Finite Group Databasebeta

← #44560717

Subgroups — #44560717

Order 32128.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#30632318C211yesmaximal, characteristic, complemented
2#30632254C211yesmaximal, characteristic
3#30632050C212yesmaximal, complemented
4#30632045C212yesmaximal, complemented
5#30632040C212yesmaximal, complemented
6#30632032C212yesmaximal, complemented
7#30631692C211yesmaximal, characteristic, complemented
8#30631686C211yesmaximal, characteristic, complemented
9#30631684C211yesmaximal, characteristic, complemented
10#30631642C211yesmaximal, characteristic, complemented
11C2×#102×C251C211yesmaximal, characteristic, complemented
12C2×#16860810C2211yescharacteristic
13C2×#16860794C2211yescharacteristic
14#22548714—21no—
15#22548714C2212yescomplemented
16#22548711C2212yes—
17#22548706—21no—
18#22548706C2212yescomplemented
19#22548702C2212yes—
20#22548694—22no—
21#22548686—21no—
22#22548686—21no—
23#22548684—22no—
24#22548674C2211yescharacteristic
25#22548674C2211yescharacteristic
26#22548640C2211yescharacteristic, complemented
27#22548640C2211yescharacteristic, complemented
28#22548638C2212yescomplemented
29#22548637C2211yescharacteristic, complemented
30#22548637C2211yescharacteristic, complemented
31#22548636C2212yes—
32#22548636C2212yes—
33#22548635C2212yescomplemented
34#22548625C2212yescomplemented
35C8×Dic251C2212yescomplemented
36C22×D4×C251C2211yescharacteristic
37C22×C8×C251C2211yescharacteristic
38C2×C4⋊C4×C251C2211yescharacteristic
39#102×C251C2212yes—
40#102×C251C2212yes—
41#16860810—22no—
42#16860810—22no—
43C2×Dic502—22no—
44C2×Dic502C2311yescharacteristic
45C2×D1004C2311yescharacteristic
46C2×C22⋊D251—22no—
47C2×C22⋊D251—22no—
48C2×C22⋊D251—21no—
49C2×C4×D251—21no—
50#16860794—22no—
51C2×C251⋊C8C2312yes—
52C4⋊Dic251—22no—
53C4⋊Dic251D411yescharacteristic
54C4⋊Dic251D411yescharacteristic
55C4⋊Dic251C2311yescharacteristic
56C4⋊Dic251C2311yescharacteristic
57Dic251⋊C4—22no—
58Dic251⋊C4—22no—
59C22⋊Dic251—22no—
60C22⋊Dic251—22no—
61C22⋊Dic251—21no—
62C22⋊Dic251D412yes—
63C4×Dic251—21no—
64C4×Dic251C2312yes—
65C24×C251—21no—
66C2×D4×C251—21no—
67C2×D4×C251—22no—
68C2×D4×C251—22no—
69C2×D4×C251C2312yes—
70C22×C4×C251—21no—
71C22×C4×C251C2311yescharacteristic
72C2×C8×C251—22no—
73C2×C8×C251C2311yescharacteristic
74C2×C8×C251C2311yescharacteristic
75C4⋊C4×C251—21no—
76C4⋊C4×C251C2312yes—
77C22×D251—21no—
78Dic502—21no—
79Dic502—21no—
80Dic502—82no—
81D1004—21no—
82D1004—21no—
83C22⋊D251—42no—
84C22⋊D251—82no—
85C22⋊D251—82no—
86C2×Dic251—21no—
87C2×Dic251—21no—
88C2×Dic251—21no—
89C2×Dic251C2×D412yes—
90C2×Dic251C2×D412yes—
91C4×D251—42no—
92C251⋊C8—22no—
93C23×C251—22no—
94C23×C251—42no—
95C23×C251—81no—
96C23×C251—81no—
97C23×C251C2×D411yescharacteristic
98D4×C251—22no—
99D4×C251—22no—
100D4×C251—42no—
101C2×C4×C251—22no—
102C2×C4×C251—81no—
103C2×C4×C251C2×D411yescharacteristic
104C2×C4×C251#4112yes—
105C2×C4×C251#4112yes—
106C2×C4×C251C2411yescharacteristic
107C8×C251—21no—
108C8×C251—21no—
109C22×C251—21no—
110C22×C251—21no—
111C22×C251—42no—
112C22×C251—42no—
113C22×C251—42no—
114C22×C251—84no—
115C22×C251#12412yes—
116C22×C251C22×D411yescharacteristic
117C2×D251—41no—
118C2×D251—41no—
119C2×D251—41no—
120C4×C251—42no—
121C4×C251#12412yes—
122C4×C251C2×#4111yescharacteristic
123C4×C251C2×#4111yescharacteristic
124Dic251—22no—
125Dic251—22no—
126Dic251—41no—
127Dic251—41no—
128Dic251—41no—
129C2×C251—22no—
130C2×C251—82no—
131C2×C251C2×#12411yescharacteristic
132C2×C251#57611yescharacteristic
133C2×C251#57711yescharacteristic
134D251—81no—
135C251#313611yescharacteristic, complemented, Sylow
136#3136—2511nomaximal, Sylow
137#544—2511no—
138#535—2511no—
139#490—2512no—
140#489—2512no—
141#488—2512no—
142#486—2512no—
143#418—2511no—
144#417—2511no—
145#416—2511no—
146C2×#102D25111yescharacteristic, complemented
147#405—2511no—
148C2×#41—2511no—
149C22×D4C2×D25111yescharacteristic
150C2×M16—2511no—
151C22×C8C2×D25111yescharacteristic
152C4⋊Q8—5021no—
153C4⋊Q8—2512no—
154#127—2512no—
155#125—5021no—
156#125—2512no—
157#124—2512no—
158C22⋊Q8—5022no—
159#121—5021no—
160#121—5021no—
161#120—5022no—
162#117—2511no—
163#117—2511no—
164C2×C4⋊C4C2×D25111yescharacteristic
165#103—2511no—
166#103—2511no—
167#103—2512no—
168#102—2512no—
169#102—2512no—
170#102—2511no—
171#102—2511no—
172#102—2512no—
173#102C2×D25112yes—
174#102C2×D25112yes—
175#97—2512no—
176C4×C8—2512no—
177C24—21no—
178#41—5022no—
179#41—5022no—
180C2×Q8—5022no—
181C2×Q8—2511no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—5021no—
186C2×D4—5022no—
187C2×D4—5022no—
188C2×D4—2511no—
189C2×D4C22×D25112yes—
190C22×C4—21no—
191C22×C4—5021no—
192C22×C4C22×D25111yescharacteristic
193M16—5022no—
194C2×C8—22no—
195C2×C8—2512no—
196C2×C8C22×D25111yescharacteristic
197C2×C8C22×D25111yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—5022no—
200C4⋊C4—5022no—
201C4⋊C4—5022no—
202C4⋊C4—2511no—
203C4⋊C4—2511no—
204C4⋊C4—2511no—
205C4⋊C4—2511no—
206C4⋊C4C22×D25112yes—
207C22⋊C4—5021no—
208C22⋊C4—5022no—
209C22⋊C4—5022no—
210C22⋊C4—2512no—
211C42—5021no—
212C42—2512no—
213C23—22no—
214C23—42no—
215C23—81no—
216C23—81no—
217C23—5021no—
218C23D4×D25111yescharacteristic
219Q8—20082no—
220Q8—5021no—
221Q8—5021no—
222D4—22no—
223D4—22no—
224D4—42no—
225D4—20082no—
226D4—20082no—
227D4—10042no—
228D4—5021no—
229D4—5021no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—10042no—
233C2×C4—5021no—
234C2×C4—5021no—
235C2×C4—5021no—
236C2×C4—2512no—
237C2×C4—2512no—
238C2×C4D4×D25111yescharacteristic
239C2×C4#1686080912yes—
240C2×C4#1686081012yes—
241C2×C4C23×D25111yescharacteristic
242C8—21no—
243C8—21no—
244C8—5022no—
245C22—21no—
246C22—21no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—84no—
251C22—10041no—
252C22—10041no—
253C22—10041no—
254C22#2254870112yes—
255C22C2×D4×D25111yescharacteristic
256C4—42no—
257C4—10041no—
258C4—10041no—
259C4—10041no—
260C4—5022no—
261C4—5022no—
262C4#2254870112yes—
263C4C2×#1686080911yescharacteristic
264C4C2×#1686081011yescharacteristic
265C2—22no—
266C2—82no—
267C2—20081no—
268C2C2×#2254870111yescharacteristic
269C2#3063248811yescharacteristic
270C2#3063249311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#102×C251C2
socleC22×C251C22×D4
prime radical#30632254C2
hypercenterC2×#102D251
hypocenterC251#3136
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #30632318C2semidirect
3. #30632050C2semidirect
4. #30632045C2semidirect
5. #30632040C2semidirect
6. #30632032C2semidirect
7. #30631692C2semidirect
8. #30631686C2semidirect
9. #30631684C2semidirect
10. #30631642C2semidirect
11. C2×#102×C251C2semidirect
15. #22548714C22semidirect
18. #22548706C22semidirect
26. #22548640C22semidirect
27. #22548640C22semidirect
28. #22548638C22semidirect
29. #22548637C22semidirect
30. #22548637C22semidirect
33. #22548635C22semidirect
34. #22548625C22semidirect
35. C8×Dic251C22semidirect
135. C251#3136semidirect
146. C2×#102D251semidirect

Maximal subgroups

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

subgroupindex
#306323182
#306322542
#306320502
#306320452
#306320402
#306320322
#306316922
#306316862
#306316842
#306316422
C2×#102×C2512
#3136251