Finite Group Databasebeta

← #138526

Subgroups — #138526

Order 512.

521 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 521 conjugacy classes of subgroups — too many to draw. Only its 86 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#29733C211yesmaximal, characteristic
2#21938C211yesmaximal, characteristic
3#11889C212yesmaximal, complemented
4#11418C211yesmaximal, characteristic
5#7233C212yesmaximal, complemented
6C2×#467C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2848—21no—
9C2×#380—21no—
10#2112C2212yes—
11#2075C412yescomplemented
12#2035—22no—
13#2001C412yescomplemented
14#1892—21no—
15#1853—24no—
16#1807—21no—
17#1807C2211yescharacteristic
18#1807C2212yes—
19#1348—24no—
20#1314—22no—
21C24×C4D411yescharacteristic
22C22×Q16C2×C411yescharacteristic, complemented
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8—22no—
27C22×C4⋊C4—22no—
28C22×C22⋊C4—41no—
29C2×M32—22no—
30#467—22no—
31#467—22no—
32C2×#107C2×C411yescharacteristic, complemented
33C2×#106—22no—
34C2×#106—22no—
35C2×#106—21no—
36C2×#103—21no—
37C2×#103—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C412yes—
41C2×C22⋊C8D412yes—
42#385—41no—
43#385—42no—
44#380—22no—
45#380—22no—
46#380—22no—
47#380—21no—
48#380—41no—
49#380—41no—
50#379—41no—
51C2×#95—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#340—22no—
55#340—22no—
56#340C2×C412yescomplemented
57#340D414yes—
58C25D811yescharacteristic
59C22×Q8—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—42no—
65C23×C4—81no—
66C23×C4—81no—
67C23×C4C22⋊C412yes—
68C23×C4D811yescharacteristic
69C23×C4SD1612yes—
70C23×C4C2×D411yescharacteristic
71C2×Q16—22no—
72C2×Q16—81no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—81no—
76C22×C8—81no—
77C22×C8C22⋊C412yes—
78C22×C8C22×C411yescharacteristic
79C22×C8C2×D412yes—
80C22×C8C2×D411yescharacteristic
81C22×C8#4112yes—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84C22⋊Q8—41no—
85C22⋊Q8—41no—
86C22⋊Q8—82no—
87C22⋊Q8—82no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—82no—
95C2×C4⋊C4—82no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C22⋊C4—84no—
103M32—82no—
104C2×C16—42no—
105#107—22no—
106#106—82no—
107#106—82no—
108#106—81no—
109#103—42no—
110#103—82no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113#95—42no—
114#95—42no—
115#95—41no—
116#95—42no—
117C24—21no—
118C24—24no—
119C24—21no—
120C24—42no—
121C24—81no—
122C24#10212yes—
123C24C2×D811yescharacteristic
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—41no—
127C2×Q8—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—44no—
144C22×C4—81no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4—82no—
148C22×C4—82no—
149C22×C4—82no—
150C22×C4—84no—
151C22×C4—84no—
152C22×C4—84no—
153C22×C4#10212yes—
154C22×C4C2×C22⋊C411yescharacteristic
155C22×C4#12011yescharacteristic
156C22×C4#12311yescharacteristic
157C22×C4C2×SD1611yescharacteristic
158C22×C4AGL(1,ℤ/8)11yescharacteristic
159C22×C4#13711yescharacteristic
160Q16—82no—
161C2×C8—22no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—81no—
169C2×C8C4×D412yes—
170C2×C8#12112yes—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—44no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—84no—
178C4⋊C4—84no—
179C4⋊C4—81no—
180C4⋊C4—81no—
181C22⋊C4—82no—
182C22⋊C4—84no—
183C22⋊C4—84no—
184C22⋊C4—84no—
185C22⋊C4—84no—
186C22⋊C4—81no—
187C22⋊C4—81no—
188C16—82no—
189C23—22no—
190C23—22no—
191C23—24no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—81no—
202C23—81no—
203C23—88no—
204C23#36014yes—
205C23C2×#10211yescharacteristic
206C23#44711yescharacteristic
207C23#48011yescharacteristic
208Q8—82no—
209Q8—81no—
210Q8—81no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—44no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—81no—
229C2×C4—84no—
230C2×C4—82no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—84no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—82no—
237C2×C4—82no—
238C2×C4—84no—
239C2×C4—168no—
240C2×C4#38611yescharacteristic
241C2×C4#41711yescharacteristic
242C2×C4#44811yescharacteristic
243C2×C4#48111yescharacteristic
244C8—22no—
245C8—22no—
246C8—42no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—24no—
253C22—21no—
254C22—21no—
255C22—44no—
256C22—44no—
257C22—44no—
258C22—44no—
259C22—42no—
260C22—42no—
261C22—82no—
262C22—84no—
263C22#187411yescharacteristic
264C22C2×#36012yes—
265C22#219112yes—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—42no—
270C4—162no—
271C4—164no—
272C4—161no—
273C4—161no—
274C4#191111yescharacteristic
275C4#191311yescharacteristic
276C2—22no—
277C2—22no—
278C2—41no—
279C2—44no—
280C2#920311yescharacteristic
281C2#1300011yescharacteristic
282C2#1300111yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11889C2semidirect
5. #7233C2semidirect
11. #2075C4semidirect
13. #2001C4semidirect
22. C22×Q16C2×C4semidirect
32. C2×#107C2×C4semidirect
56. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
#297332
#219382
#118892
#114182
#72332