Finite Group Databasebeta

← #239022

Subgroups — #239022

Order 512.

985 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 190 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#60968C211yesmaximal, characteristic
2#20826C211yesmaximal, characteristic, complemented
3#17750C212yesmaximal, complemented
4C2×#1492C212yesmaximal, complemented
5#10531C214yesmaximal, complemented
6#10530C214yesmaximal, complemented
7#8129C211yesmaximal, characteristic, complemented
8C2×#532C2212yes—
9#3253C2214yes—
10#3227—22no—
11#3205—22no—
12#3192—22no—
13#3184—22no—
14#3109—24no—
15#2920C2211yescharacteristic
16#2920C2211yescharacteristic
17C2×#423C2212yes—
18C22×C4×C8C2211yescharacteristic
19#1848C2212yescomplemented
20#1837C2212yescomplemented
21#1810—22no—
22C4×#108—22no—
23#1560C2218yescomplemented
24#1551—24no—
25#1548—22no—
26#1545—22no—
27#1492—24no—
28#1492C2214yescomplemented
29#1492C2214yescomplemented
30C2×D4⋊C8—24no—
31#1309C2214yescomplemented
32C22×#41—22no—
33C22×M16—22no—
34C23×C8—21no—
35#532—22no—
36#532—22no—
37#532—22no—
38#532—22no—
39#532—22no—
40#532—22no—
41#532D414yes—
42#532C2314yes—
43C2×C4⋊Q8D412yes—
44C2×#127D412yes—
45C2×#124—22no—
46C2×#121—22no—
47C2×#121—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C22×C42C2311yescharacteristic
51#498—22no—
52#493—22no—
53#486—24no—
54#481—28no—
55#476—28no—
56#469—28no—
57#459—28no—
58#447—28no—
59#442—28no—
60#432—22no—
61#432—22no—
62#431—22no—
63#431—22no—
64C2×#108—24no—
65#423—22no—
66#423—22no—
67#423—22no—
68#423C2312yes—
69#423C2314yes—
70#423C2312yes—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73#407—21no—
74#407D412yes—
75C4×M16—21no—
76C4×M16D412yes—
77C2×C4×C8—21no—
78C2×C4×C8—22no—
79C2×C4×C8D414yes—
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82#341—24no—
83#333—28no—
84#331—28no—
85D4⋊C8—28no—
86D4⋊C8—28no—
87C2×#41—24no—
88C2×#41—24no—
89C2×#41—42no—
90C2×#41—42no—
91C2×#41—84no—
92C2×#41—84no—
93C22×Q8—22no—
94C22×D4—22no—
95C22×D4—22no—
96C22×D4—22no—
97C22×D4—42no—
98C23×C4—42no—
99C23×C4C2×D411yescharacteristic
100C23×C4#4112yes—
101C2×D8—28no—
102C2×M16—22no—
103C2×M16—24no—
104C2×M16—42no—
105C2×M16—84no—
106C2×M16C2×D414yes—
107C22×C8—21no—
108C22×C8—24no—
109C22×C8—21no—
110C22×C8—44no—
111C22×C8—42no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8C2×D414yes—
116#127—22no—
117#127—22no—
118#127—22no—
119#127C2×D414yes—
120#124—24no—
121#124—24no—
122#124—42no—
123#124—42no—
124#121—24no—
125#121—24no—
126#121—24no—
127#121—24no—
128#121—42no—
129#121—42no—
130#121—42no—
131#121—42no—
132C4×D4—24no—
133C4×D4—24no—
134C4×D4—24no—
135C4×D4—24no—
136C4×D4—42no—
137C4×D4—42no—
138C4×D4—42no—
139C4×D4—42no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42#4112yes—
149C2×C42#4112yes—
150C2×C42C2411yescharacteristic
151#108—48no—
152#106—28no—
153C4⋊C8—24no—
154C4⋊C8—24no—
155C4⋊C8—24no—
156C4⋊C8—24no—
157C4⋊C8—42no—
158C4⋊C8—42no—
159#102—216no—
160C22⋊C8—24no—
161C22⋊C8—42no—
162#97—22no—
163#97—41no—
164C4×C8—22no—
165C4×C8—24no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—41no—
169C4×C8C2×D412yes—
170C4×C8C2×D412yes—
171C24—42no—
172C24#12411yescharacteristic
173#41—88no—
174#41—88no—
175#41—84no—
176#41—84no—
177#41—84no—
178#41—84no—
179C2×Q8—24no—
180C2×Q8—42no—
181C2×Q8—84no—
182C2×D4—24no—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—44no—
186C2×D4—44no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—84no—
193C2×D4—84no—
194C2×D4—88no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—84no—
202C22×C4—84no—
203C22×C4#12012yes—
204C22×C4#12112yes—
205C22×C4#12112yes—
206C22×C4#12112yes—
207C22×C4#12312yes—
208C22×C4#12312yes—
209C22×C4#12412yes—
210C22×C4#12411yescharacteristic
211C22×C4C22×D411yescharacteristic
212C22×C4C2×#4112yes—
213D8—88no—
214M16—44no—
215M16—44no—
216M16—84no—
217M16—84no—
218C2×C8—24no—
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—22no—
222C2×C8—48no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—44no—
226C2×C8—84no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C42#12011yescharacteristic
236C42#12011yescharacteristic
237C42#12112yes—
238C42#12112yes—
239C42#12112yes—
240C42#12312yes—
241C42#12312yes—
242C42C22×D412yes—
243C42C22×D411yescharacteristic
244C42C2×#4111yescharacteristic
245C23—24no—
246C23—41no—
247C23—44no—
248C23—84no—
249C23—84no—
250C23—84no—
251C23#39412yes—
252C23C2×#12411yescharacteristic
253Q8—84no—
254Q8—84no—
255D4—48no—
256D4—44no—
257D4—44no—
258D4—84no—
259D4—84no—
260D4—88no—
261D4—88no—
262D4—84no—
263D4—84no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—88no—
277C2×C4—84no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4#39412yes—
281C2×C4#39412yes—
282C2×C4#39412yes—
283C2×C4#47114yes—
284C2×C4#47214yes—
285C2×C4C2×#12011yescharacteristic
286C2×C4C2×#12111yescharacteristic
287C2×C4C2×#12112yes—
288C2×C4C2×#12312yes—
289C8—24no—
290C8—42no—
291C8—44no—
292C8—82no—
293C8—84no—
294C22—22no—
295C22—22no—
296C22—21no—
297C22—22no—
298C22—44no—
299C22—84no—
300C22—84no—
301C22—84no—
302C22—168no—
303C22#205114yes—
304C22C2×#39411yescharacteristic
305C4—21no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—84no—
310C4#205114yes—
311C4#301812yes—
312C4C2×#47112yes—
313C4C2×#47212yes—
314C2—24no—
315C2—41no—
316C2—164no—
317C2#2349811yescharacteristic
318C2C2×#205112yes—
subgroupG/H
center, socleC22C2×#394
commutator, Frattini, cosocleC2×C42C24
prime radical#60968C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20826C2semidirect
3. #17750C2semidirect
4. C2×#1492C2semidirect
5. #10531C2semidirect
6. #10530C2semidirect
7. #8129C2semidirect
19. #1848C22semidirect
20. #1837C22semidirect
23. #1560C22semidirect
28. #1492C22semidirect
29. #1492C22semidirect
31. #1309C22semidirect

Maximal subgroups

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

subgroupindex
#609682
#208262
#177502
C2×#14922
#105312
#105302
#81292