Finite Group Databasebeta

← #44560828

Subgroups — #44560828

Order 32128.

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

normal subgroups only

This group has 572 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×#22548747C212yesmaximal, complemented
2C2×#22548742C212yesmaximal, complemented
3#30632238C211yesmaximal, characteristic, complemented
4#124×D251C211yesmaximal, characteristic, complemented
5#30632079C211yesmaximal, characteristic, complemented
6#30632077C211yesmaximal, characteristic, complemented
7#30632076C211yesmaximal, characteristic, complemented
8#30632072C212yesmaximal, complemented
9#30632068C212yesmaximal, complemented
10#30631639C211yesmaximal, characteristic, complemented
11#497×C251C211yesmaximal, characteristic, complemented
12C2×#16860811C2212yes—
13C2×Q8×D251C2211yescharacteristic
14C2×D4×D251C2211yescharacteristic
15#22548747—28no—
16#22548742—28no—
17C2×#16860800C2212yescomplemented
18C2×#16860799C2212yescomplemented
19C2×#16860797C2212yes—
20C2×#16860797C2212yescomplemented
21C2×D2008C2212yes—
22C2×#16860795C2212yes—
23C2×#16860793C2212yes—
24#22548712C2211yescharacteristic, complemented
25#22548704C2211yescharacteristic
26#22548704C2212yescomplemented
27#22548703C2211yescharacteristic
28#22548702C2211yescharacteristic
29#22548701—22no—
30#22548690—22no—
31#22548677—22no—
32C22⋊C4×D251—22no—
33C42×D251C2211yescharacteristic
34#22548626C2211yescharacteristic, complemented
35C8×Dic251C2212yescomplemented
36C2×Q16×C251C2212yescomplemented
37C2×SD16×C251C2212yes—
38C4⋊Q8×C251C2211yescharacteristic, complemented
39#124×C251C2211yescharacteristic
40#97×C251C2211yescharacteristic, complemented
41C23×D251—21no—
42#16860811—24no—
43#16860811—24no—
44C2×Dic502—21no—
45C2×Dic502C2311yescharacteristic
46Q8×D251—24no—
47C2×D1004—22no—
48C2×D1004—22no—
49C2×D1004C2311yescharacteristic
50C2×D1004C2312yes—
51C2×C22⋊D251—21no—
52D4×D251—24no—
53C2×C4×D251—21no—
54C2×C4×D251—22no—
55C2×C4×D251D412yes—
56C2×C4×D251C2311yescharacteristic
57#16860800—24no—
58#16860799—24no—
59#16860797—24no—
60#16860797—24no—
61D2008—24no—
62#16860795—24no—
63#16860793D418yescomplemented
64C2×C251⋊C8C2312yes—
65C22⋊Dic251—22no—
66(C2×D251)⋊C4—22no—
67(C2×D251)⋊C4—24no—
68(C2×D251)⋊C4—22no—
69C4×Dic251D412yes—
70C4×Dic251C2311yescharacteristic
71C2×Q8×C251C2311yescharacteristic
72C2×Q8×C251C2312yes—
73C2×D4×C251C2311yescharacteristic
74Q16×C251—24no—
75SD16×C251—24no—
76C2×C8×C251C2312yes—
77C4⋊C4×C251—22no—
78C22⋊C4×C251—22no—
79C42×C251C2311yescharacteristic
80C22×D251—21no—
81C22×D251—22no—
82C22×D251—44no—
83C22×D251—42no—
84C22×D251C2×D411yescharacteristic
85Dic502—22no—
86Dic502—42no—
87D1004—22no—
88D1004—22no—
89D1004—22no—
90D1004—44no—
91D1004—44no—
92C22⋊D251—42no—
93C2×Dic251—21no—
94C2×Dic251C2×D412yes—
95C2×Dic251C2×D411yescharacteristic
96C4×D251—42no—
97C4×D251—42no—
98C4×D251—44no—
99C4×D251C2×D414yes—
100C251⋊C8C2×D414yes—
101C23×C251—21no—
102Q8×C251—22no—
103Q8×C251—22no—
104Q8×C251—22no—
105D4×C251—22no—
106C2×C4×C251—21no—
107C2×C4×C251—22no—
108C2×C4×C251C2×D412yes—
109C2×C4×C251C2411yescharacteristic
110C8×C251C2×D414yes—
111C22×C251—41no—
112C22×C251—42no—
113C22×C251C22×D411yescharacteristic
114C2×D251—22no—
115C2×D251—42no—
116C2×D251—41no—
117C2×D251—44no—
118C2×D251—42no—
119C2×D251—84no—
120C2×D251#12712yes—
121C4×C251—22no—
122C4×C251—42no—
123C4×C251—41no—
124C4×C251C22×D412yes—
125Dic251—22no—
126Dic251—41no—
127Dic251#12712yes—
128C2×C251—81no—
129C2×C251C2×#12711yescharacteristic
130C2×C251#57512yes—
131D251—22no—
132D251—82no—
133D251—81no—
134C251#315311yescharacteristic, complemented, Sylow
135#3153—2511nomaximal, Sylow
136C2×#137—2512no—
137C2×AGL(1,ℤ/8)—2512no—
138#532—2511no—
139C2×#124—2511no—
140#497—2511no—
141#497D25111yescharacteristic, complemented
142#496—2511no—
143#496—2511no—
144#495—2512no—
145#495—2512no—
146C4×M16—2511no—
147C2×#41—2512no—
148C22×Q8—2511no—
149C22×D4—2511no—
150#137—5028no—
151AGL(1,ℤ/8)—5028no—
152C2×Q16—2512no—
153C2×Q16C2×D25112yescomplemented
154C2×SD16—2512no—
155C2×SD16—2512no—
156C2×SD16—2512no—
157C2×SD16C2×D25112yes—
158C2×D8—2512no—
159C2×D8—2512no—
160C2×M16—2512no—
161C4⋊Q8C2×D25111yescharacteristic, complemented
162#127—2511no—
163#124—5022no—
164#124—2511no—
165#124—2512no—
166#124—2511no—
167#124—2511no—
168#124C2×D25111yescharacteristic
169#121—5022no—
170C4×D4—5022no—
171C2×C22⋊C4—5022no—
172C2×C42—2511no—
173#97—2511no—
174#97C2×D25111yescharacteristic, complemented
175C4×C8—2512no—
176C24—5021no—
177#41—5024no—
178#41—5024no—
179C2×Q8—5021no—
180C2×Q8—5024no—
181C2×Q8—2511no—
182C2×Q8C22×D25111yescharacteristic
183C2×Q8C22×D25112yes—
184C2×D4—5022no—
185C2×D4—5024no—
186C2×D4—5021no—
187C2×D4—5022no—
188C2×D4—2511no—
189C2×D4—2512no—
190C2×D4C22×D25111yescharacteristic
191C22×C4—5021no—
192C22×C4—5022no—
193C22×C4—2512no—
194C22×C4—2511no—
195Q16—24no—
196Q16—5024no—
197SD16—24no—
198SD16—5024no—
199SD16—5024no—
200SD16—5024no—
201D8—5024no—
202D8—5024no—
203M16—2518no—
204C2×C8—2512no—
205C2×C8C22×D25112yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—5022no—
209C22⋊C4—5024no—
210C22⋊C4—5022no—
211C22⋊C4—5022no—
212C42—2512no—
213C42—2511no—
214C42C22×D25111yescharacteristic
215C23—21no—
216C23—10044no—
217C23—10042no—
218C23—5021no—
219C23—5022no—
220C23—2511no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—10042no—
225Q8—5022no—
226D4—22no—
227D4—10044no—
228D4—10044no—
229D4—10042no—
230D4—5022no—
231D4—5022no—
232D4—5022no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—10042no—
236C2×C4—10042no—
237C2×C4—10044no—
238C2×C4—5021no—
239C2×C4—2512no—
240C2×C4—2511no—
241C2×C4—2514no—
242C2×C4D4×D25112yes—
243C2×C4C23×D25111yescharacteristic
244C8—2514no—
245C8D4×D25114yes—
246C22—41no—
247C22—42no—
248C22—20084no—
249C22—10042no—
250C22—10041no—
251C22—10044no—
252C22—10042no—
253C22—5022no—
254C22—2512no—
255C22C2×D4×D25111yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—10041no—
260C4—5022no—
261C4—2512no—
262C4C2×D4×D25112yes—
263C2—81no—
264C2—20081no—
265C2—20082no—
266C2—5022no—
267C2#127×D25111yescharacteristic
268C2#3063248612yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#497×C251C2
socleC22×C251C22×D4
hypercenter#497D251
hypocenterC251#3153
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#22548747C2semidirect
2. C2×#22548742C2semidirect
3. #30632238C2semidirect
4. #124×D251C2semidirect
5. #30632079C2semidirect
6. #30632077C2semidirect
7. #30632076C2semidirect
8. #30632072C2semidirect
9. #30632068C2semidirect
10. #30631639C2semidirect
11. #497×C251C2semidirect
17. C2×#16860800C22semidirect
18. C2×#16860799C22semidirect
20. C2×#16860797C22semidirect
24. #22548712C22semidirect
26. #22548704C22semidirect
34. #22548626C22semidirect
35. C8×Dic251C22semidirect
36. C2×Q16×C251C22semidirect
38. C4⋊Q8×C251C22semidirect
40. #97×C251C22semidirect
63. #16860793D4semidirect
134. C251#3153semidirect
141. #497D251semidirect
153. C2×Q16C2×D251semidirect
161. C4⋊Q8C2×D251semidirect
174. #97C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487472
C2×#225487422
#306322382
#124×D2512
#306320792
#306320772
#306320762
#306320722
#306320682
#306316392
#497×C2512
#3153251