Finite Group Databasebeta

← #138524

Subgroups — #138524

Order 512.

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

normal subgroups only

This group has 560 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#29732C211yesmaximal, characteristic
2#21938C211yesmaximal, characteristic, complemented
3#11888C212yesmaximal, complemented
4#11418C211yesmaximal, characteristic, complemented
5#7233C212yesmaximal, complemented
6C2×#466C2211yescharacteristic
7C22×C22⋊C8C2211yescharacteristic
8#2847—21no—
9C2×#380—21no—
10#2112C2212yes—
11#2076C412yescomplemented
12#2034—22no—
13#1999C412yescomplemented
14#1893—21no—
15#1853—24no—
16#1807—21no—
17#1807C2211yescharacteristic
18#1807C2212yes—
19#1347—24no—
20#1314—22no—
21C24×C4D411yescharacteristic
22C22×D8C2×C411yescharacteristic, complemented
23C23×C8D411yescharacteristic
24C23×C8C2311yescharacteristic
25C2×#121—21no—
26C2×#121—22no—
27C2×#120—41no—
28C22×C4⋊C4—22no—
29C22×C22⋊C4—41no—
30C2×M32—22no—
31#466—22no—
32#466—22no—
33C2×#107C2×C411yescharacteristic, complemented
34C2×#106—22no—
35C2×#106—22no—
36C2×#106—21no—
37C2×#102—21no—
38C2×#102—22no—
39C2×C22⋊C8—22no—
40C2×C22⋊C8—22no—
41C2×C22⋊C8C2×C412yes—
42C2×C22⋊C8D412yes—
43#386—41no—
44#386—42no—
45#380—22no—
46#380—22no—
47#380—22no—
48#380—21no—
49#380—41no—
50#380—41no—
51C2×#95—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54#340—22no—
55#340—22no—
56#340C2×C412yescomplemented
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—42no—
67C23×C4—81no—
68C23×C4—81no—
69C23×C4C22⋊C412yes—
70C23×C4D811yescharacteristic
71C23×C4SD1612yes—
72C23×C4C2×D411yescharacteristic
73C2×D8—22no—
74C2×D8—81no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—81no—
78C22×C8—81no—
79C22×C8C22⋊C412yes—
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D412yes—
82C22×C8C2×D411yescharacteristic
83C22×C8#4112yes—
84#121—41no—
85#121—41no—
86#121—41no—
87#121—41no—
88#121—82no—
89#121—82no—
90#120—84no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—82no—
96C2×C4⋊C4—82no—
97C2×C22⋊C4—42no—
98C2×C22⋊C4—42no—
99C2×C22⋊C4—42no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—42no—
102C2×C22⋊C4—41no—
103C2×C22⋊C4—84no—
104M32—82no—
105C2×C16—42no—
106#107—22no—
107#106—82no—
108#106—82no—
109#106—81no—
110#102—42no—
111#102—82no—
112C22⋊C8—24no—
113C22⋊C8—24no—
114#95—42no—
115#95—42no—
116#95—41no—
117#95—42no—
118C24—21no—
119C24—24no—
120C24—21no—
121C24—42no—
122C24—41no—
123C24—81no—
124C24#10212yes—
125C24C2×D811yescharacteristic
126C2×D4—42no—
127C2×D4—42no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—82no—
131C2×D4—82no—
132C2×D4—84no—
133C2×D4—84no—
134C2×D4—84no—
135C2×D4—81no—
136C2×D4—81no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—44no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4—84no—
161C22×C4#10212yes—
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4#12011yescharacteristic
164C22×C4#12311yescharacteristic
165C22×C4C2×SD1611yescharacteristic
166C22×C4AGL(1,ℤ/8)11yescharacteristic
167C22×C4#13711yescharacteristic
168D8—82no—
169C2×C8—22no—
170C2×C8—41no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—41no—
174C2×C8—42no—
175C2×C8—42no—
176C2×C8—81no—
177C2×C8C4×D412yes—
178C2×C8#12112yes—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—84no—
186C22⋊C4—84no—
187C22⋊C4—84no—
188C22⋊C4—84no—
189C22⋊C4—82no—
190C22⋊C4—84no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C16—82no—
194C23—22no—
195C23—22no—
196C23—24no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—81no—
207C23—81no—
208C23—82no—
209C23—81no—
210C23—81no—
211C23—88no—
212C23—81no—
213C23—82no—
214C23#36014yes—
215C23C2×#10211yescharacteristic
216C23#44711yescharacteristic
217C23#48011yescharacteristic
218D4—82no—
219D4—81no—
220D4—81no—
221D4—162no—
222D4—168no—
223D4—162no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—81no—
242C2×C4—84no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—82no—
246C2×C4—84no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—82no—
250C2×C4—84no—
251C2×C4—168no—
252C2×C4#38611yescharacteristic
253C2×C4#41711yescharacteristic
254C2×C4#44811yescharacteristic
255C2×C4#48111yescharacteristic
256C8—22no—
257C8—22no—
258C8—42no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—24no—
265C22—21no—
266C22—21no—
267C22—44no—
268C22—44no—
269C22—44no—
270C22—44no—
271C22—42no—
272C22—42no—
273C22—82no—
274C22—84no—
275C22—162no—
276C22—162no—
277C22—161no—
278C22—161no—
279C22—161no—
280C22#187411yescharacteristic
281C22C2×#36012yes—
282C22#219012yes—
283C4—21no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—164no—
288C4—162no—
289C4—161no—
290C4#191111yescharacteristic
291C4#191311yescharacteristic
292C2—22no—
293C2—22no—
294C2—41no—
295C2—44no—
296C2—321no—
297C2#920311yescharacteristic
298C2#1299911yescharacteristic
299C2#1300111yescharacteristic
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. #21938C2semidirect
3. #11888C2semidirect
4. #11418C2semidirect
5. #7233C2semidirect
11. #2076C4semidirect
13. #1999C4semidirect
22. C22×D8C2×C4semidirect
33. C2×#107C2×C4semidirect
56. #340C2×C4semidirect
57. #340D4semidirect

Maximal subgroups

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

subgroupindex
#297322
#219382
#118882
#114182
#72332