Finite Group Databasebeta

← #44560847

Subgroups — #44560847

Order 32128.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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
1C2×#22548742C212yesmaximal, complemented
2C2×#22548738C211yesmaximal, characteristic, complemented
3C2×D8×D251C211yesmaximal, characteristic, complemented
4#30632255C212yesmaximal, complemented
5#30632077C212yesmaximal, complemented
6#30632073C212yesmaximal, complemented
7#30632072C211yesmaximal, characteristic, complemented
8#30632064C211yesmaximal, characteristic, complemented
9#30632062C211yesmaximal, characteristic, complemented
10#30631641C211yesmaximal, characteristic, complemented
11#495×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13C2×D4×D251C2212yes—
14#22548742—28no—
15#22548738—24no—
16C2×#16860797C2212yescomplemented
17C2×#16860797C2212yescomplemented
18C2×D2008C2211yescharacteristic
19C2×D2008C2211yescharacteristic, complemented
20C2×D2008C2212yescomplemented
21C2×#16860795C2212yes—
22C2×#16860795C2212yescomplemented
23D8×D251—24no—
24C2×#16860793C2211yescharacteristic
25C2×C8×D251C2211yescharacteristic
26#22548712C2212yescomplemented
27#22548711C2212yescomplemented
28#22548704C2212yescomplemented
29#22548688—24no—
30#22548685—24no—
31#22548672C2211yescharacteristic
32#22548626C2211yescharacteristic, complemented
33#22548625C2211yescharacteristic, complemented
34C8×Dic251C2211yescharacteristic, complemented
35C2×Q16×C251C2211yescharacteristic, complemented
36C2×SD16×C251C2212yescomplemented
37C2×D8×C251C2211yescharacteristic
38#124×C251C2212yescomplemented
39C4×C8×C251C2211yescharacteristic, complemented
40C23×D251—22no—
41#16860811—24no—
42#16860811—24no—
43C2×D1004—22no—
44C2×D1004—24no—
45C2×D1004C2312yes—
46C2×D1004C2312yes—
47C2×C22⋊D251—22no—
48C2×C22⋊D251—24no—
49D4×D251—24no—
50D4×D251—24no—
51C2×C4×D251—22no—
52C2×C4×D251C2311yescharacteristic
53#16860797—24no—
54#16860797—24no—
55D2008—22no—
56D2008—24no—
57D2008—22no—
58#16860795—24no—
59#16860795—24no—
60#16860793D414yescomplemented
61C2×C251⋊C8C2311yescharacteristic
62C2×C251⋊C8C2311yescharacteristic
63C8×D251D414yescomplemented
64Dic251⋊C4D412yes—
65(C2×D251)⋊C4—24no—
66(C2×D251)⋊C4D412yes—
67C4×Dic251C2311yescharacteristic
68C2×Q8×C251C2312yes—
69C2×D4×C251C2312yes—
70Q16×C251—22no—
71SD16×C251—24no—
72D8×C251—22no—
73C2×C8×C251C2311yescharacteristic
74C2×C8×C251C2311yescharacteristic
75C22⋊C4×C251—24no—
76C42×C251C2311yescharacteristic
77C22×D251—22no—
78C22×D251—22no—
79C22×D251—44no—
80C22×D251—84no—
81C22×D251C2×D411yescharacteristic
82D1004—22no—
83D1004—22no—
84D1004—22no—
85D1004—22no—
86D1004—44no—
87D1004—84no—
88C22⋊D251—44no—
89C22⋊D251—84no—
90C2×Dic251C2×D412yes—
91C2×Dic251C2×D411yescharacteristic
92C4×D251—44no—
93C4×D251C2×D412yes—
94C4×D251C2×D412yes—
95C251⋊C8C2×D412yes—
96C251⋊C8C2×D412yes—
97C23×C251—22no—
98Q8×C251—22no—
99Q8×C251—22no—
100D4×C251—22no—
101D4×C251—22no—
102C2×C4×C251—22no—
103C2×C4×C251C2×D412yes—
104C2×C4×C251C2411yescharacteristic
105C8×C251C2×D412yes—
106C8×C251C2×D412yes—
107C22×C251—42no—
108C22×C251—44no—
109C22×C251C22×D411yescharacteristic
110C2×D251—22no—
111C2×D251—42no—
112C2×D251—42no—
113C2×D251—44no—
114C2×D251—44no—
115C2×D251—88no—
116C2×D251#12712yes—
117C4×C251—22no—
118C4×C251—42no—
119C4×C251C22×D411yescharacteristic
120C4×C251C22×D411yescharacteristic
121Dic251—22no—
122Dic251#12712yes—
123C2×C251—82no—
124C2×C251C2×#12711yescharacteristic
125C2×C251#57612yes—
126D251—22no—
127D251—82no—
128D251—82no—
129C251#315511yescharacteristic, complemented, Sylow
130#3155—2511nomaximal, Sylow
131C2×AGL(1,ℤ/8)—2512no—
132C2×#135—2511no—
133C22×D8—2511no—
134#535—2512no—
135#496—2512no—
136#496—2512no—
137#495—2511no—
138#495D25111yescharacteristic, complemented
139#493—2511no—
140#493—2511no—
141#405—2511no—
142C2×#41—2512no—
143C22×D4—2512no—
144AGL(1,ℤ/8)—5028no—
145#135—5024no—
146C2×Q16C2×D25111yescharacteristic, complemented
147C2×SD16—2512no—
148C2×SD16—2512no—
149C2×SD16C2×D25112yescomplemented
150C2×D8—5024no—
151C2×D8—2512no—
152C2×D8—2512no—
153C2×D8—2511no—
154C2×D8—2511no—
155C2×D8—2512no—
156C2×D8C2×D25111yescharacteristic
157C2×M16—2511no—
158C22×C8—2511no—
159#127—2512no—
160#127—2512no—
161#124—2512no—
162#124C2×D25112yescomplemented
163#121—5024no—
164#120—5024no—
165#117—2511no—
166#97—2511no—
167#97—2511no—
168C4×C8—2511no—
169C4×C8C2×D25111yescharacteristic, complemented
170C24—5022no—
171#41—5024no—
172#41—5024no—
173C2×Q8C22×D25112yes—
174C2×D4—5022no—
175C2×D4—5024no—
176C2×D4—5024no—
177C2×D4—5024no—
178C2×D4—5022no—
179C2×D4—5024no—
180C2×D4—2512no—
181C2×D4—2512no—
182C2×D4C22×D25112yes—
183C22×C4—5022no—
184C22×C4—2511no—
185Q16—22no—
186SD16—24no—
187SD16—5024no—
188SD16—5024no—
189D8—22no—
190D8—5024no—
191D8—5024no—
192D8—5022no—
193D8—5022no—
194D8—5024no—
195M16—2514no—
196C2×C8—2511no—
197C2×C8—2511no—
198C2×C8—2514no—
199C2×C8C22×D25111yescharacteristic
200C2×C8C22×D25111yescharacteristic
201C4⋊C4—2512no—
202C22⋊C4—24no—
203C22⋊C4—5024no—
204C22⋊C4—2512no—
205C42—2511no—
206C42C22×D25111yescharacteristic
207C23—22no—
208C23—20084no—
209C23—10044no—
210C23—5022no—
211C23—5022no—
212C23—2511no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—20084no—
218D4—20084no—
219D4—10044no—
220D4—10044no—
221D4—5022no—
222D4—5022no—
223D4—5022no—
224D4—5022no—
225C2×C4—22no—
226C2×C4—10044no—
227C2×C4—2512no—
228C2×C4—2511no—
229C2×C4—2512no—
230C2×C4—2512no—
231C2×C4D4×D25112yes—
232C2×C4C23×D25111yescharacteristic
233C8—2512no—
234C8—2512no—
235C8D4×D25112yes—
236C8D4×D25112yes—
237C22—42no—
238C22—44no—
239C22—20088no—
240C22—10042no—
241C22—10042no—
242C22—10044no—
243C22—10044no—
244C22—5022no—
245C22—2512no—
246C22C2×D4×D25111yescharacteristic
247C4—22no—
248C4—42no—
249C4—5022no—
250C4—2512no—
251C4C2×D4×D25111yescharacteristic
252C4C2×D4×D25111yescharacteristic
253C2—82no—
254C2—20082no—
255C2—20082no—
256C2—5022no—
257C2#127×D25111yescharacteristic
258C2#3063248912yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#495×C251C2
socleC22×C251C22×D4
hypercenter#495D251
hypocenterC251#3155
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#22548742C2semidirect
2. C2×#22548738C2semidirect
3. C2×D8×D251C2semidirect
4. #30632255C2semidirect
5. #30632077C2semidirect
6. #30632073C2semidirect
7. #30632072C2semidirect
8. #30632064C2semidirect
9. #30632062C2semidirect
10. #30631641C2semidirect
11. #495×C251C2semidirect
16. C2×#16860797C22semidirect
17. C2×#16860797C22semidirect
19. C2×D2008C22semidirect
20. C2×D2008C22semidirect
22. C2×#16860795C22semidirect
26. #22548712C22semidirect
27. #22548711C22semidirect
28. #22548704C22semidirect
32. #22548626C22semidirect
33. #22548625C22semidirect
34. C8×Dic251C22semidirect
35. C2×Q16×C251C22semidirect
36. C2×SD16×C251C22semidirect
38. #124×C251C22semidirect
39. C4×C8×C251C22semidirect
60. #16860793D4semidirect
63. C8×D251D4semidirect
129. C251#3155semidirect
138. #495D251semidirect
146. C2×Q16C2×D251semidirect
149. C2×SD16C2×D251semidirect
162. #124C2×D251semidirect
169. C4×C8C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487422
C2×#225487382
C2×D8×D2512
#306322552
#306320772
#306320732
#306320722
#306320642
#306320622
#306316412
#495×C2512
#3155251