaximal, characteristic
5#7278C212yesmaximal, complemented
6C2×#499C2211yescharacteristic
7C2×#405C2211yescharacteristic
8#2629—21no—
9#2164—24no—
10#2150C2212yes—
11#2112—21no—
12C4×M32—21no—
13#1992—21no—
14#1944C411yescharacteristic
15#1944C411yescharacteristic
16#1941—21no—
17#1844C2211yescharacteristic
18#1809C411yescharacteristic
19#1809C411yescharacteristic
20#1807—21no—
21#1564C2212yes—
22#1458—21no—
23#1363—24no—
24#1275—21no—
25C22×M16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#125—21no—
28C2×#117D411yescharacteristic
29C2×M32—22no—
30#499—22no—
31#499—22no—
32#431—22no—
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×#106—21no—
36C2×#106—21no—
37C2×#106C2×C411yescharacteristic
38C2×#106C2×C411yescharacteristic
39#405—22no—
40#405—22no—
41C2×#97C2×C411yescharacteristic
42C2×#97D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#398—41no—
46#398—41no—
47#396—41no—
48#396—41no—
49#389—21no—
50#389—21no—
51#384—21no—
52#380—21no—
53#380—21no—
54C4⋊C16—22no—
55C4⋊C16D414yes—
56C22⋊C16—24no—
57#346—22no—
58C4×C16—22no—
59#340—21no—
60#340—21no—
61#340—21no—
62#340—21no—
63#335—22no—
64#334—22no—
65#324—41no—
66#323—41no—
67C23×C4C22⋊C411yescharacteristic
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—41no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22×C411yescharacteristic
78C22×C8C2×D411yescharacteristic
79#125—42no—
80#125—42no—
81#117—22no—
82#117—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4D811yescharacteristic
94C2×C4⋊C4SD1611yescharacteristic
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4D811yescharacteristic
98C2×C22⋊C4SD1611yescharacteristic
99C2×C42C22⋊C411yescharacteristic
100C2×C42C2×D411yescharacteristic
101M32—42no—
102M32—42no—
103C2×C16—24no—
104#107—22no—
105#107—22no—
106#106—41no—
107#106—41no—
108#106—42no—
109#106—41no—
110#106—41no—
111C4⋊C8—42no—
112C22⋊C8—42no—
113#97—22no—
114#97—22no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—42no—
118C4×C8C2×D412yes—
119C4×C8#4112yes—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124C24#10211yescharacteristic
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—41no—
135C22×C4#10211yescharacteristic
136C22×C4#10211yescharacteristic
137C22×C4#10211yescharacteristic
138C22×C4C2×C22⋊C411yescharacteristic
139C22×C4C2×D811yescharacteristic
140C22×C4C2×SD1611yescharacteristic
141M16—44no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—82no—
152C2×C8#10112yes—
153C2×C8C4×D412yes—
154C2×C8#12011yescharacteristic
155C2×C8#12311yescharacteristic
156C2×C8#13512yes—
157C2×C8#13712yes—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—82no—
169C4⋊C4—82no—
170C4⋊C4—161no—
171C4⋊C4—161no—
172C4⋊C4—161no—
173C4⋊C4—161no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—161no—
177C22⋊C4—161no—
178C42—21no—
179C42—21no—
180C42#12112yes—
181C42#13512yes—
182C16—44no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—41no—
188C23C2×#10211yescharacteristic
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
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#38611yescharacteristic
215C2×C4C2×#10111yescharacteristic
216C2×C4#41311yescharacteristic
217C2×C4#41611yescharacteristic
218C2×C4#41611yescharacteristic
219C2×C4#41911yescharacteristic
220C2×C4#46512yes—
221C2×C4#46612yes—
222C8—22no—
223C8—22no—
224C8—41no—
225C8—162no—
226C8#45212yes—
227C8#48512yes—
228C22—21no—
229C22—21no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—81no—
235C22#154011yescharacteristic
236C22#154611yescharacteristic
237C22#189311yescharacteristic
238C4—22no—
239C4—22no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—161no—
244C4—161no—
245C4—161no—
246C4—161no—
247C4#154411yescharacteristic
248C4#155411yescharacteristic
249C4#187311yescharacteristic
250C4#190111yescharacteristic
251C2—21no—
252C2—21no—
253C2—81no—
254C2#913011yescharacteristic
255C2#1307611yescharacteristic
256C2#1307711yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#405 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11935C2semidirect
5. #7278C2semidirect

Maximal subgroups

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

subgroupindex
#334192
#332142
#119352
#86012
#72782