Finite Group Databasebeta

← #44560957

Subgroups — #44560957

Order 32128.

408 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 119 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#30632235C211yesmaximal, characteristic, complemented
2#30632216C211yesmaximal, characteristic, complemented
3#30632096C211yesmaximal, characteristic, complemented
4#30632051C211yesmaximal, characteristic, complemented
5#30632046C211yesmaximal, characteristic, complemented
6#30631937C212yesmaximal, complemented
7#30631930C212yesmaximal, complemented
8SD16×Dic251C212yesmaximal, complemented
9D8×Dic251C212yesmaximal, complemented
10#30631639C211yesmaximal, characteristic, complemented
11#496×C251C211yesmaximal, characteristic, complemented
12C2×#16860809—21no—
13C2×#16860793C2212yes—
14#22548715C2211yescharacteristic
15#22548708—21no—
16#22548707C2211yescharacteristic
17#22548702—21no—
18#22548697—21no—
19#22548691C2211yescharacteristic
20#22548686—21no—
21#22548686C2211yescharacteristic
22#22548686C2212yes—
23Q8×Dic251C2211yescharacteristic
24D4×Dic251—21no—
25D4×Dic251C2211yescharacteristic
26D4×Dic251C2212yes—
27(C22⋊D251)⋊C4—21no—
28#22548671—21no—
29C42×D251C2211yescharacteristic
30#22548649C2212yes—
31#22548647C2212yes—
32#22548639C2212yes—
33#22548636C2212yes—
34#22548636C2212yes—
35#22548636C2212yes—
36#22548626C2211yescharacteristic
37C8×Dic251C2212yes—
38C2×SD16×C251C2212yes—
39C2×D8×C251C2212yes—
40#127×C251C2211yescharacteristic
41#124×C251C2211yescharacteristic
42#97×C251C2211yescharacteristic
43#16860809—44no—
44C2×Dic502—21no—
45C2×C22⋊D251—22no—
46C2×C22⋊D251—21no—
47C22×Dic251—22no—
48C22×Dic251—21no—
49C2×C4×D251D411yescharacteristic
50C2×C4×D251D411yescharacteristic
51C2×C4×D251C2311yescharacteristic
52#16860793—24no—
53C2×C251⋊C8C2312yes—
54C4⋊Dic251—21no—
55C4⋊Dic251—21no—
56C4⋊Dic251C2311yescharacteristic
57C4⋊Dic251C2311yescharacteristic
58C4⋊Dic251C2312yes—
59Dic251⋊C4—21no—
60Dic251⋊C4—21no—
61Dic251⋊C4—21no—
62C22⋊Dic251—21no—
63C22⋊Dic251—22no—
64C22⋊Dic251—22no—
65C22⋊Dic251—21no—
66(C2×D251)⋊C4—21no—
67C4×Dic251—21no—
68C4×Dic251D411yescharacteristic
69C4×Dic251D411yescharacteristic
70C4×Dic251C2311yescharacteristic
71C2×Q8×C251C2311yescharacteristic
72C2×D4×C251—21no—
73C2×D4×C251—21no—
74C2×D4×C251C2311yescharacteristic
75C2×D4×C251C2312yes—
76SD16×C251—24no—
77D8×C251—24no—
78C2×C8×C251C2312yes—
79C22⋊C4×C251—21no—
80C22⋊C4×C251—21no—
81C42×C251C2311yescharacteristic
82C22×D251C2×D411yescharacteristic
83Dic502—42no—
84C22⋊D251—42no—
85C22⋊D251—44no—
86C2×Dic251—21no—
87C2×Dic251—21no—
88C2×Dic251—22no—
89C2×Dic251—42no—
90C2×Dic251—44no—
91C2×Dic251C2×D411yescharacteristic
92C2×Dic251C2×D411yescharacteristic
93C2×Dic251C2×D411yescharacteristic
94C4×D251—22no—
95C4×D251—22no—
96C4×D251—41no—
97C251⋊C8—22no—
98C23×C251—21no—
99C23×C251—22no—
100Q8×C251—22no—
101D4×C251—22no—
102D4×C251—22no—
103D4×C251—22no—
104D4×C251—42no—
105D4×C251—42no—
106C2×C4×C251—21no—
107C2×C4×C251C2×D411yescharacteristic
108C2×C4×C251C2×D411yescharacteristic
109C2×C4×C251C2411yescharacteristic
110C8×C251#4114yescomplemented
111C22×C251—41no—
112C22×C251—42no—
113C22×C251—42no—
114C22×C251—41no—
115C22×C251—42no—
116C22×C251—41no—
117C22×C251C22×D411yescharacteristic
118C2×D251—21no—
119C2×D251—21no—
120C2×D251—21no—
121C4×C251—21no—
122C4×C251—21no—
123C4×C251—41no—
124C4×C251C2×#4112yes—
125Dic251—21no—
126Dic251—21no—
127Dic251—41no—
128Dic251—42no—
129Dic251—41no—
130Dic251AGL(1,ℤ/8)12yes—
131C2×C251—82no—
132C2×C251—81no—
133C2×C251#53211yescharacteristic
134C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
135C2×C251#57511yescharacteristic
136D251—41no—
137C251#317111yescharacteristic, complemented, Sylow
138#3171—2511nomaximal, Sylow
139#532—2511no—
140#529—2511no—
141#501—2511no—
142#496D25111yescharacteristic, complemented
143#490—2511no—
144#489—2511no—
145#469—2512no—
146#468—2512no—
147C4×SD16—2512no—
148C4×D8—2512no—
149C4×M16—2511no—
150C2×#41—5021no—
151C2×SD16C2×D25112yes—
152C2×D8C2×D25112yes—
153C2×M16—2512no—
154C4⋊Q8—2511no—
155#127C2×D25111yescharacteristic
156#126—5021no—
157#125—2511no—
158#124—5021no—
159#124C2×D25111yescharacteristic
160#123—5021no—
161C22⋊Q8—2511no—
162#121—5021no—
163#121—2511no—
164#121—2512no—
165C4×Q8—2511no—
166C4×D4—5021no—
167C4×D4—5021no—
168C4×D4—2511no—
169C4×D4—2512no—
170#117—5021no—
171C2×C42—2511no—
172#107—2512no—
173#106—2512no—
174#103—2512no—
175#102—2512no—
176#102—2512no—
177#102—2512no—
178#97—2511no—
179#97C2×D25111yescharacteristic
180C4×C8—2512no—
181#41—10044no—
182C2×Q8—5021no—
183C2×Q8C22×D25111yescharacteristic
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—5021no—
187C2×D4—5022no—
188C2×D4C22×D25111yescharacteristic
189C2×D4C22×D25112yes—
190C22×C4—5021no—
191C22×C4—5022no—
192C22×C4—2511no—
193C22×C4—2511no—
194C22×C4—2511no—
195SD16—24no—
196D8—24no—
197M16—5024no—
198C2×C8—2512no—
199C2×C8C22×D25112yes—
200C4⋊C4—5021no—
201C4⋊C4—5021no—
202C4⋊C4—5021no—
203C4⋊C4—5021no—
204C4⋊C4—5021no—
205C4⋊C4—2511no—
206C4⋊C4—2511no—
207C4⋊C4—2512no—
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—5021no—
211C22⋊C4—5021no—
212C22⋊C4—5022no—
213C22⋊C4—5022no—
214C22⋊C4—5021no—
215C42—5021no—
216C42—2511no—
217C42—2511no—
218C42—2511no—
219C42C22×D25111yescharacteristic
220C23—21no—
221C23—22no—
222C23—2511no—
223Q8—22no—
224Q8—10042no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4—42no—
229D4—42no—
230D4—10044no—
231D4—10042no—
232C2×C4—21no—
233C2×C4—10041no—
234C2×C4—10044no—
235C2×C4—10042no—
236C2×C4—5022no—
237C2×C4—5021no—
238C2×C4—5021no—
239C2×C4—5022no—
240C2×C4—5022no—
241C2×C4—2511no—
242C2×C4—2511no—
243C2×C4—2511no—
244C2×C4D4×D25111yescharacteristic
245C2×C4D4×D25111yescharacteristic
246C2×C4C23×D25111yescharacteristic
247C8—5022no—
248C8#1686080914yescomplemented
249C22—42no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—5021no—
256C22—5021no—
257C22—5021no—
258C22C2×D4×D25111yescharacteristic
259C4—21no—
260C4—21no—
261C4—41no—
262C4—10042no—
263C4—10041no—
264C4—10041no—
265C4—5021no—
266C4—5021no—
267C4—2512no—
268C4C2×#1686080912yes—
269C2—82no—
270C2—81no—
271C2—10041no—
272C2#3063223511yescharacteristic
273C2AGL(1,ℤ/8)×D25111yescharacteristic
274C2#3063248311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#496×C251C2
socleC22×C251C22×D4
hypercenter#496D251
hypocenterC251#3171
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #30632235C2semidirect
2. #30632216C2semidirect
3. #30632096C2semidirect
4. #30632051C2semidirect
5. #30632046C2semidirect
6. #30631937C2semidirect
7. #30631930C2semidirect
8. SD16×Dic251C2semidirect
9. D8×Dic251C2semidirect
10. #30631639C2semidirect
11. #496×C251C2semidirect
110. C8×C251#41semidirect
137. C251#3171semidirect
142. #496D251semidirect
248. C8#16860809semidirect

Maximal subgroups

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

subgroupindex
#306322352
#306322162
#306320962
#306320512
#306320462
#306319372
#306319302
SD16×Dic2512
D8×Dic2512
#306316392
#496×C2512
#3171251