Finite Group Databasebeta

← #138525

Subgroups — #138525

Order 512.

524 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 82 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#29732C211yesmaximal, characteristic
2#21939C211yesmaximal, characteristic, complemented
3#11888C212yesmaximal, complemented
4#11418C211yesmaximal, characteristic, complemented
5#7233C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2847—21no—
9#2280—21no—
10#2112C2212yes—
11#2076C412yescomplemented
12#2034—22no—
13#1999C412yescomplemented
14#1893—21no—
15#1854—24no—
16#1808—21no—
17#1807C2211yescharacteristic
18#1807C2212yes—
19#1347—24no—
20#1314—22no—
21C24×C4D411yescharacteristic
22C22×D8C2×C411yescharacteristic
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#121—21no—
26C2×#121—22no—
27C2×#120—41no—
28C2×#117—22no—
29C2×M32—22no—
30#466—22no—
31#466—22no—
32C2×#107—22no—
33C2×#107C2×C411yescharacteristic
34C2×#106—22no—
35C2×#106—21no—
36C2×#102—21no—
37C2×#102—22no—
38C2×C22⋊C8—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8C2×C412yes—
41C2×C22⋊C8D412yes—
42#386—41no—
43#386—42no—
44#381—41no—
45#380—22no—
46#380—22no—
47#380—22no—
48#380—21no—
49#379—41no—
50C4×C22⋊C4—41no—
51C4×C22⋊C4—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#340—22no—
55#340—22no—
56#340C2×C412yes—
57#340D414yescomplemented
58C25D811yescharacteristic
59C22×D4—21no—
60C22×D4—42no—
61C22×D4—41no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—81no—
67C23×C4—81no—
68C23×C4C22⋊C412yes—
69C23×C4D811yescharacteristic
70C23×C4SD1612yes—
71C23×C4C2×D411yescharacteristic
72C2×D8—22no—
73C2×D8—81no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—81no—
77C22×C8—81no—
78C22×C8C22⋊C412yes—
79C22×C8C22×C411yescharacteristic
80C22×C8C2×D412yes—
81C22×C8C2×D411yescharacteristic
82C22×C8#4112yes—
83#121—41no—
84#121—41no—
85#121—41no—
86#121—41no—
87#121—82no—
88#121—82no—
89#120—84no—
90#117—82no—
91#117—82no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—24no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C22⋊C4—41no—
102C2×C42—42no—
103M32—82no—
104C2×C16—42no—
105#107—22no—
106#107—82no—
107#106—82no—
108#106—81no—
109#102—42no—
110#102—82no—
111C22⋊C8—24no—
112C22⋊C8—24no—
113#95—42no—
114#95—42no—
115#95—41no—
116#95—42no—
117C24—21no—
118C24—21no—
119C24—42no—
120C24—42no—
121C24—41no—
122C24—81no—
123C24#10212yes—
124C24C2×D811yescharacteristic
125C2×D4—42no—
126C2×D4—42no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—82no—
130C2×D4—82no—
131C2×D4—84no—
132C2×D4—84no—
133C2×D4—84no—
134C2×D4—81no—
135C2×D4—81no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—44no—
151C22×C4—81no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—82no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4#10212yes—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4#12011yescharacteristic
160C22×C4#12311yescharacteristic
161C22×C4C2×SD1611yescharacteristic
162C22×C4AGL(1,ℤ/8)11yescharacteristic
163C22×C4#13711yescharacteristic
164D8—82no—
165C2×C8—22no—
166C2×C8—41no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—81no—
173C2×C8C4×D412yes—
174C2×C8#12112yes—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C22⋊C4—84no—
182C22⋊C4—84no—
183C22⋊C4—82no—
184C22⋊C4—84no—
185C22⋊C4—84no—
186C22⋊C4—84no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C42—84no—
190C16—82no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—81no—
204C23—81no—
205C23—82no—
206C23—81no—
207C23—81no—
208C23—88no—
209C23—81no—
210C23—82no—
211C23C2×#10211yescharacteristic
212C23#44711yescharacteristic
213C23#48011yescharacteristic
214D4—82no—
215D4—81no—
216D4—81no—
217D4—162no—
218D4—168no—
219D4—162no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—44no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4—81no—
238C2×C4—84no—
239C2×C4—82no—
240C2×C4—81no—
241C2×C4—82no—
242C2×C4—84no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—84no—
247C2×C4#38611yescharacteristic
248C2×C4#41711yescharacteristic
249C2×C4#44811yescharacteristic
250C2×C4#48111yescharacteristic
251C8—22no—
252C8—22no—
253C8—42no—
254C22—21no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—44no—
262C22—44no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—82no—
267C22—84no—
268C22—84no—
269C22—162no—
270C22—162no—
271C22—161no—
272C22—161no—
273C22—161no—
274C22#187411yescharacteristic
275C22#214512yes—
276C22#219012yes—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—42no—
281C4—164no—
282C4—162no—
283C4—161no—
284C4#191111yescharacteristic
285C4#191311yescharacteristic
286C2—22no—
287C2—22no—
288C2—41no—
289C2—82no—
290C2—321no—
291C2#920311yescharacteristic
292C2#1299911yescharacteristic
293C2#1300211yescharacteristic
subgroupG/H
center, socleC22#1874
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radical#29732C2
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
2. #21939C2semidirect
3. #11888C2semidirect
4. #11418C2semidirect
5. #7233C2semidirect
11. #2076C4semidirect
13. #1999C4semidirect
57. #340D4semidirect

Maximal subgroups

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

subgroupindex
#297322
#219392
#118882
#114182
#72332